What is N–Cross-Polytope

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.