CAD Tools for Creating Space-filling 3D Escher Tiles

We discuss the design and implementation of CAD tools for creating decorative solids that tile 3-space in a regular, isohedral manner. Isohedral tilings of the plane, as popularized by M. C. Escher, can be constructed by hand or using existing tools on the web. Specialized CAD tools have also been developed for tiling other 2-manifolds. This work addresses the question: How can we generate interesting tilings of 3-space? To generate boundary representations of 3D tiles, we have implemented an interactive constrained Delaunay triangulation algorithm. In addition, we have designed a specialized meshcutting algorithm used in layering extruded 2D tiles to create intricate spacefilling designs. We describe visual debugging methods used during the implementation of these two geometric algorithms, and also explain user-interface decisions we made in designing the CAD tools. Finally, we show examples of 3D tilings that are derived from extruded 2D shapes and from 3D cubic lattices.