dodecahedrons

A dodecahedron is any polyhedron with twelve faces. The most familiar example is the regular dodecahedron, one of the five Platonic solids, whose faces are twelve regular pentagons meeting three at each vertex. The regular dodecahedron has 12 faces, 30 edges, and 20 vertices, satisfying Euler's formula V − E + F = 2, and is denoted by the Schläfli symbol {5,3}. Its dual polyhedron is the icosahedron {3,5}.

The regular dodecahedron exhibits icosahedral symmetry: the full symmetry group has order 120 (including reflections), while the rotational symmetry subgroup has order 60 and is isomorphic to A5. Coordinates for a centered regular dodecahedron can be expressed using the golden ratio; it can also be constructed from three mutually orthogonal golden rectangles. Variants include the nonconvex Kepler–Poinsot dodecahedral star polyhedra and numerous irregular dodecahedra with twelve nonregular faces.

Other well-known twelve-faced solids include the rhombic dodecahedron, a Catalan solid with rhombic faces that serves as the Voronoi cell of the face-centered cubic lattice and tiles space, and the trapezoidal and deltoidal dodecahedra which appear in crystallography and architectural design. Dodecahedral forms appear in mathematics, art, crystallography, and gaming (notably twelve-sided dice), valued for their symmetry and aesthetic properties.