The Detailed Structure of n - Cross - Polytopes and Polytopes With Their Participation

The Detailed Structure of n - Cross - Polytopes and Polytopes With Their Participation

Copyright: © 2019 |Pages: 23
DOI: 10.4018/978-1-5225-6968-8.ch006

Abstract

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.
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The N – Cross - Polytope Is Simplicial Polytope

In Chapter 2 it was noted that the n - cross – polytope is simplicial polytope (Ziegler, 1995; Grunboum, 1967). For n = 4 this is clear, so the facets of the 4 – cross - polytope are the tetrahedrons which are simplexes and all faces of the tetrahedron are simplexes. For n > 4 this statement is not evident. Let is consider the 5 – cross – polytope for example. The image of this polytope it was presented in Chapter 2 (Figure 10). According to equation (6) of Chapter 2 for the 5 – cross - polytope there are

Thus, the 5 – cross – polytope has 10 vertices, 40 edges, 80 flat faces (triangles), 80 tetrahedrons and 32 polytopes with dimension 4. However, from equation (6) of Chapter 2 no follow which form of the polytopes with dimension 4. One listed this elements with dimension 0, 1, 2, 3, used Figure 10 of Chapter 2. The vertices are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The edges are 10, 19, 18, 17, 15, 14, 13, 12, 20, 29, 28, 26, 25, 24, 23, 30, 39, 37, 35, 34, 40, 48, 47, 46, 45, 59, 58, 57, 56, 60, 69, 68, 67, 70, 79, 78, 80, 89, 90.

The triangles are

109, 108, 107, 104, 103, 102, 198, 197, 195, 193, 192, 187, 185, 184, 182, 175, 174, 173, 154, 153, 152, 143, 142, 132, 209, 208, 206, 204, 203, 298, 296, 295, 293, 286, 285, 284, 265, 264, 263, 254, 253, 243, 309, 307, 306, 304, 397, 396, 395, 376, 375, 374, 365, 364, 354, 408, 407, 406, 487, 485, 476, 465, 486, 475, 598, 597, 596, 587, 586, 576, 609, 608, 607, 698, 697, 687, 709, 708, 798, 809.

The tetrahedrons with listed triangle faces are

1098, 1097, 1093, 1092, 1087, 1084, 1082, 1074, 1073, 1043, 1042, 1032, 1987, 1985,1982, 1975, 1974, 1973, 1953, 1952, 1932, 1875, 1874, 1854, 1852, 1842, 1754, 1753,1743, 1543, 1542, 1532, 1432, 2098, 2096, 2093, 2086, 2084, 2064, 2063, 2043, 2986,2985, 2965, 2963, 2953, 2865, 2864, 2854, 2654, 2653, 2643, 2543, 3097, 3096, 3076,3074, 3064, 3976, 3975, 3965, 3765, 3764, 3754, 3654, 3654, 4087, 4086, 4076, 4876,4875, 4765, 4865, 5987, 5986, 5976, 6098, 6097, 6087, 6987, 7098.

The number of vertices, edges, triangles and tetrahedrons defined according to Figure 10 of Chapter 2 exactly coincide with the number of elements of dimension zero, one, two and three, calculated according to formula (6) of Chapter 2. The number of these figures is limited by the condition that there are no edges connecting the vertices opposite in Figure 10.

Key Terms in this Chapter

Polytope with Factor – Structure: Polytope in which vertices are polytopes with same dimension k , and edges are polytopes with dimension k – 1.

Simplicial Polytope: Polytope with dimension n in which all faces with dimension less n are simplexes.

Incidence in Polytopes: Incidence in polytopes define the number of elements of higher dimension the given element of lower dimension belongs.

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