Etymologie, Etimología, Étymologie, Etimologia, Etymology
US Vereinigte Staaten von Amerika, Estados Unidos de América, États-Unis d'Amérique, Stati Uniti d'America, United States of America
Topologie, Topología, Topologie, Topologia, Topology

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

Im Unterschied zu realen Kugeln (Sphären) können die Topologen Sphären mit idealen Eigenschaften definieren. Und so einigten sie sich darauf, dass eine Sphäre eine dünne Folie sein soll, die eine Kugeloberfläche ohne Löcher bildet, die sich jedoch beliebig dehnen läßt und sich selbst durchdringen kann - und zwar ohne zu reißen. Die Aufgabe bestand nun darin eine Kugel bestehend aus dieser idealen Folie umzustülpen ohne dabei scharfe Kanten oder Knicke zu verursachen. Mit realen Kugeln ist dies nicht möglich, aber bei idealen "Folien-Kugeln" der Topologen sollte dies möglich sein. Davon waren die Topologen überzeugt. Der amerikanische Mathematiker Stephen Smale hatte dies im Jahr 1959 bewiesen.

Der Mathematiker und Grafikexperte Nelson Max stellte im Jahr 1977 einen Film her, mit dem Titel "Turning a Sphere Inside Out". Theoretische Arbeiten dazu stammten von Bernard Morins, einem blinden französischen Topologen.

Im Jahr 1990 entdeckten Mathematiker einen Weg, einen "geometrisch optimalen Pfad", der die Energie minimiert, die beim Umstülpen einer Sphäre benötigt wird. Ein kleiner Film, der diese optimale Umstülping veranschaulicht ist mittlerweile unter der Bezeichnung "The Optiverse" verfügbar.

Ich muß gestehen: Selbst durch die optische Darstellung kann ich mir die knickfreie Umstülpung einer idealisierten Kugel nicht vorstellen.

Die Bezeichnung "Optiverse" (etwa "Sichtbare Wendung") setzt sich dabei zusammen aus lat. "optica", griech. "optike" = dt. "das Sehen betreffend", griech. "optikós" = dt. "optisch" und lat. "versus" = dt. "Umwenden" zu lat. "vertere" = dt. "wenden", "drehen".

Zur großen Familie von lat. "vertere" zählen auch "Aversion", "Kontroverse", "Konversation", "Konvertit", "pervers", "Prosa", "Revers", "universal", "Universität", "Universum", "versiert", "Version".

Über die Wurzel ide. "*uer-" = dt. "drehen", "biegen", "winden", "flechten" findet man als weitere Verwandte "-wärts", "Gewürm", "reiben", "renken", "Rist", "unwirsch", "verwirren", "wahren", "wehren", "werfen", "Werk", "wert", "Wirtel", "wringen", "Wurf", "Wurm" (engl. "worm"), "wurmen", "Wurst", "Würde", "Würfel", "würgen".

(E?)(L?) http://arxiv.org/pdf/math/9905020v2.pdf

“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.
...


(E?)(L?) http://www.gogeometry.com/videos/computer_graphic_optiverse.htm

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.


(E?)(L?) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.137.9166

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.
...


(E?)(L?) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.9313

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.
...


(E?)(L?) http://link.springer.com/chapter/10.1007%2F978-3-662-04909-9_22

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.
...


(E?)(L?) http://torus.math.uiuc.edu/jms/Papers/isama/

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


(E?)(L?) http://torus.math.uiuc.edu/jms/Papers/isama/bw/

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


(E?)(L?) http://torus.math.uiuc.edu/jms/Papers/isama/color/

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


(E?)(L?) http://www.youtube.com/watch?v=cdMLLmlS4Dc

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.


(E?)(L?) http://youtu.be/cdMLLmlS4Dc




Erstellt: 2014-04

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