tag:theconversation.com,2011:/id/topics/mathematics-and-statistics-awareness-month-37083/articlesMathematics and Statistics Awareness Month – The Conversation2017-04-19T22:35:25Ztag:theconversation.com,2011:article/749092017-04-19T22:35:25Z2017-04-19T22:35:25ZThe extraordinary return of sea otters to Glacier Bay<figure><img src="https://images.theconversation.com/files/165688/original/image-20170418-32716-dgt62a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A sea otter floats in Kachemak Bay, Alaska.</span> <span class="attribution"><a class="source" href="http://www.apimages.com/metadata/Index/Sea-Otter-Bounty/d17ce8d14b9f4427a9a917fe952b0233/13/0">AP Photo/Laura Rauch</a></span></figcaption></figure><p>Human beings have a long history of persecuting apex predators such as wolves, tigers and leopards. The loss of these predators – animals at the top of the food chain – has resulted in ecological, economic and social <a href="http://science.sciencemag.org/content/333/6040/301">impacts around the globe</a>. Rarely do the predators fully recover from human oppression, and, when they do, we often lack data or tools to assess their recovery.</p>
<p>The sea otters in Glacier Bay, Alaska, are an exception. In <a href="http://dx.doi.org/10.1002/ecy.1643">a recent study</a>, our team chronicled the incredible return of sea otters to an area where they’ve been absent for at least 250 years. </p>
<p>Our approach – which fuses mathematics, statistics and ecology – can help us better understand the role of sea otters in marine ecosystems and the ability of apex predators to return to an ecosystem after they’ve been absent. It may even help us learn what a changing climate means for many other species.</p>
<h2>Return to Glacier Bay</h2>
<p>Although not typically viewed in the same vein as wolves, tigers and leopards, sea otters are an apex predator of the nearshore marine ecosystem - the narrow band between terrestrial and oceanic habitat.</p>
<p>During the commercial maritime fur trade in the 18th and 19th centuries, sea otters were nearly hunted to extinction across their range in the North Pacific Ocean. By 1911, only a handful of small isolated populations remained.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=458&fit=crop&dpr=1 600w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=458&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=458&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=576&fit=crop&dpr=1 754w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=576&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=576&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Historic range (gray shading) and 1911 remnant populations (red icons) of sea otters. The populations at Queen Charlotte Islands and San Benito Islands went extinct by 1920.</span>
<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>But sea otter populations have recovered in many areas, thanks to a few changes. The <a href="http://celebrating200years.noaa.gov/events/fursealtreaty/welcome.html">International Fur Seal Treaty</a> in 1911 protected sea otters from most human harvest. Wildlife agencies also made an effort to aid sea otter recolonization.</p>
<p>Eventually, sea otters began to increase in abundance and distribution, and they made their way to Glacier Bay, a tidewater glacier fjord and national park in southeastern Alaska. Glacier Bay is functionally one of the largest marine <a href="https://www.nps.gov/glba/index.htm">protected areas</a> in the northern hemisphere. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=1246&fit=crop&dpr=1 600w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=1246&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=1246&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Glacier Bay National Park, southeastern Alaska.</span>
<span class="attribution"><span class="source">Map used with permission from the National Park Service</span></span>
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</figure>
<p>Glacier Bay was completely covered by glacier ice until approximately 1750 – about the same time sea otters vanished from the surrounding area due to over-harvest. It then endured the most rapid and extensive tidewater glacier retreat in recorded history. After glacier retreat, a rich environment emerged. This new environment supported high concentrations of wildlife, including sea otter prey species – such as crabs, mollusks and sea urchins – that were able to increase in size and abundance in the absence of sea otters. </p>
<p>Sea otters first reappeared at the mouth of Glacier Bay in 1988. Here they encountered vast habitat, abundant prey populations and protection from all human harvest.</p>
<h2>Our approach</h2>
<p>It’s challenging to estimate how populations grow and spread, due to their dynamic nature. Each year, animals move to new areas, increasing the amount of area and effort required to find them. Airplanes searching for sea otters have to cover more ground, usually with the same amount of time and money. Additionally, individuals may move from one area to the next during any time period for a number of reasons, including sea otter social behavior and their reaction to the environment. Because these challenges can interfere with accurate population estimates, it’s important to understand and address them. </p>
<p>Shortly after sea otters arrived in Glacier Bay, scientists from the U.S. Geological Survey began collecting data to document their return. Although the data clearly indicated that sea otters were increasing, we needed novel statistical methods to unveil the extent of this increase.</p>
<p>First, we developed a mathematical model using partial differential equations to describe the growth and spread of sea otters. Partial differential equations are commonly used to describe phenomena such as fluid dynamics and quantum mechanics. Therefore, they were a natural choice to describe how a mass – in our case, the sea otter population – spreads through space and time. </p>
<p>The new approach allowed us to incorporate our current understanding of sea otter ecology and behavior, including habitat preferences, maximum growth rates and where sea otters were first observed in Glacier Bay. </p>
<p>Second, we incorporated our equations within a hierarchical statistical model. Hierarchical models are used to draw conclusions from data that arise from complex processes. They provide flexibility to describe and distinguish among various sources of uncertainty, such as uncertainty in data collection and ecological processes.</p>
<p>Partial differential equations are not new to the field of ecology, dating back to <a href="http://www.math.wm.edu/%7Eshij/math490-2006/skellam.pdf">at least 1951</a>. However, by fusing these equations with formal statistical models, we can reliably infer dynamic ecological processes, while appropriately quantifying the uncertainty associated with our findings. It provides a data-driven way to analyze surveys of sea otter abundance for the past 25 years. </p>
<p>This gave us rigorous and honest estimates of colonization dynamics that incorporated our understanding of the ecological system. </p>
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<a href="https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.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">Group of sea otters in Glacier Bay National Park, 2016.</span>
<span class="attribution"><span class="source">Photo by Jamie Womble</span></span>
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</figure>
<h2>A record-breaking recovery</h2>
<p>Using our new approach, we discovered that the Glacier Bay sea otter population grew more than 21 percent per year between 1993 and 2012. </p>
<p>By comparison, the estimated growth rates of sea otters in other populations in Alaska, who were also recovering, have been limited to 17 to 20 percent. Furthermore, the maximum biological reproductive rate – the fastest rate sea otters can reproduce – is between 19 to 23 percent per year. That means that the Glacier Bay sea otter growth rate was near or at maximum, and greater than any recorded sea otter population in history. </p>
<p>In the wake of glacier retreat, sea otters went from nonexistent to colonizing nearly all of Glacier Bay in a span of 20 years. Today, they are one of the most abundant marine mammals in Glacier Bay. Recent observations have documented large groups of more than 500 sea otters in some parts of lower Glacier Bay, suggesting that prey resources are abundant. </p>
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<p>The fusion of state-of-the-art statistical and mathematical methods depicted, for the first time, just how extraordinary the growth and spread of this population was. </p>
<p>Sea otters had great success in the wake of tidewater glacier retreat in Glacier Bay. While climate-induced loss of sea ice can negatively affect some wide-ranging apex predators – such as polar bears or walruses – other species may benefit from the emergence of newly available habitat and prey resources. </p>
<p>Humans have caused the global decline of apex predators, and these declines are often difficult to reverse. However, our results suggest that, when there is minimal human interference, apex predators can be widely successful at recolonizing suitable habitat.</p><img src="https://counter.theconversation.com/content/74909/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mevin Hooten receives funding from the National Park Service, U.S. Geological Survey, National Oceanic and Atmospheric Administration, and the National Science Foundation. </span></em></p><p class="fine-print"><em><span>Perry Williams 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>Sea otters had been absent from this Alaskan national park for at least 250 years. By marrying math and statistics, scientists map this animal’s successful comeback.Perry Williams, Postdoctoral Fellow in Statistics and Fish, Wildlife, and Conservation Biology, Colorado State UniversityMevin Hooten, Assistant Unit Leader, U.S. Geological Survey, Colorado Cooperative Fish and Wildlife Research Unit; Associate Professor, Fish, Wildlife, and Conservation Biology and Statistics, Colorado State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/748062017-04-19T01:10:05Z2017-04-19T01:10:05ZWhat Netflix can teach us about treating cancer<figure><img src="https://images.theconversation.com/files/162238/original/image-20170323-4948-hcdwjq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A tumor under the microscope.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/76113756@N07/7087945939/in/photolist-bNkzot-pFEv1-bNkzkV-KQFoi-pTSHEo-6p4KHu-aDhQf9-8T8qYG-6G4SDG-d2qB81-9t97Lv-5vpM9D-au4amg-piqevs-9zZio8-au6Qco-sNkfq-qd8pow-6TwsHb-d2qGo7-6ZwhSG-axz84L-atMLeP-axwqfe-baXQEz-6YGqVj-5vKdvj-D5WbG4-24ESD-G2P9bc-24EST-6dAdJh-6Bbn2K-6ZdEGR-p54VMo-9bJHw2-33cdAj-6ZsdgE-343cj1-6riY8e-9zZivB-uEE11-9wVTTD-9wVU1K-nZV1Zb-6dw5uc-76fTxT-fA4xJj-axwqd8-8rrS5W">Cropped from cnicholsonpath/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>Two years ago, former President Barack Obama announced the <a href="https://obamawhitehouse.archives.gov/precision-medicine">Precision Medicine initiative</a> in his State of the Union Address. The initiative aspired to a “new era of medicine” where disease treatments could be specifically tailored to each patient’s genetic code. </p>
<p>This resonated soundly in cancer medicine. Patients can already manage their cancer with therapies that target the specific genes that are altered in their particular tumor. For example, women with a type of breast cancer caused by the amplification of gene HER2 are often treated with a therapeutic called herceptin. Because these targeted therapeutics are specific to cancer cells, they tend to have fewer side effects than traditional cancer treatments with chemotherapy or radiation. </p>
<p>However, such treatments are not available for most cancer patients. In many cancers, the specific genetic alterations that are responsible for a cancer remain unknown. To create individualized cancer treatments, we must know more about the functional genetic alterations.</p>
<p>With data on cancer genetics growing rapidly, mathematics and statistics can now help unlock the hidden patterns in this data to find the genes that are responsible for an individual’s cancer. With this knowledge, physicians can select appropriate treatments that block the action of these genes to personalize therapies for individual patients. My research aims to improve precision medicine in cancer – by building on the same methods that have been used to find patterns in Netflix movie ratings.</p>
<h2>Sifting through the data</h2>
<p>Today, there is unprecedented public access to cancer genetics data. These data come from generous patients who donate their tumor samples for research. Scientists then apply sequencing technologies to measure the mutations and activity in each of the 20,000 genes in the human genome.</p>
<p>All these data are a direct result of the <a href="http://www.nature.com/nature/journal/v431/n7011/full/nature03001.html">Human Genome Project</a> in 2003. That project determined the sequence for all the genes that make up healthy human DNA. Since the completion of that project, the cost of sequencing the human genome has <a href="http://www.nature.com/news/technology-the-1-000-genome-1.14901">more than halved every year</a>, surpassing the growth of computing power described in <a href="https://theconversation.com/moores-law-is-50-years-old-but-will-it-continue-44511">Moore’s Law</a>. This cost reduction enables researches to collect unprecedented genetics data from cancer patients.</p>
<p>Most scientific studies on cancer genetics performed worldwide release their data to a centralized, public database provided by the U.S. National Institutes of Health (NIH) National Library of Medicine. The NIH National Cancer Institute and National Human Genome Research Institute have also freely released genetic data from over 11,000 tumors in 33 cancer types through a project called <a href="https://cancergenome.nih.gov/">The Cancer Genome Atlas.</a></p>
<p>Every biological function – from extracting energy from food to healing a wound – results from activity in different combinations of genes. Cancers hijack the genes that enable people to grow to adulthood and that protect the body from the immune system. Researchers dub these the <a href="http://www.cell.com/abstract/S0092-8674(11)00127-9">“hallmarks of cancer.”</a> This so-called gene dysregulation enables a tumor to grow uncontrollably and form metastases in distant organs from the original tumor site.</p>
<p>Researchers are actively using these public data to find the set of gene alterations that are responsible for each tumor type. But this problem is not as simple is identifying a single dysregulated gene in each tumor. Hundreds, if not thousands, of the 20,000 genes in the human genome are dysregulated in cancer. The group of dysregulated genes varies in each patient’s tumor, with smaller sets of commonly reused genes enabling each cancer hallmark. </p>
<p>Precision medicine relies on finding the smaller groups of dysregulated genes that are responsible for biological function in each patient’s tumor. But, genes may have multiple biological functions in different contexts. Therefore, researchers must uncover a set of “overlapping” genes that have common functions in a set of cancer patients. </p>
<p>Linking gene status to function requires complex mathematics
and immense computing power. This knowledge is essential to predict of outcome to therapies that would block the function of these genes. So, how can we uncover those overlapping features to predict individual outcomes for patients? </p>
<h2>What Netflix can teach us</h2>
<p>Fortunately for us, this problem has already been solved in computer science. The answer is a class of techniques called “matrix factorization” – and you’ve likely already interacted with these techniques in your everyday life. </p>
<p>In 2009, <a href="http://www.netflixprize.com/index.html">Netflix held a challenge</a> to personalize movie ratings for each Netflix user. On Netflix, each user has a distinct set of ratings of different movies. While two users may have similar tastes in movies, they may vary wildly in specific genres. Therefore, you cannot rely on comparing ratings from similar users. </p>
<p>Instead, a matrix factorization algorithm finds movies with similar ratings among a smaller group of users. The group of users will vary for each movie. The computer associates each user with a group of movies to a different extent, based upon their individual tastes. The relationships among users are referred to as “patterns.” These patterns are learned from the data, and may find common rankings unforeseen by movie genre alone – for example, users may share a preference for a particular director or actor. </p>
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<a href="https://images.theconversation.com/files/164115/original/image-20170405-20472-c764c6.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/164115/original/image-20170405-20472-c764c6.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/164115/original/image-20170405-20472-c764c6.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=464&fit=crop&dpr=1 600w, https://images.theconversation.com/files/164115/original/image-20170405-20472-c764c6.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=464&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/164115/original/image-20170405-20472-c764c6.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=464&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/164115/original/image-20170405-20472-c764c6.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=583&fit=crop&dpr=1 754w, https://images.theconversation.com/files/164115/original/image-20170405-20472-c764c6.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=583&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/164115/original/image-20170405-20472-c764c6.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=583&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="attribution"><span class="source">Genevieve Stein-O'Brien</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
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</figure>
<p>The same process can work in cancer. In this case, the measurements of gene dysregulation are analogous to movie ratings, movie genres to biological function and users to patients’ tumors. The computer searches across patient tumors to find patterns in gene dysregulation that cause the malignant biological function in each tumor. </p>
<h2>From movies to tumors</h2>
<p>The analogy between movie ratings and cancer genetics breaks down in the details. Unless they are minors, Netflix users are not constrained in the movies they watch. But, our bodies instead prefer to minimize the number of genes used for any single function. There are also substantial redundancies between genes. To protect a cell, one gene may easily substitute for another to serve a common function. Gene functions in cancer are even more complex. Tumors are also highly complex and rapidly evolving, depending upon random interactions between the cancer cells and the adjacent healthy organ. </p>
<p>To account for these complexities, we have developed a matrix factorization approach called <a href="https://www.ncbi.nlm.nih.gov/pubmed/20810601">Coordinated Gene Activity in Pattern Sets – or CoGAPS for short</a>. Our algorithm accounts for biology’s minimalism by incorporating as few genes as possible into the patterns for each tumor. </p>
<p>Different genes can also substitute for one another, each serving a similar function in a different context. To account for this, CoGAPS simultaneously estimates a statistic for the so-called “patterns” of gene function. This allows us to compute the probability of each gene being used in each biological function in a tumor. </p>
<p>For example, many patients take a targeted therapeutic called cetuximab to prolong survival in colorectal, pancreatic, lung and oral cancers. Our recent work found that these patterns can distinguish gene function in cancer cells that respond to the targeted therapeutic agent cetuximab from those that do not. </p>
<h2>The future</h2>
<p>Unfortunately, cancer therapies that target genes usually cannot cure a patient’s disease. They can only delay progression for a few years. Most patients then relapse, with tumors that are no longer responsive to the treatment. </p>
<p><a href="http://www.impactjournals.com/oncotarget/index.php?journal=oncotarget&page=article&op=view&path%5B%5D=12075">Our own recent work</a> found that the patterns that distinguish gene function in cells that are responsive to cetuximab include the very genes that give rise to resistance. Emerging immunotherapies are promising and appear to cure some cancers. Yet, far too often, patients with these treatments also relapse. New data that track the cancer genetics after treatment is essential to determine why patients no longer respond. </p>
<p>Along with these data, cancer biology also requires a new generation of scientists who can bridge mathematics and statistics to determine the genetic changes occurring over time in drug resistance. In other fields of mathematics, computer programs are able to forecast long-term outcomes. These models are used commonly in weather prediction and investment strategies. </p>
<p>In these fields and <a href="http://journals.ametsoc.org/doi/abs/10.1175/2010MWR3515.1">my own previous research</a>, we have found that updates to the models from large datasets – such as satellite data in the case of weather – improve long-term forecasts. We have all seen the effect of these updates, with weather predictions improving the closer that we are to a storm. </p>
<p>Just as tools from computer science used can be adapted to both movie recommendations and cancer, the future generation of computational scientists will adopt prediction tools from an array of fields for precision medicine. Ultimately, with these computational tools, we hope to predict tumors’ response to therapy as commonly as we predict the weather, and perhaps more reliably.</p><img src="https://counter.theconversation.com/content/74806/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Elana Fertig receives funding from The National Institutes of Health, The Johns Hopkins University Discovery Award, and The Johns Hopkins University IDIES Award. She is related to a staff member of The Conversation US. </span></em></p>Cancer researchers dream of offering personalized treatments to patients. Can they get there using the same math that drives Netflix recommendations?Elana Fertig, Assistant Professor of Oncology Biostatistics and Bioinformatics, Johns Hopkins UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/612542017-04-05T01:11:26Z2017-04-05T01:11:26ZWith new technology, mathematicians turn numbers into art<figure><img src="https://images.theconversation.com/files/163969/original/image-20170404-5739-1joyq1v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Mathematical visualization techniques led the author to create this virtual scene, showing shapes from the realm of mathematics bursting into the physical world.</span> <span class="attribution"><span class="source">Frank Farris</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>Once upon a time, mathematicians imagined their job was to discover new mathematics and then let others explain it.</p>
<p>Today, digital tools like <a href="https://theconversation.com/explainer-what-is-3d-printing-and-whats-it-for-9456">3-D printing</a>, animation and virtual reality are more affordable than ever, allowing mathematicians to investigate and illustrate their work at the same time. Instead of drawing a complicated surface on a chalkboard, we can now hand students a physical model to feel or invite them to fly over it in virtual reality.</p>
<p>Last year, a workshop called “Illustrating Mathematics” at the Institute for Computational and Experimental Research in Mathematics (ICERM) brought together an eclectic group of mathematicians and digital art practitioners to celebrate what seems to be a golden age of mathematical visualization. Of course, visualization has been central to mathematics since Pythagoras, but this seems to be the first time it had a workshop of its own.</p>
<p>The atmosphere was electric. Talks ran the gamut, from wildly creative thinkers who apply mathematics in the world of design to examples of pure mathematical results discovered through computer experimentation and visualization. It shed light on how powerful visualization has become for studying and sharing mathematics.</p>
<h2>Reimagining math</h2>
<p>Visualization plays a growing role in mathematical research. According to <a href="http://page.math.tu-berlin.de/%7Esullivan/">John Sullivan</a> at the Technical University of Berlin, <a href="https://www.researchgate.net/publication/252220599_MATHEMATICAL_THINKING_STYLES_-_AN_EMPIRICAL_STUDY">mathematical thinking styles</a> can be roughly categorized into three groups: “the philosopher,” who thinks purely in abstract concepts; “the analyst,” who thinks in formulas; and “the geometer,” who thinks in pictures.</p>
<p>Mathematical research is stimulated by collaboration between all three types of thinkers. Many practitioners believe teaching should be calibrated to connect with different thinking styles. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/133296/original/image-20160807-473-5booxs.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/133296/original/image-20160807-473-5booxs.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/133296/original/image-20160807-473-5booxs.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=578&fit=crop&dpr=1 600w, https://images.theconversation.com/files/133296/original/image-20160807-473-5booxs.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=578&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/133296/original/image-20160807-473-5booxs.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=578&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/133296/original/image-20160807-473-5booxs.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=727&fit=crop&dpr=1 754w, https://images.theconversation.com/files/133296/original/image-20160807-473-5booxs.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=727&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/133296/original/image-20160807-473-5booxs.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=727&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Borromean Rings, the logo of the International Mathematical Union.</span>
<span class="attribution"><span class="source">John Sullivan</span></span>
</figcaption>
</figure>
<p>Sullivan’s own work has benefited from images. He studies geometric knot theory, which involves finding “best” configurations. For example, consider his Borromean rings, which won the logo contest of the International Mathematical Union several years ago. The rings are linked together, but if one of them is cut, the others fall apart, which makes it a nice symbol of unity. </p>
<p>The “bubble” version of the configuration, shown below, is minimal, in the sense that it is the shortest possible shape where the tubes around the rings do not overlap. It’s as if you were to blow a soap bubble around each of the rings in the configuration. Techniques for proving that configurations like this are optimal often involve concepts of flow: If a given configuration is not the best, there are often ways to tell it to move in a direction that will make it better. This topic has great potential for visualization. </p>
<p>At the workshop, Sullivan dazzled us with a video of the three bands flowing into their optimal position. This animation allowed the researchers to see their ideas in action. It would never be considered as a substitute for a proof, but if an animation showed the wrong thing happening, people would realize that they must have made an error in their mathematics.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/133295/original/image-20160807-473-1twehbb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/133295/original/image-20160807-473-1twehbb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/133295/original/image-20160807-473-1twehbb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/133295/original/image-20160807-473-1twehbb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/133295/original/image-20160807-473-1twehbb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/133295/original/image-20160807-473-1twehbb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/133295/original/image-20160807-473-1twehbb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/133295/original/image-20160807-473-1twehbb.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">In this version of the Borromean Rings, a virtual ‘soap bubble’ is blown around the wire-frame configuration.</span>
<span class="attribution"><span class="source">John Sullivan</span></span>
</figcaption>
</figure>
<h2>The digital artists</h2>
<p>Visualization tools have helped mathematicians share their work in creative and surprising ways – even to rethink what the job of a mathematician might entail. </p>
<p>Take <a href="http://fabienne.us/">mathematician Fabienne Serrière</a>, who raised US$124,306 through Kickstarter in 2015 to buy an industrial knitting machine. Her dream was to make custom-knit scarves that demonstrate cellular automata, mathematical models of cells on a grid. To realize her algorithmic design instructions, Serrière hacked the code that controls the machine. She now works full-time on custom textiles from a Seattle studio.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/163668/original/image-20170403-21979-gbyz2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/163668/original/image-20170403-21979-gbyz2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=372&fit=crop&dpr=1 600w, https://images.theconversation.com/files/163668/original/image-20170403-21979-gbyz2j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=372&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/163668/original/image-20170403-21979-gbyz2j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=372&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/163668/original/image-20170403-21979-gbyz2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=468&fit=crop&dpr=1 754w, https://images.theconversation.com/files/163668/original/image-20170403-21979-gbyz2j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=468&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/163668/original/image-20170403-21979-gbyz2j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=468&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">In this sculpture by Edmund Harriss, the drill traces are programmed to go perpendicular to the growth rings of the tree. This makes the finished sculpture a depiction of a concept mathematicians know as ‘paths of steepest descent.’</span>
<span class="attribution"><span class="source">Edmund Harriss</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p><a href="http://directory.uark.edu/people/eharriss">Edmund Harriss</a> of the University of Arkansas hacked an architectural drilling machine, which he now uses to make mathematical sculptures from wood. The control process involves some deep ideas from differential geometry. Since his ideas are basically about controlling a robot arm, they have wide application beyond art. According to his website, Harriss is “driven by a passion to communicate the beauty and utility of mathematical thinking.”</p>
<p>Mathematical algorithms power the products made by <a href="http://n-e-r-v-o-u-s.com/">Nervous System</a>, a studio in Massachusetts that was founded in 2007 by Jessica Rosenkrantz, a biologist and architect, and Jess Louis-Rosenberg, a mathematician. Many of their designs, for things like custom jewelry and lampshades, look like naturally occurring structures from biology or geology. </p>
<p>Their first <a href="http://n-e-r-v-o-u-s.com/blog/?p=7162">3-D printed dress</a> consists of thousands of interlocking pieces designed to fit a particular model. In order to print the dress, the designers folded up their virtual version, using protein-folding algorithms. A selective laser sintering process fused together parts of a block of powder to make the dress, then let all the unwanted powder fall away to reveal its shape. </p>
<p>Meanwhile, a delightful collection called <a href="http://www.geometrygames.org/">Geometry Games</a> can help everyone, from elementary school students to professional mathematicians, explore the concept of space. The project was founded by mathematician Jeff Weeks, one of the rock stars of the mathematical world. The iOS version of his “Torus Games” teaches children about multiply-connected spaces through interactive animation. According to Weeks, the app is verging on one million downloads. </p>
<h1>Mathematical wallpaper</h1>
<p>My own work, described in my book <a href="http://press.princeton.edu/titles/10435.html">“Creating Symmetry: The Artful Mathematics of Wallpaper Patterns</a>,” starts with a visualization technique called the domain coloring algorithm. </p>
<p>I developed this algorithm in the 1990s to visualize mathematical ideas that have one dimension too many to see in 3-D space. The algorithm offers a way to use color to visualize something seemingly impossible to visualize in one diagram: a complex-valued function in the plane. This is a formula that takes one complex number (an expression of the form <em>a</em>+_b_i, which has two coordinates) and returns another. Seeing both the 2-D input and the 2-D output is one dimension more than ordinary eyes can see, hence the need for my algorithm. Now, I use it to <a href="https://theconversation.com/patterns-are-math-we-love-to-look-at-44390">create patterns and mathematical art</a>.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/133297/original/image-20160807-493-11nfe2x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/133297/original/image-20160807-493-11nfe2x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/133297/original/image-20160807-493-11nfe2x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/133297/original/image-20160807-493-11nfe2x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/133297/original/image-20160807-493-11nfe2x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/133297/original/image-20160807-493-11nfe2x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/133297/original/image-20160807-493-11nfe2x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/133297/original/image-20160807-493-11nfe2x.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">A curve with pleasing 5-fold symmetry, constructed using Fourier techniques.</span>
<span class="attribution"><span class="source">Frank A Farris</span></span>
</figcaption>
</figure>
<p>My main pattern-making strategy relies on a branch of mathematics called Fourier theory, which involves the superposition of waves. Many people are familiar with the idea that the sound of a violin string can be broken down into its fundamental frequencies. My “wallpaper functions” break down plane patterns in just the same way. </p>
<p>My book starts with a lesson in making symmetric curves. Taking the same idea into a new dimension, I figured out how to weave polyhedral solids – think cube, dodecahedron, and so on – from symmetric bands made from these waves. I staged three of these new shapes, using Photoshop’s 3-D ray-tracing capacity, in the “Platonic Regatta” shown below. The three windsails display the symmetries of Platonic solids: the icosahedron/dodecahedron, cube/octahedron and tetrahedron.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/133298/original/image-20160807-484-dokih9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/133298/original/image-20160807-484-dokih9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/133298/original/image-20160807-484-dokih9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=480&fit=crop&dpr=1 600w, https://images.theconversation.com/files/133298/original/image-20160807-484-dokih9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=480&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/133298/original/image-20160807-484-dokih9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=480&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/133298/original/image-20160807-484-dokih9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=603&fit=crop&dpr=1 754w, https://images.theconversation.com/files/133298/original/image-20160807-484-dokih9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=603&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/133298/original/image-20160807-484-dokih9.jpg?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"></a>
<figcaption>
<span class="caption">A Platonic Regatta. Mathematical art by Frank A. Farris shows off three types of polyhedral symmetry: icosahedral/dodecahedral, cube/octaheral and tetrahedral.</span>
<span class="attribution"><span class="source">Frank Farris</span></span>
</figcaption>
</figure>
<p>About an hour after I spoke at the workshop, mathematician <a href="http://blog.mikael.johanssons.org/">Mikael Vejdemo-Johansson</a> had posted <a href="https://twitter.com/symmetric_curve">a Twitter bot</a> to animate a new set of curves every day! </p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"843250874927513600"}"></div></p>
<p>Mathematics in the 21st century has entered a new phase. Whether you want to crack an unsolved problem, teach known results to students, design unique apparel or just make beautiful art, new tools for visualization can help you do it better.</p>
<p><em>This article was updated on April 5, 2017 with the full name of Mikael Vejdemo-Johansson.</em></p><img src="https://counter.theconversation.com/content/61254/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Frank A. Farris received funding from ICERM to cover travel to the "Illustrating Mathematics" workshop.</span></em></p>It’s a golden age for visualization in mathematics. How tools like 3-D printing, animation and even knitting machines are reimagining the way mathematicians study and share their work.Frank A. Farris, Associate Professor of Mathematics, Santa Clara UniversityLicensed as Creative Commons – attribution, no derivatives.