András Lengyel

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.

On the Roundness of Polyhedra for Soccer Ball Design

Most soccer balls are made of stitched leather or synthetic flat panels with a bladder inside. The initial flat configuration is represented by polyhedra in 3-space. This paper studies polyhedra related to different symmetry groups in order to find the optimal topology and the optimal dimensions for soccer ball design. A number of polyhedra obtained from regular ones by truncation are investigated. Two mathematical quantities are introduced to measure the sphericity of the ball. They are surface integrals of expressions of the coordinates: the first one expresses moments around the origin of the coordinate system, and the other measures the deviation of a surface from the perfect sphere. We set up a ranking for different ball designs and the results are compared to those of previous studies in this field. Our mathematical approach is applicable to inflated balls with curved surface.

The Roundest Polyhedra with Symmetry Constraints

Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132.