Toroidal polyhexes, tripartite graphs, and double groups

Abstract

The concept of regular Platonic polyhedra (namely the tetrahedron, octahedron, icosahedron, and their duals) can be generalized to regular polygonal networks on surfaces of non-zero genus known as Platonic tessellations. Such tessellations are useful in molecular physics as geometrical models for double groups as well as for structures of low-density carbon and boron nitride allotropes. The symmetries of such tessellations are described by automorphism groups, which generally are non-Euclidean groups going sometimes far beyond the standard symmetry point groups. A special type of Platonic tessellation, namely the duals of the regular tripartite graphs K n , n , n , can either be depicted as a tessellation of regular 2n-gons on a surface of genus (n − 2)(n − 1)/2 or as an n × n rhomboidal array of n 2 regular hexagons on a torus, called a toroidal polyhex. The simplest such tessellation is the cube considered as the dual of the octahedron (the K 2,2,2 graph), which has a point group symmetry O h with a rotational subgroup O isomorphic to the tetrahedral group T d having operations of periods 2, 3, and 4. The 2 × 2 toroidal polyhex corresponding to the cube likewise has O h as its automorphism group, but its proper rotation subgroup is isomorphic to the tetrahedral group T h having operations of periods 2, 3 and 6. A related more complicated case, recently studied by Ceulemans, Lijnens, and Fowler, is the Dyck tessellation of 12 octagons on a genus 3 surface of interest as a model for a possible carbon allotrope containing six- and eight-membered rings. The Dyck tessellation is the dual of the K 4,4,4 graph embedded in a genus 3 surface and corresponds to a 4 × 4 toroidal polyhex. Related p × q (p ≠ q) toroidal polyhex embeddings also appear to be at least feasible for Platonic tessellations corresponding to double and higher multiple groups of certain Platonic polyhedra and other tessellations. Thus the 2 × 4 toroidal polyhex is seen to correspond to the genus 2 {4 + 4, 3} tessellation, which is the dual for the model of the standard double octahedral group 2 O h , generated by adding the new period 2 operation . However, the group O h can also be doubled with a different new period 2 operation to give a different double group 2 W h with periods of 2, 4, and 6 rather than the periods 2, 3, and 8 in 2 Oh .