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Symmetry 2017, 9(3), 41; doi:10.3390/sym9030041

The Roundest Polyhedra with Symmetry Constraints

1
Department of Structural Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary
2
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Received: 5 December 2016 / Revised: 3 March 2017 / Accepted: 8 March 2017 / Published: 15 March 2017
(This article belongs to the Special Issue Polyhedral Structures)
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Abstract

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. View Full-Text
Keywords: polyhedra; isoperimetric problem; point group symmetry polyhedra; isoperimetric problem; point group symmetry
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Lengyel, A.; Gáspár, Z.; Tarnai, T. The Roundest Polyhedra with Symmetry Constraints. Symmetry 2017, 9, 41.

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