Creation of Dihedral Escher-like Tilings Based on As-Rigid-As-Possible Deformation

Abstract

An Escher-like tiling is a tiling consisting of one or a few artistic shapes of tile. This paper proposes a method for generating Escher-like tilings consisting of two distinct shapes (dihedral Escher-like tilings) that are as similar as possible to the two goal shapes specified by the user. This study is an extension of a previous study that successfully generated Escher-like tilings consisting of a single tile shape for a single goal shape. Building upon the previous study, our method attempts to exhaustively search for which parts of the discretized tile contours are adjacent to each other to form a tiling. For each configuration, two tile shapes are optimized to be similar to the given two goal shapes. By evaluating the similarity based on as-rigid-as possible deformation energy, the optimized tile shapes preserve the local structures of the goal shapes, even if substantial deformations are necessary to form a tiling. However, in the dihedral case, this approach is seemingly unrealistic because it suffers from the complexity of the energy function and the combinatorial explosion of the possible configurations. We developed a method to address these issues and show that the proposed algorithms can generate satisfactory dihedral Escher-like tilings in a realistic computation time, even for somewhat complex goal shapes.