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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.




KUSNER'S MINIMAL SURFACE



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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.






FLYTHROUGH



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This image is from the flythrough scene, showing interior views of the 2-fold eversion around the time the triple points appear.





TRANSPARENT EVERSIONS



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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.



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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 projective planes.






THE 3-FOLD EVERSION



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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).

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





INTERIOR VIEWS



   


These two views show the gastrula stage of the 2-fold eversion from inside--the second looks almost along the symmetry axis.



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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 left).






MISCELLANY


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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).





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