On the Roundness of Polyhedra for Soccer Ball Design

Abstract

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.

Keywords

Soccer balls polyhedra optimal dimensions roundness criteria.

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