On the completenes of the list of convex $RR$-polyhedra

V. I. Subbotin

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 297-309.

Résumé

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.

Détail

Keywords: symmetric rhombic vertex, the star of the vertex, the belt of regular faces, $ RR$-polyhedron.

Bibliographie

[1] Coxeter H. S., Regular polytopes, London–NY, 1963 | MR | Zbl

[2] Deza M., Grishukhin V. P., Shtogrin M. I., Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices, MCNMO, M., 2008

[3] Emelichev V. A., Kovalev M. M., Kravzov M. K., Polyhedra. Graph. Optimization, Nauka, M., 1981 | MR

[4] Cromwell P. R., Polyhedra, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[5] Grunbaum B., Convex Polytopes, John Wiley Sons, New York, 1967 | MR | Zbl

[6] Johnson N. W., “Convex polyhedra with regular faces”, Can. J. Math., 18:1 (1966), 169–200 | DOI | MR | Zbl

[7] Jurij Kovic, “Centrally symmetric convex polyhedra with regular polygonal faces”, Math. Commun., 18 (2013), 429–440 | MR | Zbl

[8] Zalgaller V. A., “Convex polyhedra with regular faces”, Zapiski nauchnych seminarov LOMI, 2, 1967, 1–220 | MR

[9] Timofeenko A. V., “On convex polyhedra with equiangular and parquet faces”, Chebyshevskiy sbornik, 12:2 (2011), 118–126 | MR | Zbl

[10] Gurin A. M., “On the history of studying convex polyhedra with regular faces”, Sib. electron. math. izv., 7 (2010), A5–A23

[11] Subbotin V. I., “On convex $RR$-polyhedrons with non-triangular faces”, Proc. Int. Conf. “Algebra, number theory and discrete geometry: modern problems, applications and problems of history” (Tula, 2019), 277–278

[12] Субботин В. И., “Об одном классе сильно симметричных многогранников”, Чебышевский сборник, 17:4 (2016), 132–140 | MR | Zbl

[13] Subbotin V. I., “Some generalizations strongly symmetric polyhedra”, Chebyshevskiy sbornik, 16:2 (2015), 222–230 | MR | Zbl

[14] Subbotin V. I., “On Completely symmetrical polyhedra”, Proc. Int. Conf. on discrete geometry and its applications (Moskow, 2001), 88–89

[15] Subbotin V. I., “Convex polyhedra with delthoidal vertices”, Chebyshevskiy sbornik, 19:2 (2018), 489–498

[16] Subbotin V. I., “Characterization of polyhedral partitioning a space”, Voronoy conference on analytic number theory and spatial tessellations (Kiev, September, 22–28 2003), 46

[17] Subbotin V. I., “Polyhedra with the maximum number of asymmetrical faces”, Proc. Int. Conf. "Metric geometry of surfaces and polyhedra" (Moskow, 2010), 60–61

[18] Subbotin V. I., “On symmetric polyhedra with asymmetrical faces”, Proc. Int. Seminar "Discrete Mathematics and Its Applications" (Moskow, 2012), 398–400

[19] Subbotin V. I., “On two classes of polyhedra with rhombic vertices”, Zapiski nauchnych seminarov POMI, 476, 2018, 153–164

[20] Subbotin V. I., “On polyhedra with symmetric rhombic vertices and regular faces”, Proc. Int. Conf. "Classical and modern geometry" (Moskow, 2019), 139–140

[21] Farris S. L., “Completely classifying all vertex-transitive and edge-transitive polyhedra.”, Geometriae Dedicata, 26:1 (1988), 111–124 | DOI | MR | Zbl

[22] Grunbaum B., “Regular polyhedra — old and new”, Aequationes mathematicae, 16:1–2 (1977), 1–20 | DOI | MR | Zbl

[23] Wills J. M., “On polyhedra with transitivity properties”, Discrete and Computational Geometry, 1:3 (1986), 195–199 | DOI | MR | Zbl

[24] Berman M., “Regular-faced Convex Polyhedra”, Journal of The Franklin Institute, 291:5 (1971), 329–352 | DOI | MR | Zbl

[25] Coxeter H. S., “Regular and semi-regular polytopes. II”, Mathematische Zeitschrift, 188:4 (1985), 559–591 | DOI | MR | Zbl

[26] Coxeter H. S., “Regular and semi-regular polytopes. III”, Mathematische Zeitschrift, 200:1 (1988), 3–45 | DOI | MR | Zbl