Abstract
The problem treated here is: how must N antipodal pairs of equal circles (spherical caps) of given angular radius r be arranged on the surface of a sphere so that the area covered by the circles will be as large as possible? Conjectured solutions of this problem for N = 4,5,7 are given where r varies from the maximum antipodal packing radius to the minimum antipodal covering radius. These cases exhibit sequences of symmetry– and edge–number transitions in the contact polyhedra, which, in general, differ from those of the unconstrained problem where centrosymmetry is not enforced. New minimum antipodal covering arrangements are conjectured for 4, 5 and 7 pairs: they are a hexagonal bipyramid, a D 2h deltahedron and the cube+octahedron compound, respectively.
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Schedule a callMore From: Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
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