Final version, 20 May 1998
- Transparent eversion (22 sec):
An eversion turns a sphere inside-out without ripping or pinching.
However, the surface is allowed to pass through itself.
Our sphere eversion differs from earlier ones:
it is computed automatically by minimizing elastic bending energy.
The computer uses polyhedral approximations,
adaptively updating the triangulation.
- Solid eversion (22 sec):
The sphere, with its two sides painted yellow-to-red and blue-to-purple,
increases energy as it contorts itself into the symmetric halfway stage.
Then, it relaxes down again to a round sphere.
The minimax eversion constitutes an optimal route
over the lowest saddle for the bending energy.
- Framework eversion (31 sec):
The computation begins with this immersed sphere
whose four-fold symmetry interchanges inside and out.
Using second-order information about the energy, we push off the saddle.
Then we proceed downhill by gradient descent.
- Double locus (44 sec):
When the north pole pushes past the south pole,
the sphere ceases to be embedded, and the first double-curve is born.
Lobes near the neck form a second double-curve.
Arcs converge to spawn four triple points,
the corners of a tetrahedron, which shrinks to the quadruple point.
On the other side of the energy saddle,
the triple-points disappear in pairs and the double-curves shrink and vanish.
The whole process is inverted in time, with a 90-degree twist.
At all times, the everting sphere maintains a two-fold rotational symmetry.
- Topological Events (37 sec):
A tangency of two surface patches
creates, annihilates, or reconnects the double curve.
This reconnection was called the "isthmus point"
in the computer-animation classic by Nelson Max.
Four additional reconnections happen at the so-called "ears".
At the halfway stage, two ears open, just as the other two are closing.
When three surface patches meet along a common tangent line,
a pair of triple points is born, or dies.
Four surface patches cross at a quadruple point.
- Möbius transformations (65 sec):
A Morin surface is an elaborately immersed sphere
whose pleasing four-fold symmetry reverses orientation,
interchanging the orange and blue sides.
This halfway model is a Morin surface of least Willmore bending energy.
Willmore's energy is invariant under the Möbius group,
so any of these conformally equivalent surfaces
could have been our halfway model.
When the quadruple point reaches infinity,
we have Kusner's minimal surface,
with this explicit Weierstrass representation.
Kusner's surface has four flat ends,
which cut the sphere at infinity in a cuboctahedral pattern.
Continuing our conformal tour, we send the isthmus point to infinity,
so that the surface now has two wavy ends,
with a quadruple point at the center.
- Flythrough (23 sec):
To evert a sphere it suffices to describe a regular homotopy
from the halfway model to the round sphere.
Morin classified the simplest of these homotopies.
Brakke's computer program, the Evolver,
automatically finds one of these strategies,
purely by minimizing bending energy.
- Apery eversion (22 sec):
This triangulated cuboctahedron was everted by Apery and Denner.
Their halfway model is a polyhedral Morin surface.
Relaxing it in the Evolver, keeping the symmetry,
we descend to the Minimax halfway model.
- Four-fold eversion (25 sec):
The Minimax principle applies to all eversions in an infinite sequence
with increasing rotational symmetry, called the "tobacco-pouch" eversions.
Those of even symmetry, like this order-four example,
generalize Morin's eversion:
Their half-way models reverse orientation by a rotational symmetry.
- Three-fold eversion (45 sec):
The eversions of odd symmetry use doubly-covered projective planes
at the halfway stage.
This order-three eversion passes through Boy's surface,
an immersed projective plane with three-fold symmetry and a single triple point.
Converging to Boy's surface, two oppositely oriented sheets overlap
and compete to show their orange and blue colors,
accounting for the twinkling facets.
Then, Boy's propeller-shaped double-curve splits into its four-fold covering;
and a cube grows from the sextuple point.
- Transparent Dance (30 sec):
Our ensemble of spheres now join in performing eversions
with two-, three-, four-, and five-fold symmetry.