О некоторых обобщениях сильно симметричных многогранников

V. I. Subbotin

Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 222-230.

Résumé

The paper deals with the symmetry properties of the associated closed convex polyhedra in three-dimensional Euclidean space. Themes work relates in part to the problem of generalization class of regular (Platonic) polyhedra. Historically, the first such generalizations are equiangularly-semiregular (Archimedean) polyhedra. The direction of generalization of regular polyhedra, considered by the author in this paper due to the symmetry axes of the convex polyhedron. A convex polyhedron is called symmetric if it has at least one non-trivial symmetry axis. All the axis of symmetry of the polyhedron intersect at one point called the center of the polyhedron. All considered the polyhedra are polyhedra with the center. Previously we listed all polyhedra, strongly symmetrical with respect to rotation of faces, as well as their dual-metrically polyhedra strongly symmetric with respect to the rotation polyhedral angles [9]–[15]. It is interesting to note that among the highly symmetric polyhedra there are exactly eight of which are not even equivalent to combinatorial Archimedean or equiangularly semiregular polyhedra. By definition, the property of strong symmetry polyhedron requires a global symmetry of the polyhedron with respect to each axis of symmetry perpendicular to the faces of the polyhedron. It is therefore of interest to find weaker conditions on the symmetry elements of the polyhedron. We give a new proof of the local criterion of strong symmetry of the polyhedron, which is based on the properties of the axes of two consecutive rotations. We also consider two classes of polyhedra that generalize the concept of a strongly rotationally symmetrical faces of: a class of polyhedra with isolated asymmetrical faces and the class of polyhedra with isolated asymmetrical zone. It is proved that every polyhedron with isolated asymmetrical faces can be obtained by cutting off the vertices or edges of a polyhedron, highly symmetrical with respect to rotation faces; and each polyhedron with isolated asymmetrical zone by build axially symmetric truncated pyramids on some facets of one of the highly symmetric with respect to rotation of the faces of the polyhedron.In each of these classes there are the largest number of polyhedron faces excluding two infinite series: truncated prisms; truncated on two apex and elongated bipyramid. Bibliography: 15 titles.

Détail

Mots-clés : strongly symmetric polyhedrons, the axis of rotation, isolated asymmetrical face, locally symmetric faces, isolated asymmetrical zone.

Bibliographie

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[9] Subbotin V. I., “On Completely symmetrical polyhedra”, Proc. Int. Conf. on discrete geometry and its applications, Materialy Megdunarudnoy conferencii po diskretnoy geometrii i eyo prilogeniyam, posviash. 70-letiju prof. S. S. Ryshkova (Moskow, 2001), 88–89

[10] Subbotin V. I., “The enumeration of polyhedra, strongly symmetrical with respect to rotation”, Proc. Int. School-Seminar on Geometry and Analysis, Trudy megdunarodnoy shkoly-seminara po geometrii i analizu pamyti N. V. Efimova, Rostov-on-Don, 2002, 77–78 | MR

[11] Subbotin V. I., “On two classes of symmetric polyhedra”, Izvestiya vuzov. Sev.-kavk. region. Estestv. nauki, 2003, no. 3, 17–19 | MR

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

[13] Subbotin V. I., “About polyhedra strongly symmetric with respect to rotation”, Chebyshevskiy sbornik, 7:2(18) (2006), 168–171 | MR | Zbl

[14] Subbotin V. I., “Polyhedra with the maximum number of asymmetrical faces”, Proc. Int. Conf. “Metric geometry of surfaces and polyhedra”, Materialy megdunarodnoy konferencii, posvjachennoi 100-letiyu N. V. Efimova, M., 2010, 60–61

[15] Subbotin V. I., “On symmetric polyhedra with asymmetrical faces”, Diskretnaja matematika i ejo prilogeniya, Proc. Int. Seminar “Discrete Mathematics and Its Applications”, Materialy megdunarodnogo seminara, posvjachennogo 80-letiyu O. B. Lupanova, M., 2012, 398–400