Method for generating constant curvature surfaces by Truchet tiling on Parallelohedron

Abstract

This study proposes a method for generating constant curvature surfaces through Truchet tiling on parallelohedra. The method extends a two-dimensional Truchet tiling technique to three dimensions by applying Truchet tiling composed of arcs with the same radius to the faces of five types of parallelohedra: cube, hexagonal prism, rhombic dodecahedron, elongated rhombic dodecahedron, and truncated octahedron. Spherical or hyperbolic surfaces are arranged along the arcs. It was confirmed that some of these shapes can fill space through combinations, demonstrating that the proposed method is effective for creating space-filling forms with complex curved surfaces.