with original score by Camille Goudeseune.

*The Optiverse* is a 6.5-minute
computer-animated video showing an entirely
new way to turn a sphere inside out.
The video captures scenes that can also be viewed as real-time interactive
computer animations, on a workstation console or
in the CAVE^{TM}
immersive virtual environment. The narration is accompanied by
parambiences, which are novel experiments in scientific sonification.

*The Optiverse* was premiered at
the VideoMath
festival at ICM'98,
the International Congress of Mathematicians, August 1998 in Berlin.
A special two-minute cut was shown in the Electronic Theater
at SIGGRAPH 98, July 1998 in Orlando.

It was a surprising consequence of an abstract mathematical theorem of Steve Smale [Sma] that a spherical surface can be turned inside out without tearing or creasing, if we do allow the surface to pass through itself. Over the intervening forty years, mathematicians have designed different ways to see explicitly how such a "sphere eversion" can be achieved.

One of the first was designed by the blind topologist Bernard Morin
[Mor],
and rendered in the film *Turning a Sphere Inside Out* by Nelson
Max [Max]. More recently, Bill Thurston's scheme for creating eversions
has been illustrated in the video *Outside In* from the Geometry
Center [LMM]. For a survey of different sphere
eversions and their relation to *The Optiverse*, see also
Sullivan's article [Sul].

We have computed a family of sphere eversions which have rotational symmetry of different orders. In this way they are like the tobacco-pouch eversions suggested by Morin and pictured by Francis [Fra]. But our idea, originally suggested by Kusner [Ku1,Ku2], is to implement these eversions in a geometrically optimal way.

Our eversions are computed automatically by minimizing an elastic bending energy for surfaces in space (called the Willmore energy [Wil]). We start with a complicated self-intersecting sphere, which has the desired rotational symmetry, and is also halfway inside-out in the sense of having its inside and outside equally exposed. This halfway model is a saddle critical point for the Willmore energy. When we push off the saddle in two opposite directions and then flow downhill in energy to the ordinary round sphere, it is inside-out in one direction, but not in the other.

*The Optiverse* shows the first few eversions in our family,
with 2-, 3-, 4-, and 5-fold symmetry. The mathematics behind these
eversions is described more fully in our papers [FS+, FSH].
The two-fold eversion goes over the lowest possible saddle-point for the energy,
so we call it the "minimax eversion". Topologically, it is the same
as the Morin/Max eversion.
The odd-order eversions use double-covered projective planes for
their halfway models, as in Tony Phillips' famous eversion [Phi].

In any sphere eversion, the changing "double-locus" of self-intersections is a key item to watch as the sphere turns inside out. We examine in detail the sequence of events in the minimax eversion, that is, the times when there are reconnections in the double-locus. At the halfway stage in the three-fold eversion, we see the propeller-shaped double-locus of Boy's surface break apart as the oppositely oriented sheets separate.

We compute discrete polyhedral approximations to our optimal eversions
using Brakke's computer program, the *Surface Evolver*
[Bra]. This models
surfaces using triangulations, and moves the vertices to minimize
a geometric energy. We wrote scripts to automate the process of
evolving downhill while selectively refining the triangulation to
keep a good approximation to a smooth surface, even as small necks develop.

Because the computation of an eversion takes ten minutes to an hour, even on a fast SGI workstation, we have the evolver save a few hundred stages in the homotopy, which we call topes. We compute the double-locus, or curve of self-intersections, for each tope, and save this information as well.

Our interactive viewing program for these saved homotopies
works in real-time on an SGI console, or in the CAVE virtual-reality
room, displaying up to 24 frames per second. Because it is important
to distinguish the inside and outside of the surface, we use different
color ranges for the two sides: one is blue-to-purple, and the other
yellow-to-red. We have interactive control of the rendering style:
we can introduce gaps between the triangles, or holes in the middle
of the triangles; we can also cut away parts of the surface away from
the double-locus. Many scenes in *The Optiverse* were rendered
directly to disk from this real-time viewer.

One scene shows the halfway model undergoing conformal (Möbius) transformations. These are distortions of space (like reflections in a round sphere) which change distances but preserve angles and local shapes. It turns out that the mathematical bending energy we use to drive our sphere eversion is unchanged by these particular distortions. From a certain mathematical point of view, all the surfaces in the family shown have the same shape, though they look very different to the human eye. The "conformal lens" in our software, allowing interactive control of these distortions, may let us learn to see these as forms of a single shape, just as we recognize different views of a single 3D object.

The scenes with bubble-like transparency were rendered with
Pixar's *Renderman*. Sullivan wrote a custom shader
[AS]
for *Renderman*, which uses the laws of thin-film optics to
simulate the colored highlights seen in real soap films. For
self-intersecting surfaces, as in *The Optiverse*, we find
it best to modify these equations to introduce some opacity in
the bubbles, and a small amount of diffuse lighting.

Parambiences can be used in navigating through a fixed dataset (like
our precomputed family of surfaces) or in exploring a running process.
In *The Optiverse*, the parambience is controlled by a single
knob (set from the index into the family of surfaces), together
with buttons triggered when we pass certain stages like
self-intersections and formation of triple points.
In fact, the control signal in this model does not even include
these buttons; instead they are generated internally by the audio
system when the knob passes certain values.
This results in very low bandwidth for communication.

Parambiences which do not use pulsed rhythms can often be quickly implemented through techniques of granulation, although the source material for the grains requires more disk space with increasing number of parameters and their fineness of resolution. Music with a regular pulse (a striated rhythmic space, to use Boulez's general term) requires deeper structure and may require shortcuts to software synthesis such as MIDI sequences.

The composer of a parambience faces a problem similar to that of composing for contemporary dance: it cannot be so interesting as to distract from the visual or environmental experience; but it cannot be so overly simple as to be tedious. A second challenge in composing such music is making a forgery of a large-scale structure. Through the classical and romantic periods, composers found various ways to lengthen the span of an unbroken piece of music. But as a parambience is driven by an inherently unpredictable environment, classical techniques of large-scale form are ill-suited to its composition. The parambience must be constructed as a parametrized family of small "moments" of sound which can in some sense be chained into a sequence which fools the ear into imagining large-scale structures. Of course, one might use newer (indeterminate or "free") techniques of large-scale musical form; but in the context of providing music for VR applications, this may overly attract the attention of viewers.

- [AS]
- "Visualization of Soap-Bubble Geometries",
F. Almgren, J.M. Sullivan,
in
**The Visual Mind**, (M. Emmer, ed.), Leonardo/MIT Press, 1993, pp 75-83. - [Bra]
- "The Surface Evolver",
K. Brakke,
*Experimental Math.***1**, pp 141-165. - [Fra]
**A Topological Picturebook**, G. Francis, Springer, 1987.- [FSH]
- "Computing Sphere Eversions", G. Francis,
J.M. Sullivan, C. Hartman,
to appear in
**Mathematical Visualization**, (H.-C. Hege, K. Polthier, eds.), Springer, 1998. - [FS+]
- "The Minimax Sphere Eversion",
G. Francis, J.M. Sullivan,
R.B. Kusner, K. Brakke, C. Hartman,
in
**Visualization and Mathematics**, (H.-C. Hege, K. Polthier, eds.), Springer, 1997, pp 3-20. - [FE+]
- "Stepping into Alpha Shapes", P. Fu, H. Edelsbrunner, U. Axen, R. Bargar, I. Choi, VROOM presentation at SIGGRAPH '94.
- [Ku1]
- "Comparison Surfaces for the Willmore Problem", R. Kusner,
*Pacific J. Math.***138**:2, 1989, pp 317-345. - [Ku2]
- "Conformal Geometry and Complete Minimal Surfaces", R. Kusner,
*Bull. Amer. Math. Soc.***17**, 1987, pp 291-295. - [LMM]
*Outside In*, S. Levy, D. Maxwell, T. Munzner, A K Peters, 1995, video.- [Max]
*Turning a Sphere Inside Out*, N. Max, Int'l Film Bureau, 1977, video.- [Mor]
- "Le retournement de la sphère",
B. Morin, J.P. Petit,
*Pour la Science***15**, 1979, pp 34-41. - [Phi]
- "Turning a Surface Inside Out",
A. Phillips,
*Scientific American***214**, May 1966, 112-120. - [Sma]
- "A Classification
of Immersions of the Two-Sphere", S. Smale,
*Trans. Amer. Math. Soc.***90**, 1959, pp 281-290. - [Sul]
- "'The Optiverse' and Other Sphere Eversions",
*Bridges 1999*, Southwestern Coll., Kansas, 1999, pp 265-274; also online at http://www.math.uiuc.edu/~jms/Papers/isama/. - [Wil]
- "A Survey on Willmore Immersions",
T.J. Willmore,
in
**Geometry and Topology of Submanifolds, IV (Leuven, 1991)**, (F. Dillen, L. Verstraelen, eds.), World Scientific, 1992, pp 11-16.