XXVI. On the regular or platonic solids

The Platonic Solids

The tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. From a first glance, one immediately notices that the Platonic Solids exhibit remarkable symmetry. They are the only convex polyhedra for which the same same regular polygon is used for each face, and the same number of faces meet at each vertex. Their symmetries are aesthetically pleasing, like those of stones cu...

3-manifolds from Platonic solids

The problem of classifying, up to isometry, the orientable spherical and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. This allows us to complete the classification begun by Best [Canad. J. Math. 23 (1971) 451], Lorimer [Pacific J. Math. 156 (1992) 329], Richardson and Rubinstein [Hyperbolic manifolds from a regul...

Symmetry Groups of Platonic Solids

Hypergroups associated to random walks on the Platonic solids

We explicitly construct the convolution hypergroups of spheres in the vertex graphs of the classical Platonic solids, their character tables, and the structure of their dual signed hypergroups. This allows us to easily determine certain aspects of the random walks on these graphs. The character tables share many of the pleasant properties of character tables of finite groups.

Constructing Finite Frames via Platonic Solids

Finite tight frames have many applications and some interesting physical interpretations. One of the important subjects in this area is the ways for constructing such frames. In this article we give a concrete method for constructing finite normalized frames using Platonic solids.