Ingeometry, anicosahedron (/ˌaɪkɒsəˈhiːdrən,-kə-,-koʊ-/ or/aɪˌkɒsəˈhiːdrən/[1]) is apolyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" (/-drə/) or "icosahedrons".
There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non-stellated)regular icosahedron—one of thePlatonic solids—whose faces are 20equilateral triangles.
There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20equilateral triangle faces with five meeting at each of its twelve vertices. Both haveicosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called agreat icosahedron.
The convex regular icosahedron is usually referred to simply as theregular icosahedron, one of the five regularPlatonic solids, and is represented by itsSchläfli symbol {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex.
Itsdual polyhedron is theregular dodecahedron {5, 3} having three regular pentagonal faces around each vertex.
Thegreat icosahedron is one of the four regular starKepler-Poinsot polyhedra. ItsSchläfli symbol is {3,5/2}. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is apentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges.
Itsdual polyhedron is thegreat 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 bookThe Fifty-Nine Icosahedra, Coxeter et al. enumerated 59 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.
| Notablestellations 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 relatedpolyhedra andcompounds withicosahedral symmetry. | |||||||||
| Pyritohedral and tetrahedral symmetries | |||||
|---|---|---|---|---|---|
| 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 | Th, [4,3+], (3*2), order 24 | ||||
| Rotation group | Td, [3,3]+, (332), order 12 | ||||
| Dual polyhedron | Pyritohedron | ||||
| Properties | convex | ||||
 Net | |||||
| |||||
Aregular icosahedron can be distorted or marked up as a lowerpyritohedral symmetry,[2][3] and is called asnub octahedron,snub tetratetrahedron,snub tetrahedron, andpseudo-icosahedron.[4] This can be seen as analternatedtruncated octahedron. If all the triangles areequilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently.
Pyritohedral symmetry has the symbol (3*2), [3+,4], 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 congruentisosceles triangles.
These symmetries offerCoxeter diagrams: and respectively, each representing the lower symmetry to theregular icosahedron
, (*532), [5,3]icosahedral symmetry of order 120.
TheCartesian 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 thetruncated octahedron withalternated vertices deleted.
This construction is called asnub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (ϕ, 1, 0), whereϕ is thegolden ratio.[3]
In Jessen's icosahedron, sometimes calledJessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently so that the figure is non-convex and hasrightdihedral angles.
It isscissors congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.
A regular icosahedron is topologically identical to acuboctahedron with its 6 square faces bisected on diagonals with pyritohedral symmetry. The icosahedra with pyritohedral symmetry constitute an infinite family of polyhedra which include the cuboctahedron, regular icosahedron, Jessen's icosahedron, anddouble cover octahedron. Cyclical kinematic transformations among the members of this family exist.
Therhombic icosahedron is azonohedron made up of 20 congruent rhombs. It can be derived from therhombic triacontahedron by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is notface-transitive.
Common icosahedra with pyramid and prism symmetries include:
SeveralJohnson solids are icosahedra:[5]
| 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 |
