Construction of topological representation of geometric patterns using Cantor self-similar set

Abstract

Universal representation of geometric patterns of disordered matters is investigated with the aid of general topology. By utilizing the result obtained in the previous study (S. Ohmori, et.al., Phys. Scr. 94, 105213 (2019)) that any patterns can be represented by a specific topological space, a construction of topological representation of patterns using Cantor set is shown. The obtained topological representations are then demonstrated by the contractions that characterize the self-similarity of Cantor set. For some practical geometric patterns, e.g., network, dendritic, and clusterized patterns, their topological representations are focused on.