On the existence of $RR$-polyhedra associated with the icosahedron

V. I. Subbotin

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 253-264.

Résumé

The work refers to the direction in the theory of polyhedra in $ E ^ 3 $, in which classes of convex polytopes are studied that extend the class of regular (Platonic) polyhedra: polyhedra of such classes retain only some properties of regular polyhedra. Earlier, the author found new classes of polyhedra united by such symmetry conditions under which the conditions for the regularity of the faces were not assumed in advance. At the same time, the completeness of the lists of the considered classes was proved. Further, the author considered the class of so-called $ RR $ -polyhedra. A $ RR $-polyhedron (from the words rombic and regular) is a convex polyhedron that has symmetric rhombic vertices and there are faces that do not belong to any star of these vertices; moreover, all faces that are not included in the star of the rhombic vertex are regular polygons. If a faceted star $ Star (V) $ of a vertex $ V $ of a polyhedron consists of $ n $ equal and equally spaced rhombuses (not squares) with a common vertex $ V $, then $ V $ is called rhombic. If the vertex $ V $ belongs to the axis of rotation of the 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 $ converge at their obtuse angles. An example of an $RR$-polyhedron is an elongated rhombododecahedron. Previously, the author found all $ RR $-polyhedra with two symmetric rhombic vertices. In this paper, we consider the question of the existence of closed convex $ RR $-polyhedra in $ E ^ 3 $ with one symmetric obtuse rhombic vertex and regular faces of the same type. A theorem is proved that there are only two such polyhedra, a $13$-faced and a $19$-faced. Both of these polyhedra are obtained from the regular — icosahedron. The proof of the existence of a $19$-hedron is based, in particular, on A.D. Aleksandrov's theorem on the existence of a convex polyhedron with a given unfolding.

Détail

Keywords: symmetric rhombic vertices, $ RR $-polyhedron, rhombic vertex star, unfolding.

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