This article is cited in 1 paper

Geometry and topology

Euclidean realization of the product of cycles without hidden symmetries

S. Lawrencenko^a, A. Yu. Shchikanov<sup>b

a</sup> Russian State University of Tourism and Service, ul. Glavnaya, 99, 141221, Cherkizovo, Pushkino District, Moscow Region, Russia
^b University of Technology, ul. Gagarin, 42, 141070, Korolev, Moscow Region, Russia

Abstract: It is shown that any graph G that is the Cartesian product of two cycles can be realized in four-dimensional Euclidean space in such a way that every edge-preserving permutation of the vertices of G extends to a symmetry of the Euclidean realization of G. As a corollary, there exists an infinite series of regular toroidal two-dimensional polyhedra inscribed in the Clifford torus just like the five regular spherical polyhedra are inscribed in a sphere.

Keywords: quadrangulation, torus, Cartesian product of graphs, geometric realization, symmetry group, regular polyhedron.

UDC: 514.1

MSC: 51M20

Received April 10, 2015, published November 5, 2015

DOI: 10.17377/semi.2015.12.063