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| Contents |
| Icosahedron | |
|---|---|
Click on picture for large version. Click for spinning version. | |
| Type | Platonic |
| Face polygon | triangle |
| Faces | 20 |
| Edges | 30 |
| Vertices | 12 |
| Faces per vertex | 5 |
| Vertices per face | 3 |
| Symmetry group | icosahedral (Ih) |
| Dual polyhedron | dodecahedron |
| Properties | regular, convex |
An icosahedron [ˌaikəsə'hiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces. The faces of a regular icosahedron are equilateral triangles. [Etymology: 16th Century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], "icosa'hedral adjective
In geometry, the regular icosahedron is one of the five Platonic solids. It is a convex regular polyhedron composed of twenty triangular faces, with five meeting at each of the twelve vertices. It has 30 edges. Its dual polyhedron is the dodecahedron.
The area A and the volume V of a regular icosahedron of edge length a are:
There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cubeThe snub cube or snub cuboctahedron is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a cube or an octahedron. The snub cube has 38 faces, of which 6 are squares and the other 32 are equilateral triangles. and snub dodecahedronThe snub dodecahedron or snub icosidodecahedron is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a dodecahedron or an icosahedron. The snub dodecahedron has 92 faces, of which 12 are pentagons and the other, including some forms which are chiral and some with Th-symmetry, i.e. have different planes of symmetry than the tetrahedron. The icosahedron has a large number of stellations, including one of the Kepler-Poinsot solids and some of the regular compounds, which could be discussed here.
Many viruses, including HIV and herpes, have the shape of an icosahedron. Viral structures are built of repeated identical protein subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome.
In several roleplaying games, such as D&D, the twenty-sided dice (short d20) plays a vital role in determining success or failure of an action.