Derivation of one of the fundamental formulas of the doctrine of symmetry

Abstract

The formula in question here is a logical consequence of two already known formulas, which were given in full detail by the author of the doctrine of symmetry, specifically in the section published under the title “Symmetry of Finite Figures.” The formula, applicable to any group of symmetry axes (type of alignment symmetry), though, of course, not applicable to a single axis taken separately, makes it possible to directly derive the order of symmetry from the number of symmetry axes. From it, incidentally, it follows that the order of a symmetry operation must be even (which is understandable given the obligatory presence of twofold axes of symmetry in the aggregates), and therefore the order of symmetry of those types where, in addition to the axes of symmetry, elements of reflection symmetry are present, must necessarily be divisible by four.

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