SURFACES IN THE STYLE OF J. STEINER ET AL.

Wilhelm M. Pieper (Germany)

Received December 24, 2007

Abstract

The surfaces of J. Steiner, E. Kummer, E. Goursat et al. are invariant against the interchange of the variables x, y and z in their defining implicit functions  They show the hexahedral symmetry of a cube. This article describes a method to generalize them to the symmetries of other Platonic bodies, i.e., to tetrahedral up to icosahedral symmetries and displays the surfaces.

Keywords and phrases: implicit surfaces, surface of Steiner, surface of Kummer, surface of Goursat, Platonic bodies, central planes, polynomials of higher order, radial projection.