On Maximal Homogeneous 3-Geometries—A Polyhedron Algorithm for Space Tilings

In this paper we introduce a polyhedron algorithm that has been developed for finding space groups. In order to demonstrate the problem and the main steps of the algorithm, we consider some regular plane tilings in our examples, and then we generalize the method for 3-dimensional spaces. Applying the algorithm and its computer implementation we investigate periodic, face-to-face, regular polyhedron tilings in 3-spaces of constant curvature and of the other homogeneous 3-geometries, too. We summarize and visualize the most important results, emphasizing the fixed-point-free space groups which determine 3-dimensional manifolds.