Abstract
It is shown that known classification of convex polyhedrons with sides from correct polygons offered by V.A. Zallgaller as full transfer of all possible convex figures of this type is far from completeness. There exist nine new convex polyhedrons with sides from correct polygons and the impossibility of existence of two figures from Zallgeller classification. Thus, the general number of regularly faceted polyhedrons except for the regular polyhedrons, known semiregular polyhedrons, prisms, and antiprisms make 100 figures instead of 92 as Johnson earlier assumed. The existence of a convex polyhedron with faces of regular polygons was discovered, which differs as much as possible from the bodies of Archimedes and Plato by the presence of six different types of gonohedrons in it (polytypichedron-max).
TopIntroduction
Regularly faceted three-dimensional polytopes are polytopes whose faces are regular polygons. Moreover, in the same polyhedron there can be faces with a different number of sides. The question of determining all possible convex regularly faceted polyhedrons and their classification (in what follows we will only talk about convex polyhedrons and therefore there is no need to emphasize this each time) has a long history. A definite generalization of these works can be considered the monograph by V.A. Zalgaller (Zalgaller, 1966). This monograph contains a bibliography of works in which, to one degree or another, the question of the existence of convex regularly faceted polyhedrons is considered. It, in essence, is devoted to proving one theorem: apart from prisms and antiprisms, there are only 28 simple convex regularly faceted polyhedrons. (A regularly faceted polyhedron is called simple if it does not allow cutting by a plane into two other regularly faceted polyhedrons.) Schematic representations of these polyhedrons are given. In addition, it is proved that the assertion, expressed earlier (Johnson, 1960), that (apart from prisms and antiprisms) there are only 92 regularly faceted polytopes, which have 28 simple regularly faceted polytopes in their composition (Zalgaller, 1966 a).
Currently, interest in convex regularly faceted polyhedrons is associated with the analysis of various dissipative structures that arise in various natural phenomena and in the processes of human activity (Zhizhin, 2005 a, b; 2006). In many cases, a dissipative structure is a spatial network, the unit cells of which are regularly faceted polyhedrons. Examples are biological tissues, the cell shapes of which are often close to regularly faceted polyhedrons (Lewis, 1923; D`Arsy, 1942; Petrov, 2007), and as it was recently established based on the analysis of photographs of the Universe, the large-scale structure of the Universe can be considered as a spatial network with a unit cell in the form of a truncated octahedron (Zhizhin, 2008). The shapes of regular polyhedrons are of interest in construction as elements playing the role of space fillers, as well as shapes of clusters in nanotechnology (Zhizhin, 2006, 2021).
Key Terms in this Chapter
Regular Polyhedron: Convex polyhedron with faces of regular polygons of one type.
Convex Regularly Faceted Polyhedrons: Convex three-dimensional polytopes whose faces are regular polygons.
Mechanical Operations on a Polyhedron: Conceivable transformations of a polyhedron, including cutting a polyhedron (cutting off vertices), deforming a section of a cut to obtain a regular polygon in a section, rotating parts of a polyhedron after cutting relative to each other.
Semiregular Polyhedrons: Convex polyhedrons with faces of regular polygons of two or more types, if all gonohedrons of the polyhedron have the same composition and arrangement of faces.
Gonohedron: A polyhedral corner whose faces are regular polygons.