The Cube: Its Relatives, Geodesics, Billiards, and Generalisations

Starting with a cube and its symmetry group one can get related polyhedrons via corner and edge trimming and dualizing processes or via adding congruent polyhedrons to its faces. These processes deliver a subset of Archimedian solids and their duals, but also starshaped solids. Thereby, polyhedrons with congruent faces are of special interest, as these faces could be used as tiles in mosaics, either in a Euclidean or at least in a non-Euclidean plane. Obviously this approach can be applied when taking a regular tetrahedron or a regular pentagon-dodecahedron as start figure. But a hypercube in R^n (an “n-cube”), too, suits as start object and gives rise to interesting polytopes. The cube’s geodesics and (inner) billiards, especially the closed ones, are already well-known (see references). Hereby, a ray’s incoming angle equals its outcoming angle. There are many practical applications of reflections in a cube’s corner, as e.g. the cat’s eye and retroreflectors or reflectors guiding ships through bridges. Geodesics on a cube can be interpreted as billiards in the circumscribed rhombi-dodecahedron. This gives a hint, how to treat geodesics on arbitrary polyhedrons. Generalising reflections to refractions means that one has to apply Snellius’ refraction law saying that the sine-ratio of incoming and outcoming angles is constant. Application of this law (or a convenient modification of it) to geodesics on a polyhedron will result in trace polygons, which might be called “quasi-geodesics”. The concept “pseudo-geodesic”, coined for curves c on smooth surfaces Φ, is defined by the property of c that its osculating planes enclose a constant angle with the normals n of Φ. Again, this concept can be modified for polyhedrons, too. We look for these three types of traces of rays in and on a 3-cube and a 4-cube.