On the enumeration of convex $RR$-polyhedra

V. I. Subbotin

Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 194-207.

Résumé

The problem of enumerating a class of polyhedra with given symmetry conditions is one of the important problems of the modern theory of convex polyhedra. There are many works in which the problem of a complete enumeration of polyhedra with symmetry conditions is posed. If we limit ourselves to polyhedra in $E^3$, then examples of such polyhedra are regular, regular stellate, Archimedean polyhedra, the Johnson-Zalgaller class, polyhedra with conditional edges, and polyhedra with parquet faces. Specifically, the symmetry conditions for the class of closed convex regular polyhedra consist in the conditions that the polyhedron's equal faces are regular and its vertices are of the same type. For Johnson-Zalgaller polyhedra — under the condition that the faces of a closed convex polyhedrons are regular. It is known that the last class, in addition to regular and Archimedean polyhedra, is an infinite series prisms and antiprisms, contains 92 polyhedra with regular faces. Previously, the author found new classes of polyhedra (for example, the so-called polyhedra that are strongly symmetric with respect to rotation); they have such a symmetry of elements in which the conditions for the regularity of the faces are not presupposed. At the same time, the completeness of the lists of the considered classes was proven. Returning to such symmetry conditions, which include the conditions of regularity of faces, the author introduced a class of closed convex $RR$-polytopes (from the words rhombic and regular). Briefly, this class is defined by the following symmetry conditions. The faces of an $RR$-polytope can be divided into two non-empty disjoint sets — a set of equal symmetrical rhombic stars that do not have common edges, and a set of regular faces. Moreover, a vertex $V$ is called rhombic if the faceted star $Star(V)$ of a vertex $V$ of the polyhedron consists of $n$ equal and identically spaced rhombuses (not squares) having a common vertex $V$. If the vertex $V$ belongs to the rotation axis of order $n$ of the star $Star(V)$, then $V$ is called symmetric. A symmetric rhombic vertex $V$ is called obtuse if the rhombuses of the star $Star(V)$ at the vertex $V$ meet at their obtuse angles. An example of a $RR$-polyhedron is the elongated rhombic dodecahedron. This paper provides a modified proof of the theorem on the existence and uniqueness of a closed convex $RR$-polyhedron associated with the icosahedron and proves the existence of a twenty-fourth $RR$-polyhedron with triangular faces and four obtuse rhombic vertices. Theorems on the non-existence of certain polyhedra with regular faces of various types, “close” to $RR$-polytopes, have also been proven.

Détail

Keywords: symmetry conditions, rhombic vertices, $ RR $-polyhedron, rhombic vertex star.

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