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In geometry, an icosahedron (// or //) is a polyhedron with 20 faces. The name comes from Greek είκοσι (eíkosi), meaning "twenty", and έδρα (hédra), meaning "seat". The plural can be either "icosahedra" (-//) or "icosahedrons".
Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. It is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure.
In their book The fifty nine icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron.
Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them.
Other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are often referred to as such.
|Notable stellations of the icosahedron|
|Regular||Uniform duals||Regular compounds||Regular star||Others|
|(Convex) icosahedron||Small triambic icosahedron||Medial triambic icosahedron||Great triambic icosahedron||Compound of five octahedra||Compound of five tetrahedra||Compound of ten tetrahedra||Great icosahedron||Excavated dodecahedron||Final stellation|
|The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.|
|Pyritohedral and tetrahedral symmetries|
Four views of an icosahedron with tetrahedral symmetry, with eight equilateral triangles (red and yellow), and 12 blue isosceles triangles. Yellow and red triangles are the same color in pyritohedral symmetry.
|Coxeter diagrams|| (pyritohedral) |
|Edges||30 (6 short + 24 long)|
|Symmetry group||Th, [4,3+], (3*2), order 24|
|Rotation group||Td, [3,3]+, (332), order 12|
A regular icosahedron can be constructed with pyritohedral symmetry, and is called a snub octahedron or snub tetratetrahedron or snub tetrahedron. this can be seen as an alternated truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently.
Pyritohedral symmetry has the symbol (3*2), [4,3+], with order 24. Tetrahedral symmetry has the symbol (332), [3,3]+, with order 12. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles.
The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.
It is scissors congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.
The rhombic icosahedron is a zonohedron made up of 20 congruent rhombs. It can be derived from the rhombic triacontahedron by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is not face-transitive.
20 triangles can also be arranged with tetrahedral symmetry (332), [3,3]+, seen as the 8 triangles marked (colored) in alternating pairs of four, with order 12. These symmetries offer Coxeter diagrams: and respectfully, each representing the lower symmetry to the regular icosahedron , (*532), [5,3] icosahedral symmetry of order 120.
Common icosahedra with pyramid and prism symmetries include:
Gyroelongated triangular cupola
Elongated triangular orthobicupola
Elongated triangular gyrobicupola