Abstract
A direct construction of regular polytopes of dimension four was carried out by connecting three-dimensional figures along whole flat faces included in the polytope. It was found that the images of a 24-cell cell known from Coxeter's works do not correspond to reality. It is proved that the 600-cell and 120-cell polytopes cannot exist, since the process of their construction leads to a contradiction with the necessary conditions for the existence of polytopes. The existence of a new class of polytopes has been discovered: polyincidental quasi-regular polytopes having edges with different incidence values within the same polytope. Semi-regular four-dimensional polytopes are constructed using the operations of cutting off the neighborhood of the vertices of regular polytopes. The images of semi-regular polytopes of the highest dimension are presented.
TopIntroduction
In the first chapter, regular and semi-regular three-dimensional polyhedrons (Plato's bodies and Archimedes' bodies) were considered. As the dimension increases, the class of semi-regular polytopes changes sharply and the composition of regular polytopes changes. This chapter is devoted to the study of these polytopes. Moreover, in accordance with the conditions necessary for the existence of polytopes, considered in the previous chapter, an analysis was made of the possibility of the existence of four-dimensional polytopes, for which a complete proof of the existence has not been previously carried out. Direct construction of the 24-cell, with a list of all its components, proved the possibility of its existence. Direct construction of a 600-cell led to a contradiction with the necessary conditions for the incidence of elements of different dimensions in this polytope. This proves that the 600 - cell cannot exist. Consequently, the 120-cell, which is dual to it, cannot exist either. Semi-regular polytopes of higher dimension are obtained from regular polytopes of higher dimension as a result of single and double cutting of the vertices of the polytopes. Images of a series of semi-regular polytopes of dimension 4 are shown.
Key Terms in this Chapter
N–Cross-Polytope: The convex polytope of dimension n in which opposite related of centrum edges not have connection of edge.
N–Simplex: The convex polytope of dimension n in which each vertex is joined by edges with all remain vertices of polytope.
Nanocluster: A nanometric set of connected atoms, stable either in isolation state or in building unit of condensed matter.
N–Cube: The convex polytope of dimension n in which each vertex incident to n edges.
Polytope: Polyhedron in the space of higher dimension.
Incidence Coefficients of Elements of Higher Dimension With Respect to Elements of Lower Dimension: The number of elements of a given lower dimension that are included in a particular element of a higher dimension.
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
Incidence Coefficients of Elements of Lower Dimension With Respect to Elements of Higher Dimension: The number of elements of a certain higher dimension to which the given element of a lower dimension belongs.