Quantity of non-congruent struts in alternate division (Q265624)

The discretization of the \(2\)-dimensional sphere, in particular geodesic domes and Fuller's division of the regular polyhedra, has been studied extensively in the literature. However, the problem of determining how many groups of non-congruent edges there are for a given frequency of a given division is not so well documented. The main result of this article states necessary and sufficient conditions for the existence of congruent edges and the relation between the quantity of non-congruent edges and the frequency of the division of an equilateral triangle inscribed into the unit \(2\)-sphere. It is shown that the algebraicity of the side \(l\) of this triangle determines the possible existence of congruent edges. As a byproduct, an expression for the quantity of non-congruent struts/edges in terms of the frequency of the Fuller's alternate division for the regular icosahedron is obtained.