dodekaeder

Dodekaeder, or dodecahedron, is a regular polyhedron in the family of Platonic solids. It consists of twelve regular pentagonal faces, and its dual polyhedron is the icosahedron. The dodecahedron and the icosahedron are closely related through icosahedral symmetry.

Geometrically, the dodecahedron has twelve faces, thirty edges and twenty vertices. Three pentagonal faces meet at each vertex, giving the vertex configuration 5.5.5. The Schläfli symbol for the regular dodecahedron is {5,3}, indicating pentagonal faces with three meeting at every vertex. The dihedral angle between adjacent faces is approximately 116.565 degrees. The shape does not tile three-dimensional space on its own.

The symmetry of the regular dodecahedron is complex: the rotational symmetry group is isomorphic to the alternating group A5 and has order 60, while the full symmetry group including reflections has order 120. This icosahedral symmetry underlies many of its geometric properties and connections to other regular solids.

A standard way to describe its vertices is by coordinates using the golden ratio φ = (1+√5)/2. Centered at the origin, the 20 vertices can be taken as all permutations of (±1, ±1, ±1) together with (0, ±1/φ, ±φ), (±1/φ, ±φ, 0), and (±φ, 0, ±1/φ). With these coordinates, the edges are all of equal length, forming the regular dodecahedron.

In summary, the dodecahedron is a highly symmetric regular polyhedron with twelve pentagonal faces, rich geometric relationships with the icosahedron, and a standard coordinate representation based on the golden ratio.