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Covering a sphere by equal circles, and the rigidity of its graph

Published online by Cambridge University Press:  24 October 2008

T. Tarnai  and
Zs. Gáspár
T. Tarnai
Affiliation:
University of Cambridge, Engineering Department, Trumpington Street, Cambridge CB2 1PZ
Zs. Gáspár
Affiliation:
Research Group for Applied Mechanics, Hungarian Academy of Sciences, Budapest, Műegyetem rkp. 3, H-1521, Hungary
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Abstract

How must a sphere be covered by n equal circles so that the angular radius of the circles will be as small as possible? In this paper, conjectured solutions of this problem for n = 15 to 20 are given and some sporadic results for n > 20 (n = 22, 26, 38, 42, 50) are presented. The local optima are obtained by using a ‘cooling technique’ based on the theory of bar-and-joint structures. Thus the graph of the coverings by circles is considered as a spherical cable net in which the edge lengths are uniformly decreased, e.g. due to a uniform decrease in the temperature, until the graph becomes rigid and tensile stresses appear in the cables.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 71 - 89
Copyright
Copyright © Cambridge Philosophical Society 1991

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