Abstract
Methods for the formation of irregular polytopes of the highest dimension and their properties are investigated. It was found that a distinctive feature of irregular polytopes of higher dimension is the existence of a continuous outer boundary of the polytope and a jump in dimension on this boundary. This feature makes it possible to consider irregular polytopes of higher dimension as carriers of higher dimension in low-dimensional space, for example, in three-dimensional space. The possibility of creating objects with a steadily increasing dimension has been opened when deepening into the object. Various options for the implementation of the creation of incorrect polytopes of the highest dimension with the calculation of the dimension in specific cases are considered in detail.
TopAdding A Center To Simplex Polytopes
In chemical compounds, tetrahedral coordination is widespread. It's easy to see how the higher dimensional space appears when you add a vertex to the center of the tetrahedron.
Suppose we have an arbitrary tetrahedron (correct or incorrect, it does not matter). Place a vertex in the center of the tetrahedron and connect it with edges to other vertices of the tetrahedron (Figure 1).
Figure 1. Polytope a tetrahedron with center
Key Terms in this Chapter
N–Cube: The convex polytope of dimension n in which each vertex incident to n edges.
Polytope: Polyhedron in the space of higher dimension.
N–Simplex: The convex polytope of dimension n in which each vertex is joined by edges with all remain vertices of polytope.
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
N–Cross-Polytope: The convex polytope of dimension n in which opposite related of centrum edges not have connection of edge.
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.