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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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.
The minimal surface inverted in the unit sphere
These many images show the inversion of Kusner's minimal surface which sends the isthmus point (the double tangent point) to infinity and leaves the quadruple point at the origin. This surface has two wavy ends with cos(2t) behavior.