Selected Images from The Optiverse
The following images are stills taken from
a video by John Sullivan, George Francis, and Stuart Levy,
about the minimax sphere eversion.
KUSNER'S MINIMAL SURFACE
These images show a minimal surface with four flat ends.
Applying a Möbius transformation (along the z-axis)
to this surface gives the halfway model for the 2-fold eversion;
the point at infinity becomes the quadruple point.
This image is from the flythrough scene, showing interior views
of the 2-fold eversion around the time the triple points appear.
These images show the 2-fold eversion in an early "gastrula" stage,
at a stage between the creation of the first and second double curves,
and finally almost at the halfway stage.
These images all show the 2-, 3-, 4-, and 5-fold eversions
in the upper left, lower left, upper right, and lower right cornerse,
respectively. First we see an early stage, with p fingers
growing in the p-fold everion. Next we see an intermediate
stage when the fingers have mostly overlapped. Finally we see the
four halfway models. For p odd, these are doubly covered
THE 3-FOLD EVERSION
The halfway model for the 3-fold eversion the double cover of a projective
plane, a Boy's surface which minimizes Willmore bending energy.
The fact that two oppositely oriented sheets overlap means the computer
graphics system randomly picks which color to draw (left).
Just after the halfway stage, the propellor-shaped double locus of
Boy's surface breaks apart into a four-fold cover (since the two
sheets of surface have pulled apart). Where Boy's surface had
a triple point, we see a small growing cube in the double locus (right).
Near the end of the 3-fold eversion, three fingers retract to
form a gastrula which then moves to a round sphere.
These two views show the gastrula stage of the 2-fold
eversion from inside--the second looks almost
along the symmetry axis.
These pictures view the 2-fold eversion from inside, looking up along
the symmetry axis, with a wide-angle lens. First we see the stage just after
creation of the second double curve. Then we see
a stage shortly after the creation of two pairs
of triple points (towards the upper right and lower
These images show the 2-fold eversion, first in
the gastrula stage (left), then between the times
the first and second double curves appear (middle),
and finally at the halfway model (right).