Skeletal Geometric Complexes and Their Symmetries

Abstract

Polyhedra and polyhedron-like structures have been studied since the early days of geometry (Coxeter [5]). The most well-known are the five Platonic solids -the tetrahedron, octahedron, cube, icosahedron, and dodecahedron. Book XIII of Euclid's "Elements" [10] was entirely devoted to their mathematical properties.In modern terminology, the Platonic solids are precisely the regular convex polyhedra in ordinary Euclidean space E 3 . Historically, as the name suggests, these figures were viewed as solids bounded by a collection of congruent regular polygons fitting together in a highly symmetric fashion. With the passage of time, the perception of what should constitute a polyhedron has undergone fundamental changes. The famous Kepler-Poinsot star polyhedra no longer are solids but instead form self-intersecting polyhedral structures that feature regular star polygons as faces or vertex-figures. These four "starry" figures are precisely the regular star polyhedra in E 3 [5], which along with the five Platonic solids might be called the classical regular polyhedra [18] and are shown in Figure 1 [31, 32].Why say that a polyhedron must be finite? In the 1930's, Petrie and Coxeter discovered three more regular polyhedra in E 3 , each forming a non-compact polyhedral surface which is tessellated by regular convex polygons and bounds two unbounded (congruent) "solids" (infinite "handlebodies"). The three polyhedra are shown in Figure 2 [31, 32]. The trick here is to allow the vertex-figures to be skew polygons rather than convex polygons. The vertex-figure of a polyhedron P at a vertex v is the polygon whose vertices are the vertices *

Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids

“…Here we generalise Ringel's approach in two directions. If we do not insist that each symbol appears exactly twice, we may use such schemes to describe the combinatorial structure of more general polygonal complexes in the sense of Schulte et al [53,54,55,72]. On the other hand, if we allow symbols with a single appearance, we may describe chemical structures, such as benzenoids as graphs embedded in a surface with a boundary.…”

Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids

“…Here we generalise Ringel's approach in two directions. If we do not insist that each symbol appears exactly twice, we may use such schemes to describe the combinatorial structure of more general polygonal complexes in the sense of Schulte et al [56][57][58]75] On the other hand, if we allow symbols with a single appearance, we may describe chemical structures, such as benzenoids as graphs embedded in a surface with a boundary.…”

“…[28] Example 1. A typical example of a polygonal complex in the sense of Schulte et al [56][57][58]75] is a 2-dimensional skeleton of the tesseract (the 4-dimensional cube, see Figure 1). This skeleton is composed of 16 vertices, 32 edges and 24 quadrilateral faces.…”

Parallelohedra, old and new

“…The polygons and polyhedra we discuss in this paper are skeletal. Our definitions are adapted from Gru ¨nbaum (1977, 2010a) and from Schulte & Weiss (2017).…”

“…Would it suffice to translate the edges of a skeletal polyhedron P, creating a web in three-dimensional space whose edges and vertices are all translates of P? And parallelohedra need not be finite (see Schulte & Weiss, 2017). Infinite parallelohedra are an interesting chapter in this stilldeveloping field.…”