Topology of compact space forms from Platonic solids. I.

Tiling Space by Platonic Solids, I

There exist precisely 914, 58 and 46 equivariant types of tile-transitive tilings of 3-dimensional euclidean space by topological cubes, oc-tahedra and tetrahedra, that fall into 11, 3, and 9 topological families, respectively. Representatives are described for all topological families. A general method for obtaining results of this kind is introduced.

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...

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...

Symmetry Groups of Platonic Solids

Faces of Platonic solids in all dimensions.

This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n - 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types A(n), B(n), C(n), F4, also called the Weyl groups or, ...