Improved packing of equal circles on a sphere and rigidity of its graph

Abstract

How must n equal non-overlapping circles be packed on a sphere so that the angular diameter of the circles will be as great as possible? In the paper, the conjectured solutions of this problem for n = 18, 27, 34, 35, 40 are improved on the basis of an idea of Danzer. Using the theory of bar structures it is ascertained that, in these cases, the edge-length of the graphs of the circle-packings can be increased till, in the graphs, additional edges appear which prevent further motions apart from rigid motions. The cases of n = 17 and 32 are also dealt with and there are references to the possibilities of further applications of the method applied in this paper (n = 59,80,110,122).

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

“…In this paper we use the same terms and relationships of the theory of bar structures as in [27]. For the sake of convenience of the reader we present the most important ones here also.…”

“…The edges of the graph of a covering by circles do not intersect [21]. Therefore, due to proposition (5l) of [27] the graph of a covering of the sphere by circles cannot be infinitesimally rigid. If the graph can have a state of self-stress, then the graph is rigid, that is, has no finite free motion (but still has infinitesimal free motion).…”

“…Let us denote this submatrix by G. The properties of matrix G are identical to that of the compatibility matrix of structures belonging to class (iii). Therefore, for G, the statements in §4 (iii) of [27] are valid, but instead of an increase, a decrease of temperature should be taken into account. After executing the abovementioned modification in (3) we can determine ds.…”

“…As mentioned in the Introduction the problem of packing n equal circles on a spherical surface and the problem of covering a sphere by n equal circles are dual counterparts of each other. This is also expressed by the fact that for the packing problem a heating algorithm [27], while for the covering problem a cooling algorithm can be used. For packing we have the modified Robinson conjecture [31]: CONJECTURE 1.…”

“…The graph of a covering of the sphere by circles may be considered theoretically as a cable net, but it turns out in practice to be more convenient to investigate a net of bars instead. Thus, on the analogy with spherical circle packings [27] the graph of a covering is modelled as a structure consisting of straight bars and frictionless pin joints (of types (A) and (B)) sliding on the surface of the sphere of a fixed diameter; and in the analysis the theory of bar structures [24] is applied. Change in length of the bars is attributed to a change in 'temperature'.…”