Symmetric Polyhedra with Rhombic Vertices

Subbotin, V. I.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 3.

Abstract:
Closed convex polyhedra in three-dimensional Euclidean space, some vertices of which are simultaneously isolated, symmetric and rhombic are considered in this paper. The rhombicity of the vertex means that all the faces of the polyhedron incident to this vertex are \(n\) rhombi equal to each other. The symmetry of a vertex means that it is located on a nontrivial rotation axis of order \(n\) of the polyhedron. Taking into account that the set of all rhombi of a vertex \(P\) is called a rhombic star of a vertex \(P\), the isolation of a vertex \(P\) means that its rhombic star has no common points with rhombic stars of other vertices of a polyhedron. Suppose that in a polyhedron there are also faces \(F_i\) that do not belong to a single rhombic star, and each of \(F_i\) has a rotation axis, which is the local axis of rotation of a star of this face. Polyhedra with such conditions are called in the paper \(RS\)-polyhedra (from the first letters of the words rombic, symmetry). \(RS\)-polyhedrons are related to polyhedra that are strongly symmetric with respect to rotation. Polyhedra, strongly symmetric with respect to rotation were previously introduced and are completely listed by the author; they are a generalization of the class of regular (Platonic) polyhedra. We note that among strongly symmetric polyhedra there are seven such that are not combinatorically equivalent to either regular or equilateral semiregular (Archimedean). In the present paper, all \(RS\)-polyhedrons are found. It is shown that some of them are related to parallelohedra in three-dimensional Euclidean space.

Keywords: strongly symmetrical polyhedron, rhombic vertex, \(RS\)-polyhedron, \(TE\)-transformation, parallelohedron.

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