Based on the results of the study of the geometry of polytopes of higher dimension, a number of conditions necessary for the existence of polytopes have been formulated. Each of these conditions and their combination are necessary, but not sufficient for the existence of a polytope of the highest dimension. A sufficient condition for the existence of a polytope of the highest dimension is its complete construction (i.e., determination of its complete structure with a listing of all elements of the polytope and their contacts with each other). The importance of the concept of incidence in the polytope of the highest dimension is emphasized on the basis of which the law of conservation of incidents is formulated, which has a fundamental information value.
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When passing from three-dimensional polytopes (i.e., polyhedrons) to polytopes of higher dimensions, the question of the necessary and sufficient conditions for the existence of these polytopes should be raised. To answer this question, it is required to take into account the properties of multidimensional figures and refrain from attempts to mechanically transfer the properties of three-dimensional figures to the properties of multidimensional figures, arguing, as is often done, that when moving to a higher-dimensional space, nothing changes significantly, only the number of spatial coordinates increases. Such statements lead to important negative consequences. An example is the history of attempts to solve the 18th problem of Hilbert (Hilbert, 1901; Alexsandrov, 1969) on the construction of a space of higher dimension from congruent figures. The answer to this proposal was a series of works by various researchers in subsequent years, unfortunately, the properties of figures of higher dimensions were not taken into account (Minkowski, 1911; Delaunay, 1929,1937, 1961; Delaunay, & Sandakova, 1961; Voronoi, 1952; Alexsandrov, 1934, 1954; Venkov, 1954, 1959; Stogrin, 1973).
In particular, when analyzing four-dimensional parallelotopes (Delone, 1929), a table of parallelohedrons of 51 figures was compiled. Later, another figure was added to this table (Shtogrin, 1973). However, it turned out (Zhizhin, 2019 b, 2021) that none of these figures was tested for the fulfillment of the Euler-Poincaré equation, which is necessary to prove the existence of a polytope of higher dimension (Poincaré, 1895). The Euler-Poincaré equation is the result of the continuation of the Euler equation, which is valid in three-dimensional space, to a space of higher dimension. A recent check (Zhizhin, 2019b 2021) showed that this equation does not hold for Delaunay parallelotopes. In this regard, the theory of four-dimensional stereohedrons developed by Delaunay (Delaunay, 1961; Delaunay, Sandakova, 1961), which also does not use the Euler-Poincaré equation, remains incorrect. It should be noted that in many works on polytopes of higher dimensions (Coxeter, 1961, 1973), when proving the existence of these polytopes, they are limited to calculating the possibility of locating a certain number of figures in a polytope around an element of a certain dimension, for example, an edge. Schläfli's symbols are built on this basis (Schläfli, 1901). But it cannot be argued that if a polytope satisfies the numbers in Schläfli's symbols, then it exists. This condition can be considered only as the minimum necessary condition for the existence of a polytope. The proof of the existence of a polytope of higher dimension can only be its construction with a list of all the elements included in it and with an indication of all contacts of these elements with each other. This is how the figures in three-dimensional space were studied in the works of Fedorov (Fedorov, 1885, 1889, 1891).