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Convex regular icosahedron A tensegrity icosahedron. In geometry, an icosahedron (/ ˌ aɪ k ɒ s ə ˈ h iː d r ən,-k ə-,-k oʊ-/ or / aɪ ˌ k ɒ s ə ˈ h iː d r ən / [1]) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'.
The icosahedron has a large number of stellations. Coxeter et al. (1938) in their work stated fifty-nine stellations were identified for the regular icosahedron. [22] Regular icosahedron itself is the first stellation of an icosahedron, and the subsequent stellation obtained by radiating spikes from the faces of a regular icosahedron.
Icosahedral symmetry fundamental domains A soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry. Rotations and reflections form the symmetry group of a great icosahedron. In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron.
The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.
It can be seen as the compound of an icosahedron and dodecahedron.It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual.. It has icosahedral symmetry (I h) and the same vertex arrangement as a rhombic triacontahedron.
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3, 5 ⁄ 2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.
triakis icosahedron : truncated dodecahedron: 60 90 32 isosceles triangle: pentakis dodecahedron : truncated icosahedron: 60 90 32 isosceles triangle: deltoidal hexecontahedron : rhombicosidodecahedron: 60 120 62 kite: disdyakis triacontahedron or hexakis icosahedron truncated icosidodecahedron: 120 180 62 scalene triangle
Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, and perhaps only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron. [3]