Perturbation of self-similar sets and some regular configurations and comparison of fractals

TL;DR

This paper explores modified distances between point sets to analyze fractals and geometric patterns, extending classical results to quasi-self-similar sets and examining the nature of non-crystalline solids.

Contribution

It introduces new metrics for comparing sets, applies them to fractals and tilings, and extends classical theorems to quasi-self-similar structures.

Findings

Same results as Moran's theorem for δ-quasi-self-similar sets under open set condition

Proposed new distances to describe fractal form and regular configurations

Discussed the behavior of non-crystalline solids approximating crystals

Abstract

We consider several distances between two sets of points, which are modifications of the Hausdorff metric, and apply them to describe some fractals such as $\delta$-quasi-self-similar sets, and some other geometric notions in Euclidean space, such as tilings with quasi-prototiles and patterns with quasi-motifs. For the $\delta$-quasi-self-similar sets satisfying the open set condition we obtain the same result as a classical theorem due to P. A. P. Moran. In this paper we try to gaze on fractals in an aspect of their "form" and suggest a few of related questions. Finally, we attempt to inquire an issue -- what nature and behavior do non-crystalline solids that approximate to crystals show?