icosahedrons

The icosahedron is a polyhedron with 20 faces, all of which are equilateral triangles. It is one of the five Platonic solids, characterized by its high degree of symmetry, with 12 vertices and 30 edges. The regular icosahedron has the Schläfli symbol {3,5}, indicating triangular faces with five meeting at each vertex. Its dual polyhedron is the dodecahedron, which has 12 pentagonal faces.

In a regular icosahedron, five triangular faces meet at every vertex. The dihedral angle between adjacent faces is approximately 138.19 degrees. The rotational symmetry group is isomorphic to A5 and has order 60; including reflections, the full symmetry group has order 120.

A common coordinate realization uses the golden ratio φ = (1+√5)/2. The twelve vertices can be placed at (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1), up to overall scale. For edge length a, the circumradius is R = (a/4)√(10+2√5) and the volume is V = (5(3+√5)/12) a^3.

The icosahedron appears in a variety of contexts, including molecular chemistry and virology, where icosahedral symmetry describes virus capsids. It also features in architecture, games, and design, notably in icosahedral dice used in tabletop gaming.