dodekaedron

The dodekaedron, or dodecahedron, is a regular polyhedron and one of the five Platonic solids. It is composed of twelve regular pentagonal faces, twenty vertices, and thirty edges. At each vertex three pentagons meet. The Schläfli symbol is {5,3}, indicating pentagonal faces with three meeting at each vertex. Its dual polyhedron is the icosahedron.

The full symmetry group of the regular dodecahedron is the icosahedral group Ih, of order 120. The dodecahedron and icosahedron form a dual pair: the centers of the dodecahedron’s faces correspond to the vertices of an icosahedron, and vice versa.

In practice, the dodecahedron appears in geometry, art, and education, and is also used as a 12-faced die in some tabletop games. It is often presented with coordinate models that exploit the golden ratio.

A common centered coordinate model uses the golden ratio φ = (1+√5)/2. The vertices can be taken as the eight points (±1, ±1, ±1) together with the twelve points obtained by permuting (0, ±1/φ, ±φ). With edge length a, the circumscribed radius is R = (a/4)√3 (1+√5). This reflects the close relationship between the dodecahedron and the icosahedron.

The name derives from Greek dodeka- “twelve” and hedron “face.” The shape has been known since antiquity and figures prominently in discussions of symmetry and geometry.

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