# Scaling in the Process of Hierarchical Filling of n - Dimensional Space

DOI: 10.4018/978-1-5225-6968-8.ch007

## Abstract

The hierarchical filling of the n - dimensional space with geometric figures is studied, accompanied by a process of discrete similar changes in their dimensions, i.e. 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.
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## Hierarchical Filling Of Two - Dimensional Flat

Let the figure that one fill the plane be a regular pentagon (Figure 1).

Figure 1.

The regular pentagon

Continuing its sides before crossing into the outer region, can to get a pentagon similar to the original one . Having done the same with a pentagon , can to get an even larger regular pentagon , and so on. By joining the adjacent vertices of a polygon, skipping one next vertex, can to 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 to conclude that ∆ , and hence

(1) but

## Key Terms in this Chapter

Denominator of Geometric Progression: The relationship between the previous and subsequent elements of a geometric progression.

Scaling: Scale change of shape.

Hierarchical Filling of Space: The filling of space by a figure with its discrete resizing, preserving the resemblance of figures at each step of its change.

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