Faces of Platonic solids in all dimensions

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, ...

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

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...

Athermal jamming of soft frictionless Platonic solids.

A mechanically based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless Platonic solids. Systems of these regular convex polyhedra exhibit mechanically stable phases with density substantially less than optimal for a given shape, revealing that thermal motion is necessary to access high-density phases. We confirm that the large system j...