tag:theconversation.com,2011:/us/topics/non-euclidean-geometry-24199/articlesNon-Euclidean geometry – The Conversation2023-01-01T19:39:45Ztag:theconversation.com,2011:article/1960532023-01-01T19:39:45Z2023-01-01T19:39:45ZExploring the mathematical universe – connections, contradictions, and kale<figure><img src="https://images.theconversation.com/files/502042/original/file-20221220-22-x7dzft.png?ixlib=rb-4.1.0&rect=443%2C71%2C3293%2C1886&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock/The Conversation</span></span></figcaption></figure><p>Science and maths skills are widely celebrated as keys to economic and technological progress, but abstract mathematics may seem bafflingly far from industrial optimisation or medical imaging. Pure mathematics often yields unanticipated applications, but without a time machine to look into the future, how do mathematicians like me choose what to study?</p>
<p>Over Thai noodles, I asked some colleagues what makes a problem interesting, and they offered a slew of suggestions: surprises, contradictions, patterns, exceptions, special cases, connections. These answers might sound quite different, but they all support a view of the mathematical universe as a structure to explore. </p>
<p>In this view, mathematicians are like anatomists learning how a body works, or navigators charting new waters. The questions we ask take many forms, but the most interesting ones are those that help us see the big picture more clearly. </p>
<h2>Making maps</h2>
<p>Mathematical objects come in many forms. Some of them are probably quite familiar, like numbers and shapes. Others might seem more exotic, like equations, functions and symmetries.</p>
<p>Instead of just naming objects, a mathematicians might ask how some class of objects is organised. Take prime numbers: we know there are infinitely many of them, but we need a structural understanding to work out how frequently they occur or to identify them in an efficient way.</p>
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<a href="https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A grid of blue dots" src="https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=602&fit=crop&dpr=1 600w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=602&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=602&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=756&fit=crop&dpr=1 754w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=756&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=756&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 ‘Ulam spiral’ reveals some structure in the primes. If you arrange the counting numbers in squares spiralling outward, it becomes clear that many prime numbers fall on diagonal lines.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Ulam_spiral#/media/File:Spirale_Ulam_150.jpg">Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>Other good questions explore relationships between apparently different objects. For example, shapes have symmetry, but so do the solutions to some equations. </p>
<p>Classifying objects and finding connections between them help us assemble a coherent map of the mathematical world. Along the way, we sometimes encounter surprising examples that defy the patterns we’ve inferred. </p>
<p>Such apparent contradictions reveal where our understanding is still lacking, and resolving them provides valuable insight.</p>
<h2>Consider the triangle</h2>
<p>The humble triangle provides a famous example of an apparent contradiction. Most people think of a triangle as the shape formed by three connecting line segments, and this works well for the geometry we can draw on a sheet of paper. </p>
<p>However, this notion of triangle is limited. On a surface with no straight lines, like a sphere or a curly kale leaf, we need a more flexible definition. </p>
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<strong>
Read more:
<a href="https://theconversation.com/pythagoras-revenge-humans-didnt-invent-mathematics-its-what-the-world-is-made-of-172034">Pythagoras’ revenge: humans didn’t invent mathematics, it’s what the world is made of</a>
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<p>So, to extend geometry to surfaces that aren’t flat, an open-minded mathematician might propose a new definition of a triangle: pick three points and connect each pair by the shortest path between them. </p>
<p>This is a great generalisation because it matches the familiar definition in the familiar setting, but it also opens up new terrain. When mathematicians first studied these generalised triangles in the 19th century, they solved a millennia-old mystery and revolutionised mathematics.</p>
<h2>The parallel postulate problem</h2>
<p>Around 300 BC, the Greek mathematician Euclid wrote a treatise on planar geometry called The Elements. This work presented both fundamental principles and results that were logically derived from them. </p>
<p>One of his principles, called the parallel postulate, is equivalent to the statement that the sum of the angles in any triangle is 180°. This is exactly what you’ll measure in every flat triangle, but later mathematicians debated whether the parallel postulate should be a foundational principle or just a consequence of the other fundamental assumptions. </p>
<p>This puzzle persisted until the 1800s, when mathematicians realised why a proof had remained so elusive: the parallel postulate is false on some surfaces. </p>
<figure class="align-center ">
<img alt="Image showing that a triangle on the surface of a sphere will have angles that add up to more than 180°, but on a hyperbolic surface will add up to less than 180°." src="https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=265&fit=crop&dpr=1 600w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=265&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=265&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=334&fit=crop&dpr=1 754w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=334&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=334&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
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<p>On a sphere, the sides of a triangle bend away from each other and the angles add up to more than 180°. On a rippled kale leaf, the sides bow in towards each other and the angle sum is less than 180°. </p>
<p>Triangles where the angle sum breaks the apparent rule led to the revelation that there are kinds of geometry Euclid never imagined. This is a deep truth, with applications in physics, computer graphics, fast algorithms, and beyond. </p>
<h2>Salad days</h2>
<p>People sometimes debate whether mathematics is discovered or invented, but both points of view feel real to those of us who study mathematics for a living. Triangles on a piece of kale are skinny whether or not we notice them, but selecting which questions to study is a creative enterprise. </p>
<p>Interesting questions arise from the friction between patterns we understand and the exceptions that challenge them. Progress comes when we reconcile apparent contradictions that pave the way to identify new ones. </p>
<p>Today we understand the geometry of two-dimensional surfaces well, so we’re equipped to test ourselves against similar questions about higher-dimensional objects.</p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/corals-crochet-and-the-cosmos-how-hyperbolic-geometry-pervades-the-universe-53382">Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe</a>
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<p>In the past few decades we’ve learned that three-dimensional spaces also have their own innate geometries. The most interesting one is called hyperbolic geometry, and it turns out to act like a three-dimensional version of curly kale. We know this geometry exists, but it remains mysterious: in my own research field, there are lots of questions we can answer for any three-dimensional space … except the hyperbolic ones.</p>
<p>In higher dimensions we still have more questions than answers, but it’s safe to say that study of four-dimensional geometry is entering its salad days.</p><img src="https://counter.theconversation.com/content/196053/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Joan Licata 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>Mathematicians are like anatomists learning how a body works, or navigators charting new waters.Joan Licata, Associate Professor, Mathematics, Australian National UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/811432017-07-17T23:56:59Z2017-07-17T23:56:59ZMaryam Mirzakhani was a role model for more than just her mathematics<p>On July 14, 2017, Maryam Mirzakhani, Stanford professor of mathematics and the only female winner of the prestigious Fields Medal in Mathematics, died at the age of 40. </p>
<p>In just a few hours, her name, both in her native Farsi (#مریم میرزاخانی) and English (#maryammirzakhani), was trending on Twitter and Facebook. <a href="http://ifpnews.com/exclusive/iran-newspaper-front-page-july-16-2017/">Most major news agencies</a> were <a href="http://www.bbc.com/news/science-environment-40617094">covering the news</a> of <a href="https://www.nytimes.com/2017/07/16/us/maryam-mirzakhani-dead.html">her death</a> as well as recounting <a href="http://news.stanford.edu/2017/07/15/maryam-mirzakhani-stanford-mathematician-and-fields-medal-winner-dies/">her many achievements</a>.</p>
<p>The grief was especially hard-hitting for a generation of younger academics like me who have always held Maryam as a role model whose example is helping redefine women’s status in science and especially mathematics. </p>
<p>The irony was that Maryam always tried to avoid the media’s spotlight. Her modesty and simplicity despite being the only woman to gain such high status in the world of mathematics – winning what’s often called the “Nobel Prize of math” – stood out to those who knew her.</p>
<p>Unfortunately, I did not get the chance to meet Maryam personally. But like many of my Iranian peers in academia, I looked to her example as proof that the world would welcome us and our scientific contributions no matter our skin color, nationality or religion. </p>
<p>As people around the globe grieve the loss of this talented mathematician, Maryam’s life stands as an inspiration for young girls and boys from all walks of life the world over.</p>
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<iframe width="440" height="260" src="https://www.youtube.com/embed/swLWqlKMl5M?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Maryam Mirzakhani in her own words in a video by the Simons Foundation and the International Mathematical Union.</span></figcaption>
</figure>
<h2>Steady advances of a hardworking genius</h2>
<p>Despite her calm expression and warm smile, Maryam was a warrior. She and her family, alongside many other Iranians, lived through the hard economic and social transformations after the Iran revolution in 1979 and also survived the eight years of the Iran-Iraq war a few years after that.</p>
<p>Maryam originally wanted to be a writer, a passion of hers that never faded away even during her postgraduate studies. However, she found an even greater joy in how rewarding it felt to solve mathematical problems. As a student, she was the first female member of Iran’s national team to participate in the International Math Olympiad, and she <a href="https://www.imo-official.org/participant_r.aspx?id=926">won two gold medals</a> in two consecutive years – still a record.</p>
<p>She received her bachelor’s degree from Sharif University of Technology in Iran and later a doctorate from Harvard. In 2014, Maryam was <a href="https://theconversation.com/meet-the-winners-of-the-fields-medal-the-nobel-prize-of-maths-30411">recognized with the Fields Medal</a>, the highest-ranking award in mathematics, for her efforts in what’s known as <a href="https://theconversation.com/corals-crochet-and-the-cosmos-how-hyperbolic-geometry-pervades-the-universe-53382">hyperbolic geometry</a>. Her work focused on curved surfaces – such as spheres or donut shapes – and how to understand their properties. Her achievements have applications in other fields of science including quantum field theory, engineering and material science, and could even influence theories around how our universe was born.</p>
<p>Maryam was a “hall of fame” all by herself. She modestly attributed her own success to her perseverance, hard work and patience. <a href="https://www.theguardian.com/science/2014/aug/13/interview-maryam-mirzakhani-fields-medal-winner-mathematician">As she put it</a>:</p>
<blockquote>
<p>“The beauty of mathematics only shows itself to more patient followers.”</p>
</blockquote>
<p>Unfortunately, when she was honored with the Fields Medal, she was already tackling her last challenge, the breast cancer that eventually killed her.</p>
<h2>Who she was, not just what she did, matters</h2>
<p><a href="https://theconversation.com/maryam-mirzakhanis-success-showed-us-the-challenges-women-in-maths-still-face-81193">Maryam’s contributions</a> to the field of mathematics will long be remembered. But just as important is her legacy as a role model. </p>
<p>Maryam was an Iranian, a woman and an immigrant to the United States. Unfortunately, these three words together raise red flags for some in Western countries, particularly in the U.S., in the time of <a href="https://theconversation.com/us/topics/trump-travel-ban-35583">Trump’s proposed travel ban</a>. </p>
<p>Against all odds, Maryam’s talent was nurtured in Iran and later flourished in the U.S. Her successes discredit the xenophobic stereotypes that are encouraged by a politics of fear. Maryam defied expectations and rose above all the labels that make it easy to judge others who are not like “us.”</p>
<p>Maryam’s legend may continue to grow after her early death. Still only 20 percent of full-time math faculty at U.S. universities are women, <a href="http://www.ams.org/profession/data/annual-survey/demographics">according to a 2015 demographic survey</a> of 213 departments by the American Mathematical Society. Research shows that <a href="https://doi.org/10.1177/0361684312459328">stereotyped role models can influence</a> whether people “see themselves” in certain STEM careers. The example of a woman who rose to the top of this still very male field may help inspire math’s next generation. </p>
<p>In the same way people think of Marie Curie or Jane Goodall as scientific pioneers, Maryam Mirzakhani will go down in history as a trailblazer as well as a mathematical genius.</p><img src="https://counter.theconversation.com/content/81143/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mehrdokht (Medo) Pournader 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>Mirzakhani blazed to the top of her field due to her talent. But who she was and where she came from also make her a role model for those from underrepresented demographics in the world of math.Mehrdokht (Medo) Pournader, Senior lecturer at The University of Melbourne, The University of MelbourneLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/533822016-01-27T19:10:12Z2016-01-27T19:10:12ZCorals, crochet and the cosmos: how hyperbolic geometry pervades the universe<figure><img src="https://images.theconversation.com/files/108537/original/image-20160119-29758-q26y1k.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The frilly forms of corals and sponges are biological variations of hyperbolic geometry, as seen here on the Great Barrier Reef, near Cairns, Queensland, Australia.</span> <span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Folded_Coral_Flynn_Reef.jpg">Wikimedia/Toby Hudson</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>We have built a world of largely straight lines – the houses we live in, the skyscrapers we work in and the streets we drive on our daily commutes. Yet outside our boxes, nature teams with frilly, crenellated forms, from the fluted surfaces of lettuces and fungi to the frilled skirts of sea slugs and the gorgeous undulations of corals. </p>
<p>These organisms are biological manifestations of what we call <a href="http://mathworld.wolfram.com/HyperbolicGeometry.html">hyperbolic geometry</a>, an alternative to the <a href="http://www.britannica.com/topic/Euclidean-geometry">Euclidean geometry</a> we learn about in school that involves lines, shapes and angles on a flat surface or plane. In hyperbolic geometry the plane is not necessarily so flat.</p>
<p>Yet while nature has been playing with hyperbolic forms for hundreds of millions of years, mathematicians spent hundreds of years trying to prove that such structures were impossible.</p>
<p>But these efforts led to a realisation that hyperbolic geometry is logically legitimate. And that, in turn, led to the revolution that produced the kind of maths now underlying general relativity, and thus the structure of the universe. </p>
<h2>Non-Euclidean clause</h2>
<p>Hyperbolic geometry is radical because it violates one of the <a href="https://plus.maths.org/content/maths-minute-euclids-axioms">axioms of Euclidean geometry</a>, which long stood as a model for reason itself. </p>
<p>The fifth and final axiom of Euclid’s system – the so-called parallel postulate – turns out not to be correct. Or at least not necessarily so. If we accept it, we get Euclidean geometry, but if we abandon it, other geometries become possible, most famously the hyperbolic variety.</p>
<p>Here’s how the parallel postulate works. Consider a simple question: if I have a straight line, and a point outside the line, how many straight lines can I draw through the point that never meet the original line? Euclid said the answer is <em>one</em> and there couldn’t be any more, which feels intuitively right.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/108513/original/image-20160119-29783-u56qiy.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/108513/original/image-20160119-29783-u56qiy.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/108513/original/image-20160119-29783-u56qiy.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=150&fit=crop&dpr=1 600w, https://images.theconversation.com/files/108513/original/image-20160119-29783-u56qiy.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=150&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/108513/original/image-20160119-29783-u56qiy.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=150&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/108513/original/image-20160119-29783-u56qiy.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=189&fit=crop&dpr=1 754w, https://images.theconversation.com/files/108513/original/image-20160119-29783-u56qiy.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=189&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/108513/original/image-20160119-29783-u56qiy.png?ixlib=rb-4.1.0&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">Euclid could only see one possible straight line through a point that does not meet the original line.</span>
<span class="attribution"><span class="source">Margaret Wertheim</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Mathematicians, being sticklers, wanted to prove this was true, but in the end such efforts led them to see that there is a logically consistent geometric system in which the answer is infinity. We can represent the situation as follows.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/108514/original/image-20160119-29777-6ybwm8.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/108514/original/image-20160119-29777-6ybwm8.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/108514/original/image-20160119-29777-6ybwm8.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=300&fit=crop&dpr=1 600w, https://images.theconversation.com/files/108514/original/image-20160119-29777-6ybwm8.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=300&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/108514/original/image-20160119-29777-6ybwm8.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=300&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/108514/original/image-20160119-29777-6ybwm8.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=377&fit=crop&dpr=1 754w, https://images.theconversation.com/files/108514/original/image-20160119-29777-6ybwm8.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=377&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/108514/original/image-20160119-29777-6ybwm8.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=377&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">What if the straight lines look curved?</span>
<span class="attribution"><span class="source">Margaret Wertheim</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>This seems impossible and a first reaction is to say it’s cheating because the lines look curved. But they only look curved because we’re trying to project an image of a curved surface onto a flat plane.</p>
<p>It’s the same as when we’re trying to project an image of the surface of the Earth onto a flat map; the relationships get distorted. To really see countries relative to one another we have to look at a globe.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/JlZIU5E8UC8?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">How to turn our home planet into a flat Earth.</span></figcaption>
</figure>
<p>So also with hyperbolic geometry. To really see what’s going on we have to look at the curved surface itself, and here the lines are straight. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/108806/original/image-20160121-9732-1mv0bqx.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/108806/original/image-20160121-9732-1mv0bqx.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/108806/original/image-20160121-9732-1mv0bqx.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=237&fit=crop&dpr=1 600w, https://images.theconversation.com/files/108806/original/image-20160121-9732-1mv0bqx.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=237&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/108806/original/image-20160121-9732-1mv0bqx.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=237&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/108806/original/image-20160121-9732-1mv0bqx.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=298&fit=crop&dpr=1 754w, https://images.theconversation.com/files/108806/original/image-20160121-9732-1mv0bqx.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=298&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/108806/original/image-20160121-9732-1mv0bqx.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=298&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 image shows straight lines drawn on a paper model of a hyperbolic plane. All the pencil lines that appear to be curved were drawn with a ruler so they are actually straight.</span>
<span class="attribution"><span class="source">Margaret Cagyle, Institute For Figuring</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>One way of understanding different geometries is in terms of their curvature. A flat, or Euclidean plane has zero curvature. The surface of a sphere (like a beach ball) has positive curvature, and a hyperbolic plane has negative curvature. It’s a geometric analogue of a negative number. </p>
<p>When mathematicians discovered this aberrant geometry in the early 19th century they were nearly driven mad. “For God’s sake please give it up,” said the <a href="http://www.math.cornell.edu/%7Edwh/papers/crochet/crochet.html">Hungarian mathematician Wolfgang Bolyai</a> to his son János Bolyai, urging him abandon to work on hyperbolic geometry.</p>
<h2>Nature’s work</h2>
<p>Yet critters who’d never studied non-Euclidean geometry had meanwhile just been doing it. Along with corals, many other species of reef organisms have hyperbolic forms, including sponges and kelps. </p>
<p>Wherever there is an advantage to maximising surface area – such as for filter feeding animals – hyperbolic shapes are an excellent solution. There are hyperbolic structures in cells, hyperbolic cacti and hyperbolic flowers, such as calla lilies. In the film Avatar, there is a fabulous CGI grove of giant hyperbolic blooms that curl up when touched.</p>
<p>Hyperbolic surfaces can also be built at the molecular scale from carbon atoms. These carbon nano-foams were discovered in 1997 by physicist <a href="https://physics.anu.edu.au/people/profile.php?ID=356">Andrei Rode</a> and his colleagues at the Australian National University.</p>
<p>That year Cornell mathematician <a href="http://www.math.cornell.edu/%7Edtaimina/">Daina Taimina</a> also worked out how to model such surfaces using crochet, which was a big deal because it’s actually hard for humans to construct these forms. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/zGEDHMF4rLI?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>For the past 10 years, I’ve been spearheading a project where we use hyperbolic crochet to make woolly simulations of coral reefs. Our <a href="http://www.crochetcoralreef.org/">Crochet Coral Reefs</a> are an artistic response to the devastation of living reefs due to global warming and have been exhibited at art galleries and science museums around the world, including the Smithsonian. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/108809/original/image-20160121-9746-jcpy86.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/108809/original/image-20160121-9746-jcpy86.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/108809/original/image-20160121-9746-jcpy86.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=237&fit=crop&dpr=1 600w, https://images.theconversation.com/files/108809/original/image-20160121-9746-jcpy86.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=237&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/108809/original/image-20160121-9746-jcpy86.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=237&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/108809/original/image-20160121-9746-jcpy86.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=297&fit=crop&dpr=1 754w, https://images.theconversation.com/files/108809/original/image-20160121-9746-jcpy86.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=297&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/108809/original/image-20160121-9746-jcpy86.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=297&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 crochet coral reef based on hyperbolic geometry.</span>
<span class="attribution"><span class="source">Institute For Figuring</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Here, a ball of wool and a crochet hook become pedagogical tools bringing mathematics out of textbooks, and taking it to people as a living tactile experience.</p>
<p>More than 8,000 women in a dozen countries (including Australia, the United States of America, and the United Arab Emirates) have participated in making these installations, which reside at the intersection of mathematics, marine biology, community art practice and environmentalism. </p>
<h2>The shape of the universe</h2>
<p>Once mathematicians realised that different geometrical spaces are possible, a question arose as to which one is realised in physical space. What is the shape of our universe?</p>
<p>Galileo Galilei and Isaac Newton founded modern physics on the assumption that space is Euclidean, but Albert Einstein’s equations of <a href="https://theconversation.com/au/topics/general-relativity">general relativity</a> describe a universe that can have complex curved forms.</p>
<p>One of the major questions astronomers are trying to resolve, with instruments such as the Hubble Space Telescope, is <a href="http://www.space.com/24309-shape-of-the-universe.html">what shape our universe</a> has. While most of the large-scale evidence points to a Euclidean structure, there is some tantalising evidence that we might just live in a hyperbolic world.</p><img src="https://counter.theconversation.com/content/53382/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Margaret Wertheim gave a talk for the Australian Mathematical Sciences Institute at their 2016 annual Summer School.</span></em></p>The magic and wonder of the mathematics of straight lines in curved spaces is best explained when you look to nature for examples.Margaret Wertheim, Vice-Chancellor’s Fellow in Science Communication, The University of MelbourneLicensed as Creative Commons – attribution, no derivatives.