Hyperseeing, Hypersculptures, Knots, and Minimal Surfaces
by
Nat Friedman
University at Albany (New York)

In order to see a two-dimensional painting on a wall, we step back in a third dimension. We can then see the painting completely from one viewpoint. By analogy, in order to see a three-dimensional sculpture from one viewpoint would theoretically require stepping back in a fourth spatial dimension. Four-dimensional space is hyperspace and I refer to seeing in hyperspace as hyperseeing. Thus theoretically one could hypersee a three-dimensional object completely from one viewpoint in hyperspace. Cubist painters such as Picasso applied hyperseeing in order to show objects from multiple viewpoints in the same painting.

Although we do not live in hyperspace, it is still desirable to attain a type of hyperseeing. This is facilitated by viewing a hypersculpture defined as follows. First a sculpture is defined as an object in a given orientation relative to a fixed horizontal plane (the base). Two sculptures are said to be related if they consist of the same object. Note that related sculptures may look quite different. A HYPERSCULPTURE is a set of related sculptures. Thus viewing a hypersculpture allows one to see multiple views from one viewpoint, which therefore helps to develop a type of hyperseeing in our own three-dimensional world. In general, I refer to HYPERSEEING as an all-around seeing from multiple viewpoints. To hypersee a sculpture requires walking around it and integrating the multiple views so that one can gain a more complete understanding of the three-dimensional structure of the sculpture. Examples of hypersculptures by Arthur Silverman and Charles Ginnever will be presented in slides. A three-dimensional model of a knot can look completely different from different viewpoints. Thus knots are excellent examples on which to practice hyperseeing. Examples of knots and their corresponding soap film minimal surfaces will be shown. In particular, the n-(n+1) torus knot has a minimal surface consisting of n Mobius bands that alternately share edges and cross over each other, n = 2, 3, ...

Date received: September 12, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caek-13.