Escherization with Generalized Distance Functions Focusing on Local Structural Similarity

TL;DR

This paper introduces new local-structure-focused distance functions for the Escherization problem, enabling efficient search algorithms to generate tile shapes similar to complex goal figures within reasonable time.

Contribution

It proposes novel local-structure-based similarity measures and integrates them into an existing framework, improving the efficiency and effectiveness of Escherization shape generation.

Findings

Proposed distance functions effectively capture local structural similarity.

Algorithms successfully generate satisfactory tiles for complex figures.

Achieved reasonable computation times with the new methods.

Abstract

The Escherization problem involves finding a closed figure that tiles the plane that is most similar to a given goal figure. In Koizumi and Sugihara's formulation of the Escherization problem, the tile and goal figures are represented as $n$-point polygons where the similarity between them is measured based on the difference in the positions between the corresponding points. This paper presents alternative similarity measures (distance functions) suitable for this problem. The proposed distance functions focus on the similarity of local structures in several different manners. The designed distance functions are incorporated into a recently developed framework of the exhaustive search of the templates for the Escherization problem. Efficient exhaustive and incomplete search algorithms for the formulated problems are also developed to obtain results within a reasonable computation time. Experimental results showed that the proposed algorithms found satisfactory tile shapes for fairly complicated goal figures in a reasonable computation time.