Concentric stacking of symmetrical sets of equal spheres

Abstract

If we are given a type of symmetry and, in accordance with it, on one given sphere we place equal spheres in layers at the distance of their centers from the center of a given sphere and, moreover, in such a way that these spheres fit into the recesses between the previous ones and form a regular set, then the number of spheres of the layer will be quite definite, namely, it will be equal to the value 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, neither coincides with the axes of symmetry, nor is in the planes of symmetry), and will a certain divisor of this number in particular cases. I will consider three sets of spheres of hexakisoctahedral type of symmetry, corresponding to the three possible systems of parallelohedrons in this case: triparallelohedrons, hexaparallelohedrons and heptaparallelohedrons, as well as a set of dihexonic-bipyramidal type of symmetry (and a system of tetraparallelohedrons).