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Regular Icosahedron  

(Click here for rotating model)  
Type  Platonic solid 
Elements  F = 20, E = 30 V = 12 (χ = 2) 
Faces by sides  20{3} 
Schläfli symbols  {3,5} 
s{3,4}, sr{3,3}  
Wythoff symbol  5  2 3 
Coxeter diagram  
Symmetry  I_{h}, H_{3}, [5,3], (*532) 
Rotation group  I, [5,3]^{+}, (532) 
References  U_{22}, C_{25}, W_{4} 
Properties  Regular convex deltahedron 
Dihedral angle  138.189685° = arccos(√5/3) 
3.3.3.3.3 (Vertex figure)  Dodecahedron (dual polyhedron) 
Net 
In geometry, an icosahedron (/ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/) is a polyhedron with 20 triangular faces, 30 edges and 12 vertices. A regular icosahedron with identical equilateral faces is often meant because of its geometrical significance as one of the five Platonic solids.
It has five triangular faces meeting at each vertex. It can be represented by its vertex figure as 3.3.3.3.3 or 3^{5}, and also by Schläfli symbol {3,5}. It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex.
A regular icosahedron is a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations.
The name comes from the Greek: εικοσάεδρον, from είκοσι (eíkosi) "twenty" and εδρα (hédra) "seat". The plural can be either "icosahedrons" or "icosahedra" (/drə/).
If the edge length of a regular icosahedron is a, the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is
and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is
while the midradius, which touches the middle of each edge, is
where φ (also called τ) is the golden ratio.
The surface area A and the volume V of a regular icosahedron of edge length a are:
The latter is F=20 times the volume of a general tetrahedron with apex at the center of the inscribed sphere, where the volume of the tetrahedron is one third times the base area √3a^{2}/4 times its height r_{i}.
The volume filling factor of the circumscribed sphere is
The following Cartesian coordinates define the vertices of an icosahedron with edgelength 2, centered at the origin:^{[1]}
where φ = (1 + √5) / 2 is the golden ratio (also written τ). Note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings.
If the original icosahedron has edge length 1, its dual dodecahedron has edge length , one divided by the golden ratio.
The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly subdividing each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound.
The locations of the vertices of a regular icosahedron can be described using spherical coordinates, for instance as latitude and longitude. If two vertices are taken to be at the north and south poles (latitude ±90°), then the other ten vertices are at latitude ±arctan(1/2) ≈ ±26.57°. These ten vertices are at evenly spaced longitudes (36° apart), alternating between north and south latitudes.
This scheme takes advantage of the fact that the regular icosahedron is a pentagonal gyroelongated bipyramid, with D_{5d} dihedral symmetry—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal antiprism.
The icosahedron has three special orthogonal projections, centered on a face, an edge and a vertex:
Centered by  Face  Edge  Vertex 

Coxeter plane  A_{2}  A_{3}  H_{3} 
Graph  
Projective symmetry  [6]  [2]  [10] 
Graph  Face normal  Edge normal  Vertex normal 
Icosahedron H_{3} Coxeter plane  6orthoplex D_{6} Coxeter plane 
This construction can be geometrically seen as the 12 vertices of the 6orthoplex projected to 3 dimensions. This represents a geometric folding of the D_{6} to H_{3} Coxeter groups: Seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. 
The following construction of the icosahedron avoids tedious computations in the number field necessary in more elementary approaches.
The existence of the icosahedron amounts to the existence of six equiangular lines in . Indeed, intersecting such a system of equiangular lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of a regular icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system.
In order to construct such an equiangular system, we start with this 6×6 square matrix:
A straightforward computation yields A^{2} = 5I (where I is the 6×6 identity matrix). This implies that A has eigenvalues and , both with multiplicity 3 since A is symmetric and of trace zero.
The matrix induces thus a Euclidean structure on the quotient space which is isomorphic to since the kernel of has dimension 3. The image under the projection of the six coordinate axes in forms thus a system of six equiangular lines in intersecting pairwise at a common acute angle of . Orthogonal projection of ±v_{1}, ..., ±v_{6} onto the eigenspace of A yields thus the twelve vertices of the icosahedron.
A second straightforward construction of the icosahedron uses representation theory of the alternating group A_{5} acting by direct isometries on the icosahedron.
The rotational symmetry group of the regular icosahedron is isomorphic to the alternating group on five letters. This nonabelian simple group is the only nontrivial normal subgroup of the symmetric group on five letters. Since the Galois group of the general quintic equation is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and nonabelian, the general quintic equation does not have a solution in radicals. The proof of the Abel–Ruffini theorem uses this simple fact, and Felix Klein wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation, (Klein 1888). See icosahedral symmetry: related geometries for further history, and related symmetries on seven and eleven letters.
The full symmetry group of the icosahedron (including reflections) is known as the full icosahedral group, and is isomorphic to the product of the rotational symmetry group and the group C_{2} of size two, which is generated by the reflection through the center of the icosahedron.
Pseudoicosahedron  

Four views of the pseudoicosahedron, with eight equilateral triangles (red and yellow), and 12 blue isosceles triangles.  
Coxeter diagrams  (pyritohedral) (tetrahedral) 
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 pseudoicosahedron is an icosahedron with pyritohedral symmetry. The 20 triangular faces are divided into two groups of 8 equilateral triangles and 12 isosceles triangles. If all the triangles are equilateral triangles, the symmetry can also be distinguished by coloring the 8 and 12 triangle sets differently. The pseudoicosahedron is an alternated truncated octahedron.^{[citation needed]}
If the 8 equilateral triangles are geometrically identical, the pseudoicosahedron has pyritohedral symmetry (3*2), [4,3^{+}], with order 24. A lower tetrahedral symmetry (332), [3,3]^{+}, exists as well, seen as the 8 triangles marked (colored) in alternating pairs of four, with order 12. These symmetries offer CoxeterDynkin diagrams: and respectfully, 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.^{[8]}
The regular star polyhedron, great icosahedron, , or can be considered a special case of parametric pseudoicosahedron with a vertex figure that overlaps into a regular pentagram. The tetrahedral symmetry of this form is called a retrosnub tetrahedron:
Iron pyrites have been observed to have formed crystals in the form of pseudoicosahedra.^{[9]}
According to specific rules defined in the book The FiftyNine Icosahedra, 59 stellations were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regular Kepler–Poinsot polyhedron. Three are regular compound polyhedra.^{[10]}
The faces of the icosahedron extended outwards as planes intersect, defining regions in space as shown by this stellation diagram of the intersections in a single plane.  
There are distortions of the icosahedron that, while no longer regular, are nevertheless vertexuniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cube and snub dodecahedron, including some forms which are chiral and some with T_{h}symmetry, i.e. have different planes of symmetry from the tetrahedron. The icosahedron has a large number of stellations, including one of the Kepler–Poinsot polyhedra and some of the regular compounds, which could be discussed here.
The icosahedron is unique among the Platonic solids in possessing a dihedral angle not less than 120°. Its dihedral angle is approximately 138.19°. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in n dimensions, at least three facets must meet at a peak and leave a positive defect for folding in nspace). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semiregular polychora (for example the snub 24cell), just as hexagons can be used as faces in semiregular polyhedra (for example the truncated icosahedron). Finally, nonconvex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral 120cell, one of the ten nonconvex regular polychora.
An icosahedron can also be called a gyroelongated pentagonal bipyramid. It can be decomposed into a gyroelongated pentagonal pyramid and a pentagonal pyramid or into a pentagonal antiprism and two equal pentagonal pyramids.
There are 3 uniform colorings of the icosahedron. These colorings can be represented as 11213, 11212, 11111, naming the 5 triangular faces around each vertex by their color.
The icosahedron can be considered a snub tetrahedron, as snubification of a regular tetrahedron gives a regular icosahedron having chiral tetrahedral symmetry. It can also be constructed as an alternated truncated octahedron, having pyritohedral symmetry. The pyritohedral symmetry version is sometimes called a pseudoicosahedron, and is dual to the pyritohedron.
Name  Regular icosahedron  snub octahedron  Snub tetratetrahedron  Pentagonal gyroelongated bipyramid 

CoxeterDynkin  
Schläfli symbol  {3,5}  s{3,4}  sr{3,3}  
Wythoff symbol  5  3 2   3 3 2  
Symmetry  I_{h} [5,3] (*532)  T_{h} [3^{+},4] (3*2)  T [3,3]^{+} (332)  D_{5d} [2^{+},10] (2*5) 
Symmetry order  60  24  12  10 
Uniform coloring  (11111)  (11212)  (11213)  (11122)&(22222) 
The icosahedron can be transformed by a truncation sequence into its dual, the dodecahedron:
Symmetry: [5,3], (*532)  [5,3]^{+}, (532)  

{5,3}  t{5,3}  r{5,3}  2t{5,3}=t{3,5}  2r{5,3}={3,5}  rr{5,3}  tr{5,3}  sr{5,3} 
Duals to uniform polyhedra  
V5.5.5  V3.10.10  V3.5.3.5  V5.6.6  V3.3.3.3.3  V3.4.5.4  V4.6.10  V3.3.3.3.5 
As a snub tetrahedron, and alternation of a truncated octahedron it also exists in the tetrahedral and octahedral symmetry families:
Symmetry: [3,3], (*332)  [3,3]^{+}, (332)  

{3,3}  t{3,3}  r{3,3}  t{3,3}  {3,3}  rr{3,3}  tr{3,3}  sr{3,3} 
Duals to uniform polyhedra  
V3.3.3  V3.6.6  V3.3.3.3  V3.6.6  V3.3.3  V3.4.3.4  V4.6.6  V3.3.3.3.3 
Symmetry: [4,3], (*432)  [4,3]^{+} (432)  [1^{+},4,3] = [3,3] (*332)  [3^{+},4] (3*2)  

{4,3}  t{4,3}  r{4,3} r{3^{1,1}}  t{3,4} t{3^{1,1}}  {3,4} {3^{1,1}}  rr{4,3} s_{2}{3,4}  tr{4,3}  sr{4,3}  h{4,3} {3,3}  h_{2}{4,3} t{3,3}  s{4,3} s{3^{1,1}} 
=  =  =  = or  = or  =  
Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
Finite  Euclidean  Compact hyperbolic  Paracompact  

{3,2}  {3,3}  {3,4}  {3,5}  {3,6}  {3,7}  {3,8}  {3,9}  ...  (3,∞} 
The regular icosahedron, seen as a snub tetrahedron, is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.
Symmetry n32 [n,3]^{+}  Spherical  Euclidean  Compact hyperbolic  Paracompact  

232 [2,3]^{+} D_{3}  332 [3,3]^{+} T  432 [4,3]^{+} O  532 [5,3]^{+} I  632 [6,3]^{+} P6  732 [7,3]^{+}  832 [8,3]^{+}...  ∞32 [∞,3]^{+}  
Snub figure  3.3.3.3.2  3.3.3.3.3  3.3.3.3.4  3.3.3.3.5  3.3.3.3.6  3.3.3.3.7  3.3.3.3.8  3.3.3.3.∞ 
Coxeter Schläfli  sr{2,3}  sr{3,3}  sr{4,3}  sr{5,3}  sr{6,3}  sr{7,3}  sr{8,3}  sr{∞,3} 
Snub dual figure  V3.3.3.3.2  V3.3.3.3.3  V3.3.3.3.4  V3.3.3.3.5  V3.3.3.3.6  V3.3.3.3.7  V3.3.3.3.8  V3.3.3.3.∞ 
Coxeter 
Spherical  Hyperbolic tilings  

{2,5}  {3,5}  {4,5}  {5,5}  {6,5}  {7,5}  {8,5}  ...  {∞,5} 
The icosahedron shares its vertex arrangement with three Kepler–Poinsot solids. The great dodecahedron also has the same edge arrangement.
Picture  Great dodecahedron  Small stellated dodecahedron  Great icosahedron 

CoxeterDynkin 
The icosahedron can tessellate hyperbolic space in the order3 icosahedral honeycomb, with 3 icosahedra around each edge, 12 icosahedra around each vertex, with Schläfli symbol {3,5,3}. It is one of four regular tessellations in the hyperbolic 3space.
It is shown here as an edge framework in a Poincaré disk model, with one icosahedron visible in the center. 
Many viruses, e.g. herpes virus, have icosahedral shells.^{[11]} Viral structures are built of repeated identical protein subunits known as capsomeres, and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome.
Various bacterial organelles with an icosahedral shape were also found.^{[12]} The icosahedral shell encapsulating enzymes and labile intermediates are built of different types of proteins with BMC domains.
In 1904, Ernst Haeckel described a number of species of Radiolaria, including Circogonia icosahedra, whose skeleton is shaped like a regular icosahedron. A copy of Haeckel's illustration for this radiolarian appears in the article on regular polyhedra.
The closocarboranes are chemical compounds with shape very close to isosahedron. Icosahedral twinning also occurs in crystals, especially nanoparticles.
Many borides and allotropes of boron contain boron B_{12} icosahedron as a basic structure unit.
In several roleplaying games, such as Dungeons & Dragons, the twentysided die (d20 for short) is commonly used in determining success or failure of an action. This die is in the form of a regular icosahedron. It may be numbered from "0" to "9" twice (in which form it usually serves as a tensided die, or d10), but most modern versions are labeled from "1" to "20". See d20 System.
An icosahedron is the threedimensional game board for Icosagame, formerly known as the Ico Crystal Game.
An icosahedron is used in the board game Scattergories to choose a letter of the alphabet. Six letters are omitted (Q, U, V, X, Y, and Z).
Inside a Magic 8Ball, various answers to yesno questions are inscribed on a regular icosahedron.
R. Buckminster Fuller and Japanese cartographer Shoji Sadao^{[13]} designed a world map in the form of an unfolded icosahedron, called the Fuller projection, whose maximum distortion is only 2%.
The "Sol de la Flor" light shade consists of twenty panels, which meet at the corners of an icosahedron in rosettes resembling the overlapping petals of a frangipani flower.
If each edge of an icosahedron is replaced by a one ohm resistor, the resistance between opposite vertices is 0.5 ohms, and that between adjacent vertices 11/30 ohms.^{[14]}
The company logo of the TDK Corporation contains a geometric figure which is based on the stellation diagram of the icosahedron.
A icosahedron was used for a logo for the Australian TV company; Grundy Television.
The skeleton of the icosahedron—the vertices and edges—form a graph. The high degree of symmetry of the polygon is replicated in the properties of this graph, which is distancetransitive, distanceregular, and symmetric. The automorphism group has order 120. The vertices can be colored with 4 colors, the edges with 5 colors, and the diameter is 3.^{[15]}
The icosahedral graph is Hamiltonian: there is a cycle containing all the vertices. It is also a planar graph.
Wikimedia Commons has media related to Icosahedron. 
Wikisource has the text of the 1911 Encyclopædia Britannica article Icosahedron. 
Look up icosahedron in Wiktionary, the free dictionary. 


Notable stellations of the icosahedron  
Regular  Uniform duals  Regular compounds  Regular star  Others  
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. 