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MATHÉMATICS & ARTS |
Exhibit & Lecture |
Initialy
presented at the Institut Henri Poincaré,
Paris. France. |
January
22 - June 30, 2005 |
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- BANCHOFF Thomas
- Mathematician
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- Thomas Banchoff received his Ph.D. from the University of California, Berkeley, in 1964 under the direction of Prof. Shiing-Shen Chern. He was a Benjamin Peirce Instructor for two years at Harvard and one year as a Research Associate at the University of Amsterdam before coming to Brown University in 1967. He has been collaborating with computer scientists since 1968 investigating visualizations of phenomena in three- and four-dimensional space. In 1978, his film "The Hypercube: Projections and Slicing" with Computer Scientist Charles Strauss was awarded the Prix de la Recherche Fondamentale at the International Festival of Scientific and Technical Films in Brussels, and he gave the first presentation of computer animated geometric films in an invited lecture at the International Congress of Mathematicians in Helsinki in 1978. In 2002 in Beijing at the International Congress of Mathematical Software he presented an interactive poster describing the progress in mathematical visualization over twenty-five years of research and teaching. This poster features original wire-frame images compared with their modern fully-rendered counterparts. The accompanying text available online describes the geometric phenomena associated with surfaces in four-space, as used in mathematical research and in presentations before general audiences and art galleries.
- http://www.math.brown.edu/TFBCON2003/art/wecome.html
- http://www.math.union.edu/~dpvc/professional/brief.html
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| The following images reactualised by Davide Cervone, were created by Huseyin Kocak, Fred Bisshopp, David Laidlaw, David Margolis, et Thomas Banchoff. |
Click thumbnail to enlarge picture |
Stereographic Projection of a Sphere
Stereographic projection starts from a point of a sphere (the center of projection) and projects the object on a plan in the following way: the circles passing by the point of projection become lines of the plan, the other circles traced on the sphere become circles of the plan. One can see on the sphere the circular colored bands and their projection on a level located under the sphere. Projections of a hollow torus (the surface of a mass in the shape of crown) which appear in In and Outside the Torus and Hopf-Link Torus are of the same type. |
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Pendulum Tori
The name owes to the fact that these tori can be used with the representation of the physical system known as "double pendulum": it consists of a second pendulum placed and balancing itself at the end of a first pendulum. For a given relationship between the lengths of the two pendulums, the various positions of the system correspond to the points of a fixed torus inside the family represented here, dependent on the constitution of the sphere in the 4 dimensional space . |
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In- and Outside the Torus
One can manufacture the sphere in 4 -dimension space starting from two full tori (two masses having each the shape of a crown), by joining them by their surface, the torus (hollow). This connecting hollow torus is located in a 4-dimension space. Its form is not very visible but better identified by his stereographic projections in the work space, the center of projection being located on this torus. Circles layouts (known as of Hopf) on this same torus are represented by colored bands. |
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Hopf links
A full torus can be broken up into fine plates of hollow tori of increasingly small rays, the final stage being a circle. The sphere of 4-dimension space can be designed as the association of two tori fully welded by the hollow torus in connection to their surface. The one represented here is stereographic projections in a 3-dimensional space of two hollow tori, one green pertaining to the one of the full tori, the other red pertaining to the other full torus. Intermediate hollow tori are represented by blue bands painted on these tori. |
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