Selected Images from The Optiverse

The following images are stills taken from The Optiverse, a video by John Sullivan, George Francis, and Stuart Levy, about the minimax sphere eversion.

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The 4-fold eversion

The 4-fold eversion has four fingers at an early stage. Later, they overlap. At the halfway stage, we have an octuple point for an instant. The second half of the eversion is like the first with time reversed and the other side of the surface showing.
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.

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.

Near the end of the 3-fold eversion, three fingers retract to form a gastrula which then moves to a round sphere.