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In geometry, an icosahedron (/ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/) 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" (/drə/) or "icosahedrons".
There are many kinds of icosahedron, with some being more symmetrical than others. The most well known is the regular convex or Platonic icosahedron.
The most symmetrical are the two kinds of regular icosahedron. Each has 20 equilateral triangle faces with five meeting at each of its twelve vertices.
The convex regular icosahedron is one of the five regular Platonic solids and is represented by its Schläfli symbol {3, 5}.
The dual polyhedron is the regular dodecahedron {5, 3} having three regular pentagonal faces around each vertex.
The great icosahedron is one of the four regular star KeplerPoinsot polyhedra. Its Schläfli symbol is {3, 5/2}.
The dual polyhedron is the great stellated dodecahedron {5/2, 3}, having three regular star pentagonal faces around each vertex.
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) (tetrahedral) 
Schläfli symbol  s{3,4} sr{3,3} or 
Faces  20 triangles: 8 equilateral 12 isosceles 
Edges  30 (6 short + 24 long) 
Vertices  12 
Symmetry group  T_{h}, [4,3^{+}], (3*2), order 24 
Rotation group  T_{d}, [3,3]^{+}, (332), order 12 
Dual polyhedron  Pyritohedron 
Properties  convex 
Net 
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 signflips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.
This construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (φ, 1, 0), where φ is the golden ratio.^{[1]}
In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently such that the figure is nonconvex. It has right dihedral angles.
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 facetransitive.
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:
Several Johnson solids are icosahedra:^{[2]}
J22  J35  J36  J59  J60  J92 

Gyroelongated triangular cupola  Elongated triangular orthobicupola  Elongated triangular gyrobicupola  Parabiaugmented dodecahedron  Metabiaugmented dodecahedron  Triangular hebesphenorotunda 
16 triangles 3 squares 1 hexagon  8 triangles 12 squares  8 triangles 12 squares  10 triangles 10 pentagons  10 triangles 10 pentagons  13 triangles 3 squares 3 pentagons 1 hexagon 

