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

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?