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In geometry, an icosahedron ( /ˌaɪkɒsəˈhiːdrən,kə,koʊ/ or /aɪˌkɒsəˈhiːdrən/ ^{ [1] }) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty',andἕδρα (hédra) '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.
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  (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 distorted or marked up as a lower pyritohedral symmetry,^{ [2] }^{ [3] } 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: and respectively, each representing the lower symmetry to the regular icosahedron , (*532), [5,3] icosahedral symmetry of order 120.
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.^{ [3] }
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.
A regular icosahedron is topologically identical to a cuboctahedron 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, and double cover octahedron. Cyclical kinematic transformations among the members of this family exist.
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:^{ [4] }
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 
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 radially equilateral.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, 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 faces.
In geometry, an octahedron is a polyhedron with eight faces. 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.
In geometry, a Platonic solid is a convex, regular polyhedron in threedimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square, it is an Archimedean solid. 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". Stellation is the reciprocal or dual process to faceting.
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.
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 rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirtyfaced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.
In geometry, a uniform polyhedron has regular polygons as faces and is vertextransitive. It follows that all vertices are congruent.
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra, with Schläfli symbol {3,5⁄2} and CoxeterDynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.
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).
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertextransitive and edgetransitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertextransitive.
Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a nonconvex polyhedron with the same numbers of vertices, edges, and faces as the regular icosahedron. It is named for Børge Jessen, who studied it in 1967. In 1971, a family of nonconvex polyhedra including this shape was independently discovered and studied by Adrien Douady under the name sixbeakedshaddock; later authors have applied variants of this name more specifically to Jessen's icosahedron.
In geometry, 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.
The skeleton of a cuboctahedron, considering its edges as rigid beams connected at flexible joints at its vertices but omitting its faces, does not have structural rigidity and consequently its vertices can be repositioned by folding at edges and face diagonals. The cuboctahedron's kinematics is noteworthy in that its vertices can be repositioned to the vertex positions of the regular icosahedron, the Jessen's icosahedron, and the regular octahedron, in accordance with the pyritohedral symmetry of the icosahedron.