Visualizing a Sphere Eversion

IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. XX, NO. Y, MMM 2003

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Visualizing a Sphere Eversion George Francis, John M. Sullivan Abstract— The mathematical process of everting a sphere (turning it inside-out allowing self-intersections) is a grand challenge for visualization because of the complicated, ever changing internal structure. We have computed an optimal minimax eversion, requiring the least bending energy. Here we discuss techniques we used to help visualize this eversion for visitors to virtual environments and viewers of our video The Optiverse. Keywords— Visualization, Sphere eversion, Boy surface, Morin surface, Regular homotopy, Immersions, Willmore energy, The CAVE.

I. Introduction

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ISUALIZATION is especially challenging when the objects to be viewed have complicated internal structure. Our human visual system is trained mostly on opaque objects whose insides cannot be seen; most transparent objects in our everyday experience are like glass windows, intended to be looked through rather than looked at. But increasingly, numerical simulations of real-world phenomena are being made in 3D×Time. The results of these simulations have internal structure that changes in time, and methods are needed for its visualization. In mathematics, perhaps the most common objects with internal structure are immersed surfaces. This means surfaces which might be intrinsically simple (like a sphere or torus) but which are mapped into space in a complicated way (like the immersed spheres in Fig. 1). Locally, every small patch of an immersed surface is embedded smoothly into the ambient space, so that an immersed surface never has rips, corners or creases. But globally, different sheets of an immersed surface are allowed to pass through each other, unlike any kind of physical surface. Unlike the sphere and the torus, which are orientable, a closed nonorientable surface must have self-intersection when immersed in 3-space. The real projective plane (also known as the cross surface) is the simplest of these, and its simplest immersion in 3-space (meaning the one with the least complicated self-intersection) is Boy’s surface, shown in Fig. 1(left). (For a visual introduction to the topology of surfaces see Appendix C of The Shape of Space [1].) However, immersions also play an important role in the deformations of surfaces (like the sphere) which can be embedded in 3D. Clearly it is impossible to turn a sphere inside out through a succession of embeddings. To turn a physical sphere inside out, one must cut a hole, pull the rest of the surface through the hole, and then patch the hole; the surface being turned inside-out is not a sphere but a disk (a sphere with a hole). Mathematically, it is most natural to consider the question for immersed spheres: Can a sphere be turned inside out via a smooth, one-parameter family of Department of Mathematics, University of Illinois, Urbana, IL, USA 61801. E-mail: [email protected], [email protected]

Fig. 1. These are the halfway models for the two simplest minimax eversions. The Boy’s surface (left), an immersed projective plane with three-fold symmetry and a single triple point (where three sheets of the surface cross each other), minimizes Willmore’s elastic bending energy. The figure actually shows an immersed sphere doubly covering Boy’s surface, with its two (oppositely-oriented) sheets pulled apart slightly. The Morin surface shown (right) also minimizes Willmore energy; it has a four-fold rotational symmetry which reverses orientation, exchanging the lighter and darker sides of the surface.

(smooth) immersions, called a regular homotopy by Whitney. Both Boy and Whitney [2] independently showed that this was not possible one dimension lower: a circle cannot be turned inside-out in the plane by a regular homotopy. It would be extremely hard to settle this question by trial and error. An abstract mathematical theorem [3] by Steve Smale in 1959 classified regular homotopies for general surfaces and had the surprising consequence that a sphere eversion was possible, without any clues as to what it might look like. It took many years for other mathematicians to construct explicit eversions; at first these were illustrated by hand-drawn pictures. In 1977 Nelson Max made a computer-animated film [4], [5] realizing Bernard Morin’s 1967 vision of a particularly persuasive eversion. Over four decades, everting the sphere has remained a rewarding problem in mathematical visualization and computer graphics, especially because of the challenge of animating a self-intersecting surface. For more information about the history of sphere eversions, see [6], [7] and the references there, especially [8] and [9, Chap. 6]. II. The Minimax Eversion Our 1998 video The Optiverse [10] illustrates an optimal eversion, computed automatically by minimizing an elastic bending energy for surfaces. Our computations [11], [12] of the sphere eversion were performed in Ken Brakke’s Evolver [13] using code to minimize the Willmore energy [14]. A. Tobacco-pouch eversions Following Morin’s tradition, we use the concept of a halfway model, an immersed sphere (like those in Fig. 1)

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which is halfway inside-out in the following sense: Some symmetry of the surface—a rigid motion of space bringing the surface back to itself—will turn it inside out by reversing the surface orientation. More precisely, for an appropriate choice of parameterization, antipodal points on the abstract domain sphere are mapped by the immersion to points related by this symmetry. Given such a halfway model, any regular homotopy which simplifies it down to the round sphere can be extended by symmetry to a sphere eversion. The halfway models that have been used in this way are of two types The first is based on an immersion, such as Boy’s surface, of the projective plane, which topologically is the nonorientable surface obtained by identifying antipodal points on the sphere. A double covering of such a projective plane is an immersed sphere. Here the orientation-reversing symmetry is simply the identity map in space: antipodal points of the sphere lie at the same place in the halfway model. The early sphere eversions of Shapiro [15], Phillips [16] and Kuiper [17] used Boy’s surface as a halfway model. The second type of halfway model has 2p-fold rotational symmetry reversing orientation (and thus p-fold symmetry preserving orientation). The original Morin-Froissart halfway model was of this type, with p = 2. In the early seventies, Morin built tiny plasticine models of a family of sphere eversions for integers p > 1, later called the tobacco-pouch eversions. He inspired Charles Pugh to build large, chicken wire models for the p = 2 case, and these formed the database for Nelson Max’s masterpiece computer graphics sphere eversion. In 1977, students in Francis’s freshman honors topology seminar, after seeing Max’s film [4], helped design an accurate combinatorial description of the tobacco-pouch eversions [18]. For p even, the halfway model used in these eversions is of the second class, with 2p-fold rotational symmetry (reversing orientation). For p odd, it is of the first class, a projective plane with p-fold rotational symmetry. In both cases, the entire eversion can proceed maintaining p-fold rotational symmetry. Morin had found analytic expressions for the essential steps of these eversions [19] (see [9, p. 116f]), which were further developed by Ap´ery [20]. Although these formulas are analytically elegant, they do not lead to well shaped, easily apprehended pictures. Thus we are led to look for more optimal geometric forms for these eversions. B. Willmore-critical spheres An elastic bending energy for surfaces should be quadratic in the principal curvatures; by the Gauss-Bonnet theorem it can Rbe reduced to the integral of mean curvature squared, W = H 2 dA, known as the Willmore energy [21]. (See [14] for more about the history of this energy, and some early computer experiments minimizing it.) This energy is invariant under M¨ obius transformations, and Bryant [22] showed that all critical points among immersed spheres arise as M¨ obius transformations of minimal surfaces in R3 with flat ends. These can be described ex-

plicitly by the Weierstrass representation. The Willmore energy of such a critical point is W = 4πk, where k is the number of ends; aside from the round sphere (a global minimum at W = 4π) the lowest energy examples occur with k = 4. Kusner [23], [24] soon found particular examples of such critical spheres with rotational symmetry, which he proposed as particularly nice geometric realizations of the halfway models for the tobacco-pouch eversions. He described a minimal surface Sp as the image of the (puncˆ under the map tured) Riemann sphere z ∈ C
! 2p + 1) i(z 2p−1 − z), z 2p−1 + z, i p−1 p (z √ . Sp (z) = < 2p−1 p z 2p + 2 p−1 z −1 To get a halfway model Mp with the same rotational symmetry, we apply a M¨obius transformation to Sp by inverting in a sphere centered at some point (0, 0, s) along the z-axis. Because Sp passes through the origin (but no other point of the z-axis) we must choose s 6= 0 to get a compact image. We chose s ≈ 13 purely for aesthetic reasons. C. Minimax symmetric eversions Each of the halfway models Mp described above is a critical point for the bending energy W , with orientationreversing symmetry of order 2p. In general, we expect the (Morse) index of a critical point to decrease as more symmetry is imposed. Here the index is not known theoretically, but the numerical experiments we have performed support a reasonable conjecture: Mp is a local minimum for W among spheres with its 2p-fold symmetry, but is unstable (with Morse index one) if we enforce just p-fold symmetry. Each halfway model thus seems to be a saddle critical point for the bending energy; perturbing it off the saddle one way or the other, and letting the surface flow downhill, we reach the round sphere, the global minimum for energy. To get an eversion, we start with the round sphere, play this descent backwards to reach the halfway model, then apply the symmetry and play the descent forwards, now down the other side of the saddle. These eversions form minimax paths from the round sphere to the inside-out round sphere, in the sense that it minimizes (over all possible paths) the maximum value of the bending energy along that path (achieved at the halfway model). (For p = 2 we have the global minimum, giving the optimal minimax eversion. For p > 2 the minimax point is merely a local minimum over all nearby paths.) Any path in time from the energy-minimizing round sphere up to the halfway model and back downhill to the round sphere could be called a minimax path. But for our minimax eversions we use gradient descent to find the most direct downhill path. The gradient flow for W is a fourth-order parabolic flow, which is not yet well understood. Recent papers of Kuwert and Sch¨atzle [25], [26] have shown that, if we start close enough, the flow leads to the round sphere. In other cases, however, the flow can start with a smooth surface

FRANCIS AND SULLIVAN: VISUALIZATION OF A SPHERE EVERSION

and pinch off a neck. Such a change in the topology of the surface would prevent us from obtaining a regular homotopy of the sphere. Other events which would spoil the regularity, such as the birth of pinch-points, should not occur because at such singularities the bending energy would become infinite. Certainly for our eversion computations, if the halfway model is indeed a saddle and if the flow remains smooth, it must converge to the round sphere, since that is the only critical point with lower energy. The computer simulations we have performed with Brakke’s Evolver give clear evidence that, for our eversion, the topology of the approximated surface remains an immersed sphere at all times. The minimax eversion that we compute in this way (with p = 2) turns out to be equivalent to one of Morin’s original eversions. Equivalence means they have the same sequence of topological events, as illustrated in Fig. 2. Our energyminimization procedure gives all of the tobacco-pouch eversions more pleasing shapes than in their earlier realizations, which were all designed by hand even if executed on a computer. In general, shapes that mathematically optimize some geometric energy are often aesthetically pleasing to the human eye. [27]. III. Issues in Visualization The challenge of visualizing a sphere eversion rests in the fact that the interesting stages have complicated internal structure, which the externally visible structure does not predict. This is not a new problem. Anatomists have, over the centuries, developed a series of artistic conventions: the cut away, the window, the excised organ, the skeleton, vascular tree, the musculature, false but descriptive coloring, and even semi-transparency (though the latter is seen more often in automotive shop manuals). To meet our challenge, we use analogous topological techniques: conformal warping and shaping, geometrically meaningful rendering and coloring, surreal optical transparency, isolating details, structural textures and materials, and, finally, also sculptured models. A. Warping and shaping by inversion in spheres One mathematically meaningful method of seeing the inside of an object is to apply a M¨ obius transformation to invert it in a sphere. Of course, this hides what was originally the external structure, and for the sphere eversion, we want to see both at once. We did, however, use M¨ obius transformations to good effect in one scene in The Optiverse. As we mentioned above, the minimax halfway model Mp is only determined up to choice of a parameter s. The M¨ obius invariance of the Willmore energy means that any conformally equivalent surface will still be a critical point for the bending energy. There is a one-parameter family of critical points with the p-fold symmetry we want; different M¨ obius-equivalent surfaces with this same symmetry are selected by varying s. If we include s = ∞ as giving the original minimal surface Sp , then there is a circle’s worth of different surfaces. For s = 0 or ∞, the surface we get is not compact, but other-

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wise we found no reasons other than esthetic ones to prefer any particular value for s; for low p, we have found s ≈ 13 gives an appealing halfway model. The sculpture at Oberwolfach described in [28] is the Boy’s surface that Bryant obtained from these formulas, with p = 3 and s = 12 . The M¨obius scene in The Optiverse animated this whole circular family, stopping to pay special attention to the minimal surface Sp at s = ∞ and its inversion at s = 0, which was nicknamed the “cosmic taco” by our postproduction engineer. These are shown in Fig. 3, clipped to show only the parts inside a large ball.

Fig. 3. This minimal surface S2 (left), with four flat ends, gives rise to Kusner’s Morin surface of least Willmore bending energy, when a conformal M¨ obius transformation is applied to compactify it. There is a one-parameter family of compactifications M2 (s), but for the particular value s = 0, the resulting surface (right) is noncompact; the double-tangent point is sent to infinity, and the surface resembles a “cosmic taco”.

B. Rendering, lighting and color Our video The Optiverse was created with our custom software AVN [29], a real-time interactive computer animator (RTICA) which runs on a wide range of platforms [30], from laptops to immersive environments like the CAVE, the CUBE [31] and the Hayden Planetarium. Our sphere eversions are computed using triangulated approximations to smooth surfaces. We like to display these triangles, to emphasize the discrete nature of the computations. In binocular stereo, this also provides more edges for the viewer’s eyes to lock onto. We thus usually avoid smooth shading. When turning a sphere inside-out, we want to distinguish the two sides of the surface, so we use different color ranges (orange and blue). But in order to most clearly see self-intersections, the exact color of any triangle within its range is determined by its normal vector, in particular the vertical component along the symmetry axis. These colors do not stay fixed to particular triangles through the homotopy – indeed the triangulation of the sphere varies – but instead reflect the spatial orientation of the triangle at a given time. When two triangles intersect, even if we are seeing the same side (say the blue side) of both, they are likely to have different shades within the range of blue . Thus, at the halfway stage in the Morin eversion, the outward-facing sheets near the quadruple point (the place where, for one instant, four sheets of surface cross each other) are colored yellow and purple, while those near the opposite isthmus point are red and indigo. An individual

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Fig. 2. This minimax sphere eversion is a geometrically optimal way to turn a sphere inside out, minimizing the elastic bending energy needed in the middle of the eversion. We start at the top with a round sphere, and proceed clockwise. In the upper right, we see that the north pole has pushed inwards to form a gastrula. In the next image, two double curves (surface self-intersections) have been created, one at the bottom, and a small one at the neck. At the lower right, a pair of triple points (where three sheets of surface cross each other) is formed when the double curves come together. (Another pair is created at the same time in back; the eversion always has two-fold rotational symmetry.) Across the bottom, we go through the Morin halfway model, a critical point for the Willmore bending energy whose four-fold rotation symmetry interchanges its inside and outside. The roles of the dark and light sides of the surface are then interchanged, and up the left column, we see the double curves disappear one after the other, leading to the inside-out round sphere. In the center, to better examine the self-intersection curves just when pairs of triple points are being created, we shrink each triangle in the surface to a quarter of its normal size.

triangle’s two sides range in color from yellow/indigo to red/purple. C. Transparency We find that complicated immersed surfaces are best viewed with a number of different techniques for revealing the internal structure. It is cumbersome to handle transparency correctly with the alpha channel in OpenGL: we would have to depth-sort the facets and subdivide those that intersect. Thus for the transparent opening and closing scenes in The Optiverse (see Fig. 4), we instead use the custom soap-film shader for Renderman originally described in [32]. The soap-film shader has the very important feature that, as for physical films, the surfaces are less transparent when viewed edge-on. This is unlike most computergraphics transparency, using an alpha channel, which tends to generate very nonintuitive results that are hard for humans to parse correctly. Physical thin films are more

opaque when viewed obliquely, and more transparent when viewed directly. Our shader does not exactly duplicate soap-film optics. In the sphere eversion, it is important to note the surface self-intersections. True soap-films may meet along triple junction lines (Plateau borders or Y junctions), which show up naturally in a rendered view. But soap films never cross, and indeed an intersection (or X junction) of fully transparent surfaces is almost invisible to the eye. Thus the soap film in the transparent scenes in The Optiverse was artificially darkened: it absorbs about half the incident light rather than none. D. Isolating details, cutaways and 3D-windows In other scenes of the video, as well as in our interactive software, we use a highly maneuverable, versatile clipping box as a 3-D window and probe into the internal structure of the shape. A 2D window in a wall, a porthole for example, lets

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Fig. 4. This is the same minimax eversion shown in Fig. 2, but now rendered transparently like darkened soap film. Again starting from the round sphere (top, moving clockwise), we push the north pole down, then push it through the south pole (upper right) to create the first double curve of surface self-intersection. Two sides of the neck then bulge up, and these bulges push through each other (right) to give the second double curve. The two double curves approach each other, and when they cross (lower right) pairs of triple points are created. In the halfway model (bottom) all four triple points merge at the quadruple point, and five isthmus events happen simultaneously. The second half of the eversion (left) proceeds through exactly the same stages in reverse order, after making a ninety-degree twist. The large central image belongs between the two lowest ones on the right, slightly before the birth of the triple points.

us look into a world, while blocking out the distracting periphery. Since we are 3D beings, a 3D window cannot be a simple dimensional analogy. That would be an empty volume, like the clipping box we create about the viewers in a virtual environment. Simply moving into a object creates such negative 3D-window. A positive 3D clipping box instead blocks out the surrounding material, enabling us to look at convoluted internal structure and processes, one part at a time, and from all sides. Guiding this 3D probe about—much like an pre-literate child will follow words with her finger—we can explore the sphere eversion. E. Gaps, frameworks and custom textures

Fig. 5. We can use AVN to display internal structure in different ways. If we shrink each triangle, as in the halfway model (left), we can focus on the elaborate double locus (self-intersection curve). Instead, we can highlight the triangulation itself by drawing just a triangular framework, as in the late gastrula stage of the eversion (right).

AVN provides other ways to see internal structure. It can shrink each triangle (towards its barycenter) by any factor, leaving gaps between the triangles. Conversely, AVN can draw just the borders of each triangle, leaving triangular windows in a framework of mullions with a distinctive, aesthetic appeal, as in Fig. 5. A topological understanding of the sphere eversion re-

quires seeing how the double locus (the set of selfintersection curves of the sphere) changes in time. Our RTICA renders a smoothed tube around the double locus, whose radius is controllable. When the gaps between the displayed triangles are large, the double locus becomes more prominent visually. Additionally, AVN can turn off

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the display of the triangles altogether, or can display only those triangles sufficiently near to or far from the double locus. Further, Stuart Levy has extended AVN to allow mapping a texture onto each triangle of the everting sphere. In this manner, we can show floating dots at the centers of the triangles, or other pleasing patterns. The resulting smoothly flowing shapes with other-worldly patterns are used at the new Hayden Planetarium in New York to suggest the formless void “before” the big bang, or the space-time quantum fluctuations in the early universe, in the Big Bang exhibit which opened in summer 2001 in the Hayden bowl. See Fig. 6.

series of models is needed to capture a homotopy. We provided Stewart Dickson with the datafiles for several stages of our minimax eversion, and he used stereolithography to make sculptural models, as part of his tactile mathematics project [33]. We had the opportunity to present some of these models to Bernard Morin in France; Fig. 7 shows how he enjoyed getting to know their shapes.

Fig. 7. Bernard Morin learned the shapes of stages in the minimax eversion not from the video The Optiverse (he has been blind since age five) but from models produced from our data by Stewart Dickson.

Fig. 6. The Hayden planetarium uses scenes from an AVN rendering of the eversion with five-fold symmetry, rendered with smooth textures on the triangles, to suggest space-time quantum fluctuations in the early universe. Photo courtesy of Stuart Levy.

F. Sculptural models Virtual models certainly have many advantages over real models: they are easy to create and can easily change in time to represent a homotopy. But despite many advances in virtual-reality, immersive environments, and holography, real physical models still have other advantages: shade and shadow come for free, and they are easier to manipulate and touch. They give a sense of concreteness which is hard to find in the virtual world. The rise of inexpensive 3dprinters (rapid-prototyping machines) makes it fairly easy to create physical models from virtual ones. Such sculptural representations in three dimension have many advantages over graphical representations in two dimensions. The viewpoint, for instance, is not fixed but can be chosen by the viewer. However, the kind of complicated internal structure present in many mathematical surfaces, like bubble clusters of the middle stages in a sphere eversion, is not easily visible in either kind of representation. Furthermore, it is extremely difficult to construct a mobile sculpture model whose shape can change, so a whole

Acknowledgments Parts of this article are based on our earlier reports: The computations of the minimax eversion were described in [11], [12], the interactive animator in [29], [30], the virtual-reality CUBE in [31], and the history of sphere eversions in [7], [6]. Sullivan is partially supported by NSF grant DMS-00-71520. Some of the work described was done jointly with Rob Kusner, Stuart Levy and others. References [1] [2] [3] [4] [5] [6]

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Jeffrey R. Weeks, The Shape of Space, vol. 249 of Pure and Applied Math., Dekker, second edition, 2002. Hassler Whitney, “On regular closed curves in the plane.,” Compos. Math., vol. 4, pp. 276–284, 1937. Stephen Smale, “A classification of immersions of the twosphere,” Trans. Amer. Math. Soc., vol. 90, pp. 281–290, 1959. Nelson L. Max, Turning a Sphere Inside Out, 1976. Narrated film (21 min). Reissued by AK Peters (2004). Nelson L. Max and William Clifford, “Computer animation of the sphere eversion,” Computer Graphics, vol. 9, no. 1, pp. 32– 39, 1975. Proceedings of SIGGRAPH ’75. John M. Sullivan, “Sphere eversions: From Smale through The Optiverse,” in Mathematics and Art: Mathematical Visualization in Art and Education, Claude P. Bruter, Ed., pp. 201–212 and 311–313, Springer, Berlin, 2002. Proceedings from Maubeuge (Sep 2000). John M. Sullivan, “The Optiverse and other sphere eversions,” in ISAMA 99, Nathaniel A. Friedman and Javier Barrallo, Eds. The International Society of The Arts, Mathematics and Architecture, pp. 491–497, Univ. of the Basque Country, 1999. arXiv:math.GT/9905020.

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[8] [9] [10]

[11]

[12]

[13] [14] [15] [16] [17] [18]

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

[30]

[31]

[32]

Silvio Levy, Making Waves: A Guide to the Ideas Behind Outside In, AK Peters, Wellesley, MA, 1995. George Francis, A Topological Picturebook, Springer, New York, 1987. John M. Sullivan, George Francis, and Stuart Levy, “The Optiverse,” in VideoMath Festival at ICM’98, Hege and Polthier, Eds., Springer, 1998. Narrated video (7 min). new.math.uiuc.edu/optiverse/. George Francis, John M. Sullivan, Robert B. Kusner, Kenneth A. Brakke, Chris Hartman, and Glenn Chappell, “The minimax sphere eversion,” in Visualization and Mathematics, Hege and Polthier, Eds., pp. 3–20. Springer, 1997. torus.math.uiuc.edu/jms/Papers/minimax/. George Francis, John M. Sullivan, and Chris Hartman, “Computing sphere eversions,” in Mathematical Visualization, Hege and Polthier, Eds., pp. 237–255. Springer, 1998. torus.math.uiuc.edu/jms/Papers/cse/. Kenneth A. Brakke, “The Surface Evolver,” Experimental Math., vol. 1, no. 2, pp. 141–165, 1992. www.susqu.edu/facstaff/b/brakke/evolver/. Lucas Hsu, Rob Kusner, and John M. Sullivan, “Minimizing the squared mean curvature integral for surfaces in space forms,” Exper. Math., vol. 1, no. 3, pp. 191–207, 1992. George Francis and Bernard Morin, “Arnold Shapiro’s eversion of the sphere,” Math. Intelligencer, vol. 2, pp. 200–203, 1979. Anthony Phillips, “Turning a sphere inside out,” Sci. Amer., vol. 214, pp. 112–120, 1966. Nicolas Kuiper, “Convex immersions of closed surfaces in E 3 ,” Comm. Helv., vol. 35, pp. 85–92, 1961. George Francis, “Drawing surfaces and their deformations: The Tobacco pouch eversions of the sphere,” Math. Modelling, vol. 1, pp. 273–281, 1980. ´ Bernard Morin, “Equations du retournement de la sph´ ere,” Comptes Rendus Acad. Sci. Paris, vol. 287, pp. 879–882, 1978. Fran¸cois Ap´ ery, “An algebraic halfway model for the eversion of the sphere,” Tohoku Math. J., vol. 44, pp. 103–150, 1992, with an appendix by Bernard Morin. Thomas J. Willmore, “Note on embedded surfaces,” An. Stiint. Univ “Al. I. Cuza” Iasi Sect. I, a Mat., vol. 11, pp. 493–496, 1965. Robert Bryant, “A duality theorem for Willmore surfaces,” J. Differential Geometry, vol. 20, pp. 23–53, 1984. Rob Kusner, “Conformal geometry and complete minimal surfaces,” Bull. Amer. Math. Soc., vol. 17, pp. 291–295, 1987. Rob Kusner, “Comparison surfaces for the Willmore problem,” Pacific J. Math., vol. 138, pp. 317–345, 1989. Ernst Kuwert and Reiner Sch¨ atzle, “The Willmore flow with small initial energy,” J. Differential Geom., vol. 57, no. 3, pp. 409–441, 2001. Ernst Kuwert and Reiner Sch¨ atzle, “Gradient flow for the Willmore functional,” Comm. Anal. Geom., vol. 10, no. 2, pp. 307– 339, 2002. John M. Sullivan, “The aesthetic value of optimal geometry,” in The Visual Mind, II, Michele Emmer, Ed. MIT Press, Cambridge, MA, 2004. To appear. Hermann Karcher and Ulrich Pinkall, “Die Boysche Fl¨ ache in Oberwolfach,” Mitteilungen der DMV, vol. 1997, no. 1, pp. 45– 47. George Francis, Stuart Levy, and John M. Sullivan, “Making the Optiverse: A mathematician’s guide to AVN, a real-time interactive computer animator,” in Mathematics, Art, Technology, Cinema, Michele Emmer and Mirella Manaresi, Eds. Springer, Berlin, 2003. Italian translation in [35]. torus.math.uiuc.edu/jms/Papers/avndoc/. John M. Sullivan, “Rescalable real-time interactive computer animations,” in Multimedia Tools for Communicating Mathematics, Borwein, Morales, Polthier, and Rodrigues, Eds. 2002, pp. 311–314, Springer, proceedings of the November 2000 conference in Lisbon. George K. Francis, Camille M.A. Goudeseune, Henry J. Kaczmarski, Benjamin J. Schaeffer, and John M. Sullivan, “ALICE on the eightfold way: Exploring curved spaces in an enclosed virtual reality theater,” in Visualization and Mathematics III, Hans-Christian Hege and Konrad Polthier, Eds., pp. 305-315 and 429. Springer, Berlin, 2003. torus.math.uiuc.edu/jms/Papers/alice8way.pdf Frederick J. Almgren, Jr. and John M. Sullivan, “Visualization

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of soap bubble geometries,” Leonardo, vol. 24, no. 3/4, pp. 267– 271, 1992. Reprinted in The Visual Mind. [33] Stewart Dickson, “Tactile mathematics,” in Mathematics and Art: Mathematical Visualization in Art and Education, Claude P. Bruter, Ed., pp. 213–222 and 314–315. Springer, Berlin, 2002. Proceedings from Maubeuge (Sep 2000). [34] John M. Sullivan, “The Optiverse and other sphere eversions,” in Bridges 1999, Reza Sarhangi, Ed., Winfield, Kansas, 1999, Bridges Conference, pp. 265–274, Southwestern College. [35] George Francis, Stuart Levy, and John M. Sullivan, “The Optiverse: una guida ai matematici per AVN, programma interattivo di animazione,” in Matematica, arte, tecnologia, cinema, Michele Emmer and Mirella Manaresi, Eds. Springer, Milano, 2002, pp. 37–51. Italian translation of [29].

George Francis is one of the pioneers of mathematical visualization, and his book A Topological Picturebook [9] is a classic in the field. He received his PhD from Michigan in 1967 and has been at Illinois ever since.

John M. Sullivan got his Ph.D. from Princeton in 1990, was a postdoc at the Geometry Center in Minnesota, and has been at Illinois since 1997. His research in optimal geometry involves a combination of mathematical theory and numerical experiments.