On the existence of 𝑅𝑅-polyhedra associated with the icosahedron
https://doi.org/10.22405/2226-8383-2021-22-4-253-264
Abstract
The work refers to the direction in the theory of polyhedra in 𝐸3, 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 𝐸3 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.
About the Author
Vladimir Ivanovich Subbotin
Don State Agrarian University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate
professor
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Review
For citations:
Subbotin V.I. On the existence of 𝑅𝑅-polyhedra associated with the icosahedron. Chebyshevskii Sbornik. 2021;22(4):253-264. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-4-253-264
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