What is Simplicial Polytope

The structure of the n – cross - polytopes for large values of n with an exact enumeration of elements of various dimensions entering their boundary complexes is studied in detail. Examples of chemical compounds with the structure of the n – cross - polytopes are given. It is shown that the polytopic prismahedrons can be polytopes, which simultaneously include the n - simplexes and the n – cross- polytopes as faces. It is established that the list of the n - simplexes and the n – cross - polytopes of different dimensions represented by Gosset as faces of this polytope contradicts the analytic laws of increasing incidence coefficients with increasing n, first established in Chapters 1, 3, 4 of this book. This testifies to the impossibility of the existence of Gosset's polytope.