Higher-Dimensional Polytopes With Regular-Faceted Polyhedrons Are Different From Archimedes Bodies

Introduction

In the second chapter of this work, the existence of three-dimensional polyhedrons with flat faces in the form of various regular polygons simultaneously present in one polyhedron with the participation of several types of gonohedrons (Fedorov, 1885, 1889, 1891) as vertices was proved. Moreover, it was found that the maximum number of different types of gonohedrons that can be simultaneously present in one polyhedron is 6. This single polyhedron PG₁₄ can name polytypichedron - max. It represents the maximum possible difference between a polyhedron and Archimedes bodies, while maintaining flat faces in the form of regular polygons. The whole family of regular-faceted polyhedrons, in which three or more of different types of genohedrons can be present shell named polytypichedrons. The question about existence of regular-faceted polyhedron was considered on 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 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). Accordingly, N. Johnson's assumption on the existence of only 92 total number regularly faceted polyhedrons is wrong too. In the proof Johnson's assumption Zalgaller used Cauchy's lemma (Cauchy, 1944; Hadamard,1938) and the theorem of A.D. Aleksandrov (Aleksandrov, 1950). In the author's work (Zhizhin, 2009), the list of regular polyhedrons proposed by Zalgaller was refined and corrected. In particular, the existence of nine previously unknown regular-faceted polyhedrons was proved. Moreover, It is these polyhedrons that have a fairly large number of types of different gonohedrons, the diversity of which has prompted to call these polyhedrons polytypichedrons.

In the sixth chapter of this work, the geometry of polytopes, which are products of polytopes, was studied in detail. A number of valuable properties of these polytopes were discovered, which made it possible, in particular, to solve the 18th Hilbert problem (Hilbert, 1901; Zhizhin, 2019 b, 2021) on the construction of n-dimensional spaces, as well as systems with a steady increase in dimension as we go deeper into the systems. Since polytopes, which are products of polytopes, form a very important class of polytopes, and polytypichedrons are interesting for the presence of different types of gonohedrons in them, the problem arises of finding polytopes of higher dimension with the simultaneous participation of different types of gonohedrons in them. This can be done by multiplying the polytypichedrons by polytopes. This chapter is devoted to this research. As objects for the study, polyhedrons were selected, the existence of which was established by the author PG₈, PG₉, PG₁₀, PG₁₁, PG₁₄ (polytypichedron - max), PG₃₅ (Chapter 2).