Abstract
The hierarchical filling of the n-dimensional space with geometric figures is studied, accompanied by a process of discrete similar changes in their sizes, that is, process of scaling. The scaling process in these fillings does not depend on time and is determined only by the geometric characteristics of the figures, which are preserved when their size is changed. Two possible ways of hierarchical filling of space are defined, under which the original figure incrementally increases its size fills the space. Investigations of the hierarchical filling of concrete geometric figures of a plane, three-dimensional space, four- and five-dimensional spaces are carried out. The denominator of geometric progressions characterizing sequences of figures in the process of scaling are determined depending on the shape of the figure and its dimension.
TopHierarchical Filling Of Two - Dimensional Flat
Let the figure that one fill the plane be a regular pentagon A0B0C0D0E0 (Figure 1).
Continuing its sides before crossing into the outer region, can get a pentagon similar the original one A1B1C1D1E1. Having done the same with a pentagon A1B1C1D1E1, can get an even larger regular pentagon A2B2C2D2E2, and so on. By joining the adjacent vertices of a polygon A0B0C0D0E0, skipping one next vertex, can obtain a polygon 
inside the original and then an even smaller polygon
, and so on. Considering such triangles formed from the intersection of lines, can conclude that ∆ 
∆E1D1C1, and hence
(9.1) but
(9.2) Key Terms in this Chapter
Poly-Incident Polytopes: A series of numbers in which a member of a series is the sum of the two previous members of the series.
Golden Ratio: The section of the segment into two parts, in which the ratio of the length of the larger part to the length of the smaller part is
Scaling: Scale change of shape.
Denominator of Geometric Progression: The relationship between the previous and subsequent elements of a geometric progression.
Poly-Incident Polytopes: Filling a space with a figure with a similar increase in the size of the figure around a certain center according to some law related to the geometric characteristics of the figure itself.