Regular dodecahedron

A Platonic solid with 12 congruent regular pentagonal faces, 30 edges, and 20 vertices. Its identical faces and vertices give it full icosahedral symmetry.

Regular pentagon: A five-sided polygon with equal sides and equal interior angles. Each of the solid’s 12 faces is a regular pentagon.
Dihedral angle: The angle between two planes, commonly measured inside a polyhedron where adjacent faces meet. The angle between neighboring pentagonal faces determines the dodecahedron’s sharpness.
Plato: An ancient Greek philosopher whose dialogues helped establish the tradition of associating regular solids with elements. The dodecahedron became one of the five solids discussed in the Platonic tradition.
Quasicrystal: A solid whose atomic arrangement has long-range order but lacks translational periodicity. Icosahedral symmetry, shared with the dodecahedron, occurs in some quasicrystals.
Platonic solid: A convex polyhedron whose faces are congruent regular polygons and whose vertices are equivalent. The dodecahedron is one of the five solids satisfying this definition.
Golden ratio: The irrational number (1 + √5)/2, arising in pentagons and many geometric constructions. Pentagonal proportions make the golden ratio appear in the dodecahedron’s coordinates and diagonals.
Timaeus: A Platonic dialogue that links four regular solids to elements and treats the dodecahedron separately. Its account helped give the dodecahedron a distinctive cosmological role.
Dodecahedral space: A proposed spherical 3-manifold obtained by identifying opposite faces of a spherical dodecahedron. The solid’s face-pairing geometry defines this model of a closed universe.
Euler characteristic: A topological invariant that, for convex polyhedra, gives the relation vertices minus edges plus faces equals two. Its counts satisfy 20 − 30 + 12 = 2.
Schläfli symbol: A notation {p,q} for a regular polyhedron whose faces are p-gons, with q faces meeting at each vertex. The dodecahedron is denoted {5,3}: pentagons meet three at each vertex.

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IcosahedronCompared with: It is the icosahedron's dual and exchanges the roles of faces and vertices.