Dodecahedron

The dodecahedron is a regular polyhedron with twelve faces, all regular pentagons. Three pentagonal faces meet at each vertex. It is one of the five Platonic solids and has Schläfli symbol {5,3}. The dual polyhedron is the icosahedron.

In summary, the dodecahedron has 12 faces, 20 vertices, and 30 edges. Its vertex figure is a triangle, reflecting that three faces meet at every vertex. The edge length is common to all edges, and the polyhedron satisfies Euler’s formula V − E + F = 2.

The dihedral angle between adjacent faces is arccos(-√5/5), approximately 116.565 degrees. For a regular dodecahedron with edge length a, the circumscribed radius (distance from the center to a vertex) is R = (a/4)√3(1+√5) ≈ 1.4013 a.

Symmetry plays a central role in its geometry. The rotational symmetry group is isomorphic to A5 and has order 60; the full symmetry group, including reflections, has order 120. This symmetry is shared with the icosahedron, its dual.

Coordinates for a centered dodecahedron can be expressed using the golden ratio φ = (1+√5)/2. One common construction places vertices at (±1, ±1, ±1) and at permutations of (0, ±1/φ, ±φ), scaled as needed to fix edge length.

In mathematics, the dodecahedron appears in graph theory as the dodecahedral graph (20 vertices, 30 edges), in crystallography, and in geometric modeling. It is a classic example of a regular polyhedron with rich connections to symmetry and the golden ratio.