Printed in Great Britain Covering a sphere by equal circles, and the rigidity of its graph

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. 1