Topological Stages in the Eversion
A generic regular homotopy has topological events occuring only
at isolated times, when the combinatorics of the selfintersection
changes. These events happen at times when the surface normals
at a point of intersection are not linearly independent. That
means either when there are two sheets of the surface tangent to each other
(and a double curve is created, annihilated, or reconnected),
or when three intersecting sheets share a tangent line (creating or
destroying a pair of triple points), or when four sheets come together
at a quadruple point.
The double tangency events come in three flavors. These can be modeled
with a rising water level across a fixed landscape. As the water
rises, we can observe creation of a new lake, the conversion between
a isthmus of land and a channel of water, or the submersion of an
island. These correspond to the creation, reconnection, or annihilation
of a doublecurve, here seen as the shoreline.
Figure 8:
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.
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 doublecurve of selfintersection.
Two sides of the neck then bulge up, and these
bulges push through each other (right) to give the second doublecurve.
The two doublecurves approach each other, and when
they cross (lower right) pairs of triple points are created.
In the halfwaymodel (bottom) all four triple points merge at
the quadruple point, and five isthmus events happen simultaneously.
This halfwaymodel is a symmetric critical point for the
Willmore bending energy for surfaces.
Its fourfold rotation symmetry interchanges
its inside and outside surfaces. Therefore, the second half of the eversion
(left) can proceed through exactly the same stages in reverse order, after
making the ninetydegree twist.
The large central image belongs between the two lowest ones on the
right, slightly before the birth of the triple points.
In the minimax eversion with twofold symmetry,
seen in Figs.
8 and 9,
the first two events create the two doublecurves. When these
twist around to intersect each other, two pairs of
triple points are created. At the halfway stage, six events happen
all at the same time. Along the symmetry axis, at one end we have
a quadruple point, while at the other end we have a double tangency
creating an isthmus event. Finally, at the four "ears", at the
inside edge of the large lobes, we have additional isthmus events:
two ears open as the other two close. (See [FSK]
for more details on these events.)
Figure 9:
This is the same eversion shown in Fig. 8, but rendered with
solid surfaces. Here we start at the top left with a round sphere, and
proceed clockwise.
We see the creation first of two doublecurves (at the top)
and then of a pair of triple points (at the top right). (Another pair is created at
the same time in back; the eversion always has twofold rotational
symmetry.) Near the bottom right we go through the halfwaymodel (also seen
in the large central image) and interchange the roles
of the redyellow and bluepurple sides of the surface.
Continuing around the left side, we see the doublecurves disappear one after the other.
The threefold minimax eversion, using the Boy's surface of
Fig. 1
as a halfwaymodel, has too many topological events to describe
easily onebyone, and its threefold symmetry means that the
events are no longer all generic. But we can still follow
the basic outline of the eversion from Fig. 10.
Figure 10:
This threefold minimax eversion starts (top row, clockwise)
with a gastrula stage like that of the twofold eversion, but
the threefold symmetry means that three fingers reach up from the neck
instead of two. They intersect each other (righthand images)
and then twist around, while complicated things are happening inside.
The lower right and central images are near the halfway model.
The two images at the bottom with gaps between the triangles
show the doublecurves; the Boy's surface halfwaymodel intersects itself in a
"propeller" curve, which then gets replaced by its fourfold cover
as we separate the two sheets.
Acknowledgments
The minimax sphere eversions described here are work
done jointly in collaboration with
Rob Kusner, Ken Brakke, George Francis, and Stuart Levy,
to whom I owe a great debt.
I would also like to thank Francis, Robert Grzeszczuk,
John Hughes, AK Peters, Nelson Max and Tony Phillips
for permission to reproduce figures from earlier sphere eversions.
This paper first appeared in the proceedings of
two summer 1999 conferences on Mathematics and Art:
ISAMA 99 (June, San Sebastián, Spain),
and Bridges (July, Kansas).
I wish to thank Nat Friedman, Reza Sarhangi, and Carlo
Séquin for the invitations to speak at these conferences.
I thank Slavik Jablan for formatting
this online web version of the paper.
My research is supported in part by NSF grant DMS9727859.
REFERENCES
