On Proofs of the Existence of Non-Composite Polyhedra with Rhombic Vertices

Subbotin, V. I.
Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 2.

Abstract:
Previously, the author considered closed convex polyhedra in \(E^3\) with symmetrical rhombic vertices and regular faces, the so-called \(RR\)-polyhedra, in two cases: 1) regular faces of such polyhedra are of the same type, 2) regular faces are of different types, but the polyhedra are composite. From the author's previously published works, it follows that in both cases the list of \(RR\)-polyhedra is complete. In the first case, there were twenty-four of them, and in the second, fifty-four. A vertex \(V\) of a polyhedron is called rhombic if the vertex star \(Star(V)\) consists of \(n\) equal and identically located rhombi (not squares) having a common vertex \(V\). If the vertex \(V\) lies on the rotation axis of order \(n\) of the star \(Star(V)\), then \(V\) is called symmetric. A symmetric rhombic vertex \(V\) is called obtuse-angled if the rhombi of the star \(Star(V)\) at the vertex \(V\) meet at their obtuse angles. The star \(Star(V)\) is then called symmetric rhombic. The faces of an \(RR\)-polyhedron can be divided into two non-empty disjoint sets: the set of equal symmetric rhombic stars that do not have common edges, and the set of regular faces. An example of an \(RR\)-polyhedron is an elongated rhombododecahedron. If an \(RR\)-polyhedron has regular faces of one type, then such an \(RR\)-polyhedron is classified as the first type. Some regular-faced polyhedra with conditional edges, such as the extended rhombododecahedron and the doubly complemented cuboctahedron, are special cases of \(RR\)-polyhedrons. In this case, the rhombic faces of regular-faced polyhedra with conditional edges, unlike the rhombi of \(RR\)-polyhedra, are composed only of regular triangles. In this paper, constructive proofs are given for the existence of four non-composite \(RR\)-polyhedra. For two of them the author previously found the angles of the rhombi that form symmetrical faceted stars of the vertices. In the case of the 25-sided polyhedron, the rhombic vertex has an acute-angled faceted star, i.e. the rhombi in it meet at their acute angles. And in the case of a 31-sided polyhedron the faceted star is obtuse. The existence of two other non-composite \(RR\)-polyhedra is proven on the basis of the regular dodecahedron and the cube.

Keywords: non-composite RR-polyhedron, faceted vertex star, symmetrical rhombic vertex.

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