Sharing the Mind's Eye: Computer Animation in Visualizing Geometry
The mind is capable of building powerful theories which do not depend on our being able to physically experience what is described. Yet often by analogy and extrapolation with what can be experienced, mathematicians can "see" in their "minds eye" non-euclidean geometries, four and higher dimensions, and other unfamiliar spaces and phenomena. [Marden 1995] But translating these ideas from personal insight into understanding for others is often quite difficult. Computer graphics and animation are one way to show some of these spaces and insights as if they were part of our everyday world.
The videos produced by the Geometry Center are part of a new genre of of visualization aimed at illustrating mathematical concepts and processes for a broad spectrum of viewers. The ability to see the process is extremely helpful for the non-specialist to understand what it is that mathematicians think about and do. It is also seen as an important companion to technical details and more formal proofs for mathematicians or serious students of mathematics .
The two videos discussed today are Not Knot and Outside In. (The third major video project was The Shape of Space, but this will be covered in Jeff Weeks' presentation.) Not Knot was the first large-scale video produced at the Geometry Center. Its aim was to introduce the viewers to the profound relationship between geometry and topology, knots and their complements, and euclidean and hyperbolic spaces, as demonstrated by the work of mathematician Bill Thurston. A fascinating and startling hyperbolic fly-through is the culmination of the video.
Less sweeping in scope, but more thorough in exposition, Outside In presents some ideas from differential topology, concentrating on the surprising discovery that it is possible to evert, or turn a sphere inside out without poking a hole in it.
Using Computer Graphics in Visualization
Computer animation forms a natural bridge from mathematical concept to visualization. Computers allow us to construct worlds that conform to mathematical rules, yet need not be constrained by the limitations of "real space" or tangible objects. For example, the sphere eversion, physically impossible to construct, can be made visible by using an abstract graphical material. And in Not Knot, we can experience the unfamiliar perspective of hyperbolic space.
Processes, even if shown visually, are often hard to see if presented as "snapshots," or from a single viewpoint. If there are not enough samples or steps, a "mental movie" may be impossible to construct. Some information may be hidden from view. Computer animation provides a continuity and flow that carries the viewer from one step to the next. And multiple views and presentations of the same process can provide more convincing visual evidence than a single view or perspective can offer.This bridge from concept to visualization could also be characterized as the link from "things thinkable and immaterial" to "what is perceptible to the senses"--or aesthetics--as defined by the Oxford English Dictionary. [Wechsler 1981] Indeed, the aesthetic inherent in Thurston's work, would scarcely be visible to a non-specialist. This sensibility is not the aesthetics or style of graphic design, animation or color, although these all contribute to the overall effect and success of the videos. It is more about having the right "fit" between ideas and the visualization process, about personal approach and the visual result this conveys. It reveals correspondences and connections, dualities and symmetries, and connects details to a bigger picture.
About Not Knot
An Experiment in Visualization
The Not Knot project was initiated to visualize some of the exciting results in modern topology. The video features the work of William Thurston, in the classification of 3 dimensional spaces. The theorem that the video illustrates is that every knot complement and every link complement (with a few exceptions) has a complete hyperbolic structure. [Thurston and Levy 1997] The correspondence and transformation to hyperbolic space is shown. But what do we mean by topology, and how is it different from geometry? What is a knot, or a link, or a complement? What is hyperbolic space, and how is it different from the space we live in?
Overview of the Terminology
A few brief definitions may be helpful here. Topology is the study of the unchanging properties of objects, even when they are deformed, shrunk or expanded. You can think of these objects as being made of an imaginary rubber sheet that can be pushed, pulled, expanded or contracted at will. In geometry, on the other hand, size matters, along with angle, area, curvature, length and volume. A sphere and a cube are topologically equivalent, since one could be pushed and pulled into looking like the other, (especially with the help of a little computer parameterization). Not so for the geometry, as you can see that the curvature of the two is quite different. That example is pretty straightforward about a doughnut and a coffee cup? Are they the same? See [Weeks 1985] below for the answer.Imagine a knot as a piece of nylon cord with its ends fused together, or perhaps an extension cord plugged into itself. In other words, knots that mathematicians care about have no loose ends. The simplest case of a knot is a simple circle or loop, otherwise known as the "unknot". Finding out whether knots that look complicated and tangled up can be manipulated until they take this simple form is one of the objectives of knot theory. Another is to find out whether knots are equivalent to one another. A link is the union of several loops or closed curves.
The complement of a knot (or link) is the space around it, the part that's not a knot, thus providing us with a title to the video. (Strictly speaking, we should have called the video "Not Link", but it doesn't have much of a ring to it.) It turns out that studying the spaces around knots is also a way of telling whether they are equivalent.
We make the assumption that our familiar everyday space is euclidean-our world looks that way to us. However, there are other systems, such as spherical and hyperbolic geometries, that are as consistent and "real" as euclidean geometry is to us. (As a matter of fact, there are theories in cosmology that explore the idea that the universe may be not be euclidean after all, but could be spherical or hyperbolic. [Weeks 1985] )
The differences between these geometries is easiest to imagine in the two dimensional case. A euclidean plane is flat and is said to have no curvature. Spherical geometry takes, not surprisingly, the form of a sphere, and is said to have positive curvature. And hyperbolic geometry is shaped like a saddle, and has negative curvature. We can see what curvature means by thinking about the angles of a triangle located in each one of these surfaces. In the euclidean case, the sum of the angles is 180 degrees, on the sphere it is greater than 180 degrees, and on the saddle it is less than 180 degrees. It is apparent that this same object built in each of these geometric systems will look quite different from one another.
Structure of Not KnotIn the video, we use these concepts (and quite a few others not mentioned here) to show how to introduce a hyperbolic structure on the complement of the Borromean rings. In order to build up the viewer's understanding, many simpler situations are shown. We color-coded the objects in the video, so that it is easier to track them throughout the process. (But is must be noted that, even with this thread to follow, the video shows extremely advanced ideas, and it is not expected for viewers to understand everything in it. It is actually used as a study guide in university level math classes.) The Not Knot supplement [Epstein and Gunn 1991] summarizes the video structure this way: "We talk about the aim of putting a structure on the space, so that the knot becomes infinitely far away. Then we turn to a point in a disk, and show how that can be pushed infinitely far away. Next we push a single axis in three-space infinitely far away. Finally we do the same thing for the Borromean rings, pushing all three rings to infinity at the same time." We also introduce the idea of the view of the insider and the outsider-what would someone living inside these spaces actually see? [Abbot 1884]. The final fly-through of hyperbolic space give us just such a glimpse into this space, one where apparent distances seem exaggerated, where a dodecahedron has 90 degree angles between every pair of adjacent faces, and where straight beams appear curved.
|The image on the far left is the complement of the Borromean rings. The image on the right was created for the Not Knot by Charlie Gunn, who also did the hyperbolic fly-through in the video. It has a different order of symmetry, which produces a more striking image.|
About Outside In
In 1957, the young mathematician Steve Smale made a startling discovery: you can turn the surface of a sphere inside out without making a hole, if you think of the surface as being made of an abstract elastic material that can pass through itself--the same "rubber sheet" that was mentioned before. Communicating how this process can be carried out has been a challenge in differential topology ever since. (Differential topology is the study of smooth objects without corners or bends.)
Smale's proof that this could be done relied on a large number of small geometric steps that, although "constructive'' in the mathematical sense, are not much help if one's goal is, for example, to make an animation of the sphere eversion. Smale's proof is somewhat like a research paper describing, in minute detail and down to the molecular chemistry, what happens to the ingredients of a souffle during the cooking process: one cannot expect a person who has never seen a souffle to follow this "recipe" in preparing the dish.
In 1961, Arnold Shapiro invented the first explicit eversion, but he did not publish or divulge it widely. The first time that most mathematicians and the public at large became aware of an explicit eversion was when Tony Phillips worked out the details of what he thought was Shapiro's eversion (though later it turned out to be a different one). Phillips published a beautifully written article in Scientific American [Phillips 1966], aimed at a wide audience and culminating with a series of pictures representing various stages of the evolution. The publication of these pictures dispelled the mysterious and paradoxical reputation of the problem; but filling in the missing levels in Phillips' pictures and tracking them through their implied deformations was not an easy task. People began searching for simpler solutions.
The French mathematician Bernard Morin devised in 1967 a new eversion that was simpler than preceding ones, in part because the sphere preserved a certain amount of symmetry throughout the process. Morin, incidentally, is blind, and the fact that he was one of the first people to understand how a sphere can turn inside out is both a tribute to his ability and a convincing proof that "visualization" goes far beyond the physical sense of sight.
Charles Pugh made wire-mesh models of various stages of Morin's eversion, and Nelson Max digitized these models and used them as the basis for the movie Turning a Sphere Inside Out [Max 1977], an early triumph of computer animation. (And one of the original inspirations for this author to get started in computer graphics!)Other methods were found later. In the mid-seventies, Bill Thurston developed his idea of corrugations, a technique that allows "mathematical materials'' like curves and surfaces to be made springy and to be moved about and bent at will. (Thurston thought of the corrugations as a way to get a "coherent story or mental movie", which he had previously found difficult to do in studying other eversions. [Thurston 1995] ) This gives another route for the eversion, as explained in the video.
|As one can see in the top view in this picture, the evolving sphere is highly symmetric. It is divided into eight strips. The actual number of pieces is not very important, except for esthetic purposes; it is the fact that the sphere can be decomposed into these pieces at all, and in a symmetric way, that makes this method relatively easy to understand.|
Still, it is a challenge to explain the process in a way that can be grasped in its entirety. A complete mental picture of any eversion is very hard to keep in mind, and clear printed or even animated images are hard to draw, because of the many layers of surface: the most successful expositions of the everting sphere concentrate on local details, and rely on the viewer's powers of synthesis to piece the details together. Symmetry is an excellent way to concentrate on just one piece--the local picture--rather than trying to visualize everything simultaneously. So that's the route that Outside In takes.
Structure of Outside In
The framework of the video is a dialogue between a female teacher and a male student. In the first scene they work out between themselves the ground rules of what it means to turn a sphere inside out. The surface can stretch and bend, and pass through itself, but cannot be ripped, punctured, or creased.
(This leads to the question, which we skirt in the movie for lack of time: why turn a sphere inside out, and why use these rules? The short answer is that these rules give a mathematical puzzle that is challenging and counterintuitive, and yet has a solution. But the rules also stem from a classification problem of surfaces and deformations, more specifically of "immersions," or smooth surfaces without kinks. In these immersions, the notion of "sidedness" is also encoded into the surfaces. This property of sidedness is preserved no matter which side of the sphere is inside or out.)
Because creases are not allowed, simply pushing the sphere's caps straight through one another won't work: a tight loop or crease will form around the equator. How can this work? The student remains skeptical that the problem can be solved under these rules. If anything his skepticism increases in subsequent scenes, as the teacher persuades him that a circle cannot be turned inside out under the same rules. However, an idea that is introduced in connection with curves--namely, adding waves, or corrugations--turns out to be useful for surfaces as well. The teacher builds up the ideas needed for the grand finale: a step-by-step demonstration of the sphere eversion, seen from many angles and in many different cross sections.
The Uses of Symmetry
Although the process looks quite complex at first glance, it can be decomposed into relatively simple pieces both in space and in time. Spatially, we divide the sphere into eight strips that go from pole to pole, and we can further cut this strip crosswise into two equal pieces. (Imagine a peeled orange with eight segments, then keep only one segment and slice it through at the "equator.") Everything that happens to one half-strip happens at the same time to the others; that is, the eversion is symmetrical, as already discussed. This one-sixteenth is the fundamental building block of the eversion--the entire sphere can be created by rotating copies of this piece.View a short clip from the video, or look at the still frames below to see the steps. In this version, we see the eight connected sections, one of the many ways that the eversion is portrayed in the video. Nine images are displayed to show some continuity, even though the eversion is divided into five main stages, as explained below.
|First, we corrugate the sphere, that is, we create bulges in the middle of each of the strips. (Images 1, 2) Second, we push the poles through each other, stopping when we create a loop at the equator. (Image 3) Third, we twist the two caps in opposite directions to convert the loops in the middle to twisting at the caps. (Images, 4, 5, 6) This is the least transparent part of the eversion; the crucial fact is that the symmetrical corrugations we have introduced make the surface very pliable, so the twisting can be carried out without introducing creases. Fourth, we push the equatorial region of the surface radially through the center. (Images 7, 8) Finally, we uncorrugate to complete the eversion. (Image 9).|
Such a short introduction can only provide a "snapshot" into the material. It is now time to provide a continuous view and show the animations, and provide a glimpse into what mathematicians think about and do. And the hope is that the viewer will then be inspired to learn more.
The videos were a collaboration between many people: mathematicians, programmers, visual and sound designers, and student volunteers. Thanks to everyone who worked on them.
I have relied heavily upon the words of Silvio Levy, Al Marden, and Bill Thurston for the background information on the mathematics behind the videos. It has been drawn from the written supplements that accompany the videos, from rough drafts we wrote about the project, and from a short talk that Silvio and I wrote, and Silvio presented, at the ISIS Symmetry conference in 1995. Thanks to Jeff Weeks for help with topological visualization. Tamara Munzner, Charlie Gunn, and Silvio Levy were the other directors, and invaluable help was provided by Stuart Levy, Nathaniel Thurston, and all the staff and students at the Geometry Center.
[Abbot 1884] Edwin Abbot, Flatland: a romance of many dimensions, Seeley, London, 1184. Reprinted by Dover, New York, 1992. This is the best place to be introduced into the concept of what insiders and outsiders see, starting with the example of A. Square in the plane.
[Epstein and Gunn 1991] David Epstein and Charlie Gunn, Supplement to Not Knot, Geometry Center, University of Minnesota. Published by Jones and Bartlett, Boston, MA, 1991. (Now published by AK Peters, Wellesley, MA) This is the booklet that accompanies the video Not Knot. There a numerous historical references and exercises in the booklet. It is essential in providing some more background on the material.
[Francis 1987] George K. Francis, A Topological Picturebook, Springer, New York, 1987. Many drawings of different sphere eversions. Silvio Levy, who wrote the brief history of sphere eversions in this paper, got a lot of his material from George Francis.
[Gunn and Maxwell 1991] Charlie Gunn and Delle Maxwell, Not Knot, (video), AK Peters, Wellesley, MA, 1991. Produced by the Geometry Center, University of Minnesota.
[Levy 1995] Levy, Silvio, Making Waves, A Guide to the Ideas behind Outside In, Geometry Center, University of Minnesota. Published by AK Peters, Wellesley MA, 1995. This is the material that supplements the video Outside In.
[Levy and Maxwell 1995] Silvio Levy and Delle Maxwell, "Symmetry and Insight: The Saga of Sphere Eversions", Geometry Center, University of Minnesota, 1995. Presented at ISIS-Symmetry Conference, 1995 by Silvio Levy,
[Levy, Maxwell, and Munzner 1994] Silvio Levy, Delle Maxwell, and Tamara Munzner, Outside In (video), AK Peters, Wellesley, MA, 1994. Produced by the Geometry center, University of Minnesota.
[Marden 1995] Al Marden, Afterword in Making Waves. See [Levy 1995]
[Max 1977] Max, Nelson, Turning a Sphere Inside Out (video), International Film Bureau, Chicago, 1977. The original video done of a sphere eversion. The rendering of the evolving sphere was done by Jim Blinn.
[Phillips 1966] Anthony Phillips, "Turning a surface inside out", Scientific American 214 (May 1966), 112-120.
[Scott 1992] Dana Scott, "Hyperbolic Space in Not Knot", essay in Der Prix Ars Electronica, publication of the Ars Electronica Festival in Linz Austria, 1992. This gives a good non-technical overview of the ideas in Not Knot, which was awarded a prize at this video festival.
[Thurston 1995] William P. Thurston, section called "Making Waves; The Theory of Corrugations", a part of the Outside In supplement. See [Levy 1995] above.
[Thurston and Levy 1997] William P. Thurston and Silvio Levy, Three-Dimensional Geometry and Topology, Princeton University Press, Princeton, 1997. All of the details!
[Wechsler 1981] Judith Wechsler, editor, On Aesthetics in Science, MIT Press, Cambridge, 1981. Interesting collection of essays by six authors, mostly scientists.
[Weeks 1985] Jeffrey R. Weeks, The Shape of Space, Marcel Dekker, Inc, New York and Basel, 1985. A good introduction to topology; lots of exercises to help build up visualization skills. There is a fun quiz on finding topologically equivalent objects in the beginning of the book. The doughnut and the coffee cup are equivalent!
This is the link to the videos page at the Geometry Center. There are many things to look at at the Geometry Center site, including interactive demos. The Shape of Space video information is here, too. There is also software, written by Nathaniel Thurston, which performs the sphere eversion. http://www.geom.umn.edu/video/
AK Peters publishes the videos and written supplements, and can be reached here: http://www.akpeters.comYou can go Jeff Weeks' site and play topology games online at : http://www.northnet.org/weeks/