Etymologie, Etimología, Étymologie, Etimologia, Etymology, (griech.) etymología, (lat.) etymologia, (esper.) etimologio
US Vereinigte Staaten von Amerika, Estados Unidos de América, États-Unis d'Amérique, Stati Uniti d'America, United States of America, (esper.) Unuigintaj Statoj de Ameriko
Topologie, Topología, Topologie, Topologia, Topology, (esper.) topologio

Optiverse (W3)
oder
Wie kehrt man das Innere einer Kugel nach außen?

“THE OPTIVERSE” AND OTHER SPHERE EVERSIONS
JOHN M. SULLIVAN

Abstract. For decades, the sphere eversion has been a classic subject for mathematical visualization. The 1998 video "The Optiverse" shows geometrically optimal eversions created by minimizing elastic bending energy. We contrast these minimax eversions with earlier ones, including those by Morin, Phillips, Max, and Thurston. The minimax eversions were automatically generated by flowing downhill in energy using Brakke’s Evolver.

1. A History of Sphere Eversions

To evert a sphere is to turn it inside-out by means of a continuous deformation, which allows the surface to pass through itself, but forbids puncturing, ripping, creasing, or pinching the surface. An abstract theorem proved by Smale in the late 1950s implied that sphere eversions were possible [Sma], but it remained a challenge for many years to exhibit an explicit eversion. Because the self-intersecting surfaces are complicated and nonintuitive, communicating an eversion is yet another challenge, this time in mathematical visualization. More detailed histories of the problem can be found in [Lev] and in Chapter 6 of [Fra].

The earliest sphere eversions were designed by hand, and made use of the idea of a halfway-model. This is an immersed spherical surface which is halfway inside-out, in the sense that it has a symmetry interchanging the two sides of the surface. If we can find a way to simplify the halfway-model to a round sphere, we get an eversion by performing this simplification first backwards, then forwards again after applying the symmetry. The eversions of Arnold Shapiro (see [FM]), Tony Phillips [Phi], and Bernard Morin [MP] can all be understood in this way.
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Video Description: The Optiverse and other minimax sphere eversion

The Optiverse and other minimax sphere eversion is an example of computer graphics animation developed by students at the Electronic Visualization Lab ("EVL"). This animation was created by John Sullivan, George Francis and Stuart Levy, with an original soundtrack by Camille Goudeseune. Post Production was handled by Dana Plepys and Jeff Carpenter. Produced at: grafiXlav, Department of Mathematics, University of Illinois, Urbana. National Center for Supercomputing Applications (NCSA), Electronic Visualization Lab (EVL), University of Illinois, Chicago.

Sphere eversions: From Smale through "The Optiverse" (2002)
by John M. Sullivan

Abstract

For decades, the sphere eversion has been a favorite subject for mathematical visualization. The 1998 video "The Optiverse" shows minimax eversions, computed automatically by minimizing elastic bending energy using Brakke’s Evolver. We contrast these geometrically optimal eversions with earlier ones, including those by Morin, Phillips, Max, and Thurston.
...

Making the Optiverse: A mathematician’s guide to AVN, a real-time interactive computer animator (2003)
by George K. Francis , Stuart Levy , John M. Sullivan

Abstract

Abstract. Our 1998 video "The Optiverse" illustrates an optimal sphere eversion, computed automatically by minimizing an elastic bending energy for surfaces. This paper describes AVN, the custom software program we wrote to explore the computed eversion. Various special features allowed us to use AVN also to produce our video: it controlled the camera path throughout and even rendered most of the frames.
...

Sphere Eversions: from Smale through “The Optiverse”
John M. Sullivan

For decades, the sphere eversion has been a favorite subject for mathematical visualization. The 1998 video "The Optiverse" shows minimax eversions, computed automatically by minimizing elastic bending energy using Brakke’s Evolver. We contrast these geometrically optimal eversions with earlier ones, including those by Morin, Phillips, Max, and Thurston.
...

"The Optiverse" and Other Sphere Eversions
John M. Sullivan

"The Optiverse" and Other Sphere Eversions
John M. Sullivan

"The Optiverse" and Other Sphere Eversions
John M. Sullivan

Hochgeladen am 27.12.2007

(1998) This is an example of computer graphics animation developed by students at the Electronic Visualization Lab. This animation was created by John Sullivan, George Francis and Stuart Levy, with an original soundtrack by Camille Goudeseune. Post Production was handled by Dana Plepys and Jeff Carpenter.