Dodecahedra

A dodecahedron is a regular polyhedron, one of the five Platonic solids. It has twelve regular pentagonal faces, with three faces meeting at each of its twenty vertices. All edges have equal length, and the arrangement is highly symmetric.

The Schläfli symbol for the solid is {5,3}. Its dual polyhedron is the icosahedron. The solid belongs to the icosahedral symmetry group, with full symmetry of order 120 and orientational symmetry of order 60. The dihedral angle between adjacent faces is approximately 116.565 degrees.

A regular dodecahedron can be constructed in various ways, including by defining coordinates using the golden ratio φ = (1+√5)/2. One common centered coordinate system places the vertices at permutations of (±1, ±1, ±1) and (0, ±1/φ, ±φ), scaled to the desired edge length.

In mathematical literature, the dodecahedron is commonly studied for its relation to the icosahedron via duality and for illustrating symmetry properties. Although aesthetically related to decorative art and design, it is not the standard die used in gaming; dice with twelve faces exist but are typically not true regular dodecahedra.