Geodesic
Symmetric polyhedra with rhombic vertices
Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 87-93 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Closed convex polyhedra in three-dimensional Euclidean space, some vertices of which are simultaneously isolated, symmetric and rhombic are considered in this paper. The rhombicity of the vertex means that all the faces of the polyhedron incident to this vertex are $n$ rhombi equal to each other. The symmetry of a vertex means that it is located on a nontrivial rotation axis of order $n$ of the polyhedron. Taking into account that the set of all rhombi of a vertex $P$ is called a rhombic star of a vertex $P$, the isolation of a vertex $P$ means that its rhombic star has no common points with rhombic stars of other vertices of a polyhedron. Suppose that in a polyhedron there are also faces $F_i$ that do not belong to a single rhombic star, and each of $F_i$ has a rotation axis, which is the local axis of rotation of a star of this face. Polyhedra with such conditions are called in the paper $RS$-polyhedra (from the first letters of the words rombic, symmetry). $RS$-polyhedrons are related to polyhedra that are strongly symmetric with respect to rotation. Polyhedra, strongly symmetric with respect to rotation were previously introduced and are completely listed by the author; they are a generalization of the class of regular (Platonic) polyhedra. We note that among strongly symmetric polyhedra there are seven such that are not combinatorically equivalent to either regular or equilateral semiregular (Archimedean). In the present paper, all $RS$-polyhedrons are found. It is shown that some of them are related to parallelohedra in three-dimensional Euclidean space. 
@article{VMJ_2018_20_3_a7,
     author = {V. I. Subbotin},
     title = {Symmetric polyhedra with rhombic vertices},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {87--93},
     year = {2018},
     volume = {20},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a7/}
}
TY  - JOUR
AU  - V. I. Subbotin
TI  - Symmetric polyhedra with rhombic vertices
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2018
SP  - 87
EP  - 93
VL  - 20
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a7/
LA  - ru
ID  - VMJ_2018_20_3_a7
ER  - 
%0 Journal Article
%A V. I. Subbotin
%T Symmetric polyhedra with rhombic vertices
%J Vladikavkazskij matematičeskij žurnal
%D 2018
%P 87-93
%V 20
%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a7/
%G ru
%F VMJ_2018_20_3_a7
V. I. Subbotin. Symmetric polyhedra with rhombic vertices. Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 87-93. http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a7/

[1] Coxeter H. S. M., Regular Polytopes, Dover, N. Y., 1973 | MR

[2] Subbotin V. I., “On a Class of Strongly Symmetric Polyhedra”, Chebyshevskii Sbornik, 17:4 (2016), 132–140 (in Russian) | DOI | MR | Zbl

[3] Subbotin V. I., “Strongly Symmetric Polyhedra”, Journal of Mathematical Sciences, 131:1 (2005), 5438–5444 | DOI | MR | Zbl

[4] Subbotin V. I., “Some Generalizations of Strongly Symmetric Polyhedra”, Chebyshevskii Sbornik, 16:2 (2015), 222–230 (in Russian) | MR

[5] Timofeenko A. V., “To the list of convex polyhedra with regular faces”, Modern Problems of Mathematics and Mechanics, 6, no. 3, 2011, 155–170 (in Russian)

[6] Emelichev V. A., Kovalev M. D., Kravtsov M. K., Polyhedra. Graphs. Optimization, M., 1981, 344 pp. | MR

[7] Ziegler G. M., Lectures on Polytopes, New York, 1995, 373 pp. | DOI | MR

[8] Artamonov V. A., “Quasicrystals and their symmetries”, Journal of Mathematical Sciences, 139:4 (2006), 6657–6662 | DOI | MR | Zbl

[9] Alexandrov A. D., Convex Polyhedra, New York, 2005, 524 pp. | DOI | MR | MR