GII Testbed Applications on the I-WAY

14 July 1996 With powerful parallel computers networked to virtual environments, mathematicians can explore previously inaccessible problems in geometry. For example, in four dimensions a surface such as a topological sphere may be knotted. When it is merely tangled, mathematicians evolve it to its familiar round shape to show it unknotted. They guide surfaces toward optimality by minimizing mathematical abstractions of physical energies like the Coulomb potential or the bending energy of bilipid membranes. Mathematical surfaces in three dimensions (like shadows from four dimensions) generally self-intersect, but the Willmore bending energy can still be used to optimize their shape. A sphere can be turned inside out, keeping the surface smooth but allowing complex selfintersections. The researchers demonstrate this bypresenting for the first time a geometrically optimal and computationally automatic eversion of the sphere. http://new.math.uiuc.edu/laterna/