On Composite \(RR\)-Polyhedra of the Second Type

Subbotin, V. I.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 1.

Abstract:
In classical and modern geometry, the problem of classifying polyhedra in \(E^3\) on the basis of the symmetry properties of the polyhedron elements is topical. The first examples of such a classification are five regular (Platonic, more precisely, Pythagorean) polyhedra, i.\,e. equiangular-semiregular (Archimedean) polyhedra. The class of equi-semiregular polytopes is characterized by the fact that all its faces are regular polygons and the symmetry group of the polytope is transitive at its vertices. Among the examples of nonconvex polytopes, one can single out four regular stellated Kepler-Poinsot polyhedra, the completeness of the list of which was proved by O. Cauchy. Among the numerous modern generalizations and developments of the above examples, we indicate a class consisting of ninety-two closed convex polyhedra in \(E^3\), whose faces are regular polygons of various types (Johnson--Zalgaller polytopes). In this paper, the author continues the study of \(RR\)-polyhedra: a complete list of composite \(RR\)-polyhedra of the second type is found. A \(RR\)-polyhedron (from the words "rhombic" and "regular") is a closed convex polyhedron in \(E^3 \), the set of faces of which can be divided into two nonempty disjoint classes - the class of faces that form faceted stars of symmetric rhombic vertices and a class of regular faces; if the regular faces of such a polyhedron are of the same type, then we will refer it to the first type; if different, to the second type of \(RR\)-polyhedra. If the star of the vertex \(V\) of the polyhedron consists of equal and equally spaced, i.e. converging at the vertex \(V\) either by their acute or obtuse angles of rhombuses (not squares), then the vertex \(V\) will be called rhombic. If the vertex \(V\) is located on such an rotation axis of the star that the order of the axis coincides with the number of rhombuses in the star, then \(V\) is called a symmetric rhombic vertex. Earlier, the author found twenty-three \(RR\)-polyhedra of the first type and proved the completeness of the list of such polyhedra.

Keywords: \(RR\)-polyhedron, composite polyhedron of the second type, symmetric rhombic vertex

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