From spherical circle coverings to the roundest polyhedra

Abstract

The problem treated here is: amongst the convex polyhedra that can be circumscribed about the unit sphere and have faces, which has the minimum surface area? A new optimization method based on mechanical analogies is worked out to solve this problem. By using this method, new computer-generated solutions are presented for and . The second of these two conjectured roundest polyhedra has icosahedral symmetry. The relation of the results of this problem to the minimum coverings of the sphere with equal circles is discussed.

Keywords:

• packing
• covering
• isoperimetric problem
• polyhedra
• tensegrity structures
• mechanical models

Acknowledgments

The research reported here was supported by the OTKA grant no. K81146 awarded by the Hungarian Scientific Research Fund. We thank Prof. P. W. Fowler for many valuable comments and suggestions.