On the completenes of the list of convex RR-polyhedra

Abstract

The article gives proof of the completeness of the list of one class of convex symmetric polyhedra in three-dimensional Euclidean spac. This class belongs to the class of so-called RR-polyhedra. The RR-polyhedra are characterized by the following symmetry conditions: each polyhedron of the class RR has symmetric rhombic vertices and there are faces that do not belong to any star of these vertices; and each face that does not belong to the star of the rhombic vertex is regular. The vertex rhombicity here means that the vertex star is composed of n equal, equally spaced rhombuses. The symmetry of the vertex means that the rotation axis of the order n of its star passes through it. Previously, the author found all polyhedra with rhombic or deltoid vertices and locally symmetric faces. Moreover, locally symmetric faces do not belong to any of the rhombic or deltoid stars. The class of RR-polyhedrons is obtained from the previously considered replacement of the local symmetry condition of the non-rombic faces by the condition of their regularity.

Thus, the considered class RR is connected with the well-known result of D. Johnson and V. Zalgaller on the enumeration of all convex polyhedra with the condition of the regularity of the faces. But as shown in this article, RR-polyhedra cannot simply be obtained from the class of regular-faced, but require a special method. The proof of completeness of the class of RR-polyhedra with two isolated symmetric rhombic vertices V, W is given in this article. Wherein, the rhombuses converge at the vertices of V, W not necessarily at its acute angles, and V, W are not necessarily separated by only one belt of regular faces.