Download PDFOpen PDF in browserThe Cube: Its Relatives, Geodesics, Billiards, and GeneralisationsEasyChair Preprint 30612 pages•Date: June 26, 2018AbstractStarting 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 nonEuclidean plane. Obviously this approach can be applied when taking a regular tetrahedron or a regular pentagondodecahedron as start figure. But a hypercube in R^n (an “ncube”), too, suits as start object and gives rise to interesting polytopes. The cube’s geodesics and (inner) billiards, especially the closed ones, are already wellknown (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 rhombidodecahedron. 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 sineratio 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 “quasigeodesics”. The concept “pseudogeodesic”, 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 3cube and a 4cube.
Keyphrases: Billiard polygon, Deltahedron, Geodesic polygon, Rhombidodecahedron, cube, polyhedron
