Concave deltahedral rings based on the geometry of the concave antiprisms of the second sort

Abstract

We have shown that there is a link between the geometry of the CA II-nM’s with bases n∈{3, 4, 5} and that of the convex antiprisms with the same bases. An integer number (K) of CA II-nM’s fragments, can be used to form a full multilaterally symmetrical ring of concave deltahedral surfaces, either flower-like (case A) or star-like (case B). The obtained rings can also be termed “of the second sort” (denoted by CDR II-n) as they inherit from the given CA II-nM the following: a) the linear and angular measurements needed for their graphic and mathematical elaboration, b) two rows of equilateral triangles in the lateral surface, and c) the high level of symmetry. The possible formation of CDR II-n’s with the highest level of symmetry (i.e. excluding the cases A), and with the number of petals/star-points in which any integer K ≥ 2 can be a subject of further research.