
Journal for Geometry and Graphics 23 (2019), No. 2, 201210 Copyright Heldermann Verlag 2019 The Cube: Its Billiards, Geodesics, and QuasiGeodesics Gunter Weiss Institute for Geometry, Technical University, 01062 Dresden, Germany and: Inst. of Discrete Mathematics, University of Technology, Wiedner Hauptstr. 810/104, 1040 Vienna, Austria weissgunter@gmx.at The cube and its higher dimensional counterparts ("ncubes") are wellknown basic polytopes with a wellstudied symmetry group, and from them one easily can derive other interesting polyhedrons and polytopes by a chamfering or adding process. The cube's geodesics and billiards, especially the closed ones, are already treated in the literature. Hereby, a ray's incoming angle must equal 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. When generalising reflections to refractions, 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 modifications to geodesics on a polyhedron will result in polygons, which might be called quasigeodesics. The concept of 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. Keywords: Polyhedron, cube, geodesic polygon, billiard polygon, Snellius' refraction law. MSC: 51M20; 52B10, 51N05 [ Fulltextpdf (1351 KB)] 