## "The Optiverse" and Other Sphere Eversions

## John M. Sullivan

Dept. of Mathematics,
Univ. of Illinois, Urbana

*jms@math.uiuc.edu*
For decades, the sphere eversion has been a classic subject for
mathematical visualization. Our 1998 video
"The Optiverse"
shows geometrically optimal eversions created by minimizing elastic bending
energy. This paper contrasts these minimax eversions with earlier ones,
including those by Morin, Phillips, Max, and Thurston. Our minimax
eversions were automatically generated by flowing downhill in
energy using Brakke's Evolver.

This paper,
*"The Optiverse" and Other Sphere Eversions*,
is available here in HTML in two versions:

(The color version was formatted by
Slavik Jablan for his online journal
VisMath, which is available from the
US
or from Yugoslavia.)
A printable version (typeset in LaTeX)
is available (in many formats) as ArXiv e-print
math.GT/9905020.

An updated LaTeX version (with a few minor corrections) was
prepared for publication by Springer in the proceedings
of the Sep'00 Maubeuge Colloquium on Math and Art.
It is now available here as a PDF file.
This version has the highest-resolution images.
The book *Mathematics and Art:
Mathematical Visualization in Art and Education*, edited
by Claude P. Bruter, will be published in June 2002.

This paper
was first published in the proceedings of two summer 1999 conferences
on mathematics and art:

*ISAMA 99*, Univ. Basque Country, 1999, pp 471-479.
*Bridges 1999*, Southwestern Coll., Kansas, 1999, pp 265-274.