Icosahedron

The icosahedron is a regular polyhedron with 20 equilateral triangular faces, 12 vertices, and 30 edges. It is one of the five Platonic solids and is described by the Schläfli symbol {3,5}, indicating triangular faces with five meeting at each vertex.

In a regular icosahedron, the arrangement of faces around each vertex is highly symmetric: five triangles meet at every vertex. The solid possesses 60 rotational symmetries and 120 total symmetries when reflections are included. It satisfies Euler’s formula, with V − E + F = 2, where V = 12, E = 30, and F = 20.

Dual to the icosahedron is the dodecahedron: the 12 vertices of the icosahedron correspond to the 12 faces of a dodecahedron, and the 20 faces of the icosahedron correspond to the 20 vertices of a dodecahedron. This dual relationship links their symmetry and geometric properties.

A common coordinate model places the 12 vertices at (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1), where φ = (1 + √5)/2 is the golden ratio. With appropriate scaling, these coordinates yield a regular icosahedron of any desired edge length.

For an edge length a, the circumscribed radius R of a regular icosahedron is R = (a/4)√(10 + 2√5), which is approximately 0.95106a. Other linear measures follow standard formulas for regular polyhedra.

Applications and appearances include gaming dice (notably the d20) and geometric models in architecture, art, and design. The icosahedron’s high degree of symmetry and uniform faces make it a classic subject in both mathematics and visual culture.