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 $δ$-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 $δ$-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?