In geometry, an icosahedron ( /ˌaɪkɒsəˈhiːdrən,kə,koʊ/ or /aɪˌkɒsəˈhiːdrən/ ^{ [1] }) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi), meaning 'twenty',andἕδρα (hédra), meaning 'seat'. The plural can be either "icosahedra" ( /drə/ ) or "icosahedrons".
There are infinitely many nonsimilar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, nonstellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.
Convex regular icosahedron  Great icosahedron 
There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a great icosahedron.
The convex regular icosahedron is usually referred to simply as the regular icosahedron, one of the five regular Platonic solids, and is represented by its Schläfli symbol {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex.
Its 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}. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges.
Its 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 FiftyNine 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  

Coxeter diagrams  
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 distorted or marked up as a lower pyritohedral symmetry,^{ [2] } and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron, and pseudoicosahedron. 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), [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 congruent isosceles triangles.
These symmetries offer Coxeter diagrams:
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.^{ [2] }
In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently so that the figure is nonconvex and 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.
Common icosahedra with pyramid and prism symmetries include:
Several Johnson solids are icosahedra:^{ [3] }
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 
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertextransitive but also edgetransitive. It is the only radially equilateral convex polyhedron.
In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most sides.
In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the threedimensional analogs of polygonal compounds such as the hexagram.
In threedimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent, regular, polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria:
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from Latin stella, "star".
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edgetransitive, vertextransitive and facetransitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
In geometry, the great icosahedron is one of four KeplerPoinsot polyhedra, with Schläfli symbol {3,^{5}⁄_{2}} and CoxeterDynkin diagram of
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U_{54}. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r{3,^{5}⁄_{2}}. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).
The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edgetruncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.
Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a nonconvex polyhedron with the same number of vertices, edges and faces as the regular icosahedron. Its faces meet only in right angles, even though they cannot all be made parallel to the coordinate planes. It is named for Børge Jessen who investigated it in 1967.
In geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.
The pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the truncated rhombic triacontahedron.
In geometry, a tetrahedrally diminished dodecahedron is a topologically selfdual polyhedron made of 16 vertices, 30 edges, and 16 faces.
In geometry, chamfering or edgetruncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.