Concentric packing of symmetrical sets of equal spheres
Abstract
If we set a type of symmetry and, in accordance with it, place equal spheres on a given sphere in layers according to the distance of their centers from the center of the given sphere and, in such a way that these spheres fit into the recesses between the previous spheres and form a regular set, then the number of spheres in the layer will be quite definite, namely, it will be equal to the multiplicity of symmetry in the general case, when the direction of the radius vector of each such sphere (starting from the center of the initial sphere) will be common (that is, it will neither coincides with the axes of symmetry, nor lie in the planes of symmetry), and will be a definite divisor of this number in particular cases. I shall consider three sets of spheres of the hexakisoctahedral type of symmetry, corresponding respectively to the three systems of parallelohedra possible in this case: triparallelohedra, hexaparallelohedra and heptaparallelohedr, as well as a set of dihexonal-bipyramidal type of symmetry (and a the system of tetraparallelohedra).
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