{"id":2501,"date":"2013-09-24T10:47:17","date_gmt":"2013-09-24T18:47:17","guid":{"rendered":"http:\/\/www.danamackenzie.com\/blog\/?p=2501"},"modified":"2013-09-24T10:47:17","modified_gmt":"2013-09-24T18:47:17","slug":"dana-blogs-math-in-love-with-geometry","status":"publish","type":"post","link":"https:\/\/danamackenzie.com\/blog\/?p=2501","title":{"rendered":"dana blogs math: In Love With Geometry"},"content":{"rendered":"
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In this animation by Keenan Crane, a surface in the shape of a bunny flows to a sphere. In a "conformal" flow, like this one, all the surface patterns remain relatively undistorted. (Image courtesy of Keenan Crane.)<\/p><\/div>\n

For generations, geometers have gotten used to not being able to see the objects that they prove theorems about. It\u2019s a somewhat sad development in a subject that began in Euclid\u2019s day with the splendidly visual concepts of points, lines, triangles, circles, conic sections and the like. But at the same time, it seemed like a necessary price for progress. Even in ancient Greece it became clear that pictures could fool you, and that abstract arguments (the \u201ctheorem-proof\u201d approach of Euclid) were the best way to avoid making mistakes. Also, in the nineteenth and twentieth centuries, geometers moved to the study of much more complicated objects: surfaces and spaces of many dimensions that could never be fully visualized in our rather modestly endowed three-dimensional universe.<\/p>\n

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Keenan Crane. Photo by Katrin Schmid, courtesy of Mathematisches Forschungsinstitut Oberwolfach.<\/p><\/div>\n

However, in the twenty-first century, mathematical computer scientists like Keenan Crane<\/a> of Columbia University are giving a new life to the study of surfaces and curves we can actually see. Crane, who was selected as one of the 200 young participants in the Heidelberg Laureate Forum, works on computer visualization of objects from differential geometry. His work has placed him squarely at the intersection of both fields. \u201cMy background is really in computer science and computer graphics, but as I tried to solve these graphics problems I realized more and more that I was in love with geometry,\u201d he says.<\/p>\n

Looking at Crane\u2019s pictures, it\u2019s easy to see why. One of his current projects involves the deformation of surfaces by conformal maps. Suppose, for example, you want to morph a rabbit into a sphere, as shown in the animation above. Real-world objects will always have some surface coloration or texture on them, and aesthetically the most pleasing and most convincing \u201cmorphs\u201d are the ones that don\u2019t distort those patterns too much. That means that in any small patch of the surface, shapes are preserved. Eyes still look like eyes, and writing still remains legible even as the surface flows toward its new shape. Such shape-preserving transformations are called conformal<\/em>.<\/p>\n

Conformal mappings have a very distinguished pedigree in mathematics as well as in computer visualization. One of the landmark theorems of nineteenth-century mathematics, Riemann\u2019s Uniformization Theorem<\/a>, guarantees that any surface that is \u201ctopologically\u201d a sphere\u2014in other words, it has no boundaries and no handles\u2014can be mapped conformally<\/em> to a sphere. The proof, done in classical mathematical style, is very abstract and challenging even for a mathematics graduate student. When I was a graduate student, every one of my classmates boned up on this proof while preparing for their qualifying exams. It was a sort of rite of passage\u2014the hardest thing we could be expected to know on the exam. (I’m not sure if anybody ever got asked about the Uniformization Theorem. It’s likely that the faculty knew about this ritual and therefore asked us about other things instead.)<\/p>\n

Crane\u2019s algorithm is essentially a visual illustration of Riemann\u2019s theorem, and it is dead simple. No matter what shape of bunny you start with\u2014it can even be a cow, or an octopus\u2014his algorithm will find a way to deform it conformally to a sphere, and fast<\/em>. The speed does not come from programming tricks. It comes from mathematical principles. Crane found a new way to describe surfaces, not in terms of the physical locations of the points on the surface, but by their curvature. The morphing process decreases the Willmore energy<\/a> of the curvature. To put it more prosaically, it irons out any wrinkles as efficiently as possible.<\/p>\n

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Two different conformal flows that map a coffee cup to a doughnut. (Image courtesy of Keenan Crane.)<\/p><\/div>\n

Although it\u2019s cool to watch a cow or a bunny turn into a sphere, things get much more interesting when you apply conformal flow to surfaces with holes or handles in them. The second series of pictures illustrates a standard joke in mathematics: \u201cA topologist can\u2019t tell a coffee cup from a doughnut.\u201d As the coffee cup deforms into a doughnut, the mathematical equations written on its surface remain highly legible, because of the conformal (shape-preserving) property of the flow.<\/p>\n

Unlike spheres, doughnuts come in many different conformal types. Nineteenth-century mathematicians discovered a way of classifying them that is beautiful but highly abstract. (It\u2019s called a complex structure<\/em>, so that already gives you an idea\u2026) It provides little visual intuition about what the roundest, most wrinkle-free doughnut in a given conformal class ought to look like. But by using his conformal flows, Crane is discovering a link between the complex structure and the appearance of the Willmore-energy-minimizing doughnut. \u201cAs the complex structure gets more twisted, the torus also gets more twisted,\u201d he says.<\/p>\n

You might say this is not surprising, perhaps even obvious. But it\u2019s something that nineteenth-century mathematicians like Riemann could never prove, could never even guess at, because they couldn\u2019t see the things they were working with. It\u2019s another example of how the \u201cSilicon Age\u201d is making new mathematics possible.<\/p>\n

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A length-preserving flow untangles a complicated curve and reveals it to be homotopic (or topologically equivalent) to a figure eight. (Image courtesy of Keenan Crane.)<\/p><\/div>\n

Even if you don\u2019t understand their mathematical meaning, Crane\u2019s visualizations are just plain pretty to look at. Take the third picture, for instance. It shows the idea of deforming a curve while preserving its turning number<\/a>. You can think of carrying a compass with you as you drive around the curve, and asking how many times the needle goes around while you make a complete loop. If you\u2019re driving around the complicated duck-shaped curve on the left, it\u2019s hard to say. But for the figure-eight on the right, it\u2019s easy. If you start in the middle, the needle goes around part way, then backs up to where it started. The net turning number is zero. Because the figure-eight curve came about from a smooth deformation of the duck curve, the duck curve must also have a turning number of zero.<\/p>\n

Crane says that he doesn\u2019t have any plans to meet or talk with specific people at the Heidelberg Laureate Forum; he is just looking forward to meeting \u201ca lot of fascinating people.\u201d Nevertheless, he would like to get a chance to meet Srinivasa Varadhan<\/a>, one of the laureates attending the meeting. \u201cI\u2019ve been working on a project that uses the classic Varadhan formula to compute geodesic distance, and in fact I\u2019m going to talk with some people at DreamWorks<\/a> about it next month,\u201d Crane says. [DreamWorks is the movie studio that brought you Kung Fu Panda<\/em>.] \u201cI\u2019d like to tell Varadhan that I\u2019m using his result to do something useful and practical. It\u2019s fun to see an idea going all the way from math to something you can see on the movie screen.\u201d<\/p>\n

This blog post\u00a0originates from the official blog of the 1st Heidelberg Laureate Forum<\/a> (HLF)\u00a0which takes place in Heidelberg, Germany, September 22 – 27, 2013. 40 Abel, Fields, and Turing Laureates will gather to meet a select group of 200 young researchers.<\/p>\n

Dana Mackenzie<\/strong><\/a>\u00a0is a member of the HLF blog team. Please find all his postings on the HLF blog<\/a>.<\/p>\n

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For generations, geometers have gotten used to not being able to see the objects that they prove theorems about. It\u2019s a somewhat sad development in a subject that began in Euclid\u2019s day with the splendidly visual concepts of points, lines, triangles, circles, conic sections and the like. But at the same time, it seemed like […]<\/p>\n","protected":false},"author":80,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[235],"tags":[2678,2680,2681,2677,2675,2674,2679,2676],"class_list":["post-2501","post","type-post","status-publish","format-standard","hentry","category-off-topic","tag-animations","tag-coffee-cup","tag-doughnut","tag-flow","tag-keenan-crane","tag-srinivasa-varadhan","tag-uniformization-theorem","tag-visualization"],"_links":{"self":[{"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/2501","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/80"}],"replies":[{"embeddable":true,"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2501"}],"version-history":[{"count":5,"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/2501\/revisions"}],"predecessor-version":[{"id":2510,"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/2501\/revisions\/2510"}],"wp:attachment":[{"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2501"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2501"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/danamackenzie.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2501"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}