Abstract
For more than 100 years in science, many researchers, when trying to solve Hilbert's 18th problem of constructing n-dimensional space, used the principles of the Delaunay geometric theory. In this book, as a result of a careful analysis of the work in this direction, it is shown that the principles of the Delaunay theory are erroneous. They do not take into account the features of figures of higher dimensionality, do not agree with modern advances in the physics of the structure of matter, and lead to erroneous results. A new approach to solving the 18th Hilbert problem is proposed, based on modern knowledge in the field of the structure of matter and the geometric properties of figures of higher dimension. The basis of the new approach to solving the 18th Hilbert problem is the theory developed by the author on polytopic prismahedrons.
TopN - Cube As A Polytopic Prismahedron
Theorem 8.1.N – cube is n – dimensional parallelohedron in n – dimensional space (n3 1).
Proof. In cases where n is equal to unity, two or three of theorem 8.1 obviously holds. No proof is required. A four - dimensional cube is the simplest version of a polytopic prismahedron, since it is the product of a three - dimensional cube and a one - dimensional segment (Figure 1). Can to introduce one of the vertices of the 4 – cube origin of the four - dimensional space (x, y, z, t). Orient the coordinates, such as indicated in Figure 8.1.
Assume that the length of each edge is equal to 1. Then, each vertex of 4 - cube can be associated with a set of integers (Figure 8.1). Then
(8.1)Representing the 4 - cubes A0÷A15 dots in three - dimensional space, can get the 4 - cube again (4 - cube A). Moreover, the edges of the 4 - cube A correspond to possible changes in the values of one of the coordinates of the vertices of the 4 - cubes unit (Figure 2).
Figure 2. The 4 - cube A from 16 4 - cubes
Key Terms in this Chapter
N-Simplex: The convex polytope of dimension n in which each vertex is joined by edges with all remain vertices of polytope.
Polytope: Polyhedron in the space of higher dimension.
N-Cross-Polytope: The convex polytope of dimension n in which opposite related of centrum edges do not have connection of edge.
Parallelogon: A polygon capable of filling a two-dimensional plane without gaps during its translation, with adjacent polygons adjacent to each other on the whole sides of the polygons.
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
Parallelohedron in Three-Dimensional Space: A polyhedron capable of filling a three-dimensional space without gaps during its translation, with adjacent polyhedrons adjacent to each other along whole flat faces of the polyhedrons.
Congruent Polyhedrons: Polyhedrons compatible with each other by movement.