George Francis and John M. Sullivan- Visualizing a Sphere Eversion

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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. XX, NO. Y, MMM 2003 1

Visualizing a Sphere EversionGeorge 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, everchanging internal structure. We have computed an optimalminimax eversion, requiring the least bending energy. Herewe discuss techniques we used to help visualize this eversionfor visitors to virtual environments and viewers of our videoThe Optiverse .

Keywords— Visualization, Sphere eversion, Boy surface,Morin surface, Regular homotopy, Immersions, Willmoreenergy, The CAVE.

I. Introduction

VISUALIZATION is especially challenging when theobjects to be viewed have complicated internal struc-

ture. Our human visual system is trained mostly on opaqueobjects whose insides cannot be seen; most transparent ob-

jects in our everyday experience are like glass windows,intended to be looked through rather than looked at .

But increasingly, numerical simulations of real-worldphenomena are being made in 3D×Time. The results of these simulations have internal structure that changes intime, and methods are needed for its visualization.

In mathematics, perhaps the most common objects withinternal structure are immersed surfaces. This means sur-faces which might be intrinsically simple (like a sphere ortorus) but which are mapped into space in a complicatedway (like the immersed spheres in Fig. 1). Locally, every

small patch of an immersed surface is embedded smoothlyinto the ambient space, so that an immersed surface neverhas rips, corners or creases. But globally, different sheetsof an immersed surface are allowed to pass through eachother, unlike any kind of physical surface.

Unlike the sphere and the torus, which are orientable,a closed nonorientable surface must have self-intersectionwhen immersed in 3-space. The real projective plane (alsoknown as the cross surface) is the simplest of these, and itssimplest immersion in 3-space (meaning the one with theleast complicated self-intersection) is Boy’s surface, shownin Fig. 1(left). (For a visual introduction to the topologyof surfaces see Appendix C of The Shape of Space [1].)

However, immersions also play an important role in thedeformations of surfaces (like the sphere) which can be em-bedded in 3D. Clearly it is impossible to turn a sphereinside out through a succession of embeddings. To turn aphysical sphere inside out, one must cut a hole, pull the restof the surface through the hole, and then patch the hole; thesurface being turned inside-out is not a sphere but a disk (asphere with a hole). Mathematically, it is most natural toconsider the question for immersed spheres: Can a spherebe 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 minimaxeversions. The Boy’s surface (left), an immersed projective plane withthree-fold symmetry and a single triple point (where three sheets of the surface cross each other), minimizes Willmore’s elastic bendingenergy. The figure actually shows an immersed sphere doubly cov-ering Boy’s surface, with its two (oppositely-oriented) sheets pulledapart slightly. The Morin surface shown (right) also minimizes Will-

more energy; it has a four-fold rotational symmetry which reversesorientation, exchanging the lighter and darker sides of the surface.

(smooth) immersions, called a regular homotopy by Whit-ney. Both Boy and Whitney [2] independently showed thatthis was not possible one dimension lower: a circle cannotbe turned inside-out in the plane by a regular homotopy.

It would be extremely hard to settle this question bytrial and error. An abstract mathematical theorem [3] bySteve Smale in 1959 classified regular homotopies for gen-eral surfaces and had the surprising consequence that asphere eversion was possible, without any clues as to what

it might look like.It took many years for other mathematicians to con-

struct explicit eversions; at first these were illustratedby hand-drawn pictures. In 1977 Nelson Max made acomputer-animated film [4], [5] realizing Bernard Morin’s1967 vision of a particularly persuasive eversion. Overfour decades, everting the sphere has remained a reward-ing problem in mathematical visualization and computergraphics, especially because of the challenge of animatinga self-intersecting surface.

For more information about the history of sphere ever-sions, 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 optimaleversion, computed automatically by minimizing an elas-tic bending energy for surfaces. Our computations [11],[12] of the sphere eversion were performed in Ken Brakke’sEvolver [13] using code to minimize the Willmore en-

ergy [14].

A. Tobacco-pouch eversions

Following Morin’s tradition, we use the concept of ahalfway model , an immersed sphere (like those in Fig. 1)



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

which is halfway inside-out in the following sense: Somesymmetry of the surface—a rigid motion of space bringingthe surface back to itself—will turn it inside out by re-versing the surface orientation. More precisely, for an ap-propriate choice of parameterization, antipodal points onthe abstract domain sphere are mapped by the immersionto points related by this symmetry. Given such a halfwaymodel, any regular homotopy which simplifies it down tothe round sphere can be extended by symmetry to a sphereeversion.

The halfway models that have been used in this wayare of two types The first is based on an immersion, suchas Boy’s surface, of the projective plane, which topolog-ically is the nonorientable surface obtained by identify-ing antipodal points on the sphere. A double covering of such a projective plane is an immersed sphere. Here theorientation-reversing symmetry is simply the identity mapin space: antipodal points of the sphere lie at the sameplace in the halfway model. The early sphere eversions of Shapiro [15], Phillips [16] and Kuiper [17] used Boy’s sur-

face as a halfway model. The second type of halfway modelhas 2 p-fold rotational symmetry reversing orientation (andthus p-fold symmetry preserving orientation). The origi-nal Morin-Froissart halfway model was of this type, with

p = 2.In the early seventies, Morin built tiny plasticine mod-

els of a family of sphere eversions for integers p > 1, latercalled the tobacco-pouch eversions. He inspired CharlesPugh to build large, chicken wire models for the p = 2 case,and these formed the database for Nelson Max’s master-piece computer graphics sphere eversion. In 1977, studentsin Francis’s freshman honors topology seminar, after see-ing 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 secondclass, with 2 p-fold rotational symmetry (reversing orienta-tion). For p odd, it is of the first class, a projective planewith p-fold rotational symmetry. In both cases, the entireeversion can proceed maintaining p-fold rotational symme-try.

Morin had found analytic expressions for the essentialsteps of these eversions [19] (see [9, p. 116f]), which werefurther developed by Apery [20]. Although these formulasare analytically elegant, they do not lead to well shaped,easily apprehended pictures. Thus we are led to look formore optimal geometric forms for these eversions.

B. Willmore-critical spheres

An elastic bending energy for surfaces should bequadratic in the principal curvatures; by the Gauss-Bonnettheorem it can be reduced to the integral of mean curvaturesquared, W =

H 2 dA, known as the Willmore energy [21].

(See [14] for more about the history of this energy, and someearly computer experiments minimizing it.)

This energy is invariant under Mobius transformations,and Bryant [22] showed that all critical points among im-mersed spheres arise as Mobius transformations of minimalsurfaces in R3 with flat ends. These can be described ex-

plicitly by the Weierstrass representation. The Willmoreenergy of such a critical point is W = 4πk, where k is thenumber of ends; aside from the round sphere (a global min-imum at W = 4π) the lowest energy examples occur withk = 4.

Kusner [23], [24] soon found particular examples of suchcritical spheres with rotational symmetry, which he pro-posed as particularly nice geometric realizations of thehalfway models for the tobacco-pouch eversions. He de-scribed a minimal surface S p as the image of the (punc-

tured) Riemann sphere z ∈ C under the map

S p(z) =

i(z2 p−1 − z), z2 p−1 + z, i

p−1 p

(z2 p + 1)

z2 p + 2√

2 p−1

p−1z p − 1

.

To get a halfway model M p with the same rotationalsymmetry, we apply a Mobius transformation to S p by in-verting in a sphere centered at some point (0, 0, s) along thez-axis. Because S p passes through the origin (but no otherpoint of the z-axis) we must choose s = 0 to get a compact

image. We chose s ≈ 13 purely for aesthetic reasons.

C. Minimax symmetric eversions

Each of the halfway models M p described above is acritical point for the bending energy W , with orientation-reversing symmetry of order 2 p. In general, we expect the(Morse) index of a critical point to decrease as more sym-metry is imposed. Here the index is not known theoret-ically, but the numerical experiments we have performedsupport a reasonable conjecture: M p is a local minimumfor W among spheres with its 2 p-fold symmetry, but isunstable (with Morse index one) if we enforce just p-foldsymmetry.

Each halfway model thus seems to be a saddle criticalpoint for the bending energy; perturbing it off the saddleone 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, playthis descent backwards to reach the halfway model, thenapply the symmetry and play the descent forwards, nowdown the other side of the saddle.

These eversions form minimax paths from the roundsphere to the inside-out round sphere, in the sense thatit minimizes (over all possible paths) the maximum valueof the bending energy along that path (achieved at thehalfway model). (For p = 2 we have the global mini-mum, giving the optimal minimax eversion. For p > 2 theminimax point is merely a local minimum over all nearbypaths.) Any path in time from the energy-minimizinground sphere up to the halfway model and back downhillto the round sphere could be called a minimax path. Butfor our minimax eversions we use gradient descent to findthe most direct downhill path.

The gradient flow for W is a fourth-order parabolic flow,which is not yet well understood. Recent papers of Kuw-ert and Schatzle [25], [26] have shown that, if we startclose enough, the flow leads to the round sphere. In othercases, however, the flow can start with a smooth surface



FRANCIS AND SULLIVAN: VISUALIZATION OF A SPHERE EVERSION 3

and pinch off a neck. Such a change in the topology of the surface would prevent us from obtaining a regular ho-motopy of the sphere. Other events which would spoil theregularity, such as the birth of pinch-points, should not oc-cur because at such singularities the bending energy wouldbecome infinite. Certainly for our eversion computations, if the halfway model is indeed a saddle and if the flow remainssmooth, it must converge to the round sphere, since thatis the only critical point with lower energy. The computersimulations we have performed with Brakke’s Evolver giveclear evidence that, for our eversion, the topology of theapproximated surface remains an immersed sphere at alltimes.

The minimax eversion that we compute in this way (with p = 2) turns out to be equivalent to one of Morin’s originaleversions. Equivalence means they have the same sequenceof topological events, as illustrated in Fig. 2. Our energy-minimization procedure gives all of the tobacco-pouch ev-ersions more pleasing shapes than in their earlier realiza-tions, which were all designed by hand even if executed on

a computer. In general, shapes that mathematically opti-mize some geometric energy are often aesthetically pleasingto the human eye. [27].

III. Issues in Visualization

The challenge of visualizing a sphere eversion rests in thefact that the interesting stages have complicated internalstructure, which the externally visible structure does notpredict. This is not a new problem. Anatomists have, overthe centuries, developed a series of artistic conventions: thecut away, the window, the excised organ, the skeleton, vas-cular 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, geometricallymeaningful rendering and coloring, surreal optical trans-parency, isolating details, structural textures and materi-als, and, finally, also sculptured models.

A. Warping and shaping by inversion in spheres

One mathematically meaningful method of seeing the in-side of an object is to apply a Mobius transformation toin vert it in a sphere. Of course, this hides what was orig-inally the external structure, and for the sphere eversion,we want to see both at once.

We did, however, use Mobius transformations to good ef-fect in one scene in The Optiverse . As we mentioned above,the minimax halfway model M p is only determined up tochoice of a parameter s. The Mobius invariance of theWillmore energy means that any conformally equivalentsurface 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 Mobius-equivalent sur-faces with this same symmetry are selected by varying s.If we include s = ∞ as giving the original minimal surfaceS p, then there is a circle’s worth of different surfaces. Fors = 0 or ∞, the surface we get is not compact, but other-

wise we found no reasons other than esthetic ones to preferany particular value for s; for low p, we have found s ≈ 1

3

gives an appealing halfway model. The sculpture at Ober-wolfach described in [28] is the Boy’s surface that Bryantobtained from these formulas, with p = 3 and s = 1

2.

The Mobius scene in The Optiverse animated this wholecircular family, stopping to pay special attention to the

minimal surfaceS

p ats

=∞

and its inversion ats

= 0,which was nicknamed the “cosmic taco” by our postpro-duction engineer. These are shown in Fig. 3, clipped toshow only the parts inside a large ball.

Fig. 3. This minimal surface S 2 (left), with four flat ends, gives riseto Kusner’s Morin surface of least Willmore bending energy, when aconformal Mobius transformation is applied to compactify it. Thereis a one-parameter family of compactifications M 2(s), but for theparticular value s = 0, the resulting surface (right) is noncompact;the double-tangent point is sent to infinity, and the surface resemblesa “cosmic taco”.

B. Rendering, lighting and color

Our video The Optiverse was created with our customsoftware AVN [29], a real-time interactive computer anima-tor (RTICA) which runs on a wide range of platforms [30],

from laptops to immersive environments like the CAVE, theCUBE [31] and the Hayden Planetarium.

Our sphere eversions are computed using triangulatedapproximations to smooth surfaces. We like to displaythese triangles, to emphasize the discrete nature of thecomputations. In binocular stereo, this also provides moreedges for the viewer’s eyes to lock onto. We thus usuallyavoid smooth shading.

When turning a sphere inside-out, we want to distin-guish the two sides of the surface, so we use different colorranges (orange and blue). But in order to most clearly seeself-intersections, the exact color of any triangle within itsrange is determined by its normal vector, in particular the

vertical component along the symmetry axis. These colorsdo not stay fixed to particular triangles through the homo-topy – indeed the triangulation of the sphere varies – butinstead reflect the spatial orientation of the triangle at agiven time. When two triangles intersect, even if we areseeing the same side (say the blue side) of both, they arelikely to have different shades within the range of blue .

Thus, at the halfway stage in the Morin eversion, theoutward-facing sheets near the quadruple point (the placewhere, for one instant, four sheets of surface cross eachother) are colored yellow and purple, while those near theopposite isthmus point are red and indigo. An individual




Fig. 2. This minimax sphere eversion is a geometrically optimal way to turn a sphere inside out, minimizing the elastic bending energyneeded 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 thenorth pole has pushed inwards to form a gastrula . In the next image, two double curves (surface self-intersections) have been created, oneat 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 rotationalsymmetry.) Across the bottom, we go through the Morin halfway model, a critical point for the Willmore bending energy whose four-foldrotation symmetry interchanges its inside and outside. The roles of the dark and light sides of the surface are then interchanged, and up theleft column, we see the double curves disappear one after the other, leading to the inside-out round sphere. In the center, to better examinethe self-intersection curves just when pairs of triple points are being created, we shrink each triangle in the surface to a quarter of its normalsize.

triangle’s two sides range in color from yellow/indigo tored/purple.

C. Transparency

We find that complicated immersed surfaces are bestviewed with a number of different techniques for revealingthe internal structure. It is cumbersome to handle trans-

parency correctly with the alpha channel in OpenGL: wewould have to depth-sort the facets and subdivide thosethat intersect. Thus for the transparent opening and clos-ing scenes in The Optiverse (see Fig. 4), we instead usethe custom soap-film shader for Renderman originally de-scribed in [32].

The soap-film shader has the very important featurethat, as for physical films, the surfaces are less transpar-ent when viewed edge-on. This is unlike most computer-graphics transparency, using an alpha channel, which tendsto generate very nonintuitive results that are hard for hu-mans to parse correctly. Physical thin films are more

opaque when viewed obliquely, and more transparent whenviewed directly.

Our shader does not exactly duplicate soap-film optics.In the sphere eversion, it is important to note the surfaceself-intersections. True soap-films may meet along triple

junction lines (Plateau borders or Y junctions), which showup naturally in a rendered view. But soap films never cross,

and indeed an intersection (or X junction) of fully transpar-ent surfaces is almost invisible to the eye. Thus the soapfilm in the transparent scenes in The Optiverse was arti-ficially darkened: it absorbs about half the incident lightrather than none.

D. Isolating details, cutaways and 3D-windows

In other scenes of the video, as well as in our interactivesoftware, we use a highly maneuverable, versatile clippingbox as a 3-D window and probe into the internal structureof the shape.

A 2D window in a wall, a porthole for example, lets




Fig. 4. This is the same minimax eversion shown in Fig. 2, but now rendered transparently like darkened soap film. Again starting fromthe round sphere (top, moving clockwise), we push the north pole down, then push it through the south pole (upper right) to create the firstdouble curve of surface self-intersection. Two sides of the neck then bulge up, and these bulges push through each other (right) to give thesecond double curve. The two double curves approach each other, and when they cross (lower right) pairs of triple points are created. In thehalfway 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 imagebelongs 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 distractingperiphery. Since we are 3D beings, a 3D window cannotbe a simple dimensional analogy. That would be an emptyvolume, like the clipping box we create about the viewers ina virtual environment. Simply moving into a object createssuch negative 3D-window.

A positive 3D clipping box instead blocks out the sur-rounding material, enabling us to look at convoluted in-ternal structure and processes, one part at a time, andfrom all sides. Guiding this 3D probe about—much like anpre-literate child will follow words with her finger—we canexplore the sphere eversion.

E. Gaps, frameworks and custom textures

AVN provides other ways to see internal structure. It canshrink each triangle (towards its barycenter) by any factor,leaving gaps between the triangles. Conversely, AVN candraw just the borders of each triangle, leaving triangularwindows in a framework of mullions with a distinctive, aes-thetic appeal, as in Fig. 5.

A topological understanding of the sphere eversion re-

Fig. 5. We can use AVN to display internal structure in differentways. 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 atriangular framework, as in the late gastrula stage of the eversion(right).

quires seeing how the double locus (the set of self-intersection curves of the sphere) changes in time. OurRTICA renders a smoothed tube around the double locus,whose radius is controllable. When the gaps between thedisplayed triangles are large, the double locus becomesmore prominent visually. Additionally, AVN can turn off




the display of the triangles altogether, or can display onlythose triangles sufficiently near to or far from the doublelocus.

Further, Stuart Levy has extended AVN to allow mappinga texture onto each triangle of the everting sphere. In thismanner, we can show floating dots at the centers of the tri-angles, or other pleasing patterns. The resulting smoothlyflowing shapes with other-worldly patterns are used at thenew Hayden Planetarium in New York to suggest the form-less void “before” the big bang, or the space-time quantumfluctuations in the early universe, in the Big Bang exhibitwhich opened in summer 2001 in the Hayden bowl. SeeFig. 6.

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

F. Sculptural models

Virtual models certainly have many advantages over realmodels: they are easy to create and can easily change intime to represent a homotopy. But despite many advancesin virtual-reality, immersive environments, and holography,real physical models still have other advantages: shade andshadow come for free, and they are easier to manipulateand touch. They give a sense of concreteness which is hard

to find in the virtual world. The rise of inexpensive 3d-printers (rapid-prototyping machines) makes it fairly easyto create physical models from virtual ones.

Such sculptural representations in three dimension havemany advantages over graphical representations in two di-mensions. The viewpoint, for instance, is not fixed but canbe chosen by the viewer. However, the kind of complicatedinternal structure present in many mathematical surfaces,like bubble clusters of the middle stages in a sphere ever-sion, is not easily visible in either kind of representation.

Furthermore, it is extremely difficult to construct a mo-bile sculpture model whose shape can change, so a whole

series of models is needed to capture a homotopy.We provided Stewart Dickson with the datafiles for sev-

eral stages of our minimax eversion, and he used stere-olithography to make sculptural models, as part of his tac-tile mathematics project [33]. We had the opportunity topresent 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 minimaxeversion not from the video The Optiverse (he has been blind since agefive) but from models produced from our data by Stewart Dickson.

Acknowledgments

Parts of this article are based on our earlier reports:The computations of the minimax eversion were describedin [11], [12], the interactive animator in [29], [30], thevirtual-reality CUBE in [31], and the history of sphere ev-ersions in [7], [6]. Sullivan is partially supported by NSFgrant DMS-00-71520. Some of the work described was done

jointly with Rob Kusner, Stuart Levy and others.

References

[1] Jeffrey R. Weeks, The Shape of Space, vol. 249 of Pure andApplied Math., Dekker, second edition, 2002.

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[5] Nelson L. Max and William Clifford, “Computer animation ofthe sphere eversion,” Computer Graphics, vol. 9, no. 1, pp. 32–39, 1975. Proceedings of SIGGRAPH ’75.

[6] John M. Sullivan, “Sphere eversions: From Smale throughThe Optiverse,” in Mathematics and Art: Mathematical Vi-sualization in Art and Education , Claude P. Bruter, Ed., pp.201–212 and 311–313, Springer, Berlin, 2002. Proceedings fromMaubeuge (Sep 2000).

[7] 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 Ar-chitecture, pp. 491–497, Univ. of the Basque Country, 1999.arXiv:math.GT/9905020.




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sphere eversion,” in Visualization and Mathematics, Hege andPolthier, Eds., pp. 3–20. Springer, 1997.torus.math.uiuc.edu/jms/Papers/minimax/.

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[27] John M. Sullivan, “The aesthetic value of optimal geometry,”in The Visual Mind, II , Michele Emmer, Ed. MIT Press, Cam-bridge, MA, 2004. To appear.

[28] Hermann Karcher and Ulrich Pinkall, “Die Boysche Flache inOberwolfach,” Mitteilungen der DMV , vol. 1997, no. 1, pp. 45–47.

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Cinema , Michele Emmer and Mirella Manaresi, Eds. Springer,Berlin, 2003. Italian translation in [35].torus.math.uiuc.edu/jms/Papers/avndoc/.

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[31] George K. Francis, Camille M.A. Goudeseune, Henry J. Kacz-marski, Benjamin J. Schaeffer, and John M. Sullivan, “ALICEon the eightfold way: Exploring curved spaces in an enclosedvirtual reality theater,” in Visualization and Mathematics III ,Hans-Christian Hege and Konrad Polthier, Eds., pp. 305-315 and429. Springer, Berlin, 2003.torus.math.uiuc.edu/jms/Papers/alice8way.pdf

[32] Frederick J. Almgren, Jr. and John M. Sullivan, “Visualization

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 Mathematicsand 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 Op-

tiverse: una guida ai matematici per AVN, programma interat-tivo 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 ofmathematical visualization, and his book ATopological Picturebook [9] is a classic in thefield. He received his PhD from Michigan in1967 and has been at Illinois ever since.

John M. Sullivan got his Ph.D. from Prince-ton in 1990, was a postdoc at the GeometryCenter in Minnesota, and has been at Illinoissince 1997. His research in optimal geometryinvolves a combination of mathematical theoryand numerical experiments.

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George Francis and John M. Sullivan- Visualizing a Sphere Eversion