tag:theconversation.com,2011:/au/topics/patterns-7487/articlesPatterns – The Conversation2024-03-26T12:39:43Ztag:theconversation.com,2011:article/2238092024-03-26T12:39:43Z2024-03-26T12:39:43ZHow AI and a popular card game can help engineers predict catastrophic failure – by finding the absence of a pattern<figure><img src="https://images.theconversation.com/files/584159/original/file-20240325-10630-wq22k6.png?ixlib=rb-1.1.0&rect=84%2C498%2C1343%2C882&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Can you find a matching set?</span> <span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/File:Set_isomorphic_cards.svg">Cmglee/Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>Humans are very good at <a href="https://doi.org/10.1016/j.neuron.2018.05.013">spotting patterns</a>, or repeating features people can recognize. For instance, ancient Polynesians navigated across the Pacific by <a href="https://nla.gov.au/nla.obj-318911458/view?partId=nla.obj-318923632#page/n90/mode/1up">recognizing many patterns</a>, from the stars’ constellations to more subtle ones such as the directions and sizes of ocean swells.</p>
<p>Very recently, <a href="https://scholar.google.com/citations?user=L1lKOGsAAAAJ&hl=en">mathematicians like me</a> have started to study large collections of objects that have no patterns of a particular sort. How large can collections be before a specified pattern has to appear somewhere in the collection? Understanding such scenarios can have significant real-world implications: For example, what’s the smallest number of server failures that would lead to the severing of the internet?</p>
<p>Research from mathematician <a href="https://scholar.google.com/citations?user=b7P6YbkAAAAJ&hl=en">Jordan Ellenberg</a> at the University of Wisconsin and researchers at <a href="https://deepmind.google/">Google’s Deep Mind</a> have proposed a novel approach to this problem. Their work <a href="https://doi.org/10.1038/s41586-023-06924-6">uses artificial intelligence to find</a> large collections that don’t contain a specified pattern, which can help us understand some worst-case scenarios.</p>
<h2>Patterns in the card game Set</h2>
<p>The idea of patternless collections can be illustrated by a popular card game <a href="https://brilliant.org/wiki/set-game/">called Set</a>. In this game, players lay out 12 cards, face up. Each card has a different simple picture on it. They vary in terms of number, color, shape and shading. Each of these four features can have one of three values.</p>
<p>Players race to look for “sets,” which are groups of three cards in which every feature is either the same or different in each card. For instance, cards with one solid red diamond, two solid green diamonds and three solid purple diamonds form a set: All three have different numbers (one, two, three), the same shading (solid), different colors (red, green, purple) and the same shape (diamond).</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/7AeEr9QtDF0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Marsha Falco originally created the game Set to help explain her research on population genetics.</span></figcaption>
</figure>
<p>Finding a set is usually possible – but not always. If none of the players can find a set from the 12 cards on the table, then they flip over three more cards. But they still might not be able to find a set in these 15 cards. The players continue to flip over cards, three at a time, until someone spots a set.</p>
<p>So what is the maximum number of cards you can lay out without forming a set?</p>
<p>In 1971, mathematician Giuseppe Pellegrino showed that the <a href="https://www.quantamagazine.org/set-proof-stuns-mathematicians-20160531/">largest collection of cards without a set is 20</a>. But if you chose 20 cards at random, “no set” would happen only <a href="https://www-cs-faculty.stanford.edu/%7Eknuth/programs/setset-all.w">about one in a trillion times</a>. And finding these “no set” collections is an extremely hard problem to solve.</p>
<h2>Finding ‘no set’ with AI</h2>
<p>If you wanted to find the smallest collection of cards with no set, you could in principle do an exhaustive search of every possible collection of cards chosen from the deck of 81 cards. But there are an enormous number of possibilities – on the order of 10<sup>24</sup> (that’s a “1” followed by 24 zeros). And if you increase the number of features of the cards from four to, say, eight, the complexity of the problem would overwhelm any computer doing an exhaustive search for “no set” collections.</p>
<p>Mathematicians love to think about computationally difficult problems like this. These complex problems, if approached in the right way, can become tractable. </p>
<p>It’s easier to find best-case scenarios – here, that would mean the fewest number of cards that could contain a set. But there were few known strategies that could explore bad scenarios – here, that would mean a large collection of cards that do not contain a set.</p>
<p>Ellenberg and his collaborators approached the bad scenario with a type of AI called <a href="https://theconversation.com/ai-to-z-all-the-terms-you-need-to-know-to-keep-up-in-the-ai-hype-age-203917">large language models, or LLMs</a>. The researchers first wrote computer programs that generate some examples of collections of many that contain no set. These collections typically have “cards” with more than four features.</p>
<p>Then they fed these programs to the LLM, which soon learned how to write many similar programs and choose the ones that give rise to the largest set-free collections to undergo the process again. Iterating that process by repeatedly tweaking the most successful programs enables them to find larger and larger set-free collections.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Square of nine circles, four of which are colored blue, connected by grey, red, green, and yellow lines" src="https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">This is another version of a ‘no set,’ where no three components of a set are linked by a line.</span>
<span class="attribution"><a class="source" href="https://doi.org/10.1038/s41586-023-06924-6">Romera-Peredes et al./Nature</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>This method allows people to explore disordered collections – in this instance, <a href="https://doi.org/10.1038/s41586-023-06924-6">collections of cards that contain no set</a> – in an entirely new way. It does not guarantee that researchers will find the absolute worst-case scenario, but they will find scenarios that are much worse than a random generation would yield.</p>
<p>Their work can help researchers understand how events might align in a way that leads to catastrophic failure. </p>
<p>For example, how vulnerable is the electrical grid to a malicious attacker who destroys select substations? Suppose that a bad collection of substations is one where they don’t form a connected grid. The worst-case scenario is now a very large number of substations that, when taken all together, still don’t yield a connected grid. The amount of substations excluded from this collection make up the smallest number a malicious actor needs to destroy to deliberately disconnect the grid.</p>
<p>The work of Ellenberg and his collaborators demonstrates yet another way that AI is a very powerful tool. But to solve very complex problems, at least for now, it still needs human ingenuity to guide it.</p><img src="https://counter.theconversation.com/content/223809/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>John Edward McCarthy is partially supported by National Science Foundation Grant
DMS 2054199. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.</span></em></p>What mathematicians call ‘disordered collections’ can help engineers explore real-world worst-case scenarios. The simple card game Set illustrates how to predict internet and electrical grid failures.John Edward McCarthy, Professor of Mathematics, Arts & Sciences at Washington University in St. LouisLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2170352023-11-08T19:05:23Z2023-11-08T19:05:23ZHow animals get their skin patterns is a matter of physics – new research clarifying how could improve medical diagnostics and synthetic materials<figure><img src="https://images.theconversation.com/files/558150/original/file-20231107-15-ksvdj8.png?ixlib=rb-1.1.0&rect=0%2C0%2C2121%2C1412&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Color patterns seen in fish and other animals evolved to serve various purposes.</span> <span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/close-up-of-the-eye-of-a-yellowspot-rabbitfish-royalty-free-image/691700228">Lagunatic Photo/iStock via Getty Images Plus</a></span></figcaption></figure><p>Patterns on animal skin, such as zebra stripes and poison frog color patches, serve various biological functions, including <a href="https://doi.org/10.1080/00222933.2019.1607600">temperature regulation</a>, <a href="https://doi.org/10.1098/rsos.160824">camouflage</a> and <a href="https://doi.org/10.1111/j.1558-5646.2011.01257.x">warning signals</a>. The colors making up these patterns must be distinct and well separated to be effective. For instance, as a warning signal, distinct colors make them clearly visible to other animals. And as camouflage, well-separated colors allow animals to better blend into their surroundings.</p>
<p>In our newly published research in Science Advances, my student <a href="https://scholar.google.com/citations?user=ZYQyHkYAAAAJ&hl=en">Ben Alessio</a> <a href="https://scholar.google.com/citations?user=oiMqxxoAAAAJ&hl=en">and I</a> propose a <a href="http://www.science.org/doi/10.1126/sciadv.adj2457">potential mechanism</a> explaining how these distinctive patterns form – that could potentially be applied to medical diagnostics and synthetic materials.</p>
<p>A thought experiment can help visualize the challenge of achieving distinctive color patterns. Imagine gently adding a drop of blue and red dye to a cup of water. The drops will slowly disperse throughout the water due to the <a href="https://bio.libretexts.org/Learning_Objects/Worksheets/Biology_Tutorials/Diffusion_and_Osmosis">process of diffusion</a>, where molecules move from an area of higher concentration to lower concentration. Eventually, the water will have an even concentration of blue and red dyes and become purple. Thus, diffusion tends to create color uniformity.</p>
<p>A question naturally arises: How can distinct color patterns form in the presence of diffusion?</p>
<h2>Movement and boundaries</h2>
<p>Mathematician Alan Turing first addressed this question in his seminal 1952 paper, “<a href="https://doi.org/10.1098/rstb.1952.0012">The Chemical Basis of Morphogenesis</a>.” Turing showed that under appropriate conditions, the chemical reactions involved in producing color can interact with each other in a way that counteracts diffusion. This makes it possible for colors to self-organize and create interconnected regions with different colors, forming what are now called Turing patterns. </p>
<p>However, in mathematical models, the boundaries between color regions are fuzzy due to diffusion. This is unlike in nature, where boundaries are often sharp and colors are well separated.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/558105/original/file-20231107-20-d6d55o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Close-up of head of moray eel with dark brown patches separated by uneven white boundaries." src="https://images.theconversation.com/files/558105/original/file-20231107-20-d6d55o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/558105/original/file-20231107-20-d6d55o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/558105/original/file-20231107-20-d6d55o.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/558105/original/file-20231107-20-d6d55o.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/558105/original/file-20231107-20-d6d55o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/558105/original/file-20231107-20-d6d55o.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/558105/original/file-20231107-20-d6d55o.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Moray eels have distinctive patterns on their skin.</span>
<span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/laced-leopard-moray-in-indian-ocean-during-a-scuba-royalty-free-image/1306632894">Asergieiev/iStock via Getty Images</a></span>
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<p>Our team thought a clue to figuring out how animals create distinctive color patterns could be found in lab experiments on micron-sized particles, such as the <a href="https://bio.libretexts.org/Bookshelves/Introductory_and_General_Biology/Biology_(Kimball)/03%3A_The_Cellular_Basis_of_Life/3.22%3A_Chromatophores">cells involved in producing the colors</a> of an animal’s skin. <a href="https://doi.org/10.1039/D0SM00899K">My work</a> and work from <a href="https://doi.org/10.1073/pnas.1511484112">other labs</a> found that micron-sized particles form <a href="https://doi.org/10.1103/PhysRevLett.117.258001">banded structures</a> when placed between a region with a high concentration of other dissolved solutes and a region with a low concentration of other dissolved solutes.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/558109/original/file-20231107-17-u1tewc.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Diagram of a large blue circle moving to the right as it's swept along with the medium-sized red circles surrounding it also moving to the right, where there is a higher concentration of small green circles" src="https://images.theconversation.com/files/558109/original/file-20231107-17-u1tewc.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/558109/original/file-20231107-17-u1tewc.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=551&fit=crop&dpr=1 600w, https://images.theconversation.com/files/558109/original/file-20231107-17-u1tewc.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=551&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/558109/original/file-20231107-17-u1tewc.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=551&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/558109/original/file-20231107-17-u1tewc.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=693&fit=crop&dpr=1 754w, https://images.theconversation.com/files/558109/original/file-20231107-17-u1tewc.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=693&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/558109/original/file-20231107-17-u1tewc.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=693&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The blue circle in this diagram is moving to the right due to diffusiophoresis, as it is swept along with the motion of the red circles moving into an area where there are more green circles.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Schematic_of_particle_illustrating_diffusiophoresis.png">Richard Sear/Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>In the context of our thought experiment, changes in the concentration of blue and red dyes in water can propel other particles in the liquid to move in certain directions. As the red dye moves into an area where it is at a lower concentration, nearby particles will be carried along with it. This phenomenon is <a href="https://doi.org/10.1039/C6SM00052E">called diffusiophoresis</a>. </p>
<p>You benefit from diffusiophoresis whenever you <a href="https://doi.org/10.1103/PhysRevApplied.9.034012">do your laundry</a>: Dirt particles move away from your clothing as soap molecules diffuse out from your shirt and into the water.</p>
<h2>Drawing sharp boundaries</h2>
<p>We wondered whether Turing patterns composed of regions of concentration differences could also move micron-sized particles. If so, would the resulting patterns from these particles be sharp and not fuzzy? </p>
<p>To answer this question, we <a href="http://www.science.org/doi/10.1126/sciadv.adj2457">conducted computer simulations</a> of Turing patterns – including hexagons, stripes and double spots – and found that diffusiophoresis makes the resulting patterns significantly more distinctive in all cases. These diffusiophoresis simulations were able to replicate the intricate patterns on the skin of the ornate boxfish and jewel moray eel, which isn’t possible through Turing’s theory alone.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/pU-EB6fa0As?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">This video shows small particles moving due to a related phenomenon called diffusioosmosis.</span></figcaption>
</figure>
<p>Further supporting our hypothesis, our model was able to reproduce the findings of a <a href="https://doi.org/10.1038/s41567-021-01213-3">lab study</a> on how the bacterium <em>E. coli</em> moves molecular cargo within themselves. Diffusiophoresis resulted in sharper movement patterns, confirming its role as a physical mechanism behind biological pattern formation. </p>
<p>Because the cells that produce the pigments that make up the colors of an animal’s skin are also micron-sized, our findings suggest that diffusiophoresis may play a key role in creating distinctive color patterns more broadly in nature.</p>
<h2>Learning nature’s trick</h2>
<p>Understanding how nature programs specific functions can help researchers design synthetic systems that perform similar tasks. </p>
<p>Lab experiments have shown that scientists can use diffusiophoresis to create <a href="https://doi.org/10.1038/ncomms15181">membraneless water filters</a> and <a href="https://doi.org/10.1002/adma.201701516">low-cost drug development tools</a>.</p>
<p>Our work suggests that combining the conditions that form Turing patterns with diffusiophoresis could also form the basis of artificial skin patches. Just like adaptive skin patterns in animals, when Turing patterns change – say from hexagons to stripes – this indicates underlying differences in chemical concentrations inside or outside the body. </p>
<p>Skin patches that can sense these changes could diagnose medical conditions and monitor a patient’s health by detecting changes in biochemical markers. These skin patches could also sense changes in the concentration of harmful chemicals in the environment.</p>
<h2>The work ahead</h2>
<p>Our simulations exclusively focused on spherical particles, while the cells that create pigments in skin come in varying shapes. The effect of shape on the formation of intricate patterns remains unclear. </p>
<p>Furthermore, pigment cells move in a complicated biological environment. More research is needed to understand how that environment inhibits motion and potentially freezes patterns in place.</p>
<p>Besides animal skin patterns, Turing patterns are also crucial to other processes such as <a href="https://doi.org/10.1042%2FBST20200013">embryonic development</a> and <a href="https://doi.org/10.1016/j.neo.2020.09.008">tumor formation</a>. Our work suggests that diffusiophoresis may play an underappreciated but important role in these natural processes.</p>
<p>Studying how biological patterns form will help researchers move one step closer to mimicking their functions in the lab – <a href="https://doi.org/10.1038/s41427-021-00322-y">an age-old endeavor</a> that could benefit society.</p><img src="https://counter.theconversation.com/content/217035/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ankur Gupta receives funding from NSF (CBET - 2238412) and ACS Petroleum Research Fund (65836 - DNI9). </span></em></p>Understanding how the intricate spots and stripes, or Turing patterns, of many animals form can help scientists mimic those processes in the lab.Ankur Gupta, Assistant Professor of Chemical and Biological Engineering, University of Colorado BoulderLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1864332022-09-19T12:23:11Z2022-09-19T12:23:11ZWhy does nature create patterns? A physicist explains the molecular-level processes behind crystals, stripes and basalt columns<figure><img src="https://images.theconversation.com/files/475662/original/file-20220722-228-pcle5n.jpg?ixlib=rb-1.1.0&rect=10%2C21%2C7182%2C4792&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Giant's Causeway in Northern Ireland features around 40,000 exposed polygonal columns of basalt in perfect horizontal sections.</span> <span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/the-unesco-world-heritage-site-giants-causeway-in-royalty-free-image/1311800218?adppopup=true">Chris Hill/Photodisc via Getty Images</a></span></figcaption></figure><figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/281719/original/file-20190628-76743-26slbc.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/281719/original/file-20190628-76743-26slbc.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=293&fit=crop&dpr=1 600w, https://images.theconversation.com/files/281719/original/file-20190628-76743-26slbc.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=293&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/281719/original/file-20190628-76743-26slbc.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=293&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/281719/original/file-20190628-76743-26slbc.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=368&fit=crop&dpr=1 754w, https://images.theconversation.com/files/281719/original/file-20190628-76743-26slbc.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=368&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/281719/original/file-20190628-76743-26slbc.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=368&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<p><em><a href="https://theconversation.com/us/topics/curious-kids-us-74795">Curious Kids</a> is a series for children of all ages. If you have a question you’d like an expert to answer, send it to <a href="mailto:curiouskidsus@theconversation.com">curiouskidsus@theconversation.com</a>.</em></p>
<hr>
<blockquote>
<p><strong>Why does nature always create a pattern? – Saloni G., age 16, Alwar, Rajasthan, India</strong></p>
</blockquote>
<hr>
<p>The reason patterns often appear in nature is simple: The same basic physical or chemical processes occur in many patterned substances and organisms as they form. Whether in plants and animals or rocks, foams and ice crystals, the intricate patterns that happen in nature come down to what’s happening at the level of atoms and molecules.</p>
<p>A pattern in nature is any regularly repeated arrangement of shapes or colors. Some of the most striking examples include the hexagonal arrays of rocks at Giant’s Causeway in the United Kingdom, the beautiful fractal arrangements of florets on a Romanesco broccoli and the colorful stripes and spots on tropical fish.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/477244/original/file-20220802-12076-qc0cix.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Close-up of Romanesco broccoli bunches, showing off the fractal pattern of the buds" src="https://images.theconversation.com/files/477244/original/file-20220802-12076-qc0cix.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/477244/original/file-20220802-12076-qc0cix.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/477244/original/file-20220802-12076-qc0cix.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/477244/original/file-20220802-12076-qc0cix.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/477244/original/file-20220802-12076-qc0cix.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/477244/original/file-20220802-12076-qc0cix.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/477244/original/file-20220802-12076-qc0cix.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Each bud of a Romanesco broccoli bunch is composed of a series of smaller buds, arranged in a consistent spiral pattern.</span>
<span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/romanesco-close-up-royalty-free-image/79560931?adppopup=true">Creativ Studio Heinemann/Westend61 via Getty Images</a></span>
</figcaption>
</figure>
<p>Patterns like these begin to form at a small scale when materials undergo processes like drying, freezing, wrinkling, diffusing and reacting. Those changes then give rise to complex patterns at a larger scale that people can see.</p>
<h2>Patterns in ice and rock</h2>
<p>Imagine delicate frozen crystals on a windowpane during a cold day. What creates that pattern? </p>
<p>When water freezes, its molecules begin clustering together. Water molecules have a particular bent shape that causes them to stack into clusters shaped like hexagons as they freeze.</p>
<p>As the cluster grows, many outside <a href="https://doi.org/10.1088/0034-4885/68/4/R03">factors, including humidity and temperature</a>, begin to affect its overall shape. If the water is freezing on a windowpane, for example, small and random imperfections on the glass surface redirect the stacking and create the larger pattern.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/477252/original/file-20220802-14-5kue1b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Frost on an old window." src="https://images.theconversation.com/files/477252/original/file-20220802-14-5kue1b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/477252/original/file-20220802-14-5kue1b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/477252/original/file-20220802-14-5kue1b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/477252/original/file-20220802-14-5kue1b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/477252/original/file-20220802-14-5kue1b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/477252/original/file-20220802-14-5kue1b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/477252/original/file-20220802-14-5kue1b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Ice crystals on an old window in Norway.</span>
<span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/frost-on-a-old-window-royalty-free-image/1191072525?adppopup=true">Baac3nes/Moment via Getty Images</a></span>
</figcaption>
</figure>
<p>This same process of stacking molecules is responsible for the striking variety of snowflake shapes.</p>
<p>What about the amazing patterns of the basalt columns at Giant’s Causeway? These formed 50 million to 60 million years ago, as lava – hot rocky fluid from deep underground – rose to the Earth’s surface and began to lose heat. The cooling caused the top layer of basalt to contract. The deeper, hot layers resisted this pulling, creating cracks in the top layer. </p>
<p>As the lava cooled, the cracks spread deeper and deeper into the rock. The particular molecular qualities of basalt, as well as the <a href="https://doi.org/10.1063/PT.3.2584">basic physics of how materials fracture apart</a> – laws of physics universal to all substances on Earth – caused the cracks to meet up with one another at certain angles to create hexagons, much like the stacking water molecules.</p>
<p>Eventually, the cooling basalt broke into the hexagon-shaped columns of rock that still create such an impressive pattern millions of years later.</p>
<h2>Patterns in animals</h2>
<p>The creation of complex patterns in living organisms also begins with simple mechanisms at the molecular level. One important pattern-making process involves the <a href="https://doi.org/10.1098/rstb.2014.0218">way diffusing chemicals react with one another</a>.</p>
<p>Imagine how a drop of food coloring spreads in a glass of water – that’s diffusion. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/478598/original/file-20220810-7093-ifsy3x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Drops of blue dye diffusing in water." src="https://images.theconversation.com/files/478598/original/file-20220810-7093-ifsy3x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/478598/original/file-20220810-7093-ifsy3x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=375&fit=crop&dpr=1 600w, https://images.theconversation.com/files/478598/original/file-20220810-7093-ifsy3x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=375&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/478598/original/file-20220810-7093-ifsy3x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=375&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/478598/original/file-20220810-7093-ifsy3x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=471&fit=crop&dpr=1 754w, https://images.theconversation.com/files/478598/original/file-20220810-7093-ifsy3x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=471&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/478598/original/file-20220810-7093-ifsy3x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=471&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Drops of blue dye at different stages of diffusion in water.</span>
<span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/diffusion-in-water-royalty-free-image/460717093?adppopup=true">Science Photo Library via Getty Images</a></span>
</figcaption>
</figure>
<p>In 1952, English mathematician Alan Turing showed that a chemical spreading like this within another chemical can lead to the <a href="https://doi.org/10.1038/s42254-022-00486-8">formation of all kinds of patterns</a> in nature.</p>
<p>Scientists have proved that this process reproduces the patterns of a leopard’s spots, a zebra’s stripes and many other animal markings.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/477259/original/file-20220802-10039-40ywgd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Wild royal bengal female tiger on prowl – her stripes blending her in with the vegetation around her." src="https://images.theconversation.com/files/477259/original/file-20220802-10039-40ywgd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/477259/original/file-20220802-10039-40ywgd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/477259/original/file-20220802-10039-40ywgd.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/477259/original/file-20220802-10039-40ywgd.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/477259/original/file-20220802-10039-40ywgd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/477259/original/file-20220802-10039-40ywgd.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/477259/original/file-20220802-10039-40ywgd.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A tiger’s stripes can help it blend in with the surrounding environment – making it harder for prey to see.</span>
<span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/wild-royal-bengal-female-tiger-on-prowl-for-royalty-free-image/1366964135?adppopup=true">Sourabh Bharti/iStock via Getty Images Plus</a></span>
</figcaption>
</figure>
<p>What makes these markings consistent from generation to generation? As animal species evolved, these chemical reactions evolved with them and became part of their genetic codes. This might be because the markings helped them survive. For example, a tiger’s stripes camouflage it while hunting in a forest or grassland, making it easier to surprise and catch its prey.</p>
<p>However, researchers are still working out the details of which particular chemicals are involved. </p>
<p>Scientists do not always know the purpose of a pattern, or even if there is one. The molecular processes involved are simple enough that they might coincidentally generate a pattern. </p>
<p>For example, in my research team’s work studying plant pollen grains, we have <a href="https://doi.org/10.1016/j.cell.2019.01.014">seen a huge variety of patterns</a>, including spikes, stripes and many more. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/479983/original/file-20220818-546-47zyn2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Colorized scanning electron microscope image of pollen grains from a variety of common plants" src="https://images.theconversation.com/files/479983/original/file-20220818-546-47zyn2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/479983/original/file-20220818-546-47zyn2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=457&fit=crop&dpr=1 600w, https://images.theconversation.com/files/479983/original/file-20220818-546-47zyn2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=457&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/479983/original/file-20220818-546-47zyn2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=457&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/479983/original/file-20220818-546-47zyn2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=574&fit=crop&dpr=1 754w, https://images.theconversation.com/files/479983/original/file-20220818-546-47zyn2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=574&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/479983/original/file-20220818-546-47zyn2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=574&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The pollen grains of various common plants like sunflower, morning glories, prairie hollyhock, oriental lily, evening primrose and castor bean – magnified 500 times and colorized in this image – display intricate patterns.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Pollen#/media/File:Misc_pollen_colorized.jpg">Dartmouth Electron Microscope Facility</a></span>
</figcaption>
</figure>
<p>We don’t yet understand why a plant produces one particular pollen pattern rather than another. Whatever the ultimate use this and other patterns in nature may have, their variety, complexity and order are amazing.</p>
<hr>
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<p class="fine-print"><em><span>Maxim Lavrentovich receives funding from the National Science Foundation. </span></em></p>Nature begins forming patterns at the molecular level – and sometimes they grow to enormous sizes.Maxim Lavrentovich, Assistant Professor of Theoretical Biophysics, University of TennesseeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1799872022-05-05T20:05:00Z2022-05-05T20:05:00ZArchitect Christopher Alexander mined mathematics to find patterns for good living<figure><img src="https://images.theconversation.com/files/461355/original/file-20220504-16-8af3ef.jpg?ixlib=rb-1.1.0&rect=0%2C13%2C1519%2C916&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The Julian Sreet Inn, Shelter for the Homeless, in San Jose, Calif., designed by Christopher Alexander. </span> <span class="attribution"><a class="source" href="https://creativecommons.org/licenses/by-nc-sa/2.0/">(David Ing/Flickr)</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span></figcaption></figure><p>Architect and mathematician Christopher Alexander died at age 85 on March 17. </p>
<p>“The end of an era,” one of my colleagues remarked. She was perhaps referring to Alexander’s <a href="https://www.architecturalrecord.com/articles/15586-tribute-christopher-alexander-1936-2022">influential trajectory</a> of over 30 years at the University of California, Berkeley.</p>
<p>Obituaries about Alexander portrayed him as a fierce critic of modern architecture and chronicled his quest for buildings and cities that displayed qualities of <a href="https://www.ribaj.com/culture/christopher-alexander-obituary-1936-2022">warmth</a> and <a href="https://www.theguardian.com/artanddesign/2022/mar/29/christopher-alexander-obituary">aliveness</a>.</p>
<p>Alongside the production of numerous papers, books and buildings, Alexander’s quest was marked by <a href="https://applied.math.utsa.edu/%7Eyxk833/Charles.html">acclaim from royalty</a>, <a href="http://www.katarxis3.com/Alexander_Eisenman_Debate.htm">architectural disputes</a> and being hailed as a <a href="https://www.nytimes.com/2022/03/29/arts/christopher-alexander-dead.html">countercultural hero</a>. </p>
<p><a href="https://www.taylorfrancis.com/chapters/edit/10.4324/9780429264306-4/bewildered-form-maker-stands-alone-theodora-vardouli">My research</a> has explored how Alexander used mathematics to help designers tackle unwieldy design requirements.</p>
<h2>‘A Pattern Language’</h2>
<p>Alexander’s countercultural reputation mainly stemmed from <a href="https://global.oup.com/academic/product/a-pattern-language-9780195019193"><em>A Pattern Language: Towns, Buildings, Construction</em></a>, which he co-authored with researchers from the <a href="https://www.patternlanguage.com/aims/intro-2.html">Center for Environmental Structure</a>, a non-profit corporation he co-founded at Berkeley in 1967.</p>
<p><em>A Pattern Language</em> featured photos, descriptions and diagrams of 253 patterns that Alexander explored as units for the design of buildings and cities. Patterns were linked to each other. The book covered patterns related to things like the distribution of towns (pattern 2), staircases (pattern 133) and chair types (pattern 251). </p>
<p>Each pattern came with detailed commentary on the principles that drove it, and the ways it would enable <a href="https://www.latimes.com/archives/la-xpm-1995-01-29-tm-25890-story.html">wholeness</a> and relationships between “<a href="http://caper.ws/patterns/apl8/apl8.htm">the great variety of human groups and subcultures which can co-exist</a>” in cities.</p>
<p>Underlying the accessible way Alexander presented the patterns was <a href="https://www.patternlanguage.com/bookstore/timeless-way-of-building.html">a rigorous mathematical logic</a> that defined their sequence and relationships. The dual nature of the book rendered it popular among <a href="https://www.patternlanguage.com/gallery/housingcommunity.html">amateur designers</a> and <a href="https://mitpress.mit.edu/books/architectural-intelligence">software engineers</a> alike. </p>
<p>The book also incited <a href="http://www.girlwonder.com/blog/wp-content/uploads/2020/12/Volume-57_Bye-Default_Molly-Steenson.pdf">critique by architects</a> who were skeptical of its claims to universality and comprehensiveness. </p>
<p>The anti-establishment American magazine <a href="https://www.newyorker.com/news/letter-from-silicon-valley/the-complicated-legacy-of-stewart-brands-whole-earth-catalog"><em>Whole Earth Catalog</em></a> <a href="https://nevalalee.wordpress.com/tag/the-next-whole-earth-catalog/">dedicated a full page to the book</a> as a tool for DIY design and building. The book even continues to inspire <a href="http://www.patternlanguage.com/">design today</a>.</p>
<h2>‘A City is Not a Tree’</h2>
<p>An important juncture in Alexander’s theoretical explorations toward <em>A Pattern Language</em> was his musing about how a city could emulate the structure of living things and beautiful works of art. He did this in his article “<a href="https://www.patternlanguage.com/archive/cityisnotatree.html">A City is Not a Tree</a>.” </p>
<p>A tree, here, is <a href="https://mathworld.wolfram.com/Tree.html">a mathematical term</a> referring to a hierarchical ordering of elements. Alexander critiqued thinking about urban systems in terms of independent parts. He proposed that instead, these parts should be more interconnected.</p>
<p>“A City is Not a Tree” was a critique of his earlier book, <a href="https://monoskop.org/images/f/ff/Alexander_Christopher_Notes_on_the_Synthesis_of_Form.pdf"><em>Notes on the Synthesis of Form</em></a>. Here, Alexander had proposed breaking down complex design problems into hierarchical trees. </p>
<p>Published in 1964, the <em>Notes</em> presented a mathematical method for breaking complex design problems into smaller ones. The book also pioneered computation in architecture and kindled worldwide efforts to bring <a href="https://monoskop.org/images/6/66/Cross_Nigel_1993_A_History_of_Design_Methodology.pdf">scientific rigour to design</a>.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/VZHb9-Y9r_E?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Christopher Alexander recorded at the University of Oregon in 1993.</span></figcaption>
</figure>
<h2>Hidden mathematical structure</h2>
<p>Throughout his career, Alexander spoke of a hidden mathematical structure underlying empirical particulars. </p>
<p>Alexander was trained as a mathematician at Cambridge University. There, he was exposed to “modern” mathematics <a href="https://press.princeton.edu/books/ebook/9781400829040/platos-ghost">that focused not on measurements or geometric shapes, but on abstract structures</a>. </p>
<p>It would seem that Alexander imported such ideals of abstraction through structures to architecture. But Alexander’s PhD progress reports in the archive of architect <a href="https://web.archive.org/web/20060516122752/http://www.artic.edu/aic/libraries/caohp/chermayeff.html">Serge Chermayeff</a>, who was a member of his doctoral committee, suggest that work on the <em>Notes</em> began with <a href="https://www.taylorfrancis.com/chapters/edit/10.4324/9780429264306-4/bewildered-form-maker-stands-alone-theodora-vardouli">practical concerns with how to design mass-industrialized housing</a>. </p>
<p><a href="https://www.journals.uchicago.edu/doi/10.1086/691132">Inspired by the flourishing field of game theory</a>, Alexander first imagined a design process as a co-operative game between architects and the public. </p>
<p>The aim was to find a middle ground between architects following public taste and architects imposing theirs. The game’s foundation would be extensive data collection about public needs and preferences, as well as architects’ own preferences and ideals. But under what categories to classify all that data? </p>
<figure class="align-center ">
<img alt="View of a home's hallway, showing wood walls and a built-in bench under a large window facing onto trees." src="https://images.theconversation.com/files/461371/original/file-20220504-15-odhl9j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/461371/original/file-20220504-15-odhl9j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/461371/original/file-20220504-15-odhl9j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/461371/original/file-20220504-15-odhl9j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/461371/original/file-20220504-15-odhl9j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/461371/original/file-20220504-15-odhl9j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/461371/original/file-20220504-15-odhl9j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The front hall of a home designed by Alexander, Sala House, in Albany, N.Y.</span>
<span class="attribution"><a class="source" href="https://creativecommons.org/licenses/by-sa/4.0/">(Ekyono/Wikimedia Commons)</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<h2>What is a house made of?</h2>
<p>In 1959, Alexander advanced this question through a project called “The Urban House,” with Chermayeff at the <a href="https://www.jchs.harvard.edu/about/history">MIT-Harvard Joint Center for Urban Studies</a>. They asked: What is a house made of? Alexander’s answer was that it depends on the data: The data one gathers about a specific design problem ought to dictate the categories for thinking about it and for designing it. </p>
<p>Instead of thinking about a house in terms of conventional categories such as kitchens, bedrooms, windows and doors, analyzing data about people’s and architects’ behaviours, needs or preferences would define an altogether different set of categories. </p>
<p>Alexander suggested thinking of the house in terms of its failures: how its physical attributes caused it to fail meeting specific needs or requirements identified during data collection. Each failure was associated with data. </p>
<p>Examining relationships between the data would help hierarchically organize these failures and indicate the order in which architects should tackle them. Alexander also co-developed a <a href="https://www.youtube.com/watch?v=v8Gu1zq4vwg">computer program</a> implementing that method. </p>
<h2>Messy data and clean algorithms</h2>
<p><a href="https://vimeo.com/340546754">In several stages of his work</a> Alexander grappled with the relationship between concrete details stemming from observation and abstract mathematical structures that he argued held everything together. As I continue to explore in my research, mathematical structures in Alexander’s work gradually took lives of their own and became severed from the data that gave rise to them in the first place.</p>
<p>Alexander’s work will no doubt continue to be important and relevant in light of burgeoning contemporary debates about <a href="https://mitpress.mit.edu/books/all-data-are-local">how data always comes from specific settings</a> and <a href="https://mitpress.mit.edu/books/your-computer-fire">algorithmic bias</a>. </p>
<p>The story of how the tree came about and evolved in Alexander’s work shows that behind algorithms lie messy and subjective processes of <a href="https://annenberg.usc.edu/news/critical-conversations/kate-crawford-maps-world-extraction-and-exploitation-atlas-ai">extracting information</a> — and that mathematical abstraction sometimes works to conceal them.</p><img src="https://counter.theconversation.com/content/179987/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Theodora Vardouli receives funding from the Social Sciences and Humanities Research Council of Canada. </span></em></p>Architect Christopher Alexander’s work will continue to be important not only for designing buildings but also in light of contemporary debates about how data always comes from specific settings.Theodora Vardouli, Assistant Professor, Peter Guo-hua Fu School of Architecture, McGill UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1760932022-02-17T15:10:12Z2022-02-17T15:10:12ZHappy Twosday! Why numbers like 2/22/22 have been too fascinating for over 2,000 years<figure><img src="https://images.theconversation.com/files/446389/original/file-20220214-25-kswfz8.jpg?ixlib=rb-1.1.0&rect=20%2C0%2C2180%2C1325&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Is "Twosday" as special as some corners of the internet seem to think?</span> <span class="attribution"><span class="source">articular/iStock via Getty Images Plus</span></span></figcaption></figure><p>This Feb. 22, the world hits an unprecedented milestone. It’s the date itself: 2/22/22. And this so-called “Twosday” falls on a Tuesday, no less. </p>
<p>It’s true the number pattern stands out, impossible to miss. But does it mean anything? Judging by the thousands of commemorative products available for purchase online, it may appear to.</p>
<p>“Twosday” carries absolutely no historical significance or any cosmic message. Yet it does speak volumes about our brains and cultures.</p>
<p><a href="https://scholar.google.com/citations?user=ZEQu09wAAAAJ&hl=en">I’m a social psychologist</a> who studies how <a href="https://journals.sagepub.com/doi/abs/10.1525/sop.2001.44.1.21">paranormal claims and pseudoscience</a> take hold as popular beliefs. They’re nearly always absurd from a scientific perspective, but they’re great for illustrating how brains, people, groups and cultures work together to create shared meaning.</p>
<h2>Seeing patterns</h2>
<p>Twosday isn’t the only date with a striking pattern. This century alone has had a couple Onesdays (1/11/11 and 11/11/11), and 11 other months with repetitions such as 01/01/01, 06/06/06 and 12/12/12. We’ll hit Threesday, 3/3/33, in 11 years, and Foursday 11 years after that.</p>
<p>The brain has <a href="https://global.oup.com/academic/product/the-adapted-mind-9780195101072?q=tooby&lang=en&cc=us">evolved</a> a fantastic capacity to find meanings and connections. Doing so once meant the difference between survival and death. Recognizing paw prints in the soil, for example, signified dangerous predators to be avoided, or prey to be captured and consumed. Changes in daylight indicated when to plant crops and when to harvest them. </p>
<p>Even when survival isn’t at stake, it’s <a href="https://www.sciencedaily.com/releases/2018/05/180531114642.htm">rewarding</a> to detect a pattern such as a <a href="https://www.sciencedaily.com/releases/2016/10/161003093240.htm">familiar face</a> or <a href="https://doi.org/10.1073/pnas.1811878116">song</a>. Finding one, the brain zaps its synapses with a little shot of dopamine, incentivizing itself to keep finding more patterns.</p>
<p>When a number sequence seems to jump out at us, this is an example of <a href="https://www.psychologytoday.com/us/blog/reality-check/201111/11-11-11-apophenia-and-the-meaning-life">apophenia</a>: perceiving meaningful connections between unrelated things. The term was <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2800156/">first developed</a> to characterize a symptom of schizophrenia.</p>
<p>Another example of apophenia is <a href="https://skepticalinquirer.org/exclusive/astrology-more-like-religion-than-science/">astrology</a>, which visually connects stars into constellations. These are the familiar Zodiac signs such as “The Ram,” Aries; or “The Archer,” Sagittarius. Each sign is linked to meanings associated with its respective object. For example, people born under the sign of Aries are believed to be stubborn like rams. But those signs don’t exist in the sky in any physical sense, and the system <a href="https://skepticalinquirer.org/2016/11/does-astrology-need-to-be-true-a-thirty-year-update/">fails scientific tests</a>.</p>
<h2>Reading into numbers</h2>
<p>The date 2/22/22, though striking, carries no inherent meaning beyond its function in our particular calendar. This is true for numbers in general: Their meanings are limited to measuring, labeling or counting things.</p>
<p>“Twosday” is a simple example of a popular form of arithmetical shenanigans: <a href="https://www.dundurn.com/books_/t22117/a9781459705371-mysteries-and-secrets-of-numerology">numerology</a>, the pseudoscientific practice of attaching supernatural significance to numbers. </p>
<p>Numerology can be <a href="https://global.oup.com/academic/product/the-mystery-of-numbers-9780195089196?q=mystery%20of%20numbers&lang=en&cc=us">traced back</a> 2,500 years to the Greek mathematician Pythagoras, with alternative systems appearing elsewhere, including China and the Middle East. </p>
<p>Numerology may look mathematical, but it’s more akin to palmistry and reading tea leaves. It has been popularized through magazines, books, movies, television programs, websites and other social media. Assessing the extent of numerology’s popularity is difficult, but the belief that certain numbers are good or bad is common. For example, nearly a quarter of Americans <a href="https://www.statista.com/statistics/297156/united-states-common-superstitions-believe/">say 7 is lucky</a>.</p>
<p>There are <a href="https://www.jstor.org/stable/15626">many kinds</a> of numerology. The most popular form assigns numbers to names or other words, and then calculates their “root,” also known as the “<a href="https://www.allure.com/story/numerology-how-to-calculate-life-path-destiny-number">destiny number</a>” or “<a href="https://www.numerology.com/articles/your-numerology-chart/expression-number/">expression number</a>”. It starts by assigning a number to each letter of the alphabet: A = 1, B = 2, up to I = 9, then the cycle repeats with J = 1, K = 2, etc. </p>
<p>For example, adding up the five numbers in my own first name – 2, 1, 9, 9, and 7 – yields 28. To find the root, add the digits in 28 to get 10, and then add up those two digits to get 1. For my middle and last names, the roots are 4 and 9. Adding the three roots returns 14; adding those digits reveals that my “destiny number” is 5, which numerology associates with being free-thinking, <a href="https://mattbeech.com/numerology/destiny-number/destiny-5/">adventurous, restless</a> and impatient.</p>
<figure class="align-center ">
<img alt="A photograph of someone's hand as they do numerology calculations at a table covered with geodes and a feather." src="https://images.theconversation.com/files/446391/original/file-20220214-25314-17x5gyk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/446391/original/file-20220214-25314-17x5gyk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=406&fit=crop&dpr=1 600w, https://images.theconversation.com/files/446391/original/file-20220214-25314-17x5gyk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=406&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/446391/original/file-20220214-25314-17x5gyk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=406&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/446391/original/file-20220214-25314-17x5gyk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=510&fit=crop&dpr=1 754w, https://images.theconversation.com/files/446391/original/file-20220214-25314-17x5gyk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=510&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/446391/original/file-20220214-25314-17x5gyk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=510&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A woman calculates a destiny number based on numerology.</span>
<span class="attribution"><span class="source">Helin Loik-Tomson/iStock via Getty Images Plus</span></span>
</figcaption>
</figure>
<h2>More than coincidence?</h2>
<p>I was 10 years old when I first encountered numerology. A fellow coin collector showed me a clear plastic case holding two gleaming specimens: a copper Lincoln penny and a silver John F. Kennedy half dollar. On the back of the case was a printed label with numerical “facts” <a href="https://www.snopes.com/fact-check/linkin-kennedy/">linking the two presidents</a>. For example:</p>
<p>6: day of the week – Friday – of both assassinations</p>
<p>7: letters in Kennedy’s and Lincoln’s last names</p>
<p>15: letters in both assassins’ names</p>
<p>60: year elected – Lincoln 1860, Kennedy 1960</p>
<p>When you compile enough of these, it gets eerie. The experience was astonishing enough that I still recall it over a half-century later.</p>
<p>Are the Lincoln-Kennedy facts just coincidences? What gets overlooked is that they’ve been drawn from a pool of hundreds or thousands of numerical possibilities. Throw away the boring ones and you’ve framed the remaining coincidences in a way that gives them more credit than they deserve.</p>
<p>[<em>More than 140,000 readers get one of The Conversation’s informative newsletters.</em> <a href="https://memberservices.theconversation.com/newsletters/?source=inline-140K">Join the list today</a>.]</p>
<p>Another way of drawing eerie coincidences from very large pools of possibilities was exploited in “<a href="https://www.simonandschuster.com/books/The-Bible-Code/Michael-Drosnin/9780684849737">The Bible Code</a>,” a best-selling book in the 1990s. The author, Michael Drosnin, took the Old Testament and arranged it into <a href="https://skepticalinquirer.org/1997/11/hidden-messages-and-the-bible-code">a grid of text</a>. A computer algorithm <a href="https://www.ams.org/notices/199708/review-allyn.pdf">highlighted skip patterns in the grid</a>, such as “every 4th character”, or “2 across, 5 down,” to produce a huge database of letter strings. These were then sifted by another algorithm that searched for words and phrases, and distances between them.</p>
<p>The method seemed to foretell many historical events, including <a href="https://www.theguardian.com/world/2020/oct/31/assassination-yitzhak-rabin-never-knew-his-people-shot-him-in-back">the murder of Israeli Prime Minister Yitzhak Rabin</a> in 1995: A particular skip pattern yielded his name near the phrase “assassin that will assassinate.”</p>
<p>Findings such as these can seem impressive. However, <a href="https://users.cecs.anu.edu.au/%7Ebdm/codes/torah.html">critics</a> have proved that the method works just as well using any <a href="https://users.cecs.anu.edu.au/%7Ebdm/codes/torah.html">sufficiently lengthy text</a>. Drosnin himself laid down this gauntlet by challenging critics to find Rabin’s assassination foretold in the novel “Moby-Dick.” <a href="http://users.cecs.anu.edu.au/%7Ebdm/">Mathematician Brendan McKay</a> <a href="http://users.cecs.anu.edu.au/%7Ebdm/dilugim/moby.html">did exactly that</a>, along with “prophecies” for many other deaths – Lincoln’s and Kennedy’s included.</p>
<p>Which coincidences people pay attention to is largely a social phenomenon. What <a href="https://www.stonybrook.edu/commcms/sociology/people/faculty/goode.php">sociologist Erich Goode</a> terms “<a href="http://prometheusbooks.com/books/search/goode,%20erich">paranormalism</a>,” a nonscientific approach to extraordinary claims, is sustained and transmitted by group customs, norms and institutions. “The Bible Code” couldn’t exist without religion, for example, and its popularity was fueled by mass media – such as its author’s <a href="https://www.nytimes.com/2020/06/19/books/michael-drosnin-dead.html">interviews</a> on “The Oprah Winfrey Show” and elsewhere. In her book “<a href="https://mcfarlandbooks.com/product/scientifical-americans/">Scientifical Americans</a>,” science writer <a href="https://sharonahill.com/">Sharon Hill</a> makes a compelling case that popular culture in the U.S. helps to foster safe havens for individual and collective belief in the pseudoscientific and paranormal.</p>
<p>As for “Twosday,” I’ll conclude by plumbing its “hidden meaning.” Take the three roots of 02, 22 and 2022. We arrive at 2 + 4 + 6 = 12, and the destiny number 3. Some numerologists <a href="https://mattbeech.com/numerology/destiny-number/destiny-3/">associate this number with</a> optimism and joy. Though I may reject the messenger, I’ll accept that message.</p><img src="https://counter.theconversation.com/content/176093/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Barry Markovsky does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Numerology ties in with how our brains work, but that doesn’t mean its claims make sense.Barry Markovsky, Distinguished Professor Emeritus of Sociology, University of South CarolinaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1527492021-02-16T18:49:53Z2021-02-16T18:49:53ZHow new design patterns can enable cities and their residents to change with climate change<figure><img src="https://images.theconversation.com/files/382919/original/file-20210208-15-1naxqur.jpg?ixlib=rb-1.1.0&rect=533%2C0%2C6067%2C3984&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">cunaplus/Shutterstock</span></span></figcaption></figure><p>Our cities, designed for one set of climatic ranges, are increasingly “out of place” as average temperatures rise. The days above 40°C and nights above 30°C are <a href="https://www.climatecouncil.org.au/resources/angry-summer-report/">increasing</a>, especially in the expanding suburbs of Australian cities. This presents us with a massive redesign project. </p>
<p>Our <a href="https://www.coolingthecommons.com/">Cooling the Commons</a> <a href="https://www.landcom.com.au/assets/Approach/Cooling-the-Commons-Report.pdf">research</a> project, funded by Landcom, has launched a new approach using design patterns to guide how we design, and redesign, how we live in response to a changing climate. </p>
<p>A design pattern is first an observation: “People in that kind of designed situation tend to do this sort of thing”. It is then possible to design an intervention that redirects those tendencies. If that intervention succeeds, it can become a recommended pattern to help other designers: “If you encounter this kind of situation, try to make these kinds of interventions”.</p>
<p>Based on an international survey of what has worked well elsewhere, we have compiled a <a href="https://www.coolingthecommons.com/pattern%20deck/">bank of patterns</a>. These range from current patterns that increase heat and discomfort, through to remedial patterns for improving existing urban areas, to ideal patterns for new developments.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/cities-could-get-more-than-4-c-hotter-by-2100-to-keep-cool-in-australia-we-urgently-need-a-national-planning-policy-152680">Cities could get more than 4°C hotter by 2100. To keep cool in Australia, we urgently need a national planning policy</a>
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<h2>The problems with current approaches</h2>
<p>Conventionally, designing is done in three ways:</p>
<ol>
<li><p>a designer can try to (re)design everything all at once and for all time, an approach closest to current planning practices, particularly of “greenfield” sites involving building from scratch</p></li>
<li><p>a designer can seek a technical solution that can be widely replicated – think of mass-produced products, from phones to cars</p></li>
<li><p>a designer can produce a bespoke design for each client, crafting context-specific solutions one at a time. This is often how architects work.</p></li>
</ol>
<p>Given the scale of our cities, we are not in a position to start again – though climate change might force <a href="https://www.nytimes.com/interactive/2020/07/23/magazine/climate-migration.html">large numbers of people to move</a>.</p>
<p>Some places in China and the Middle East are experimenting with <a href="https://www.neom.com/en-us/">building wholly new cities</a>. However, such total designs can prove unable to adapt to changing circumstances <a href="https://www.weforum.org/agenda/2019/01/the-world-s-coastal-cities-are-going-under-here-is-how-some-are-fighting-back/">like climatic shifts</a> that demand cities be remade – “<a href="https://www.bloomsbury.com/uk/remaking-cities-9781474224154/">metrofitting</a>”. It is better to have modular designs that piece together, can pull apart and aim to remain modifiable over time. </p>
<p>The second, technical approach is what many people expect of designers these days. But this often adds to the problem by missing important differences from one place or community to another. </p>
<p>Air conditioners are a good example. While they may offer immediate relief in buildings that weren’t designed to promote natural ventilation, they also <a href="https://theconversation.com/australias-rising-air-con-use-makes-us-hot-and-bothered-20258">create problems</a>. </p>
<p>Not everyone can afford to buy and run air <a href="https://theconversation.com/high-energy-costs-make-vulnerable-households-reluctant-to-use-air-conditioning-study-86624">conditioners</a>, which greatly increase energy use. And many buildings are not designed to be air-conditioned <a href="https://theconversation.com/why-bad-housing-design-pumps-up-power-prices-for-everyone-22651">efficiently</a>. There are also social impacts such as blowing heat on neighbours and pedestrians, noisy external fans, and people <a href="https://www.westernsydney.edu.au/__data/assets/pdf_file/0020/1161470/cooling-the-commons-report.pdf">being isolated in their homes</a> on hot days.</p>
<figure class="align-center ">
<img alt="row of air conditioner units outside apartments" src="https://images.theconversation.com/files/382905/original/file-20210208-19-k5d9om.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/382905/original/file-20210208-19-k5d9om.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/382905/original/file-20210208-19-k5d9om.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/382905/original/file-20210208-19-k5d9om.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/382905/original/file-20210208-19-k5d9om.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/382905/original/file-20210208-19-k5d9om.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/382905/original/file-20210208-19-k5d9om.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Running air conditioners forces people to stay indoors to remain comfortable while adding to the heat outside.</span>
<span class="attribution"><span class="source">Konstantin L/Shutterstock</span></span>
</figcaption>
</figure>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/if-planners-understand-its-cool-to-green-cities-whats-stopping-them-55753">If planners understand it's cool to green cities, what's stopping them?</a>
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</em>
</p>
<hr>
<p>We need more systemic solutions than one-size-fits-all technologies like air conditioning.</p>
<p>However, the third kind of designing – creating tailored solutions for each unique situation – is too slow in the face of already changed climates.</p>
<p>This means designers need to adopt a fourth approach, known as pattern thinking. It helps designers to see what is not working well, where and when, and so how to redirect those situations toward more preferable ones. </p>
<h2>How does pattern thinking work?</h2>
<p>One kind of pattern is a set of rules specifying something that can be repeated over and over. This is the meaning of pattern normally associated with decorative forms, or with making clothes, templates for furniture, or blueprints for buildings.</p>
<p>But the patterns we are talking about, context-specific interactions between people and things, are more like habits. They are tendencies that lead to repeated actions. For example, consider the patterns of car-oriented urban development. </p>
<p>Hard-surfaced roads and driveways are major sources of urban heat. Car-oriented planning downplays patterns of walking, through the lack of footpaths, shade and pedestrian-oriented night lighting, or the distances between shops, schools and work. This means people who can afford it might get into the habit of staying in air-conditioned houses, only occasionally going in their air-conditioned cars to air-conditioned shopping malls.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/381699/original/file-20210201-19-1gffo5j.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="single car parked in parking lot with exposed path leading to building" src="https://images.theconversation.com/files/381699/original/file-20210201-19-1gffo5j.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/381699/original/file-20210201-19-1gffo5j.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/381699/original/file-20210201-19-1gffo5j.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/381699/original/file-20210201-19-1gffo5j.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/381699/original/file-20210201-19-1gffo5j.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/381699/original/file-20210201-19-1gffo5j.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/381699/original/file-20210201-19-1gffo5j.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">New cities, old car-oriented patterns: the car park and path to this library provide no relief from the hot sun.</span>
<span class="attribution"><span class="source">Photo: Helen Armstrong</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>To counter this, we need to create patterns for street shading along footpaths and around public transport stops. Generic tree plantings to meet <a href="https://www.greenerspacesbetterplaces.com.au/">abstract canopy coverage targets</a> are not enough. They must take into account the soil and moisture conditions of different neighbourhoods, and different use patterns, including <a href="https://www.coolingthecommons.com/pattern/caring-for-trees/">patterns of tree care</a>. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/a-solution-to-cut-extreme-heat-by-up-to-6-degrees-is-in-our-own-backyards-133082">A solution to cut extreme heat by up to 6 degrees is in our own backyards</a>
</strong>
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</p>
<hr>
<p>Related adaptive patterns might shift daytime activities into cooler night times. Some places already have these patterns: night markets and night-time use of outdoor spaces. </p>
<p>If locally adapted versions of these patterns encourage people to adopt new habits, other patterns will be needed. These will include, for example, ways to remind those cooling off outdoors in the evening that others might be trying to sleep with their naturally ventilating windows open. Such interlinked patterns point to the way pattern thinking moves from the big scale to the small. </p>
<p>To make the time to adapt each pattern to its local context, and then ensure those designs establish a pattern of long-lasting practice, requires a different <a href="https://theconversation.com/if-planners-understand-its-cool-to-green-cities-whats-stopping-them-55753">pattern of planning</a>. Planners need to be thinking about “staying with” what they plan, helping what they design to adapt to changing conditions and communities. For example, developers of <a href="https://theconversation.com/build-to-rent-could-shake-up-real-estate-but-wont-take-off-without-major-tax-changes-119603">build-to-rent</a> sites could hire “community liaison officers” to help tenants establish sustainable patterns of living. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/keeping-the-city-cool-isnt-just-about-tree-cover-it-calls-for-a-commons-based-climate-response-120491">Keeping the city cool isn't just about tree cover – it calls for a commons-based climate response</a>
</strong>
</em>
</p>
<hr>
<p><em>In addition to the authors, the Cooling the Commons research team includes: Professor Katherine Gibson, Associate Professor Louise Crabtree, Dr Stephen Healy and Dr Emma Power from the Institute for Culture and Society (ICS) at Western Sydney University (WSU), and Emeritus Professor Helen Armstrong from Queensland University of Technology (QUT).</em></p><img src="https://counter.theconversation.com/content/152749/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Cameron Tonkinwise received funding from Landcom. </span></em></p><p class="fine-print"><em><span>Abby Mellick Lopes receives funding from Landcom. </span></em></p>As our cities get hotter, rebuilding whole suburbs better suited to the heat is not an option. Instead, we can draw from the best examples of how to adapt neighbourhoods and behaviours.Cameron Tonkinwise, Professor, School of Design, University of Technology SydneyAbby Mellick Lopes, Associate Professor, Design Studies, Faculty of Design, Architecture and Building, University of Technology SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1414602020-07-28T09:22:56Z2020-07-28T09:22:56ZHow mutant zebrafish helped unlock the secret to their stripes – new research<figure><img src="https://images.theconversation.com/files/348379/original/file-20200720-64504-1tn17pj.jpg?ixlib=rb-1.1.0&rect=114%2C572%2C4048%2C2236&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/danio-rerio-236082157">Shutterstock/GrigorevMikhail</a></span></figcaption></figure><p>Zebrafish are one of the most well studied animals on the planet. But how they came by their beautiful black and gold stripes is more of a mystery. Our <a href="https://elifesciences.org/articles/52998">new research</a> used mathematical modelling – and detailed observations of mutant zebrafish patterns – to get to the bottom of one of nature’s oldest secrets. </p>
<p>Estimates suggest that zebrafish are used in over <a href="https://theconversation.com/animals-in-research-zebrafish-13804">600 labs around the world</a> to study diseases that range from <a href="https://academic.oup.com/hmg/article/12/suppl_2/R265/620445">muscular dystrophy</a> to <a href="https://core.ac.uk/download/pdf/2777109.pdf">cancer</a>. It may seem hard to imagine that a tiny tropical fish can tell us anything useful about distinctive human physiology but they are more similar to us than they appear at first glance. They have spines, hearts, livers, bones, eyes and kidneys.</p>
<p>Of equal importance is the opportunity that this hardy fish presents to investigate and understand the fundamental, and beautiful, biological processes that generate the spectacular pattern diversity seen in nature. These patterns are formed by the arrangement of pigments, usually packaged in specialised cells. </p>
<figure class="align-center ">
<img alt="Close up of zebrafish scales" src="https://images.theconversation.com/files/348122/original/file-20200717-21-khndmh.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/348122/original/file-20200717-21-khndmh.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=334&fit=crop&dpr=1 600w, https://images.theconversation.com/files/348122/original/file-20200717-21-khndmh.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=334&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/348122/original/file-20200717-21-khndmh.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=334&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/348122/original/file-20200717-21-khndmh.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=419&fit=crop&dpr=1 754w, https://images.theconversation.com/files/348122/original/file-20200717-21-khndmh.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=419&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/348122/original/file-20200717-21-khndmh.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=419&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Zoom into the zebrafish’s alternating pattern and the stripes of colour resolve into individual pigment cells.</span>
<span class="attribution"><span class="source">Wikimedia/JenniferOwen (adapted by Kit Yates)</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>From a distance zebrafish stripes look like long thin blocks of black or gold pigment. But on closer inspection it can be seen that these stripes are made up of thousands of small and distinct dots of colour. Each dot is a single pigment cell. The three cell types that produce the pattern are black melanophores, yellow xanthophores and silver and blue iridophores. Our research focused on understanding how enough of these cells interacting in the right way can result in the alternating striped patterns on a zebrafish.</p>
<figure class="align-center ">
<img alt="A mutant leopardfish" src="https://images.theconversation.com/files/347779/original/file-20200715-27-67r5db.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/347779/original/file-20200715-27-67r5db.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=474&fit=crop&dpr=1 600w, https://images.theconversation.com/files/347779/original/file-20200715-27-67r5db.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=474&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/347779/original/file-20200715-27-67r5db.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=474&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/347779/original/file-20200715-27-67r5db.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=596&fit=crop&dpr=1 754w, https://images.theconversation.com/files/347779/original/file-20200715-27-67r5db.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=596&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/347779/original/file-20200715-27-67r5db.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=596&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The patterns of the leopard mutant are spots rather than stripes.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/carolineccb/4050116019/">Flickr/carolineCCB</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>The mathematical theory that has predominated explanations of how zebrafish’s stripes emerged is <a href="https://theconversation.com/how-animals-got-their-spots-and-stripes-according-to-maths-85053">Turing patterning</a>. The mechanism is named after the visionary war hero, computer pioneer and mathematician <a href="https://theconversation.com/alan-turing-visionary-war-hero-and-the-only-choice-for-the-50-note-106470">Alan Turing</a> who first suggested it. In Turing patterns two different types of “agent” (melanophores and xanthophores in most zebrafish models) move around randomly and interact with each other in a special way, giving rise to a range of possible patterns. Although the patterns look convincing, scientists have not been able to prove this theory of <a href="https://www.crg.eu/en/news/new-theory-deepens-understanding-turing-patterns-biology">animal coat patterning</a>.</p>
<p>But <a href="https://elifesciences.org/articles/52998">our study</a> has demonstrated that the pattern formation mechanism is more complicated than a simple Turing model might suggest. As well as melanophores and xanthophores, we know iridophores also play an important role. These reflective cells give zebrafish their characteristic silvery appearance. Experiments have shown that without iridophores (or either of the other two cell types) the zebrafish’s characteristic striped pattern doesn’t form properly.</p>
<h2>Mutant zebrafish</h2>
<p>We wanted to find out which biological phenomena are crucial for pattern formation and which are just incidental. These sorts of questions can be answered with mathematical modelling.</p>
<p>We built an “agent-based” model (a computer code in which each cell is represented as an individual that can move and interact with others) which includes as much of the known biology of zebrafish patterning as possible. Once we had adapted the model to show it could reproduce the patterns seen in normal zebrafish we turned to the patterns formed by mutant zebrafish (fish with a genetic defect which changes their patterning) and tweaked the model rules to make sure it could replicate those too.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/Yqcd-GEKm9k?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">The model replicates the <em>pfeffer</em> zebrafish mutant’s broken black melanophore stripes on an iridescent iridophore background. <em>Pfeffer</em> mutants are deficient in yellow xanthophores.</span></figcaption>
</figure>
<p>Other mutant zebrafish, whose patterns were not used to build the model in the first place, acted as independent tests of the model’s pattern-replicating ability. Being able to mimic these other patterns gave us confidence in the model rules we had inferred. </p>
<p>As an example, the “choker” mutant has a defect which means silvery iridophores do not migrate to the skin in the normal manner. When we implemented this aberrant delivery of iridohphores in the model (but with essentially all the same rules) it neatly recreated the striking labyrinthine pattern seen in these fish.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/-_D4sO4mMbU?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">The model replicates the labyrinthine pattern exhibited by <em>choker mutants</em>. These mutant fish lack an initial horizontal stripe of iridophores in the middle of the embryo.</span></figcaption>
</figure>
<p>And then comes the really exciting part. The beauty of mathematical modelling is that once you’re confident your model captures the biology you can start to play around with it and ask biological questions that are difficult to answer through experiments alone.</p>
<p>For example, we were able to show that part of the reason zebrafish stripes are horizontal (as opposed to its mammalian namesake’s vertical stripes) is due to the way in which the body grows as the pattern forms. Faster growth along the head-tail axis (rather than the back-belly axis) of the fish tends to elongate groups of pigment cells into horizontal stripes rather than vertical bars.</p>
<p>Since pattern formation is an important general feature of organ development, there may be medical relevance to our research. A better understanding of pigment pattern formation might give us insights into the diseases caused by defects in cell arrangements.</p>
<p>With a working mathematical model there is no end to the questions we can ask about pigment pattern formation in zebrafish and other species. In particular, our next aim is to investigate the evolutionary origins of stripe formation in the broad family of Danio fish, of which the zebrafish (or Danio rerio) is a member. And that will help us gain an even deeper insight into how the zebrafish really got its stripes.</p><img src="https://counter.theconversation.com/content/141460/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christian Yates does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>We wanted to find out which biological phenomena are crucial for pattern formation and which are just incidental. These sorts of questions can be answered with mathematical modelling.Christian Yates, Senior Lecturer in Mathematical Biology, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1235462019-09-22T09:14:36Z2019-09-22T09:14:36ZAncient humans may have made patterns and sculptures on South Africa’s beaches<figure><img src="https://images.theconversation.com/files/293200/original/file-20190919-22446-voc66e.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A rock surface containing a circular pattern with a central depression. The scale bar = 10 cm.</span> <span class="attribution"><span class="source">Images modified from: Helm, C.W.; Cawthra, H.C.; De Vynck, J.C.; Helm, C.J.; Rust, R.; Stear. W. Patterns in the Sand: A Pleistocene hominin signature along the South African coastline? Proceedings of the Geologists’ Association (2019)</span></span></figcaption></figure><p>One of the first things many kids – or even adults – may do when they are on a beach or dune is to make patterns in the sand, or sculptures in the form of sandcastles.</p>
<p>Many generations of humans have enjoyed these activities. But until now there has been no reported evidence to suggest how far back in human history this may have occurred. Now my colleagues and I believe we may have <a href="https://doi.org/10.1016/j.pgeola.2019.08.004">found such evidence</a> at sites along South Africa’s Cape south coast. </p>
<p>Southern Africa boasts <a href="https://www.mdpi.com/2076-0752/2/1/6">an extensive record of palaeo-art</a>, and South Africa’s Cape south coast, stretching eastward along the coast from Cape Town, contains one of the richest <a href="https://www.researchgate.net/publication/285273001_Stone_Age_People_in_a_Changing_South_African_Greater_Cape_Floristic_Region">Middle Stone Age archaeological records</a> in the world. This includes <a href="https://science.sciencemag.org/content/334/6053/219">an engraved piece of ochre</a> and the <a href="https://www.nature.com/articles/s41586-018-0514-3">oldest reported example of rock painting</a>. Evidence suggests that the area may have been critical to <a href="https://www.nature.com/articles/nature06204">the survival</a> of the human species. </p>
<p>This coastal region now contains extensive aeolianites (cemented dune deposits) and cemented foreshore deposits. These rocks are the cemented remains of the dune and beach surfaces that existed when our distant ancestors and many other vertebrates were <a href="https://www.livescience.com/40311-pleistocene-epoch.html">making tracks</a> in the region in the Middle-Late Pleistocene, approximately 158,000 to 70,000 years ago. We know the ages of the rocks from <a href="https://doi.org/10.1016/j.quascirev.2010.10.003">the results</a> of previous <a href="https://www.sciencedirect.com/science/article/pii/S0031018207004476?">dating studies</a>. </p>
<p>It may seem that tracks and patterns made in the sand are ephemeral, destined to be covered by the effects of the next wind storm or tide. However, perhaps surprisingly, many of these records are preserved, ready to be identified when they are re-exposed through cliff collapse or through forces of erosion. Our team has identified more than <a href="https://doi.org/10.1016/j.quascirev.2019.07.039">140 vertebrate tracksites</a> along this coastline. For example, as many as 40 footprints made by hominins travelling down a dune surface, and estimated as being 90,000 years old, <a href="https://www.nature.com/articles/s41598-018-22059-5">were identified</a> at one site by members of our research team in 2016. </p>
<p>So, given that we know humans moved across these landscapes, we wondered whether there might also be evidence of other forms of human activity on these surfaces of sand, such as patterns, symbols, sculptures, or foraging. If so, could such ancient canvases have left evidence of human activity that can be discerned and interpreted today? Indeed, could such evidence form a previously undocumented form of Middle Stone Age hominin expression and activity? Our findings suggest the answer to these questions may be “yes”. </p>
<h2>A plethora of patterns</h2>
<p>At one site we found a large almost perfectly circular groove, along with a depression in the centre of the circle. Beside this feature was a pair of oval shapes that may represent knee impressions. If this circle was generated by a human, then a possible mechanism could have involved the use of a forked stick, in the same way that a compass is used by kids in maths classes. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/293202/original/file-20190919-22408-r85cy1.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/293202/original/file-20190919-22408-r85cy1.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=722&fit=crop&dpr=1 600w, https://images.theconversation.com/files/293202/original/file-20190919-22408-r85cy1.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=722&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/293202/original/file-20190919-22408-r85cy1.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=722&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/293202/original/file-20190919-22408-r85cy1.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=908&fit=crop&dpr=1 754w, https://images.theconversation.com/files/293202/original/file-20190919-22408-r85cy1.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=908&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/293202/original/file-20190919-22408-r85cy1.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=908&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The author demonstrates how a forked stick may have been used by a kneeling human to create a circular pattern in the sand.</span>
<span class="attribution"><span class="source">Linda Helm</span></span>
</figcaption>
</figure>
<p>Other patterns included groove features beside possible human footprints, and a “hashtag” pattern that <a href="http://dx.doi.org/10.1126/science.1211535">resembles known palaeo-art in the region</a>. We also identified two possible animal images, one of which may conceivably have taken the form of a sculpture of a sting-ray. We have proposed a new term to describe patterns made in sand by humans, which become lithified over time through a natural cementation process: ammoglyph (“ammos” being Greek for “sand”, and “glyph” being Greek for a carving, image or symbol).</p>
<p>If our interpretations are accurate, these findings represent two important things. Firstly, evidence of a human presence on these ancient dunes and beaches is more substantial than has been thought. Secondly, this evidence would buttress that of other avenues <a href="https://www.nature.com/articles/nature06204">of research</a> that <a href="https://www.sciencedirect.com/science/article/pii/S0047248404001307?via%3Dihub">attest to</a> the <a href="https://science.sciencemag.org/content/325/5942/859">cognitive abilities</a> of early humans <a href="https://www.ncbi.nlm.nih.gov/pubmed/23135405">in this region</a>.</p>
<h2>Varying interpretations</h2>
<p>There is a multitude of lines, grooves, patterns and shapes on these rock surfaces. </p>
<p>One of our challenges therefore lay in identifying whether a hominin “signature” could reasonably be inferred among this plethora of forms. We outlined other possible agents that may have caused such patterns (such as wind, water, fossil roots and branches, and traces made by invertebrates, reptiles, birds and other mammals). We also considered how to distinguish between ancient patterns made in sand and more recent patterns etched in rock – that is, graffiti.</p>
<p>In some of the cases we described we simply pointed out features that appeared puzzling, that may possibly have been created by humans, but where other causes could not be reasonably excluded. One site contained patterns that we had never encountered before, and that do not appear anywhere in the ichnological (trace fossil) literature. After due consideration we interpreted this as possibly representing a seal tracksite, and will be reporting on this elsewhere. </p>
<p>In other cases, such as the circular feature with the central depression, the presence of grooves beside possible human footprints, and the “hashtag” pattern, the evidence for a human origin appeared more compelling. However, we took a cautious approach, acknowledging that absolute certainty is elusive. </p>
<h2>Next steps</h2>
<p>Samples have been taken for dating, adjacent to a number of the sites we described. We eagerly await these results. Non-invasive imaging studies may aid in the investigation of the rocks with patterns that suggest foraging behaviour. </p>
<p>We hope that other scientists will critically examine the findings and interpretations that we have presented. Recognising that ancient sand surfaces were not all “perishable”, but that some of them have preserved an extraordinary record of what transpired on them, suggests a previously under-appreciated means of interpreting ancient human expression. </p>
<p>The resulting search for ammoglyphs on the Cape south coast has the potential to become a new field of study, at a meeting point of archaeology, art, ichnology, palaeoanthropology, pattern recognition and sedimentology.</p><img src="https://counter.theconversation.com/content/123546/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Charles Helm does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Given that we know humans moved across these landscapes, we wondered whether there might also be evidence of other forms of human activity on these surfaces of sand.Charles Helm, Research Associate, African Centre for Coastal Palaeoscience, Nelson Mandela UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1222382019-09-05T11:23:39Z2019-09-05T11:23:39ZHow to get preschoolers ready to learn math<figure><img src="https://images.theconversation.com/files/290222/original/file-20190829-106504-703i62.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Those shapes may prove as constructive as the numbers.</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/little-girl-playing-puzzle-early-education-570525310?src=-1-7">NadyaEugene/Shutterstock.com</a></span></figcaption></figure><p>If you’re a parent of a preschooler, you might be wondering how you can help set your child up for success once they enter kindergarten.</p>
<p>By now, you have probably heard of the importance of <a href="https://theconversation.com/reading-to-your-child-the-difference-it-makes-57473">reading</a> and <a href="https://theconversation.com/are-some-kids-really-smarter-just-because-they-know-more-words-47819">talking</a> to your child to support their <a href="https://theconversation.com/7-ways-to-build-your-childs-vocabulary-112370">language and literacy skills</a>. You may have even made reading, talking and learning the ABCs part of your daily routine.</p>
<p>But did you know that you can also support your child’s math learning during <a href="https://sites.google.com/view/ericazippert/home">everyday interactions at home</a>?</p>
<p>We conduct research that aims to define the broad array of early skills that support <a href="https://peabody.vanderbilt.edu/departments/psych/research/research_labs/childrens_learning_lab/BRJ.php">mathematical thinking</a>. Our goal is to get parents more familiar with the important role they can play as their child’s first teacher to lay the groundwork for learning math. </p>
<p><iframe id="llbDA" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/llbDA/2/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<h2>1=one=an apple</h2>
<p>Young children need to develop several different number skills. For instance, they need to master counting aloud from one to 10 and beyond and learn to identify <a href="https://psycnet.apa.org/doi/10.1037/a0031753">written numbers</a> like 2 and 4.</p>
<p>In addition, little kids should realize that each number word and symbol represents a <a href="https://doi.org/10.1016/0010-0277(83)90014-8">specific quantity of objects</a>. That is, the spoken word “four” and the written number 4 are the same as four cookies or four apples. They need to know that they can count to determine <a href="http://dx.doi.org/10.1037/a0031753">how many of something is in a set</a>. </p>
<p>Young children are also beginning to understand the concepts of <a href="https://doi.org/10.1038/358749a0">addition and subtraction</a>, even if they cannot do the math by themselves. And they need to start seeing which numbers are bigger or smaller than others.</p>
<h2>Where the things are</h2>
<p>Little kids also need to develop <a href="http://dx.doi.org/10.1037/a0028446">spatial skills</a> to <a href="https://doi.org/10.1007/s10648-019-09470-8">get ready</a> to learn <a href="https://doi.org/10.1080/15248372.2012.725186">math</a>. </p>
<p>Examples include <a href="https://www.tandfonline.com/doi/full/10.1080/87565640801982312">remembering and reproducing a series of events</a>, such as where and in what order different parts of a toy light up.</p>
<p>A different type of spatial skill allows kids to imagine what a shape, such as a square, would look like if you <a href="https://doi.org/10.1016/j.tics.2014.05.011">broke it in half</a> and changed its orientation. </p>
<h2>Seeing patterns</h2>
<p>Another skill that may seem less directly math-related is making and understanding <a href="https://www.sciencedirect.com/science/article/pii/S0885200617301801?via%3Dihub">patterns</a> – sequences that follow a rule.</p>
<p>Simple repeating patterns are especially appropriate for young children. They follow a rule that one part of the sequence repeats over and over again. For example, in a red-blue-red-blue-red-blue pattern, that part is red-blue.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/Ot3lS8s7y4Q?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Vanderbilt University researcher Erica Zippert recommends that parents encourage young children to recognize patterns and spatial relationships – along with numbers – to get ready to learn math.</span></figcaption>
</figure>
<h2>Supporting math skills at home</h2>
<p>When families of young children do <a href="https://theconversation.com/5-math-skills-your-child-needs-to-get-ready-for-kindergarten-103194">everyday activities</a> that support early math at home, they tend to focus on <a href="https://doi.org/10.1016/j.ecresq.2018.07.009">number-related activities</a> related to directly teaching counting and naming numbers.</p>
<p>For the best results, branch out. You can support number skills through playing <a href="https://doi.org/10.1111/j.1467-8624.2007.01131.x">board games</a> and <a href="https://doi.org/10.5964/jnc.v3i3.72">card games</a>.</p>
<p>Classic board games like <a href="https://math.byu.edu/%7Ejeffh/mathematics/games/chutes/chutes.html">Chutes and Ladders</a> help children learn to recognize written numbers on the spaces and spinners. They also help children see numbers laid out in order, which can allow them to better <a href="https://doi.org/10.1111/j.1467-8624.2004.00684.x">tell which of two numbers is bigger</a>.</p>
<p>Playing cards are especially helpful for learning to recognize written numbers and counting and labeling sets of objects, such as all the spades or diamonds. Additionally, simple <a href="https://bicyclecards.com/how-to-play/war/">card games like War</a> can encourage families to directly <a href="https://ejournals.library.vanderbilt.edu/index.php/iris/article/view/4659">compare the size of numbers side by side</a>.</p>
<p>Cards can also lay the groundwork for learning subtraction when kids try to compare card quantities more exactly. For example, the 5 of hearts has two more hearts than the 3 of hearts.</p>
<p>To support spatial skills, research suggests engaging your child in play with <a href="https://doi.org/10.1037/a0025913">puzzles</a>, <a href="https://doi.org/10.1016/j.ecresq.2018.04.006">blocks</a> and <a href="https://doi.org/10.1016/j.ecresq.2018.03.015">shapes</a>. When parents and kids do these activities together, they naturally use spatial and directional words like “over,” “under” and “next to” that help children get ready to learn these concepts at school.</p>
<p>At the same time, kids can get a head start on picturing how to create shapes and other objects out of individual blocks.</p>
<p><iframe id="0DaMD" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/0DaMD/2/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>There are many fun and easy ways to emphasize patterns at home. </p>
<p>One is having children fill in a blank in a model pattern: red-blue-red…red-blue. Another task could involve keeping a pattern going. </p>
<p>Preschoolers can learn even <a href="https://www.tandfonline.com/doi/full/10.1080/15248372.2012.689897">harder patterning tasks</a>.</p>
<p>These include replicating patterns with different materials. Given a model pattern of alternating red and blue Legos, children can make the same alternating pattern using orange and green buttons or other items you happen to have handy.</p>
<p>Children can also use their pattern skills to get more adept with numbers. For example, see if they can count off odd numbers: 1, 3, 5, 7, 9. Explain that the rule of the pattern is either adding 2 each time or skipping the next number.</p>
<p>[ <em><a href="https://theconversation.com/us/newsletters?utm_source=TCUS&utm_medium=inline-link&utm_campaign=newsletter-text&utm_content=thanksforreading">Thanks for reading! We can send you The Conversation’s stories every day in an informative email. Sign up today.</a></em> ]</p><img src="https://counter.theconversation.com/content/122238/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Erica Zippert has received funding from the Heising-Simons Foundation for her research. </span></em></p><p class="fine-print"><em><span>Bethany Rittle-Johnson receives funding from the U.S. Department of Education Institute of Education Sciences, the National Science Foundation and the Heising-Simons Foundation.</span></em></p>On top of teaching them how to recognize numbers and count to 10, make sure they’re playing with puzzles.Erica Zippert, Postdoctoral Scholar of Psychology, Vanderbilt UniversityBethany Rittle-Johnson, Associate Professor of Psychology, Vanderbilt UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1158902019-05-24T10:44:53Z2019-05-24T10:44:53ZMathematics of scale: Big, small and everything in between<figure><img src="https://images.theconversation.com/files/272592/original/file-20190503-103068-1dvf5rv.jpeg?ixlib=rb-1.1.0&rect=0%2C0%2C1359%2C1098&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">How many lakes are in Alaska? Thermokarst lakes on Alaska's North Slope are self-similar and fractal.</span> <span class="attribution"><a class="source" href="https://www.gaeacherissa.com/">Painting by Cherissa Dukelow</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>Breathe. As your lungs expand, air fills 500 million tiny alveoli, each a fraction of a millimeter across. As you exhale, these millions of tiny breaths merge effortlessly through larger and larger airways into one ultimate breath.</p>
<p>These airways are fractal.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/273393/original/file-20190508-183096-1c8a0r2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/273393/original/file-20190508-183096-1c8a0r2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/273393/original/file-20190508-183096-1c8a0r2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=905&fit=crop&dpr=1 600w, https://images.theconversation.com/files/273393/original/file-20190508-183096-1c8a0r2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=905&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/273393/original/file-20190508-183096-1c8a0r2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=905&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/273393/original/file-20190508-183096-1c8a0r2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1138&fit=crop&dpr=1 754w, https://images.theconversation.com/files/273393/original/file-20190508-183096-1c8a0r2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1138&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/273393/original/file-20190508-183096-1c8a0r2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1138&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The branches within lungs are an example of self-similarity.</span>
<span class="attribution"><span class="source">Brockhaus and Efron Encyclopedic Dictionary/Wikimedia</span></span>
</figcaption>
</figure>
<p>Fractals are a mathematical tool for describing objects with detail
at every scale. Mathematicians and physicists <a href="https://lsa.umich.edu/cscs/people/post-docs-lecturers-visiting-scholars/mgnew.html">like
me</a> use fractals and related concepts to understand how things change going from small to big.</p>
<p>You and I translate between vastly different scales when we think about how our choices affect the world. Is this latte contributing to climate change? Should I vote in this election?</p>
<p>These conceptual tools apply to the body as well as landscapes, natural disasters and society.</p>
<h2>Fractals everywhere</h2>
<p>In 1967, mathematician Benoit Mandelbrot asked, <a href="https://users.math.yale.edu/%7Ebbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf">“How long is the coast of Britain?”</a> </p>
<p>It’s a trick question. The answer depends on how you measure it. If you trace the outline on a map, you get one answer, but if you walk the coastline with a meter stick, the result is quite different. Anyone who has tried to estimate the length of a rugged hiking trail from a map knows the treachery of the large-scale picture.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/272800/original/file-20190506-103057-1idtjez.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/272800/original/file-20190506-103057-1idtjez.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/272800/original/file-20190506-103057-1idtjez.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=874&fit=crop&dpr=1 600w, https://images.theconversation.com/files/272800/original/file-20190506-103057-1idtjez.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=874&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/272800/original/file-20190506-103057-1idtjez.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=874&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/272800/original/file-20190506-103057-1idtjez.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1099&fit=crop&dpr=1 754w, https://images.theconversation.com/files/272800/original/file-20190506-103057-1idtjez.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1099&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/272800/original/file-20190506-103057-1idtjez.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1099&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Satellite image of Great Britain and Northern Ireland.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Satellite_image_of_Great_Britain_and_Northern_Ireland_in_April_2002.jpg">NASA</a></span>
</figcaption>
</figure>
<p>That’s because lungs, the British coastline and hiking trails all have <a href="https://www.worldcat.org/title/fractal-geometry-of-nature/oclc/7876824">fractality</a>: their length, number of branches or some other quantity depends on the scale or resolution you use to measure them.</p>
<p>The coastline is also <a href="https://en.wikipedia.org/wiki/Self-similarity">self-similar</a> – it’s made out of smaller copies of itself. Fern fronds, trees, snail shells, landscapes, the silhouettes of mountains and river networks all look like smaller versions of themselves. </p>
<p>That’s why, when you’re looking at an aerial photograph of a landscape, it’s often hard to tell whether the scale bar should be 50 km or 500 m. </p>
<p>Your lungs are self-similar, because the body finely calibrates each branch in exact proportions, making each branch a smaller replica of the previous. This modular design makes lungs efficient at any size. Think of a child and an adult, or a mouse, a whale. The only difference between small and large is in how many times the airways branch.</p>
<p>Self-similarity and fractality appear in art and architecture, in the arches within arches of Roman aqueducts and the spires of Gothic cathedrals that mirror the forest canopy. Even ancient Chinese calligraphers Huai Su and Yan Zhenqing <a href="https://arxiv.org/pdf/0810.1242.pdf">prized the fractality</a> of summer clouds, cracks in a wall and water stains in a leaking house in 722.</p>
<h2>Scale invariance</h2>
<p>Self-similar objects have a scale invariance. In other words, some property holds regardless of how big they get, such as the efficiency of lungs.</p>
<p>In effect, scale invariance describes what changes between scales by saying what doesn’t change.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/273383/original/file-20190508-183077-1ki11a5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/273383/original/file-20190508-183077-1ki11a5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/273383/original/file-20190508-183077-1ki11a5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=621&fit=crop&dpr=1 600w, https://images.theconversation.com/files/273383/original/file-20190508-183077-1ki11a5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=621&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/273383/original/file-20190508-183077-1ki11a5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=621&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/273383/original/file-20190508-183077-1ki11a5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=780&fit=crop&dpr=1 754w, https://images.theconversation.com/files/273383/original/file-20190508-183077-1ki11a5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=780&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/273383/original/file-20190508-183077-1ki11a5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=780&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A sketch from Leonardo da Vinci’s notes on tree branches.</span>
<span class="attribution"><span class="source">Fractal Foundation</span></span>
</figcaption>
</figure>
<p><a href="https://www.insidescience.org/news/uncovering-da-vincis-rule-trees">Leonardo da Vinci observed</a> that, as trees branch, the total cross-sectional area of all branches is preserved. In other words, going from trunk to twigs, the number of branches and their diameter change with each branching, but the total thickness of all branches bundled together stays the same.</p>
<p>Da Vinci’s observation implies a scale invariance: For every branch of a certain radius, there are four downstream branches with half that radius.</p>
<p>Earthquake frequency has a similar scale invariance, <a href="https://authors.library.caltech.edu/47734/1/185.full.pdf">which was observed in the 1940s</a>. The big ones come to mind – Lisbon 1755, San Francisco 1989 – but many small earthquakes occur in California every day. The Gutenberg-Richter law says that earthquake frequency depends on the size of the earthquake. The answer is surprisingly simple. A tenfold bigger earthquake occurs roughly one-tenth as often.</p>
<h2>Society and the power law</h2>
<p>A 19th-century economist Vilifredo Pareto – famous in business school for <a href="https://betterexplained.com/articles/understanding-the-pareto-principle-the-8020-rule/">the 80/20 rule</a> – observed that the number of families with a certain wealth is inversely proportional to their wealth, raised to some exponent. Pareto measured the exponent for different years and different countries and found that it was usually around 1.5. </p>
<p>Pareto’s wealth distribution came to be known as the power law, ostensibly because of the exponent or “power.”</p>
<p>Anything self-similar <a href="https://arxiv.org/abs/cond-mat/9707012">has a corresponding power law</a>. In <a href="https://doi.org/10.1103/PhysRevLett.122.158303">an April paper</a>, my colleague and I describe the corresponding power law for lungs, blood vessels and trees. It differs from Pareto’s power law only by taking into account specific ratios between branches.</p>
<p>The sizes of fortunes then are akin to the sizes of tree twigs or blood vessels – a few trunks or large branches and exponentially more tiny twigs.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/272801/original/file-20190506-103071-1emh5qv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/272801/original/file-20190506-103071-1emh5qv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/272801/original/file-20190506-103071-1emh5qv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/272801/original/file-20190506-103071-1emh5qv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/272801/original/file-20190506-103071-1emh5qv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/272801/original/file-20190506-103071-1emh5qv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/272801/original/file-20190506-103071-1emh5qv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/272801/original/file-20190506-103071-1emh5qv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Patterns in an oak’s branches.</span>
<span class="attribution"><span class="source">Schlegelfotos/shutterstock.com</span></span>
</figcaption>
</figure>
<p>Pareto thought of his distribution of wealth as a natural law, but <a href="https://doi.org/10.1126%2Fscience.286.5439.509">many different models of social organization</a> give rise to a Pareto distribution and societies do vary in <a href="https://en.wikipedia.org/wiki/List_of_countries_by_distribution_of_wealth">wealth inequality</a>. The higher Pareto’s exponent, the more egalitarian the society.</p>
<p>From understanding how humans are made up of tiny cells to how we affect the planet, self-similarity, fractality and scale invariance often help translate from one level of organization to another.</p><img src="https://counter.theconversation.com/content/115890/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mitchell Newberry receives funding from US National Science Foundation. </span></em></p>What do earthquakes, wealthy Italian families and your circulatory system have in common? Scientists use fractals, self-similarity and power laws to translate from local to global scales.Mitchell Newberry, Assistant Professor of Complex Systems, University of MichiganLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1104902019-02-04T21:20:52Z2019-02-04T21:20:52ZHow analyzing patterns helps students spot deceptive media<figure><img src="https://images.theconversation.com/files/256920/original/file-20190203-109820-1rs7mc9.jpg?ixlib=rb-1.1.0&rect=7%2C2608%2C4920%2C2311&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">All demographics of people are suceptible to being deceived.
</span> <span class="attribution"><span class="source">www.shutterstock.com</span></span></figcaption></figure><p>With the current pervasive use of online media for personal and academic reasons, <a href="https://www.tandfonline.com/doi/abs/10.1080/00933104.2017.1416320">it’s necessary for students to have skills to confidently filter and decipher what they’re reading</a>.</p>
<p>As such, <a href="https://journals.sagepub.com/doi/10.1177/0031721718762419">educators have called for a new approach to teach students how to analyze digital media</a> and some <a href="https://www.theguardian.com/education/2018/jun/12/fake-news-schools-trump-truth">tech and media organizations are getting involved</a>. </p>
<p>Education policy-makers have added new <a href="https://curriculum.gov.bc.ca/curriculum/english-language-arts/10/new-media">media modules to recently revamped curricula that aims to help students become better-informed and critical future citizens</a>. </p>
<p>But the questions explored in public school media units are also important for people of all ages to consider. Recent studies have shown <a href="https://www.researchgate.net/publication/237579320_Research_on_Instruction_and_Assessment_in_the_New_Literacies_of_Online_Reading_Comprehension">all demographics of people are susceptible to being deceived</a>. </p>
<p>Showing students how to determine source credibility, and to analyze tone, bias and motive, is a great way to help them with their media literacy. Another helpful way to teach students to think critically about media is to teach them to examine media formats and media patterns. </p>
<p>What exactly does that mean and how do patterns work to manipulate viewers? </p>
<h2>Patterns structure our expectations</h2>
<p>People <a href="https://eric.ed.gov/?id=ED151893">have become familiar with, and expect, specific patterns within specific genres of media</a>. Canadian communications theorist Marshall McLuhan famously argued that <a href="http://web.mit.edu/allanmc/www/mcluhan.mediummessage.pdf">“the medium is the message:”</a> the media format or genre (including its patterns), can influence people’s thoughts and beliefs. </p>
<p><a href="https://wac.colostate.edu/books/referenceguides/bawarshi-reiff/">Each genre has a unique set of characteristics</a> — for example, a haiku poem’s three lines have a particular number of syllables (five, seven, five). </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/256894/original/file-20190201-42594-uwd9p9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/256894/original/file-20190201-42594-uwd9p9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/256894/original/file-20190201-42594-uwd9p9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/256894/original/file-20190201-42594-uwd9p9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/256894/original/file-20190201-42594-uwd9p9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/256894/original/file-20190201-42594-uwd9p9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/256894/original/file-20190201-42594-uwd9p9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Haiku form: five syllables, seven syllables, five syllables.</span>
<span class="attribution"><span class="source">www.shutterstock.com</span></span>
</figcaption>
</figure>
<p>The genre of documentary film often includes suspenseful music, interviews with specialists and recorded footage or re-enactments. </p>
<p>Or consider the television news broadcast: there are often two anchors, and the colours blue, red and white are seen alongside an animated globe or map. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/256947/original/file-20190203-112389-1i0458c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/256947/original/file-20190203-112389-1i0458c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=328&fit=crop&dpr=1 600w, https://images.theconversation.com/files/256947/original/file-20190203-112389-1i0458c.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=328&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/256947/original/file-20190203-112389-1i0458c.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=328&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/256947/original/file-20190203-112389-1i0458c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=412&fit=crop&dpr=1 754w, https://images.theconversation.com/files/256947/original/file-20190203-112389-1i0458c.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=412&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/256947/original/file-20190203-112389-1i0458c.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=412&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Patterns associated with news: red, blue, white and a globe.</span>
<span class="attribution"><span class="source">www.shutterstock.com</span></span>
</figcaption>
</figure>
<p>Even the news music itself has a common pattern. The <a href="https://www.gimletmedia.com/every-little-thing/six-oclock-soundtrack#episode-player">“Six O’Clock Soundtrack”</a> episode of the podcast <em>Every Little Thing</em> by <a href="https://medium.com/@Gimlet">Gimlet Media</a> documented how news music is a global genre, and is difficult to create. </p>
<p>Yet news music composers from Israel, India, England and the U.S. all agree news music has three common patterns: </p>
<ol>
<li><p>The music starts by grabbing the attention of the viewer, and is usually quite catchy.</p></li>
<li><p>The rhythm is constant, which often provides a feeling of reliability for the viewer. </p></li>
<li><p>The tone of the music conveys a sense of urgency and importance, but at the same time, allows the viewer to feel things are still under control. The music, while tense, still provides a safe and authoritative feeling. </p></li>
</ol>
<p>In my research, I have found that <a href="https://open.library.ubc.ca/cIRcle/collections/ubctheses/24/items/1.0368554">specific media forms and patterns can impact viewers’ understanding of important issues</a>. Further, I have begun exploring how the success of deceptive media relies on manipulating viewers’ understanding of patterns. “Deceptive media” encompasses all forms of media that persuades or dupes, including fake news in all its possible formats (for example, print or video or other electronic content). </p>
<p>For example: Hollywood films can easily mimic such iconic aural or <a href="http://www.tpr.org/post/kind-big-deal-gets-even-bigger-anchorman-2">visual patterns of news</a> to create a representation of reality which almost instantaneously invites particular viewer expectations. A movie like <em>Anchorman</em> is funny because it creates a satirical story based on viewers’ existing knowledge of news. </p>
<p>Producers of deceptive media use such representational patterning techniques to <a href="https://www.bbc.com/news/blogs-trending-37846860">deliberately stimulate the viewer’s expectation that they are delivering factually-based news — even when they’re not</a>. </p>
<h2>Patterns plus personal experiences</h2>
<p>In some instances, drawing upon people’s expectations of patterns in particular media formats, <em>plus</em> their lived experiences can make deception possible. </p>
<p>For example, <a href="http://www.sacred-texts.com/ufo/mars/wow.htm">Orson Welles’s <em>The War of the Worlds</em></a> radio play, an adaptation of H.G. Wells’s book <a href="https://www.mercurytheatre.info/">broadcast by CBS on Oct. 30, 1938</a> mimics an interruption of a radio broadcast: it starts with information about the weather, and later plays ballroom music, which is then abruptly interrupted by the events of an alien invasion of Earth. </p>
<p>The opening of Welles’s play actually explained to listeners that the broadcast is an adaptation of H.G. Wells’s novel. </p>
<p>Yet there were instances when <a href="https://www.pbs.org/wgbh/americanexperience/films/worlds/">some listeners believed the broadcast to be true</a>.</p>
<p>Media scholars Jefferson Pooley and Michael J. Socolow <a href="https://slate.com/culture/2013/10/orson-welles-war-of-the-worlds-panic-myth-the-infamous-radio-broadcast-did-not-cause-a-nationwide-hysteria.html">have determined most of the so-called hysteria was exaggerated</a>. Researchers have found it was immediately fuelled by some newspaper journalists’ efforts to discredit broadcast media; <a href="http://blog.commarts.wisc.edu/2013/11/11/the-legacy-of-war-of-the-worlds-upon-media-studies/">later, the event’s legacy was amplified by a questionable academic study on psychological panic</a>. </p>
<p>Still, because Welles followed the representational patterns of a radio broadcast, and drew upon people’s realities of surviving WWI and being on the brink of WWII, Welles’s broadcast successfully convinced at least a portion of his audience the events were authentic. </p>
<h2>Considerations for the classroom</h2>
<p>A number of online resources offer ways for educators to help students detect deceptive media. Some <a href="https://www.hcdsb.org/Students/Library/Documents/Checkology%2010%20questions%20for%20fake%20news%20detection.pdf">provide checklists</a> of characteristics to explore. <a href="https://www.pbslearningmedia.org/resource/4197f3a0-5b4a-432b-9dbe-e49aae81ba7b/lesson-plan-how-to-teach-your-students-about-fake-news/">PBS offers lesson plans on teaching students about fake news.</a> </p>
<p>Such resources may be helpful, but coaching students to consider patterns in genre, and to look for representational patterns is also relevant. </p>
<p><em>I would like to thank <a href="https://www.researchgate.net/profile/Ernesto_Pena3">collaborative researcher Ernesto Peña</a>, with whom I have led workshops about genre and representational patterns in the Faculty of Education at UBC, and with teachers and teacher librarians in the Vancouver School Board</em>.</p><img src="https://counter.theconversation.com/content/110490/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Claire Ahn does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Coach students to analyze the credibility of sources, but teaching them how genre and experiential patterns can be manipulated is also relevant.Claire Ahn, Assistant Professor of Multiliteracies, Queen's University, OntarioLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1020702018-09-17T10:50:55Z2018-09-17T10:50:55ZHow the zebrafish got its stripes<figure><img src="https://images.theconversation.com/files/234456/original/file-20180831-195298-yicqm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Zebrafish are known for their black and gold stripes.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/nichd/20092260041">NICHD/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>Stripes are common in our lives. It’s a pretty basic pattern, and easy to take for granted. </p>
<p>As an applied mathematician who studies how patterns form in nature, though, I am wowed by the <a href="https://doi.org/10.1016/bs.ctdb.2015.12.012">striped patterns the zebrafish wears</a> across its body and fins. </p>
<p>Take a closer look at zebrafish’s black and gold stripes, and you’ll see <a href="https://doi.org/10.1111/pcmr.12328">different-colored pigment cells</a>, tens of thousands of them. I like to envision these cells as people walking around in a crowded room: Just like us, the cells <a href="https://doi.org/10.1111/j.1755-148X.2008.00504.x">move</a> and <a href="https://doi.org/10.1073/pnas.0808622106">interact</a> with their neighbors. Stripes appear because the cells very carefully <a href="https://doi.org/10.1016/j.cub.2014.11.013">instruct and signal each other on how to behave</a>. They even “shake hands” in some sense <a href="https://doi.org/10.1242/dev.099804">by reaching</a> toward distant cells.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/235432/original/file-20180907-90571-4u471q.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/235432/original/file-20180907-90571-4u471q.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/235432/original/file-20180907-90571-4u471q.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=249&fit=crop&dpr=1 600w, https://images.theconversation.com/files/235432/original/file-20180907-90571-4u471q.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=249&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/235432/original/file-20180907-90571-4u471q.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=249&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/235432/original/file-20180907-90571-4u471q.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=313&fit=crop&dpr=1 754w, https://images.theconversation.com/files/235432/original/file-20180907-90571-4u471q.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=313&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/235432/original/file-20180907-90571-4u471q.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=313&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Look closer at zebrafish’s striped bodysuit and you’ll find the tiny pigment cells that make up its patterns.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Zebrafish_(26436913602).jpg">Images adapted by Alexandria Volkening from Oregon State University/Wikimedia and from Development 2013 (doi:10.1242/dev.096719)</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>From a mathematical perspective, zebrafish stripes fall into the field of self-organization, a phenomenon in which individuals interact to produce some pattern much bigger than any individual, without external direction. <a href="https://doi.org/10.1098/rsfs.2012.0025">Bird flocks and schooling fish</a> are also examples of self-organization in nature. No one is on a megaphone calling out directions so that birds flock or pigment cells produce fish stripes, yet remarkably, they both <a href="https://en.wikipedia.org/wiki/Self-organization">organize themselves</a> to create patterns.</p>
<p>Until recently, the research community thought only <a href="https://doi.org/10.1073/pnas.0607790104">two types of cells</a> were involved in zebrafish stripes: black and gold stripes, so black and gold cells. However, <a href="http://eb.mpg.de/emeriti/research-group-colour-pattern-formation/">experiments showed</a> that a third type of pigment cell – <a href="https://doi.org/10.1038/ncb2955">blue and silver iridophores</a> – is <a href="https://doi.org/10.1371/journal.pgen.1003561">critical to pattern formation</a>. Remove it from the skin, and zebrafish have <a href="https://doi.org/10.1242/dev.096719">spots</a>! </p>
<p>So how do thousands of different-colored cells on a growing zebrafish work together to consistently form stripes? To help answer this question, I developed a <a href="https://doi.org/10.1038/s41467-018-05629-z">mathematical model</a> in collaboration with applied mathematics professor <a href="http://www.dam.brown.edu/people/sandsted/">Bjorn Sandstede</a>. In our model, pigment cells are colored dots following prescribed rules and equations for how they move around, <a href="https://doi.org/10.1073/pnas.0808622106">interact</a> and change their color. Cells with different colors behave in different ways. There are lots of <a href="https://doi.org/10.1016/j.tig.2014.11.005">questions about zebrafish</a>, so we decided to focus on the newcomers to the scene: those pesky blue and silver cells.</p>
<p>Math offers a different perspective from typical biological experiments on fish. Biologists can watch how cells behave, but it’s trickier to deduce the signals behind their behavior. Using mathematical models, we can test lots of different possible cell interactions and suggest which ones are actually able to explain the behaviors biologists observe. Biologists can then test our predictions on real fish.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/235305/original/file-20180906-190659-1g1euj.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/235305/original/file-20180906-190659-1g1euj.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/235305/original/file-20180906-190659-1g1euj.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=215&fit=crop&dpr=1 600w, https://images.theconversation.com/files/235305/original/file-20180906-190659-1g1euj.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=215&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/235305/original/file-20180906-190659-1g1euj.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=215&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/235305/original/file-20180906-190659-1g1euj.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=270&fit=crop&dpr=1 754w, https://images.theconversation.com/files/235305/original/file-20180906-190659-1g1euj.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=270&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/235305/original/file-20180906-190659-1g1euj.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=270&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">It takes more than black and gold cells to create black and gold stripes. When a mutation causes zebrafish to lose their blue and silver pigment cells, spots form across the body.</span>
<span class="attribution"><a class="source" href="http://doi.org/10.1242/dev.096719">Development (2013)</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Our model suggests there are multiple signals at work that instruct silver and blue cells on the fish skin. All these signals are redundant. A few cues are all the instruction a cell may need in a perfect world, but the world isn’t perfect. For example, we think that nearby black cells signal iridophores to change their density and color. But if there are not any black cells around to transmit that signal, distant gold cells can fill in and provide the same instructions.</p>
<p>You can think of these redundant signals like a bunch of different alarm clocks. If you have an important meeting in the morning, you may set an alarm clock, put a notification on your phone and ask for a wake-up call. All that redundancy means that you will probably get a bunch of cues to wake up. But on the off chance that your phone dies or the front desk forgets to call, it also means you’ll still get to your meeting on time. The redundancy ensures the desired result, even if one signal fails.</p>
<p>The same idea may be at work in zebrafish. <a href="https://doi.org/10.1038/s41467-018-05629-z">Our model</a> suggests that different-colored cells are constantly instructing each other. This ensures that blue and silver iridophores are pummeled with directions from all sides on how to behave. Because there are multiple signals, occasional failures don’t disrupt patterns too much. The result: reliable stripes.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/TpKmqUkdVYI?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Our mathematical model simulates how different-colored cells interact to produce zebrafish stripes.</span></figcaption>
</figure>
<p>Why is this important? Zebrafish genes are surprisingly <a href="https://doi.org/10.1038/nature12111">similar to human genes</a>. By understanding how pigment cells interact in normal and mutated zebrafish, researchers may be able to start to link genes to their function.</p>
<p>The story of how zebrafish patterns form isn’t finished yet. For now, though, the next time you see a striped fish, consider pausing a moment to recognize all the work pigment cells put into creating that pattern. Those dependable stripes are pretty darn amazing.</p><img src="https://counter.theconversation.com/content/102070/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Alexandria Volkening was funded by the Mathematical Biosciences Institute and the National Science Foundation for this study under grants DMS-1148284, DGE-0228243, and DMS-1440386.</span></em></p>Zebrafish are known for their black and gold stripes, but researchers are still figuring out how pigment cells interact to form these patterns.Alexandria Volkening, Postdoctoral Fellow in Applied Mathematics, The Ohio State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/850532017-10-04T11:42:02Z2017-10-04T11:42:02ZHow animals got their spots and stripes – according to maths<figure><img src="https://images.theconversation.com/files/188730/original/file-20171004-6697-ze4s92.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/macro-shot-zebra-danio-tropical-fish-680522524?src=I3gqRJ4T4k5pIb2RBhE8TQ-2-47">Ian Grainger/Shutterstock</a></span></figcaption></figure><p>The natural world presents a palette of beautiful complexity. From the peacock tail and the <a href="https://www.livescience.com/2820-butterflies-eye-spots.html">eyespots of a butterfly</a>, to the evolving camouflage of the chameleon, nature loves patterns.</p>
<p>Biologists may be able to tell you why an animal has a certain pattern. For example, it may have evolved its skin pattern for <a href="https://static.pexels.com/photos/33118/peacock-animal-iridescent.jpg">mating purposes</a>, as a <a href="https://upload.wikimedia.org/wikipedia/commons/c/cb/Micrurus_tener.jpg">warning sign</a>, or for <a href="https://en.wikipedia.org/wiki/Deception_in_animals#/media/File:Peacock_Flounder_Bothus_mancus_in_Kona.jpg">defence purposes</a>. However, we are still in the dark when it comes to how the patterns are produced.</p>
<p>Although we currently lack the experimental insight, mathematicians have been playing around with pattern formation equations since 1952, when <a href="https://theconversation.com/alan-turings-legacy-is-even-bigger-than-we-realise-34735">the great Alan Turing</a> published the seminal paper, <a href="http://cba.mit.edu/events/03.11.ASE/docs/Turing.pdf">The Chemical Basis of Morphogenesis</a>. In this paper, he presented a theory that said patterns could spontaneously appear using nothing more than a protein’s natural tendency to move randomly through tissue and interact with other cells and proteins.</p>
<p>The theory is incredibly counter-intuitive, and we can only wonder how Turing discovered it. Patterns, as Turing saw them, depend on two components: interacting agents and agent diffusion. Each component on its own does not create a pattern. In fact, diffusion is a well-known pattern destroyer: if you put milk in water (and don’t stir), the milk will diffuse – or spread – out across the cup. You don’t end up with spots, or stripes of milk. You just have a cup of uniform milky water. </p>
<p>Turing’s genius saw through this and he demonstrated that if you combine these two components in just the right way, diffusion could actually drive the system to form spots and stripes. This idea was so far ahead of its time that we are still working on unravelling its complexity 65 years later.</p>
<h2>Light and dark</h2>
<p>Unfortunately, biology refuses to be so simple. Diffusion assumes that the agents which create a pattern – for example, chemicals, proteins or cells – are dumb, in that they move around space randomly. However, in 2014, the experimental lab of Shigeru Kondo demonstrated that cells in particular are <a href="http://www.sciencemag.org/news/2014/01/video-zebrafish-stripes-caused-cells-chase-each-other">more cunning than we thought</a>.</p>
<p>Kondo’s lab works on understanding the black and white stripes presented on zebrafish, a tropical freshwater fish, which is native to the Himalayan region. They discovered that zebrafish skin patterns are made up of a light type of cell (xanthophore) and a dark type of cell (melanophore) that interact with each other. Specifically, the light cells spread out tendrils to investigate their environment. </p>
<p>Unexpectedly, Kondo’s team found that when the light cell touches a dark cell a chasing mechanism is instigated. The light cell slowly moves towards the dark cell while the dark cell quickly “runs away”. Complicating matters further is the fact that the chasing doesn’t occur along a straight line. The cells move at an angle to one another, resulting in a spiralling chase.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/0wECUnwgN8A?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>My work extended Turing’s theory to accommodate this new knowledge of “chasing” cells. First, I modelled the system as a set of discrete, individual cells. This mathematical model is highly accurate, but difficult to work with. I then simplified the model by assuming that there are a large number of cells. </p>
<p>Having more cells may seem to complicate the system, but by increasing the number of cells you can stop worrying about each individual component and simply consider the properties of the whole population. To put this in real world terms, it means that when you consider the Great Wall of China, you do not have to worry about a single brick, but rather see it as the whole structure.</p>
<p>Although I lost accuracy on individual cell locations, the simplification allowed me to use a whole toolbox of other techniques that mathematicians have been constructing over the past 60 years. So I am able to exactly specify the conditions under which cellular populations will produce patterns and conditions under which patterns will not exist.</p>
<p>Incredibly, with the additional complexity of chasing cells we were able to greatly expand the catalogue of available patterns. No longer does a system have to evolve to a stationary pattern of spots or stripes. These chasing cells can produce patterns of rotating hexagons, spots that shuttle past each other and, perhaps most complex of all, constantly evolving stripes that oscillate to and fro.</p>
<p>All of this complexity is wrapped up in the description of how the cells chase one another: if you change the description, you change the pattern. Critically, this confirms one of Kondo’s experimental hypotheses, as not only did he experiment on normal, or wild-type, zebrafish he also experimented on mutant fish that presented broken stripes, spots, or no pattern at all. </p>
<p>Specifically, he discovered that the mutant fish which presented different patterns also presented a different chasing strategy between the dark and light cells. He concluded that the tissue-scale pattern of the skin, could be dictated by the micro-scale pattern of the cells. Incredibly, the mathematics appears to confirm this idea, although more work needs to be done to ensure a complete comparison between theory and experiment.</p>
<p>There is little doubt in my mind that the cellular interactions will still be more complicated than we currently know. Indeed, it maybe another 65 years before we are able to truly pin down the causes of zebrafish pattern formation. In the meantime, you can be sure that mathematics will be providing biologists with a new microscope with which to examine biological problems beyond their current experimental expertise.</p><img src="https://counter.theconversation.com/content/85053/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Thomas Woolley would like to thank Cardiff University, St John's College, Oxford and the Mathematical Biosciences Institute (MBI) at Ohio State University, for financially supporting this research through the National Science Foundation grant DMS 1440386. Thomas would also like to recognise the support of BBSRC grant BKNXBKOO BK00.16.</span></em></p>A mathematician has joined the dots between Alan Turing and chasing cells to find out how skin patterns are formed.Thomas Woolley, Lecturer in Applied Mathematics, Cardiff UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/753552017-04-28T01:48:41Z2017-04-28T01:48:41ZDid artists lead the way in mathematics?<figure><img src="https://images.theconversation.com/files/166566/original/file-20170424-12645-1us9bvs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Is there a geometry lesson hidden in 'The Last Supper'?</span> <span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Leonardo_da_Vinci_(1452-1519)_-_The_Last_Supper_(1495-1498).jpg">Wikimedia Commons</a></span></figcaption></figure><p>Mathematics and art are generally viewed as very different disciplines – one devoted to abstract thought, the other to feeling. But sometimes the parallels between the two are uncanny. </p>
<p>From Islamic tiling to the chaotic patterns of Jackson Pollock, we can see remarkable similarities between art and the mathematical research that follows it. The two modes of thinking are not exactly the same, but, in interesting ways, often one seems to foreshadow the other.</p>
<p>Does art sometimes spur mathematical discovery? There’s no simple answer to this question, but in some instances it seems very likely.</p>
<h2>Patterns in the Alhambra</h2>
<p>Consider Islamic ornament, such as that found in <a href="https://www.britannica.com/topic/Alhambra-fortress-Granada-Spain">the Alhambra</a> in Granada, Spain.</p>
<p>In the 14th and 15th centuries, the Alhambra served as the palace and harem of the Berber monarchs. For many visitors, it’s a setting as close to paradise as anything on earth: a series of open courtyards with fountains, surrounded by arcades that provide shelter and shade. The ceilings are molded in elaborate geometric patterns that resemble stalactites. The crowning glory is the ornament in colorful tile on the surrounding walls, which dazzles the eye in a hypnotic way that’s strangely blissful. In a fashion akin to music, the patterns lift the onlooker into an almost out-of-body state, a sort of heavenly rapture.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/166561/original/file-20170424-12468-11uth6e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/166561/original/file-20170424-12468-11uth6e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/166561/original/file-20170424-12468-11uth6e.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/166561/original/file-20170424-12468-11uth6e.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/166561/original/file-20170424-12468-11uth6e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=502&fit=crop&dpr=1 754w, https://images.theconversation.com/files/166561/original/file-20170424-12468-11uth6e.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=502&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/166561/original/file-20170424-12468-11uth6e.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=502&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Tiles at the Alhambra.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/File:Tassellatura_alhambra.jpg">Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>It’s a triumph of art – and of mathematical reasoning. The ornament explores a branch of mathematics known as <a href="http://www.math.cornell.edu/%7Emec/2008-2009/KathrynLindsey/PROJECT/Page1.htm">tiling</a>, which seeks to fill a space completely with regular geometric patterns. Math shows that a flat surface can be regularly covered by symmetric shapes with three, four and six sides, but not with shapes of five sides. </p>
<p>It’s also possible to combine different shapes, using triangular, square and hexagonal tiles to fill a space completely. The Alhambra revels in elaborate combinations of this sort, which are hard to see as stable rather than in motion. They seem to spin before our eyes. They trigger our brain into action and, as we look, we arrange and rearrange their patterns in different configurations.</p>
<p>An emotional experience? Very much so. But what’s fascinating about such Islamic tilings is that the work of anonymous artists and craftsmen also displays a near-perfect mastery of mathematical logic. Mathematicians have identified <a href="http://www2.clarku.edu/%7Edjoyce/wallpaper/seventeen.html">17 types of symmetry</a>: bilateral symmetry, rotational symmetry and so forth. At least 16 appear in the tilework of the Alhambra, almost as if they were textbook diagrams. </p>
<p>The patterns are not merely beautiful, but mathematically rigorous as well. They explore the fundamental characteristics of symmetry in a surprisingly complete way. Mathematicians, however, did not come up with their analysis of the principles of symmetry until several centuries after the tiles of the Alhambra had been set in place.</p>
<h2>Quasicrystalline tiles</h2>
<p>Stunning as they are, the decorations of the Alhambra may have been surpassed by a masterpiece in Persia. There, in 1453, anonymous craftsmen at the Darbi-I Imam shrine in Isfahan discovered <a href="http://www.nature.com/news/2007/070219/full/news070219-9.html">quasicrystalline patterns</a>. These patterns have complex and mysterious mathematical properties that were not analyzed by mathematicians until the discovery of <a href="https://theconversation.com/the-maths-behind-impossible-never-repeating-patterns-63801">Penrose tilings</a> in the 1970s. </p>
<p>Such patterns fill a space completely with regular shapes, but in a configuration which never repeats itself – indeed, is infinitely nonrepeated – although the mathematical constant known as <a href="https://theconversation.com/the-golden-mean-a-great-discovery-or-natural-phenomenon-20570">the Golden Section</a> occurs over and over again. </p>
<p>Daniel Schectman <a href="https://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/shechtman-facts.html">won the 2001 Nobel Prize</a> for the discovery of quasicrystals, which obey this law of organization. This breakthrough forced scientists to reconsider their conception of the very nature of matter.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/165892/original/file-20170419-2410-1kdah63.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/165892/original/file-20170419-2410-1kdah63.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=497&fit=crop&dpr=1 600w, https://images.theconversation.com/files/165892/original/file-20170419-2410-1kdah63.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=497&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/165892/original/file-20170419-2410-1kdah63.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=497&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/165892/original/file-20170419-2410-1kdah63.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=625&fit=crop&dpr=1 754w, https://images.theconversation.com/files/165892/original/file-20170419-2410-1kdah63.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=625&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/165892/original/file-20170419-2410-1kdah63.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=625&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Laser-cut Girih tiles.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/38462165@N05/7947938374/in/photolist-d7kgbf-d7kgmS-d7kfMw-d7kfYm-ErSBxS">Cropped from 38462165@N05/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>In 2005, Harvard physicist <a href="http://www.peterlu.org/">Peter James Lu</a> showed that it’s possible to generate such quasicrystalline patterns relatively easily <a href="http://archive.aramcoworld.com/issue/200905/the.tiles.of.infinity.htm">using girih tiles</a>. Girih tiles combine several pure geometric shapes into five patterns: a regular decagon, an irregular hexagon, a bow tie, a rhombus and a regular pentagon. </p>
<p>Whatever the method, it’s clear that the quasicrystalline patterns at Darbi-I Imam were created by craftsmen without advanced training in mathematics. It took several more centuries for mathematicians to analyze and articulate what they were doing. In other words, intuition preceded full understanding.</p>
<h2>Perspective and non-Euclidian mathematics</h2>
<p><a href="https://en.wikipedia.org/wiki/Perspective_(geometry)">Geometric perspective</a> made it possible to portray the visible world with a new verisimilitude and accuracy, creating an artistic revolution in the Italian Renaissance. One could argue that perspective also led to a major reexamination of the fundamental laws of mathematics. </p>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/166562/original/file-20170424-27254-1kmz7kr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/166562/original/file-20170424-27254-1kmz7kr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=425&fit=crop&dpr=1 600w, https://images.theconversation.com/files/166562/original/file-20170424-27254-1kmz7kr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=425&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/166562/original/file-20170424-27254-1kmz7kr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=425&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/166562/original/file-20170424-27254-1kmz7kr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=534&fit=crop&dpr=1 754w, https://images.theconversation.com/files/166562/original/file-20170424-27254-1kmz7kr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=534&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/166562/original/file-20170424-27254-1kmz7kr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=534&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">In reality, the two rails of the track never meet. But, as they approach the horizon, they seem to converge at a distant ‘vanishing point.’</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/royluck/6907270089/in/photolist-bwnyKB-pG6At9-5LztEk-Dqj7X-GDAFX-4ZTL23-SbENVy-ogaPpC-e1Dwjc-bo9gu9-5PzHLA-33gApu-pF9sDF-acSa44-5hK5in-qACtXB-9PKymD-qGx43a-bMXqhe-nyJJJb-spzJq-MtykS-QP79iu-pTut5d-pBdudH-bc9FxB-7LZvz-7YSN9X-qJgxd8-gHwBrG-pHk9v6-5am2Et-nzztoQ-pNews5-i6AaTM-8bp1pU-kjRuYs-nwgwVY-pDgiPf-7SgnAg-KmMwd-pvYwn1-6rch3G-5nqaHa-RN46qg-Rreaob-ozVXu3-S3B4av-6AdAJ7-RMjS1L">royluck/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>According to Euclidian mathematics, two parallel lines will remain parallel into infinity and never meet. In the world of Renaissance perspective, however, parallel lines eventually do meet in the far distance at the so-called “vanishing point.” In other words, Renaissance perspective present a geometry which follows regular mathematical laws, but is non-Euclidian.</p>
<p>When mathematicians first devised non-Euclidian mathematics in the early 19th century, they imagined a world in which parallel lines meet at infinity. The geometry they explored was, in many ways, similar to that of Renaissance perspective.</p>
<p>Non-Euclidian mathematics has since moved on to explore space which has 12 or 13 dimensions, far outside the world of Renaissance perspective. But it’s worth asking whether Renaissance art may have made easier to make that initial leap.</p>
<h2>Pollock’s chaotic paintings</h2>
<p>An interesting modern case of art that broke traditional boundaries – and that has suggestive parallels with recent developments in mathematics – is that of the paintings of Jackson Pollock. </p>
<p>To those who first encountered them, the paintings of Pollock seemed chaotic and senseless. With time, however, we’ve come to see that they have elements of order, though not a traditional sort. Their shapes are simultaneously predictable and unpredictable, in a fashion similar to the pattern of dripping water from a faucet. There’s no way to predict the exact effect of the next drip. But, if we chart the pattern of drips, we find that they fall within a zone that has a clear shape and boundaries. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/166563/original/file-20170424-12468-169sst4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/166563/original/file-20170424-12468-169sst4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=337&fit=crop&dpr=1 600w, https://images.theconversation.com/files/166563/original/file-20170424-12468-169sst4.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=337&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/166563/original/file-20170424-12468-169sst4.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=337&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/166563/original/file-20170424-12468-169sst4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/166563/original/file-20170424-12468-169sst4.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/166563/original/file-20170424-12468-169sst4.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">‘Greyed Rainbow’ by Jackson Pollock.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/ancientartpodcast/8978999917">ancientartpodcast/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Such unpredictability was once out of bounds for mathematicians. But, in recent years, it has become one of the hottest areas of mathematical exploration. For example, <a href="https://theconversation.com/explainer-what-is-chaos-theory-10620">chaos theory</a> explores patterns that are not predictable but fall within a definable range of possibilities, while <a href="https://theconversation.com/explainer-what-are-fractals-10865">fractal analysis</a> studies shapes that are similar but not identical.</p>
<p>Pollock himself had no particular interest in mathematics, and little known talent in that arena. His fascination with these forms was intuitive and subjective. </p>
<p>Intriguingly, <a href="http://doi.org/10.1038/nature05398">mathematicians have not been able</a> to accurately describe what Pollock was doing in his paintings. For example, there have been attempts to use fractal analysis to create a numerical “signature” of his style, but so far the method has not worked – we can’t mathematically distinguish Pollock’s autograph work from bad imitations. Even the notion that Pollock employed fractal thoughts is probably incorrect.</p>
<p>Nonetheless, Pollock’s simultaneously chaotic and orderly patterns have suggested a fruitful direction for mathematics. At some point, it may well be possible to describe what Pollock was doing with mathematical tools, and artists will have to move on and mark out a new frontier to explore.</p><img src="https://counter.theconversation.com/content/75355/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Henry Adams does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Mathematics and art are generally viewed as very different. But a trip through history – from an Islamic palace to Pollock’s paintings – proves the parallels between the two can be uncanny.Henry Adams, Ruth Coulter Heede Professor of Art History, Case Western Reserve UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/732552017-03-31T02:03:41Z2017-03-31T02:03:41ZFractal patterns in nature and art are aesthetically pleasing and stress-reducing<figure><img src="https://images.theconversation.com/files/163337/original/image-20170330-4592-1n4ji0f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A fern repeats its pattern at various scales.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/evilbu/4576466328">Michael </a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span></figcaption></figure><p>Humans are visual creatures. Objects we call “beautiful” or “aesthetic” are a crucial part of our humanity. Even the oldest known examples of <a href="https://www.amazon.com/Aesthetics-Rock-Art-Thomas-Heyd/dp/075463924X">rock and cave art served aesthetic</a> rather than utilitarian roles. Although aesthetics is often regarded as an ill-defined vague quality, <a href="https://blogs.uoregon.edu/richardtaylor/">research groups like mine</a> are using sophisticated techniques to quantify it – and its impact on the observer.</p>
<p>We’re finding that aesthetic images can induce staggering changes to the body, including <a href="http://doi.org/10.1162/leon.2006.39.3.245">radical reductions in the observer’s stress levels</a>. Job stress alone is estimated to cost American businesses <a href="http://www.businessnewsdaily.com/2267-workplace-stress-health-epidemic-perventable-employee-assistance-programs.html">many billions of dollars annually</a>, so studying aesthetics holds a huge potential benefit to society.</p>
<p>Researchers are untangling just what makes particular works of art or natural scenes visually appealing and stress-relieving – and one crucial factor is the presence of the repetitive patterns called fractals.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/163375/original/image-20170330-4592-167yg1n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/163375/original/image-20170330-4592-167yg1n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/163375/original/image-20170330-4592-167yg1n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=309&fit=crop&dpr=1 600w, https://images.theconversation.com/files/163375/original/image-20170330-4592-167yg1n.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=309&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/163375/original/image-20170330-4592-167yg1n.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=309&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/163375/original/image-20170330-4592-167yg1n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=388&fit=crop&dpr=1 754w, https://images.theconversation.com/files/163375/original/image-20170330-4592-167yg1n.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=388&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/163375/original/image-20170330-4592-167yg1n.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=388&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Are fractals the key to why Pollock’s work captivates?</span>
<span class="attribution"><a class="source" href="http://www.apimages.com/metadata/Index/Pollock-Black-Paintings/cfc2766621b444ca83c6eb44a7bf651e/1/0">AP Photo/LM Otero</a></span>
</figcaption>
</figure>
<h2>Pleasing patterns, in art and in nature</h2>
<p>When it comes to aesthetics, who better to study than famous artists? They are, after all, the visual experts. My research group took this approach with <a href="https://www.moma.org/artists/4675">Jackson Pollock</a>, who rose to the peak of modern art in the late 1940s by pouring paint directly from a can onto horizontal canvases laid across his studio floor. Although battles raged among Pollock scholars regarding the meaning of his splattered patterns, many agreed they had an organic, natural feel to them.</p>
<p>My scientific curiosity was stirred when I learned that <a href="http://us.macmillan.com/thefractalgeometryofnature/benoitbmandelbrot/9780716711865">many of nature’s objects are fractal</a>, featuring patterns that repeat at increasingly fine magnifications. For example, think of a tree. First you see the big branches growing out of the trunk. Then you see smaller versions growing out of each big branch. As you keep zooming in, finer and finer branches appear, all the way down to the smallest twigs. Other examples of nature’s fractals include clouds, rivers, coastlines and mountains.</p>
<p>In 1999, my group used computer pattern analysis techniques to show that <a href="https://blogs.uoregon.edu/richardtaylor/files/2015/12/PollockScientificAmerican-2ees1wh.pdf">Pollock’s paintings are as fractal</a> as patterns found in natural scenery. Since then, more than 10 <a href="https://blogs.uoregon.edu/richardtaylor/2016/02/08/fractal-analysis-of-jackson-pollocks-poured-paintings/">different groups</a> have performed <a href="https://blogs.uoregon.edu/richardtaylor/2017/01/04/the-facts-about-pollocks-fractals/">various forms of fractal analysis</a> on his paintings. Pollock’s ability to express nature’s fractal aesthetics helps explain the enduring popularity of his work.</p>
<p>The impact of nature’s aesthetics is surprisingly powerful. In the 1980s, architects found that patients recovered more quickly from surgery when given <a href="http://dx.doi.org/10.1126/science.6143402">hospital rooms with windows looking out on nature</a>. Other studies since then have demonstrated that just looking at pictures of natural scenes can change the way a person’s autonomic nervous system <a href="http://dx.doi.org/10.1021/es305019p">responds to stress</a>.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/163339/original/image-20170330-4557-1tpeqb6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/163339/original/image-20170330-4557-1tpeqb6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/163339/original/image-20170330-4557-1tpeqb6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=379&fit=crop&dpr=1 600w, https://images.theconversation.com/files/163339/original/image-20170330-4557-1tpeqb6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=379&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/163339/original/image-20170330-4557-1tpeqb6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=379&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/163339/original/image-20170330-4557-1tpeqb6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=476&fit=crop&dpr=1 754w, https://images.theconversation.com/files/163339/original/image-20170330-4557-1tpeqb6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=476&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/163339/original/image-20170330-4557-1tpeqb6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=476&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Are fractals the secret to some soothing natural scenes?</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/ronancantwell/5230867503">Ronan</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>For me, this raises the same question I’d asked of Pollock: Are fractals responsible? Collaborating with psychologists and neuroscientists, <a href="https://blogs.uoregon.edu/richardtaylor/2016/02/03/human-physiological-responses-to-fractals-in-nature-and-art/">we measured people’s responses to fractals</a> found in nature (using photos of natural scenes), art (Pollock’s paintings) and mathematics (computer generated images) and discovered a universal effect we labeled “<a href="http://doi.org/10.1007/978-1-4939-3995-4_30">fractal fluency</a>.”</p>
<p>Through exposure to nature’s fractal scenery, people’s visual systems have adapted to efficiently process fractals with ease. We found that this adaptation occurs at many stages of the visual system, from the way our eyes move to which regions of the brain get activated. This fluency puts us in a comfort zone and so we enjoy looking at fractals. Crucially, <a href="http://doi.org/10.1068/p5918">we used EEG</a> to record the brain’s electrical activity and <a href="http://doi.org/10.1162/leon.2006.39.3.245">skin conductance techniques</a> to show that this aesthetic experience is accompanied by stress reduction of 60 percent – a surprisingly large effect for a nonmedicinal treatment. This physiological change even accelerates <a href="http://dx.doi.org/10.1126/science.6143402">post-surgical recovery rates</a>.</p>
<h2>Artists intuit the appeal of fractals</h2>
<p>It’s therefore not surprising to learn that, as visual experts, artists have been embedding fractal patterns in their works through the centuries and across many cultures. Fractals can be found, for example, in Roman, Egyptian, Aztec, Incan and Mayan works. My favorite examples of fractal art from more recent times include <a href="https://commons.wikimedia.org/wiki/File:Etudes_turbulences_-_L%C3%A9onard_de_Vinci.jpg">da Vinci’s Turbulence</a> (1500), <a href="https://en.wikipedia.org/wiki/File:Great_Wave_off_Kanagawa2.jpg">Hokusai’s Great Wave</a> (1830), <a href="http://mathstat.slu.edu/escher/index.php/Escher's_Circle_Limit_Exploration">M.C. Escher’s Circle Series</a> (1950s) and, of course, <a href="http://www.metmuseum.org/art/collection/search/488978?sortBy=Relevance&ft=Pollock&offset=0&rpp=20&pos=3">Pollock’s poured paintings</a>.</p>
<p>Although prevalent in art, the fractal repetition of patterns represents an artistic challenge. For instance, many people have attempted to fake Pollock’s fractals and failed. Indeed, our fractal analysis has <a href="http://doi.org/10.1038/439648a">helped identify fake Pollocks</a> in high-profile cases. Recent studies by others show that fractal analysis can <a href="http://doi.org/10.1504/IJART.2015.067389">help distinguish real from fake Pollocks</a> with a 93 percent success rate.</p>
<p>How artists create their fractals fuels the nature-versus-nurture debate in art: To what extent is aesthetics determined by automatic unconscious mechanisms inherent in the artist’s biology, as opposed to their intellectual and cultural concerns? In Pollock’s case, his fractal aesthetics resulted from an intriguing mixture of both. His fractal patterns originated from his body motions (specifically an <a href="http://doi.org/10.1007/s00221-008-1521-7">automatic process related to balance</a> known to be fractal). But he spent 10 years consciously refining his pouring technique to increase the visual complexity of these fractal patterns.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/163343/original/image-20170330-4557-1kk2sdc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/163343/original/image-20170330-4557-1kk2sdc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/163343/original/image-20170330-4557-1kk2sdc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=405&fit=crop&dpr=1 600w, https://images.theconversation.com/files/163343/original/image-20170330-4557-1kk2sdc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=405&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/163343/original/image-20170330-4557-1kk2sdc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=405&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/163343/original/image-20170330-4557-1kk2sdc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=509&fit=crop&dpr=1 754w, https://images.theconversation.com/files/163343/original/image-20170330-4557-1kk2sdc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=509&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/163343/original/image-20170330-4557-1kk2sdc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=509&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The Rorschach inkblot test relies on what you read in to the image.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Rorschach_blot_04.jpg">Hermann Rorschach</a></span>
</figcaption>
</figure>
<h2>Fractal complexity</h2>
<p>Pollock’s motivation for continually increasing the complexity of his fractal patterns became apparent recently when I studied the <a href="http://doi.org/10.1038/nature.2017.21473">fractal properties of Rorschach inkblots</a>. These abstract blots are famous because people see imaginary forms (figures and animals) in them. I explained this process in terms of the fractal fluency effect, which enhances people’s pattern recognition processes. The low complexity fractal inkblots made this process trigger-happy, fooling observers into seeing images that aren’t there.</p>
<p>Pollock disliked the idea that viewers of his paintings were distracted by such imaginary figures, which he called “extra cargo.” He intuitively increased the complexity of his works to prevent this phenomenon.</p>
<p>Pollock’s abstract expressionist colleague, <a href="https://www.moma.org/artists/3213">Willem De Kooning</a>, also painted fractals. When he was diagnosed with dementia, some art scholars called for his retirement amid concerns that that it would reduce the nurture component of his work. Yet, although they predicted a deterioration in his paintings, his <a href="http://www.sfgate.com/entertainment/article/APEX-OR-DECLINE-The-great-painter-Willem-de-3022569.php">later works</a> <a href="http://www.dekooning.org/the-artist/biography">conveyed a peacefulness</a> missing from his earlier pieces. Recently, the fractal complexity of his paintings was shown to <a href="https://www.sciencedaily.com/releases/2016/12/161229113520.htm">drop steadily as he slipped into dementia</a>. The study focused on seven artists with different neurological conditions and highlighted the potential of using art works as a new tool for studying these diseases. To me, the most inspiring message is that, when fighting these diseases, artists can still create beautiful artworks.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/163377/original/image-20170330-4576-1gise5y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/163377/original/image-20170330-4576-1gise5y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/163377/original/image-20170330-4576-1gise5y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=399&fit=crop&dpr=1 600w, https://images.theconversation.com/files/163377/original/image-20170330-4576-1gise5y.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=399&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/163377/original/image-20170330-4576-1gise5y.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=399&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/163377/original/image-20170330-4576-1gise5y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=502&fit=crop&dpr=1 754w, https://images.theconversation.com/files/163377/original/image-20170330-4576-1gise5y.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=502&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/163377/original/image-20170330-4576-1gise5y.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=502&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Recognizing how looking at fractals reduces stress means it’s possible to create retinal implants that mimic the mechanism.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/detailed-photo-halved-backlit-blue-shell-98024468">Nautilus image via www.shutterstock.com.</a></span>
</figcaption>
</figure>
<p>My main research focuses on <a href="https://around.uoregon.edu/content/uo-idea-bio-inspired-implant-wins-900000-grant">developing retinal implants to restore vision</a> to victims of retinal diseases. At first glance, this goal seems a long way from Pollock’s art. Yet, it was his work that gave me the first clue to fractal fluency and the role nature’s fractals can play in keeping people’s stress levels in check. To <a href="http://www.iop.org/news/11/april/page_50684.html">make sure my bio-inspired implants induce the same stress reduction</a> when looking at nature’s fractals as normal eyes do, they closely mimic the retina’s design.</p>
<p>When I started my Pollock research, I never imagined it would inform artificial eye designs. This, though, is the power of interdisciplinary endeavors – thinking “out of the box” leads to unexpected but potentially revolutionary ideas.</p><img src="https://counter.theconversation.com/content/73255/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Richard Taylor receives funding from The Australian Research Council, The Research Council for Science Advancement, and The WM Keck Foundation.</span></em></p>Fractals are patterns that repeat at increasingly fine magnifications. They turn up in the natural world and in artists’ work. Research suggests they contribute to making something aesthetically appealing.Richard Taylor, Director of the Materials Science Institute and Professor of Physics, University of OregonLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/651272016-09-13T15:44:40Z2016-09-13T15:44:40ZPsychotextiles could be next big thing in fabrics<figure><img src="https://images.theconversation.com/files/137242/original/image-20160909-13345-1belgfd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Torro!</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-114652312/stock-photo-humanoid-and-fabric.html?src=GVF3GmWq7Ugl2-tuokxJlw-1-0">Giovanni Cancemi</a></span></figcaption></figure><p>While most of us feel pain if we’re pricked by a needle, or taste sourness sucking a lemon, scientists understand less about how we’re affected by what we see. This is because seeing is a much more complicated activity. It involves shape, dimension and colour in a three-dimensional context with multiple object associations that are changing over time. </p>
<p>We know that certain works of art make us feel certain emotions. <a href="https://www.britannica.com/topic/Mona-Lisa-painting">The Mona Lisa</a> by Da Vinci is admired <a href="http://sciencenetlinks.com/science-news/science-updates/mona-lisas-smile/">because</a> her smile has a <a href="http://www.jcheudin.fr/pdf/2015_virtual_reality_international_conference.pdf">calming effect</a> on us, for example, while <a href="http://www.edvardmunch.org/the-scream.jsp">The Scream</a> by Munch <a href="http://www2.le.ac.uk/departments/medicine/student-staff/staff-and-student-achievements/docs/APT2015Azeem513.pdf">makes us</a> anxious. </p>
<p>We <a href="http://www.goodreads.com/book/show/13539061-unconscious-branding">also know</a> that some colours and shapes <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471285277.html">influence</a> our emotions. We already use these insights in design and in commercial advertising. The colour red arouses us for example, drawing attention to the object in question. This is why Coca Cola cans and many lipsticks are red – not to mention danger signs. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/137244/original/image-20160909-13371-6j30v0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/137244/original/image-20160909-13371-6j30v0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/137244/original/image-20160909-13371-6j30v0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=753&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137244/original/image-20160909-13371-6j30v0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=753&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137244/original/image-20160909-13371-6j30v0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=753&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137244/original/image-20160909-13371-6j30v0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=946&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137244/original/image-20160909-13371-6j30v0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=946&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137244/original/image-20160909-13371-6j30v0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=946&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Aaaaahhh.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-292337915/stock-photo-clothing-design-concept-man-in-blank-green-t-shirt-front-and-back-view.html?src=KHigYxnJKptNFZ9J6GG67Q-1-22">Syda Productions</a></span>
</figcaption>
</figure>
<p>We experience something similar with sharp angles, which is why chevrons are used in road signs. On the other hand, more rounded angles and the colour green produce a calming effect. </p>
<p>But do other visual characteristics produce the same emotions in the majority of the population? And if so, can we manipulate them to change our state of mind? Our insights into colours and shapes come mainly from neuroscientists looking for ways to treat people with psychiatric problems such as depression and schizophrenia. They have tended to be limited and not practical for using in everyday life – which is what we wanted to achieve. </p>
<h2>Spot the pattern</h2>
<p><a href="https://pureapps2.hw.ac.uk/portal/en/publications/psychotextiles-and-their-interaction-with-the-human-brain(129a59ce-22c4-4234-91f2-8f46bacd1da3)/export.html">The study</a> carried out by myself and Meixuan Chen, a research student, involved two stages. In the first stage, we tested ten pairs of patterns on 20 participants. This may not sound like a big group, but you have to appreciate that we tested each participant for a number of hours. The findings correlated across over 80% of them, a big majority, which in this kind of study is considered enough to draw conclusions about the population as a whole.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/137106/original/image-20160908-25237-1v1lfsm.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/137106/original/image-20160908-25237-1v1lfsm.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/137106/original/image-20160908-25237-1v1lfsm.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=245&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137106/original/image-20160908-25237-1v1lfsm.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=245&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137106/original/image-20160908-25237-1v1lfsm.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=245&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137106/original/image-20160908-25237-1v1lfsm.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=307&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137106/original/image-20160908-25237-1v1lfsm.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=307&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137106/original/image-20160908-25237-1v1lfsm.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=307&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Experiment in progress.</span>
<span class="attribution"><span class="source">George Stylios</span></span>
</figcaption>
</figure>
<p>As you can see from the picture above, each participant was shown the images on a computer screen. We investigated their emotional responses by measuring their brain and heart activity respectively using EEG and ECG monitors, as well as asking them how they felt about each pattern. </p>
<p>We didn’t show the participants different categories of patterns at this stage, but rather a wide selection. We deliberately made them black and white, since using colours would have risked contaminating the results. Here are the patterns:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/137104/original/image-20160908-25257-1ela6dg.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/137104/original/image-20160908-25257-1ela6dg.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/137104/original/image-20160908-25257-1ela6dg.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=261&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137104/original/image-20160908-25257-1ela6dg.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=261&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137104/original/image-20160908-25257-1ela6dg.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=261&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137104/original/image-20160908-25257-1ela6dg.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=328&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137104/original/image-20160908-25257-1ela6dg.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=328&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137104/original/image-20160908-25257-1ela6dg.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=328&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/137105/original/image-20160908-25279-1x78bj1.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/137105/original/image-20160908-25279-1x78bj1.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/137105/original/image-20160908-25279-1x78bj1.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=378&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137105/original/image-20160908-25279-1x78bj1.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=378&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137105/original/image-20160908-25279-1x78bj1.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=378&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137105/original/image-20160908-25279-1x78bj1.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=475&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137105/original/image-20160908-25279-1x78bj1.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=475&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137105/original/image-20160908-25279-1x78bj1.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=475&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">George Stylios</span></span>
</figcaption>
</figure>
<p>When we analysed the results, we discerned two trends. Our participants took more pleasure from repeating patterns than non-repeating ones, and were more excited by intense patterns than weak ones. </p>
<p>With repeating patterns, for example, we found that participants registered an increase of theta brain waves in an area called the Fz channel location at the midline of the frontal lobe. <a href="http://www.ncbi.nlm.nih.gov/pubmed/16202519">The</a> literature <a href="http://www.ncbi.nlm.nih.gov/pubmed/18725252">correlates this</a> with pleasant emotion. At the same time, our participants found non-repeating patterns less pleasant and found weaker patterns more calming. </p>
<p>It is important to appreciate that pleasure and excitement are not the same thing in neuroscience. Exciting things surprise us and make us sit up and take notice. They are not necessarily pleasant, however. Things can be exciting and unpleasant, just as they can be pleasant but not hold our attention. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/137247/original/image-20160909-13371-fysnt4.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/137247/original/image-20160909-13371-fysnt4.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/137247/original/image-20160909-13371-fysnt4.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=620&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137247/original/image-20160909-13371-fysnt4.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=620&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137247/original/image-20160909-13371-fysnt4.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=620&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137247/original/image-20160909-13371-fysnt4.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=779&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137247/original/image-20160909-13371-fysnt4.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=779&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137247/original/image-20160909-13371-fysnt4.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=779&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">‘Hey good lookin’</span>
<span class="attribution"><a class="source" href="https://upload.wikimedia.org/wikipedia/commons/5/54/IMac_G3_blueberry_side.jpg">Wikimedia</a></span>
</figcaption>
</figure>
<p>The challenge for product designers is to create products that are both pleasant and exciting at the same time. Apple is the classic example of a company that achieved this with their Mac computers in the 1990s. These machines looked so different to the rest of the market that they surprised people, yet were also extremely pleasant from a visual point of view. </p>
<h2>Wise woollens</h2>
<p>For the second stage of our study, we designed and produced four smart knitted fabrics on campus from a purpose-made electrochromic composite yarn. Each fabric – we call them psychotextiles – could toggle between two kinds of patterns on a graduating scale. </p>
<p>Two of our fabrics toggled between a repeating and non-repeating pattern, while the other two toggled between weak and intense patterns. We tested them on 20 more people in a similar way to the first stage. It confirmed what we found before. Not only that, we showed that by shifting between patterns, we could make the participant switch from one emotion to the other and back again. As the video shows, different patterns produced activity in different parts of participants’ brains that are linked to certain emotional responses. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/txoGNZzscFk?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>These results raise fascinating possibilities. By using a smart fabric, it means someone could choose a pattern to achieve a particular emotion – clothes that lift your mood or calm you down, for example; or wallpaper that can be manipulated to create a party atmosphere. It could be set to respond to the weather, to the time of day or year or whatever. </p>
<p>It could be a kind of “visual medicine” that becomes an alternative to the likes of antidepressants. Equally it might transform product manufacturing, engineering and the teaching of art and design. In future we might talk about psychoart, psychointeriors, psychomaterials and psychoarchitecture, to name only a few. </p>
<p>First there needs to be a major research push into the interaction between the human brain and the surrounding environment. Researchers might look at more chaotic patterns, patterns with lettering, mixtures of angles and curves, patterns with three-dimensional effects and so on. </p>
<p>The next step would be to start combining patterns with different colours and shapes. After that, we might look more closely at smells and sounds and start mixing these with the visual elements. If we are to make the most of the reactions that we have in common, the future starts here.</p><img src="https://counter.theconversation.com/content/65127/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>George Stylios does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Welcome to the world of cloths and materials that change depending on your mood.George Stylios, Senior Research Professor, Heriot-Watt UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/638012016-08-12T12:40:47Z2016-08-12T12:40:47ZThe maths behind ‘impossible’ never-repeating patterns<figure><img src="https://images.theconversation.com/files/133922/original/image-20160812-16339-v2g90o.png?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Kite- and dart-shaped tiles create never-repeating patterns.
</span> <span class="attribution"><span class="source">PrzemekMajewski /wikimedia</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>Remember the graph paper you used at school, the kind that’s covered with tiny squares? It’s the perfect illustration of what mathematicians call a “periodic tiling of space”, with shapes covering an entire area with no overlap or gap. If we moved the whole pattern by the length of a tile (translated it) or rotated it by 90 degrees, we will get the same pattern. That’s because in this case, the whole tiling has the same symmetry as a single tile. But imagine tiling a bathroom with pentagons instead of squares – it’s impossible, because the pentagons won’t fit together without leaving gaps or overlapping one another.</p>
<p>Patterns (made up of tiles) and crystals (made up of atoms or molecules) are typically periodic like a sheet of graph paper and have related symmetries. Among all possible arrangements, these regular arrangements are preferred in nature because they are associated with the least amount of energy required to assemble them. In fact we’ve only known that non-periodic tiling, which creates never-repeating patterns, can exist in crystals for a couple of decades. Now my colleagues and I have <a href="http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.075501">made a model that can help understand</a> how this is expressed. </p>
<p>In the 1970s, physicist Roger Penrose discovered that it was possible to make a pattern from two different shapes with the angles and sides of a pentagon. This looks the same when rotated through 72-degree angles, meaning that if you turn it 360 degrees full circle, it looks the same from five different angles. We see that many small patches of patterns are repeated many times in this pattern. For example in the graphic below, the five-pointed orange star is repeated over and over again. But in each case these stars are surrounded by different shapes, which implies that the whole pattern never repeats in any direction. Therefore this graphic is an example of a pattern that has rotational symmetry but no translational symmetry. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/133833/original/image-20160811-11006-1xfss0s.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/133833/original/image-20160811-11006-1xfss0s.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=631&fit=crop&dpr=1 600w, https://images.theconversation.com/files/133833/original/image-20160811-11006-1xfss0s.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=631&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/133833/original/image-20160811-11006-1xfss0s.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=631&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/133833/original/image-20160811-11006-1xfss0s.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=792&fit=crop&dpr=1 754w, https://images.theconversation.com/files/133833/original/image-20160811-11006-1xfss0s.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=792&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/133833/original/image-20160811-11006-1xfss0s.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=792&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Penrose tiling.</span>
<span class="attribution"><span class="source">PrzemekMajewski</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Things get more complicated in three dimensions. In the 1980s, Dan Schechtman observed an “aluminium-manganese” alloy with a <a href="http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.53.1951">non-periodic pattern</a> in all directions that still had rotational symmetry when rotated by the same 72-degree angle. Until then, crystals that had no translational symmetry but possessed rotational symmetry were in fact inconceivable – and many scientists did not believe this result. In fact, this was one of those rare occasions when the definition of “what is a crystal” had to be altered because of a new discovery. In accordance, these crystals are now called “quasicrystals”. </p>
<h2>Irrational number</h2>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/133949/original/image-20160812-16327-1t2osg.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/133949/original/image-20160812-16327-1t2osg.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=586&fit=crop&dpr=1 600w, https://images.theconversation.com/files/133949/original/image-20160812-16327-1t2osg.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=586&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/133949/original/image-20160812-16327-1t2osg.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=586&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/133949/original/image-20160812-16327-1t2osg.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=736&fit=crop&dpr=1 754w, https://images.theconversation.com/files/133949/original/image-20160812-16327-1t2osg.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=736&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/133949/original/image-20160812-16327-1t2osg.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=736&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Inscribed five-point star.</span>
</figcaption>
</figure>
<p>The never-repeating pattern of a quasicrystal arises from the irrational number at the heart of its construction. In a regular pentagon, the ratio of the side length of the five-pointed star you can inscribe on the inside of a pentagon, to the side of the actual pentagon is the famous irrational number “phi” (not to be confused with pi), which is about 1.618. This number is also known as the <a href="https://www.mathsisfun.com/numbers/golden-ratio.html">golden ratio</a> (and it also satisfies the relation phi = 1+1/phi). Consequently, when a quasicrystal is constructed with tiles that are derived from a pentagon – like the ones Penrose used – we observe rotational symmetry at 72-degree angles. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/133919/original/image-20160812-16364-11u0p6u.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/133919/original/image-20160812-16364-11u0p6u.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=511&fit=crop&dpr=1 600w, https://images.theconversation.com/files/133919/original/image-20160812-16364-11u0p6u.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=511&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/133919/original/image-20160812-16364-11u0p6u.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=511&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/133919/original/image-20160812-16364-11u0p6u.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=642&fit=crop&dpr=1 754w, https://images.theconversation.com/files/133919/original/image-20160812-16364-11u0p6u.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=642&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/133919/original/image-20160812-16364-11u0p6u.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=642&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Quasicrystal lattice structure.</span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>We see this five-fold symmetry both in the image of the quasicrystal as the ten radial lines around the central red dot (above), and also in the scale model of the central part of the quasicrystal made with <a href="http://www.zometool.com/">Zometool</a> (below). In the model, it helps to think of the white balls to be the locations where we would find the particles/atoms of the crystal structure and the red and yellow rods to indicate bonds between particles, that represent the shapes and symmetries of the structure. </p>
<p><a href="http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.075501">In our recent publication</a>, we identified the two traits that a system must have in order to form a 3D quasicrystal. The first is that patterns at two different sizes (length-scale) which are at an appropriate irrational ratio (like phi) both occur in the system. And second that these can influence each other strongly. In addition to the never-repeating quasicrystal patterns, this model can also form other observed regular crystal structures such as hexagons, body-centered cubes and so on. Such a model makes it possible to explore the competition between all these different patterns and to identify the conditions under which quasicrystals will be formed in nature. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/133843/original/image-20160811-11006-xay0sy.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/133843/original/image-20160811-11006-xay0sy.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/133843/original/image-20160811-11006-xay0sy.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/133843/original/image-20160811-11006-xay0sy.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/133843/original/image-20160811-11006-xay0sy.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/133843/original/image-20160811-11006-xay0sy.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/133843/original/image-20160811-11006-xay0sy.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The structure of a quasicrystal.</span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>The mathematics behind how such never-repeating patterns are created is very useful in understanding how they are formed and even in designing them with specific properties. That is why we at the University of Leeds, along with colleagues at other institutions, are fascinated with research into such questions. </p>
<p>However, this research isn’t just a conceptual mathematical idea (although the mathematics behind it is addictive) – it has great promise for many practical applications, including making very efficient <a href="http://www.nature.com/ncomms/2014/141219/ncomms6884/full/ncomms6884.html">quasicrystal lasers</a>. This is because, when periodic crystal patterns are used in a laser, a low-power laser beam is created by the symmetry of the repeating pattern. Having defects in the crystal pattern or alternatively using a never-repeating quasicrystal pattern at the output end of a laser, makes it possible to create an efficient laser beam with high peak output power. In other applications, some researchers are even considering the reflective finishes that quasicrystals might create if added to household paint.</p><img src="https://counter.theconversation.com/content/63801/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Priya Subramanian does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Many scientists didn’t believe that crystals made up of never-repeating patterns could exist. But they do and scientists are starting to understand the weird maths behind them.Priya Subramanian, Research Fellow Applied Mathematics, University of LeedsLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/426832015-12-10T11:18:10Z2015-12-10T11:18:10ZHow a simple observation from the 1800s about patterns in big data sets can fight fraud<figure><img src="https://images.theconversation.com/files/105131/original/image-20151209-15588-qlnj0a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Why are some pages of a book of numbers tables more dog-eared than others?</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=210190552&src=lb-29877982">Book image via www.shutterstock.com.</a></span></figcaption></figure><p>Benford’s law was first mentioned by the American scientist Simon Newcomb in the 1880s, when he <a href="http://www.jstor.org/stable/2369148">noticed that</a> in books of tables of logarithms, the pages of numbers whose leading digit was 1 were more worn than the pages of numbers whose leading digit was 9. For some reason, people seemed to be consistently looking up certain numbers more frequently than others.</p>
<p>Not much was done with this observation for 50 years. Then Frank Benford, an engineer at General Electric, <a href="http://www.jstor.org/stable/984802">rediscovered it</a> again and again as he looked at a variety of different data sets, from values of special functions to river lengths, county populations, physical constants, addresses of the first 342 listed members of the American Men of Science…, and on and on.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/104322/original/image-20151203-29636-1tp14a7.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/104322/original/image-20151203-29636-1tp14a7.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/104322/original/image-20151203-29636-1tp14a7.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=422&fit=crop&dpr=1 600w, https://images.theconversation.com/files/104322/original/image-20151203-29636-1tp14a7.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=422&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/104322/original/image-20151203-29636-1tp14a7.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=422&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/104322/original/image-20151203-29636-1tp14a7.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=530&fit=crop&dpr=1 754w, https://images.theconversation.com/files/104322/original/image-20151203-29636-1tp14a7.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=530&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/104322/original/image-20151203-29636-1tp14a7.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=530&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Distribution of leading digits from the data sets of Benford’s paper; the amalgamation of all observations is denoted by ‘Average.’ Some examples agree better with Benford’s law than others, but the amalgamation of them all is fairly close to Benford’s law.</span>
<span class="attribution"><span class="source">Frank Benford</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>What he noticed is that often the first digits of numbers in a data set are not distributed equally.</p>
<p>Here’s the mathematical way to describe it. A data set follows Benford’s law if the probability of observing a first digit of d is log(1 + 1/d) (all logs here are base 10). That makes the probability of a first digit of 1 around 30%, with a 9 happening roughly 4.6% of the time. This rapid decay is in stark contrast to the intuition many people have that all numbers would be equally likely to serve as a leading digit (with 1 or 9 or any other digit happening about 11% of the time each). </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/104389/original/image-20151204-4710-11lp5wl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/104389/original/image-20151204-4710-11lp5wl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/104389/original/image-20151204-4710-11lp5wl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=382&fit=crop&dpr=1 600w, https://images.theconversation.com/files/104389/original/image-20151204-4710-11lp5wl.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=382&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/104389/original/image-20151204-4710-11lp5wl.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=382&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/104389/original/image-20151204-4710-11lp5wl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=480&fit=crop&dpr=1 754w, https://images.theconversation.com/files/104389/original/image-20151204-4710-11lp5wl.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=480&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/104389/original/image-20151204-4710-11lp5wl.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=480&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Benford probabilities.</span>
<span class="attribution"><span class="source">Steven J Miller</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>As editor of the recently published book <a href="http://press.princeton.edu/titles/10527.html">Theory and Applications of Benford’s Law</a>, I collected detailed descriptions of why so many systems exhibit this universal behavior, and what the consequences are.</p>
<p>There are a lot of explanations for why so many systems follow this law. </p>
<p>One particularly nice illustration is the example of a geometric process, say a stock that increases 4% a year. If we start with US$1, then after one year we have $1.04. After two years, we have $1.0816, and so on, finally reaching $2 after about 17.673 years. It would take approximately 58.708 years to reach $10. If we increase by a constant multiple each time, it’ll take more time to go from 1 to 2 than from 9 to 10 because the magnitude of the increase is larger at 9 than at 1 and the distance to cover is the same.</p>
<p>Here’s that explanation in the language of mathematics. At time t we have $1.04<sup>t,</sup> so if $1.04<sup>t</sup> = $2 then t log(1.04) = log(2) or t₂ = log(2)/log(1.04). Similarly we see that we reach $10 at t₁₀ = log(10)/log(1.04) or approximately 58.708 years. Thus the fraction of the time we spend with first digit 1 is t₂/t₁₀ = log(2)/log(10) = log(2) (since our logarithms are base 10). A similar argument works for the other leading digits.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/ATqiJUUcWFg?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">The author’s lecture to his probability students on Benford’s law.</span></figcaption>
</figure>
<p>In my conversations with Frank Benford’s grandson, he’s entertainingly called Benford’s law a growth industry. It’s gone from an obscure subject with a paper or two per decade to exponentially growing in numerous fields – not unlike the stock price in our example above. More and more systems have been shown to follow Benford’s law, and <a href="http://www.benfordonline.net/">paper after paper</a> has been written on this phenomenon and explanations for its prevalence.</p>
<p>One of the more interesting uses is to detect fraud. Accounting professor <a href="http://www.be.wvu.edu/faculty_staff/mark_nigrini.htm">Mark Nigrini</a> pioneered this area when he noticed Benford’s law could be used to detect financial irregularities in many data sets. Many of these data sets should follow Benford’s law – but when people create fraudulent sets they’re often unaware of this pattern. In their phony data, they either make all first digits equally likely, or cluster in the middle.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/105147/original/image-20151209-15549-5rrbzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/105147/original/image-20151209-15549-5rrbzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/105147/original/image-20151209-15549-5rrbzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/105147/original/image-20151209-15549-5rrbzc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/105147/original/image-20151209-15549-5rrbzc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/105147/original/image-20151209-15549-5rrbzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/105147/original/image-20151209-15549-5rrbzc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/105147/original/image-20151209-15549-5rrbzc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Benford’s law can help flag credit card fraud.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-340602848/stock-photo-credit-cards-background.html">Credit cards image via www.shutterstock.com.</a></span>
</figcaption>
</figure>
<p>To give you a sense of Benford’s law’s power and utility, here’s my favorite application. It involves banking. If you lose your credit card, or have it stolen, after uttering some expletives you quickly call the bank to report the incident. The person on the phone offers kind, consoling words and reassures you that you are not liable for the charges and a new card is on its way. This ends your involvement, and starts theirs. They have assumed the responsibility to pay the charges, and have two options: they can just pay, or they can try to find the thief and make them pay.</p>
<p>It’s probably not worth it to the credit card company to track down someone who’s run up $90; if it’s $90,000, that’s a different story! Banks often have a demarcation line; anything below is written off as not worth the time to investigate, while anything higher generates a probe. For many companies, that line is $5,000, and leads to my favorite example. An investigation at one bank turned up many more stolen card totals starting with a 4 than Benford’s law would predict. Eventually they found that a large number were around $4,800 or $4,900, and attributable to one agent who was having friends run up debts just below the threshold before reporting the card stolen! Fraudsters discovered, thanks again to Benford’s law.</p>
<p>There are many other uses. University of Michigan Professor of Political Science and Statistics <a href="http://www-personal.umich.edu/%7Ewmebane/">Walter Mebane</a> has fruitfully used Benford’s law to detect voter fraud. Knowing the expected pattern can help determine whether or not a digital image has been modified. The field of <a href="https://en.wikipedia.org/wiki/Steganography">steganography</a> studies hiding images inside images, where the embedded file often contains coded messages. Benford’s law has also found use in medical statistics, in psychology of games, in computer science…. Not bad for a subject that began with some worn pages in an old logarithm table.</p><img src="https://counter.theconversation.com/content/42683/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Steven J Miller does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The first digits of numbers in a data set aren’t distributed equally. And now you know more than a lot of fraudsters do – and should – when they’re making up their phony numbers.Steven J Miller, Associate Professor of Mathematics, Williams CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/443902015-09-22T09:06:14Z2015-09-22T09:06:14ZPatterns are math we love to look at<figure><img src="https://images.theconversation.com/files/95554/original/image-20150921-31518-1xflzqv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Repeating patterns are visually intriguing.</span> <span class="attribution"><span class="source">Frank A Farris</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span></figcaption></figure><p>Why do humans love to look at patterns? I can only guess, but I’ve written a whole book about new mathematical ways to make them. In <a href="http://press.princeton.edu/titles/10435.html">Creating Symmetry, The Artful Mathematics of Wallpaper Patterns</a>, I include a comprehensive set of recipes for turning photographs into patterns. The official definition of “pattern” is cumbersome; but you can think of a pattern as an image that repeats in some way, perhaps when we rotate, perhaps when we jump one unit along.</p>
<p>Here’s a pattern I made, using the logo of The Conversation, along with some strawberries and a lemon:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/92608/original/image-20150820-7216-iel9h5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/92608/original/image-20150820-7216-iel9h5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/92608/original/image-20150820-7216-iel9h5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=305&fit=crop&dpr=1 600w, https://images.theconversation.com/files/92608/original/image-20150820-7216-iel9h5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=305&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/92608/original/image-20150820-7216-iel9h5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=305&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/92608/original/image-20150820-7216-iel9h5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=383&fit=crop&dpr=1 754w, https://images.theconversation.com/files/92608/original/image-20150820-7216-iel9h5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=383&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/92608/original/image-20150820-7216-iel9h5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=383&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Repeating forever left and right.</span>
<span class="attribution"><span class="source">Frank A Farris</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Mathematicians call this a frieze pattern because it repeats over and over again left and right. Your mind leads you to believe that this pattern repeats indefinitely in either direction; somehow you know how to continue the pattern beyond the frame. You also can see that the pattern along the bottom of the image is the same as the pattern along the top, only flipped and slid over a bit.</p>
<p>When we can do something to a pattern that leaves it unchanged, we call that a symmetry of the pattern. So sliding this pattern sideways just the right amount – let’s call that translation by one unit – is a symmetry of my pattern. The flip-and-slide motion is called a <a href="https://en.wikipedia.org/wiki/Glide_reflection">glide reflection</a>, so we say the above pattern has glide symmetry.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/95547/original/image-20150921-31508-13704x1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/95547/original/image-20150921-31508-13704x1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/95547/original/image-20150921-31508-13704x1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=200&fit=crop&dpr=1 600w, https://images.theconversation.com/files/95547/original/image-20150921-31508-13704x1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=200&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/95547/original/image-20150921-31508-13704x1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=200&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/95547/original/image-20150921-31508-13704x1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=251&fit=crop&dpr=1 754w, https://images.theconversation.com/files/95547/original/image-20150921-31508-13704x1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=251&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/95547/original/image-20150921-31508-13704x1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=251&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A row of A’s has multiple symmetries.</span>
<span class="attribution"><span class="source">Frank A Farris</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>You can make frieze patterns from rows of letters, as long as you can imagine that the row continues indefinitely left and right. I’ll indicate that idea by …AAAAA…. This row of letters definitely has what we call translational symmetry, since we can slide along the row, one A at a time, and wind up with the same pattern.</p>
<p>What other symmetries does it have? If you use a different font for your A’s, that could mess up the symmetry, but if the legs of the letter A are the same, as above, then this row has reflection symmetry about a vertical axis drawn through the center of each A. </p>
<p>Now here’s where some interesting mathematics comes in: did you notice the reflection axis between the As? It turns out that every frieze pattern with one vertical mirror axis, and hence an infinite row of them (by the translational symmetry shared by all friezes), must necessarily have an additional set of vertical mirror axes exactly halfway between the others. And the mathematical explanation is not too hard.</p>
<p>Suppose a pattern stays the same when you flip it about a mirror axis. And suppose the same pattern is preserved if you slide it one unit to the right. If doing the first motion leaves the pattern alone and doing the second motion also leaves the pattern alone, then doing first one and then the other leaves the pattern alone.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/95548/original/image-20150921-31500-jwn8ku.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/95548/original/image-20150921-31500-jwn8ku.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/95548/original/image-20150921-31500-jwn8ku.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=319&fit=crop&dpr=1 600w, https://images.theconversation.com/files/95548/original/image-20150921-31500-jwn8ku.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=319&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/95548/original/image-20150921-31500-jwn8ku.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=319&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/95548/original/image-20150921-31500-jwn8ku.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=401&fit=crop&dpr=1 754w, https://images.theconversation.com/files/95548/original/image-20150921-31500-jwn8ku.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=401&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/95548/original/image-20150921-31500-jwn8ku.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=401&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Flipping and then sliding is the same as one big flip.</span>
<span class="attribution"><span class="source">Frank A Farris</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>You can act this out with your hand: put your right hand face down on a table with the mirror axis through your middle finger. First flip your hand over (the mirror symmetry), then slide it one unit to the right (the translation). Observe that this is exactly the same motion as flipping your hand about an axis half a unit from the first.</p>
<p>That proves it! No one can create a pattern with translational symmetry and mirrors without also creating those intermediate mirror symmetries. This is the essence of the mathematical concept of <a href="https://en.wikipedia.org/wiki/Symmetry_group">group</a>: if a pattern has some symmetries, then it must have all the others that arise from combining those.</p>
<p>The surprising thing is that there are only a few different types of frieze symmetry. When I talk about types, I mean that a row of A’s has the same type as a row of V’s. (Look for those intermediate mirror axes!) Mathematicians say that the two groups of symmetries are isomorphic, meaning of the same form. </p>
<p>It turns out there are exactly seven different <a href="http://www.maa.org/sites/default/files/images/upload_library/4/vol1/architecture/Math/seven.html">frieze groups</a>. Surprised? You can probably figure out what they are, with some help. Let me explain how to name them, according to the International Union of Crystallographers.</p>
<p>The naming symbol uses the template prvh, where the p is just a placeholder, the r denotes rotational symmetry (think of a row of N’s), the v marks vertical qualities and the h is for horizontal. The name for the pattern of A’s is p1m1: no rotation, vertical mirror, no horizontal feature beyond translation. They use 1 as a placeholder when that particular kind of symmetry does not occur in the pattern.</p>
<p>What do I mean by horizontal stuff? My introductory frieze was p11g, because there’s glide symmetry in the horizontal directions and no symmetry in the other slots.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/95564/original/image-20150921-31525-oqcyk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/95564/original/image-20150921-31525-oqcyk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/95564/original/image-20150921-31525-oqcyk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=150&fit=crop&dpr=1 600w, https://images.theconversation.com/files/95564/original/image-20150921-31525-oqcyk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=150&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/95564/original/image-20150921-31525-oqcyk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=150&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/95564/original/image-20150921-31525-oqcyk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=189&fit=crop&dpr=1 754w, https://images.theconversation.com/files/95564/original/image-20150921-31525-oqcyk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=189&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/95564/original/image-20150921-31525-oqcyk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=189&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Another frieze pattern, this one based on a photo of a persimmon.</span>
<span class="attribution"><span class="source">Frank A Farris</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Write down a bunch of rows of letters and see what types of symmetry you can name. Hint: the persimmon pattern above (or that row of N’s) would be named p211. There can’t be a p1g1 because we insist that our frieze has translational symmetry in the horizontal direction. There can’t be a p1mg because if you have the m in the vertical direction and a g in the horizontal, you’re forced (not by me, but by the nature of reality) to have rotational symmetry, which lands you in p2mg.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/92948/original/image-20150825-15920-1ij63v6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/92948/original/image-20150825-15920-1ij63v6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/92948/original/image-20150825-15920-1ij63v6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=299&fit=crop&dpr=1 600w, https://images.theconversation.com/files/92948/original/image-20150825-15920-1ij63v6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=299&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/92948/original/image-20150825-15920-1ij63v6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=299&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/92948/original/image-20150825-15920-1ij63v6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=376&fit=crop&dpr=1 754w, https://images.theconversation.com/files/92948/original/image-20150825-15920-1ij63v6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=376&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/92948/original/image-20150825-15920-1ij63v6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=376&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A p2mg pattern based on some of the same raw materials as our first frieze pattern.</span>
<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>It’s hard to make p2mg patterns with letters, so here’s one made from the same lemon and strawberries. I left out the logo, as the words became too distorted. Look for the horizontal glides, vertical mirrors, and centers of twofold rotational symmetry. (Here’s a funny feature: the smiling strawberry faces turn sad when you see them upside down.)</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/95553/original/image-20150921-31528-1rrk1vp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/95553/original/image-20150921-31528-1rrk1vp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/95553/original/image-20150921-31528-1rrk1vp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/95553/original/image-20150921-31528-1rrk1vp.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/95553/original/image-20150921-31528-1rrk1vp.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/95553/original/image-20150921-31528-1rrk1vp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/95553/original/image-20150921-31528-1rrk1vp.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/95553/original/image-20150921-31528-1rrk1vp.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">One consequence of the limitation on wallpaper groups is that honeybees cannot make combs with fivefold symmetry.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/16180154@N07/4917209249">LHG Creative Photography</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>In my book, I focus more on wallpaper patterns: those that repeat forever along two different axes. I explain how to use mathematical formulas called complex wave forms to construct wallpaper patterns. I prove that every wallpaper group is isomorphic – a mathematical concept meaning of the same form – to one of only 17 prototype groups. Since pattern types limit the possible structures of crystals and even atoms, all results of this type say something deep about the nature of reality.</p>
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<a href="https://images.theconversation.com/files/95556/original/image-20150921-31521-fhms5x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/95556/original/image-20150921-31521-fhms5x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/95556/original/image-20150921-31521-fhms5x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/95556/original/image-20150921-31521-fhms5x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/95556/original/image-20150921-31521-fhms5x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/95556/original/image-20150921-31521-fhms5x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/95556/original/image-20150921-31521-fhms5x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/95556/original/image-20150921-31521-fhms5x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Ancient Roman mosaic floor in Carranque, Spain.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/amarga/6178405025">a_marga</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Whatever the adaptive reasons for our human love for patterns, we have been making them for a long time. Every decorative tradition includes the same limited set of pattern types, though sometimes there are cultural reasons for breaking symmetry or omitting certain types. Did our visual love for recognizing that “Yes, this is the same as that!” originally have a useful root, perhaps evolving from an advantage in distinguishing edible from poisonous plants, for instance? Or do we just like to look at patterns? Whyever it is, we still get pleasure from these repetitive patterns thousands of years later.</p><img src="https://counter.theconversation.com/content/44390/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Frank A. Farris does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>When you see a repeating pattern, your mind easily imagines it continuing to infinity. There are mathematical rules behind the intriguing visuals.Frank A. Farris, Associate Professor of Mathematics, Santa Clara UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/458852015-08-11T03:28:35Z2015-08-11T03:28:35ZWhy predicting a flu outbreak is like betting on football or flipping a coin<figure><img src="https://images.theconversation.com/files/91373/original/image-20150811-11110-1xokyle.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Try to predict the outcome of a single coin toss and you'll have only a 50-50 chance of being correct.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/pauliantero/4685925036/">Pauli Antero/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>We humans have an innate tendency to recognise patterns. This ability has helped us survive by learning important skills such as how to distinguish danger (predators and poisonous plants, for instance) from important resources (food sources and safe shelter) and knowing the right time of year to plant crops. </p>
<p>But the same ability can sometimes convince us we’re seeing a meaningful pattern when it isn’t there. Gamblers detect “patterns” in lottery numbers and roulette wheels, fortune tellers detect “meaning” in chance events and weave a story. As a society we carry all kinds of similar superstitions, such as “bad things happen in threes”.</p>
<p>A sudden increase in the number of patients with influenza being admitted to hospital in South Australia last week provides a good case in point. Should this event be seen as a sign of flu activity that could overwhelm hospitals and medical clinics across the nation?</p>
<h2>Illness and football</h2>
<p>Let’s take a step back. A few steps back, in fact, and think about a competition for picking the winning football team over a season.</p>
<p>There’s a lot of money in predicting winners, and surely it’s easy money, right? Pick the top teams each week, turn on the television and watch your predictions come true.</p>
<p>Of course, it’s not that easy. There are upsets. Sometimes the top teams play poorly, sometimes the bottom teams hit a patch of good form, sometimes the umpire or referee makes a game-changing call, and sometimes the prior form of the two teams doesn’t seem to matter one bit.</p>
<p>While stronger teams will tend to win more often than weaker teams over the course of a season, the outcome of each game is much less predictable. In fact, it’s stochastic, which <a href="https://en.wikipedia.org/wiki/Stochastic">according to Wikipedia</a>, means: </p>
<blockquote>
<p>… events or systems that are unpredictable due to the influence of a random variable. </p>
</blockquote>
<p>But this isn’t the whole story. The more reputable <a href="http://www.oxforddictionaries.com/definition/english/stochastic">Oxford English Dictionary</a> defines the word as: </p>
<blockquote>
<p>[a] pattern that may be analysed statistically but may not be predicted precisely. </p>
</blockquote>
<p>Being stochastic doesn’t mean that there are no patterns or rules; it means that any individual outcome is subject to unpredictable effects.</p>
<h2>Rules and randomness</h2>
<p>Perhaps the simplest example of stochastic behaviour is tossing a coin. As you know, the two possible outcomes – heads or tails – are equally likely. </p>
<p>Toss this hypothetical coin 1,000 times and you should find that you count about 500 heads and 500 tails — a clear statistical trend. But try to predict the outcome of a single coin toss and you’ll have only a 50-50 chance of being correct. While there is a pattern in the long run, each individual coin toss is unpredictable.</p>
<p>Stochastic events can be even more complicated where there are more than two possible outcomes. The probability of each outcome may be difficult – or even impossible – to calculate.</p>
<p>Each football game is a sequence of stochastic events; even if we rewound time and restarted the game under exactly the same conditions, we couldn’t expect the outcome to be the same.</p>
<p>But what does all of this have to do with an unexpected flu emergency? The spread of an infectious disease, such as influenza, is a stochastic process: a sequence of stochastic events where people who are infected with flu may or may not infect other people around them. </p>
<p>Remember, this doesn’t mean there’s no pattern, and it doesn’t mean there are no “rules” that allow us to understand how influenza spreads through the community.</p>
<p>Look at the figure below, which shows the number of confirmed influenza cases in Melbourne every week for the years 2010 to 2014. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/91268/original/image-20150810-11101-1065wg0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/91268/original/image-20150810-11101-1065wg0.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=480&fit=crop&dpr=1 600w, https://images.theconversation.com/files/91268/original/image-20150810-11101-1065wg0.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=480&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/91268/original/image-20150810-11101-1065wg0.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=480&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/91268/original/image-20150810-11101-1065wg0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=603&fit=crop&dpr=1 754w, https://images.theconversation.com/files/91268/original/image-20150810-11101-1065wg0.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=603&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/91268/original/image-20150810-11101-1065wg0.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=603&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The number of confirmed influenza cases in Melbourne every week for the years 2010 to 2014.</span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>There’s a clear bell-shaped curve every winter, where the number of influenza cases rapidly rises, peaks and then drops. But you can also see clear random variations from week to week. Why?</p>
<h2>Disease transmission</h2>
<p>Consider someone who has recently caught the flu and is currently infectious. The number of people she infects (if any) depends on an astoundingly huge number of stochastic events. </p>
<p>Does she cover her mouth when coughing, or her nose when she sneezes, and does she wash her hands afterwards? Does she feel ill enough to stay at home? How many people does she live with, and how many of these people received the flu vaccine? Are most of the people she comes into contact with already infected or immune to this strain of the flu? </p>
<p>These are just a few of the most obvious factors that will determine how likely she is to infect a particular number of other people.</p>
<p>We can’t ignore stochastic effects when we’re trying to understand something as complex and variable as the spread of an infectious disease like flu. </p>
<p>Stronger teams will generally win more games of football than weaker teams. Likewise, the number of people infected with the flu will increase in the winter months and then decrease again. </p>
<p>But, for any given week, we can and should expect to observe random variations. One game doesn’t define a season and one week doesn’t define the yearly influenza outbreak. </p>
<p>So while a week of increased influenza hospitalisation in South Australia is not good news, it’s also no cause for panic about the flu.</p><img src="https://counter.theconversation.com/content/45885/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>James McCaw receives funding from the Australian Research Council through a Future Fellowship and the National Health and Medical Research Council through a Centre for Research Excellence grant. He receives funding to study influenza outbreak detection and forecasting from the Australian Government's Defence Science Technology Organisation.</span></em></p><p class="fine-print"><em><span>Jodie McVernon is a member of the Australian Technical Advisory Group on Immunisation and has provided advice to the Australian Government Office of Health Protection on pandemic planning, including on antiviral stockpiling and distribution; she is also a Director of the Influenza Specialist Group. She receives funding support from the National Health and Medical Research Council through person support and Centre of Research Excellence schemes.</span></em></p><p class="fine-print"><em><span>Rob Moss does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Predicting infectious disease outbreaks is a tricky task to begin with. And it’s made harder still by the fact that any individual outcome is subject to unpredictable – or stochastic – effects.Rob Moss, Research Fellow, Mathematical Biology and Physiology, The University of MelbourneJames McCaw, Associate Professor in Mathematical Biology, The University of MelbourneJodie McVernon, Associate Professor, Population Health, The University of MelbourneLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/394172015-05-15T10:16:22Z2015-05-15T10:16:22ZDon’t know how to get your kid to do math? Try patterns<figure><img src="https://images.theconversation.com/files/81730/original/image-20150514-28615-ktlyr6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Helping kids learn patterns can develop math skills.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/poughkeepsiedayschool/16132844393/in/photolist-qzB3zx-rwhpfr-48AhM5-nf2i45-gWEbC8-gWFnAM-8pCzLP-7zbWw7-6gxQ6a-8UFDRT-8UFE6n-8UJHfs-8UFE2g-ouLuzi-dUbPgA-fhofnR-gWEbVc-gWEc62-gWEctL-fp3eVC-dmsT3Z-7YJDR7-5vkXCZ-dmsuf1-eYvxcZ-4schxv-gWE9P8-dmt27d-gWEsBw-dmsGrm-qQZ4ek-dmsVJD-cxraVu-cxkAK1-pUvjJx-fkWUSy-4czcHY-rJLT3m-dmsQnN-7tmR89-4ddm3k-aAbFLh-fkphVs-gWEsMS-gWEbv5-8DvqfP-o9SsYW-fkabvB-gWE9zR-oWNXFs">Poughkeepsie Day School</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span></figcaption></figure><p>Parents and teachers know that reading to their children in preschool and kindergarten is important. But how can parents and teachers support young children’s mathematics knowledge?</p>
<p>One often overlooked activity is patterning, or thinking about patterns. Patterns are predictable sequences, such as stripes (for example, a yellow-green striped shirt) and rhythms (for example, da-de-dum). Young children like to make patterns when they draw and play.</p>
<p>Patterning encourages children to look for regularity and rules – a critical component of mathematical reasoning. For example, in the color pattern red-blue-blue-red-blue-blue, the rule is the part that repeats over and over (red-blue-blue in this pattern). </p>
<p>My own research shows that early pattern knowledge can support later mathematics achievement. And parents and teachers can work with their children from an early age to get them to think more deeply about patterns. </p>
<h2>What parents and teachers typically do</h2>
<p>Parents and teachers most often ask preschool children to copy and extend <a href="http://www.sciencedirect.com/science/article/pii/S088520061500006X">patterns</a>. </p>
<p>For example, they ask children to extend a pattern by deciding what comes next in the pattern. Although a good start, these tasks do not push children to think about rules and <a href="http://www.highbeam.com/doc/1G1-53414270.html">regularities</a>.</p>
<p>In contrast, parents and teachers are less likely to encourage their children to do more sophisticated tasks that promote more attention to rules and <a href="http://www.sciencedirect.com/science/article/pii/S088520061500006X">regularities</a>.</p>
<p>For example, they rarely ask children to make the same kind of pattern using different objects or sounds (we call this abstracting a pattern) or to name the part of the pattern that repeats (identify a pattern’s rule). </p>
<p>This means parents and teachers are missing out on opportunities to support children’s pattern knowledge and mathematical reasoning.</p>
<h2>Kids get better with patterns</h2>
<p>We have <a href="http://www.tandfonline.com/doi/abs/10.1080/15248372.2012.689897">found</a> that many preschool children (ages 4 to 5) are <a href="http://peabody.vanderbilt.edu/research/pro/about_peabody_research/funded_projects/career_project_home/early_algebra_research_projects/early_algebra_publications.php">able to abstract patterns</a> when prompted to do so. Further, their ability to <a href="http://www.sciencedirect.com/science/article/pii/S088520061500006X">abstract patterns</a> over the course of the pre-kindergarten year also improves. However, most have difficulty identifying a pattern’s rule.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/81603/original/image-20150513-2452-1bhx6qr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/81603/original/image-20150513-2452-1bhx6qr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/81603/original/image-20150513-2452-1bhx6qr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/81603/original/image-20150513-2452-1bhx6qr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/81603/original/image-20150513-2452-1bhx6qr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/81603/original/image-20150513-2452-1bhx6qr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/81603/original/image-20150513-2452-1bhx6qr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Using alphabet letters to explain patterns helps kids learn.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/cat.mhtml?searchterm=pattern%20alphabets%20children&keyword_search=1&page=10&thumb_size=mosaic&inline=199364972">Alphabets image via www.shutterstock.com</a></span>
</figcaption>
</figure>
<p>One effective way to support attention to the pattern rule is for adults to label patterns using common, general terms. For example, preschool children learn better when alphabet letters are used to explain a pattern. </p>
<p>So, a red-blue-blue-red-blue-blue pattern could be labeled as an “ABB pattern,” <a href="http://onlinelibrary.wiley.com/doi/10.1111/cdev.12331/abstract">rather than</a> labeled using the color names. A new yellow-green-green-yellow-green-green pattern could be labeled as an “ABB pattern” too. This helps children see that the two patterns share a common rule.</p>
<p>If <a href="http://www.tandfonline.com/doi/abs/10.1080/10409289.2013.794448#.VVDmbmaYUio">special attention</a> is given to patterning, it can help improve pattern knowledge. For example, research has found that when preschool children were <a href="http://www.jstor.org/stable/10.5951/jresematheduc.42.3.0237">encouraged</a> to create new patterns over a period of six months, using different materials and with encouragement to identify a pattern’s rule, they were able to explain patterns better a year later. </p>
<h2>Patterning supports math achievement</h2>
<p>When special attention is given to patterning, it also improves children’s general mathematics achievement. Special attention to patterning in <a href="http://www.jstor.org/stable/10.5951/jresematheduc.42.3.0237">preschool</a> <a href="http://www.tandfonline.com/doi/abs/10.1080/10409289.2013.794448#.VVDmbmaYUio">and</a> in <a href="http://www.tandfonline.com/doi/abs/10.1080/02568543.2013.766664">first grade</a> led to better general math knowledge at the end of the school year.</p>
<p>And this early pattern knowledge matters for mathematics achievement in fifth grade as well. Children with better pattern knowledge at age seven had <a href="https://my.vanderbilt.edu/mathfollowup/reports/presentations/">better</a> mathematics achievement at age 11. Early pattern knowledge was found to be important for building later knowledge across a variety of mathematics topics, including number, algebra and geometry. </p>
<p>This also raises the issue of adding this to the <a href="http://www.corestandards.org/Math/">Common Core State Standards</a>, which currently do not include patterning as a math content standard at any grade level. </p>
<p>There was limited evidence available when the standards were written, but now that we know that patterning supports important mathematical reasoning and achievement,
I believe it should be made part of the Common Core standards.</p>
<p>At the same time, teachers and parents should consider how to support patterning in preschool and the early grades. They should help children look for regularities and rules in patterns by asking them to make the same kind of pattern using different objects or sounds and to name the part of the pattern that repeats, so as to identify its rule. </p>
<p>This will help provide a foundation for future math learning and reasoning.</p><img src="https://counter.theconversation.com/content/39417/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Bethany Rittle-Johnson receives funding from the National Science Foundation and the U.S. Department of Education</span></em></p>Patterns are simple sequences that repeat over and over again in a certain order. Supporting children’s ability to recognize patterns can improve mathematical skills.Bethany Rittle-Johnson, Associate Professor of Psychology, Vanderbilt UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/339682015-01-06T10:45:48Z2015-01-06T10:45:48ZOrigami: mathematics in creasing<figure><img src="https://images.theconversation.com/files/66776/original/image-20141209-32156-rxbbh2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Paper folding may look like art, but it's all about the math.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/minadyesherhairalot/14282756480/in/photolist-nL7RU5-ETiWH-6PagmA-9sdzwG-uxogH-aeWeM8-9rpQnv-5kyEzc-4vcWht-tw8LG-mYdT1v-tQHWo-Po8vV-76Zrt5-7tbTGF-5a78aV-aQ2Q96-7KvpU-bzvzhB-nPz6B-KFMYR-aBPtnC-bt2m6D-as37Sh-7bdTQ2-btyrv4-aBLQ9p-bZvBV-9rtZCf-4zYuC9-5iFPSz-5iL8vS-5iL8Fh-5iFPXt-5iL8fy-5iFQeZ-5iFQ7x-9r6bCH-bZHC9f-gXC7J5-Ah7JU-dKGZTe-5iL8dq-k4511-4zgvsM-uxohc-3kq88n-6LinCq-bdztnV-nNS2X5/">Mina</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Origami is the ancient Japanese art of paper folding. One uncut square of paper can, in the hands of an origami artist, be folded into a bird, a frog, a sailboat, or a <a href="https://www.flickr.com/photos/cavemanboon/3099233515/">Japanese samurai helmet beetle</a>. Origami can be extraordinarily complicated and intricate.</p>
<p>The art of origami has been going through a renaissance over the past 30 years, with <a href="http://www.langorigami.com">new designs</a> being created at ever-increasing levels of complexity. It’s no coincidence that this rise in origami complexity has emerged at the same time scientists, mathematicians and origami artists themselves have been discovering more and more of the mathematical rules that govern how paper folding works.</p>
<p>Indeed, if you take an origami model, of a bird for example, and carefully unfold it, you’ll see the pattern of creases that act as a blueprint for the model. This crease pattern contains the secret of how the paper is able to fold into the bird – and that secret is math. In theory, we could use this crease pattern to determine exactly how the paper should fold up and what shape it will form – if, that is, we understood all the secret rules of paper folding.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/64018/original/jg4bt3g9-1415413829.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/64018/original/jg4bt3g9-1415413829.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=384&fit=crop&dpr=1 600w, https://images.theconversation.com/files/64018/original/jg4bt3g9-1415413829.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=384&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/64018/original/jg4bt3g9-1415413829.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=384&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/64018/original/jg4bt3g9-1415413829.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=483&fit=crop&dpr=1 754w, https://images.theconversation.com/files/64018/original/jg4bt3g9-1415413829.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=483&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/64018/original/jg4bt3g9-1415413829.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=483&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The crease pattern for the classic flapping bird model, with mountain and valley creases indicated.</span>
<span class="attribution"><span class="source">Tom Hull</span></span>
</figcaption>
</figure>
<h2>Reading between the creases</h2>
<p>At heart, mathematics is about understanding the rules and patterns of the universe, be they patterns in numbers, in the stock market, or in nature. In the case of origami, we need to look at the geometry of the crease pattern, where the lines intersect, what angles they form, and in what direction the creases fold: are they valley creases or mountain creases?</p>
<p>Most traditional origami models fold flat, meaning you could press the model in a book without crumpling it. It turns out that the crease patterns of flat origami models have some very special properties. One of them is called Maekawa’s Theorem: at every vertex where creases intersect in a flat origami crease pattern, the difference between the number of mountain and valley creases is always two. So, at a vertex you could have 5 mountains and 3 valleys, but never 6 mountains and 2 valleys, for example.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/64020/original/wy8mpzj4-1415464693.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/64020/original/wy8mpzj4-1415464693.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=158&fit=crop&dpr=1 600w, https://images.theconversation.com/files/64020/original/wy8mpzj4-1415464693.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=158&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/64020/original/wy8mpzj4-1415464693.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=158&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/64020/original/wy8mpzj4-1415464693.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=199&fit=crop&dpr=1 754w, https://images.theconversation.com/files/64020/original/wy8mpzj4-1415464693.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=199&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/64020/original/wy8mpzj4-1415464693.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=199&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The Miura map fold crease pattern folds smoothly into a flat package.</span>
<span class="attribution"><span class="source">Tom Hull</span></span>
</figcaption>
</figure>
<h2>Beyond art to applications</h2>
<p>In the 1970s, Japanese astrophysicist Koryo Miura invented his <a href="http://en.wikipedia.org/wiki/Miura_fold">Miura map fold</a>, also known as the Miura-ori. It’s an example of an origami tesselation, where one shape is repeated over and over, with no gaps, across a whole surface. In this case, the crease pattern is a tiling of parallelograms laid out so the lines of the tiling also obey the rules of flat-folded origami. Dr. Miura chose the mountains and valleys of his crease pattern so that the model would open and close very easily.</p>
<figure>
<img src="http://upload.wikimedia.org/wikipedia/commons/5/55/Miura-ori.gif">
<figcaption><span class="caption">The Miura map fold in action.</span></figcaption>
</figure>
<p>This crease pattern makes a very good alternative for folding a map, since it opens and closes so easily. But Dr. Miura used this design as a way to deploy large solar panels into outer space. Think of each parallelogram as a solar cell, all of which are then connected by hinges. The array can then fold up into a small package to be put on a space satellite before being launched on a rocket. Once in space it could be opened by a simple expansion rod without the help of human hands.</p>
<p>The Miura map fold has inspired a lot of researchers to investigate how it works, its properties, and how it can be used. For example, <a href="http://mars.wne.edu/%7Ethull/">I’ve</a> worked with a team including researchers from the <a href="http://blogs.umass.edu/csantang/2014/08/07/new-paper-origami-meta-materials/">University of Massachusetts-Amherst</a> and <a href="https://cohengroup.lassp.cornell.edu/research.php?project=10019">Cornell University</a> to study the Miura map fold as a mechanical device; how much force is required to compress the fold, and how much does it spring back when released? In <a href="http://dx.doi.org/10.1126/science.1252876">Science</a>, we reported how we can change this behavior by introducing defects into the Miura map fold, say by poking some of the vertices the other way. An example is shown below.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/64187/original/xcj7shwd-1415672838.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/64187/original/xcj7shwd-1415672838.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=624&fit=crop&dpr=1 600w, https://images.theconversation.com/files/64187/original/xcj7shwd-1415672838.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=624&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/64187/original/xcj7shwd-1415672838.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=624&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/64187/original/xcj7shwd-1415672838.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=784&fit=crop&dpr=1 754w, https://images.theconversation.com/files/64187/original/xcj7shwd-1415672838.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=784&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/64187/original/xcj7shwd-1415672838.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=784&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The Miura map fold with defects introduced. The defects lead to fewer pleats at the bottom than at the top.</span>
<span class="attribution"><span class="source">Jesse Silverberg and the Itai Cohen Group at Cornell University</span></span>
</figcaption>
</figure>
<p>Our group has also been studying self-folding. We’ve made materials that fold themselves, which has been a topic of interest to <a href="https://www.youtube.com/watch?v=Pg8VAVWkz3k">other groups</a> <a href="http://wyss.harvard.edu/viewpressrelease/162">as well</a>. <a href="http://www.pse.umass.edu/faculty/researchgroup/hayward/group">Ryan Hayward’s group</a> at the <a href="http://www.pse.umass.edu/">Conte National Center for Polymer Research</a> has developed a way to make microscopic gel sheets swell along crease lines when heated. Their methods can make a microscopic crane:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/64196/original/kbxkz57f-1415675140.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/64196/original/kbxkz57f-1415675140.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=492&fit=crop&dpr=1 600w, https://images.theconversation.com/files/64196/original/kbxkz57f-1415675140.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=492&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/64196/original/kbxkz57f-1415675140.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=492&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/64196/original/kbxkz57f-1415675140.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=618&fit=crop&dpr=1 754w, https://images.theconversation.com/files/64196/original/kbxkz57f-1415675140.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=618&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/64196/original/kbxkz57f-1415675140.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=618&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A self-folded polymer crane, just a fraction of a millimeter in width.</span>
<span class="attribution"><span class="source">Jun-Hee Na, Hayward Research Group, UMass Amherst</span></span>
</figcaption>
</figure>
<p>This crane could be the smallest folded crane ever made! The polymer self-folding gel can make very complicated designs, like this three-dimensional <a href="http://en.wikipedia.org/wiki/Space_frame">octahedron-tetrahedron truss</a> tessellation:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/64198/original/78wkpjtr-1415675467.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/64198/original/78wkpjtr-1415675467.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/64198/original/78wkpjtr-1415675467.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/64198/original/78wkpjtr-1415675467.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/64198/original/78wkpjtr-1415675467.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/64198/original/78wkpjtr-1415675467.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/64198/original/78wkpjtr-1415675467.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Octahedron tetrahedron truss confocal microscopy image.</span>
<span class="attribution"><span class="source">Jun-Hee Na, Hayward Research Group, UMass Amherst</span></span>
</figcaption>
</figure>
<p>Such tiny self-folding gel objects might someday be used in bio-engineering. Imagine a toxic anticancer drug being enclosed in a self-folding origami ball, where the ball is programmed to unfold only when it comes in contact with a tumor. Then the drug can be delivered exactly to the tumor without poisoning other parts of the patient’s body.</p>
<p>None of these origami applications would be possible without understanding the mathematical rules behind origami. It is a great example of how math – and origami – can be found in unexpected places.</p><img src="https://counter.theconversation.com/content/33968/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Thomas Hull receives funding from the National Science Foundation grant EFRI ODISSEI-1240441.</span></em></p>Origami is the ancient Japanese art of paper folding. One uncut square of paper can, in the hands of an origami artist, be folded into a bird, a frog, a sailboat, or a Japanese samurai helmet beetle. Origami…Thomas Hull, Associate Professor of Mathematics, Western New England UniversityLicensed as Creative Commons – attribution, no derivatives.