# Icosahedron

Regular Icosahedron

TypePlatonic solid
ElementsF = 20, E = 30
V = 12 (χ = 2)
Faces by sides20{3}
Schläfli symbol{3,5}
Wythoff symbol5 | 2 3
Coxeter diagram
SymmetryIh, H3, [5,3], (*532)
Rotation groupI, [5,3]+, (532)
ReferencesU22, C25, W4
PropertiesRegular convex deltahedron
Dihedral angle138.189685° = arccos(-√5/3)

3.3.3.3.3
(Vertex figure)

Dodecahedron
(dual polyhedron)

Net

In geometry, an icosahedron ( or ) 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, all regular polyhedrons including the regular tetrahedron, the cube or regular hexahedron, the regular octahedron, and the regular dodecahedron with 4, 6, 8 and 12 identical faces, respectively.

It has five triangular faces meeting at each vertex. It can be represented by its vertex figure as 3.3.3.3.3 or 35, 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.

The name comes from the Greek: εικοσάεδρον, from είκοσι (eíkosi) "twenty" and ἕδρα (hédra) "seat". The plural can be either "icosahedrons" or "icosahedra" (-).

## Dimensions

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

$r_u = \frac{a}{2} \sqrt{\varphi \sqrt{5}} = \frac{a}{4} \sqrt{10 +2\sqrt{5}} = a\sin\frac{2\pi}{5} \approx 0.9510565163 \cdot a$

and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is

$r_i = \frac{\varphi^2 a}{2 \sqrt{3}} = \frac{\sqrt{3}}{12} \left(3+ \sqrt{5} \right) a \approx 0.7557613141\cdot a$

while the midradius, which touches the middle of each edge, is

$r_m = \frac{a \varphi}{2} = \frac{1}{4} \left(1+\sqrt{5}\right) a = a\cos\frac{\pi}{5} \approx 0.80901699\cdot a$

where φ (also called τ) is the golden ratio.

## Area and volume

The surface area A and the volume V of a regular icosahedron of edge length a are:

$A = 5\sqrt{3}a^2 \approx 8.66025404a^2,$
$V = \frac{5}{12} (3+\sqrt5)a^3 \approx 2.18169499a^3.$

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 √3a2/4 times its height ri.

The volume filling factor of the circumscribed sphere is

$f=V/(4 \pi r_u^3/3) = \frac{20(3+\surd 5)}{(2\surd 5+10)^{3/2}\pi}\approx 0.6054613829.$

## Cartesian coordinates

Icosahedron vertices form three orthogonal golden rectangles

The following Cartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin:[1]

(0, ±1, ±φ)
(±1, ±φ, 0)
φ, 0, ±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 $\tfrac{\sqrt{5}-1}{2}$, one divided by the golden ratio.

Model of an icosahedron made with metallic spheres and magnetic connectors

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.

### Spherical coordinates

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 D5d dihedral symmetry—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal antiprism.

## Orthogonal projections

The icosahedron has three special orthogonal projections, centered on a face, an edge and a vertex:

Orthogonal projections
Centered byFaceEdgeVertex
Coxeter planeA2A3H3
Graph
Projective
symmetry
[6][2][10]
Graph
Face normal

Edge normal

Vertex normal

## Other facts

• An icosahedron has 43,380 distinct nets.[2]
• To color the icosahedron, such that no two adjacent faces have the same color, requires at least 3 colors.
• When an icosahedron is inscribed in a sphere, it occupies less of the sphere's volume (60.54%) than a dodecahedron inscribed in the same sphere (66.49%).

## Construction by a system of equiangular lines

 Icosahedron H3 Coxeter plane 6-orthoplex D6 Coxeter plane This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions. This represents a geometric folding of the D6 to H3 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 $\mathbb{Q}[\sqrt{5}]$ necessary in more elementary approaches.

The existence of the icosahedron amounts to the existence of six equiangular lines in $\mathbb R^3$. 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:

$A=\left(\begin{array}{crrrrr} 0&1&1&1&1&1\\ 1&0&1&-1&-1&1\\ 1&1&0&1&-1&-1\\ 1&-1&1&0&1&-1\\ 1&-1&-1&1&0&1\\ 1&1&-1&-1&1&0\end{array}\right).$

A straightforward computation yields A2 = 5I (where I is the 6×6 identity matrix). This implies that A has eigenvalues $\scriptstyle -\sqrt{5}$ and $\scriptstyle \sqrt{5}$, both with multiplicity 3 since A is symmetric and of trace zero.

The matrix $\scriptstyle A+\sqrt{5}I$ induces thus a Euclidean structure on the quotient space $\mathbb R^6/\ker(A+\sqrt{5}I)$ which is isomorphic to $\mathbb R^3$ since the kernel $\ker(A+\sqrt{5}I)$ of $\scriptstyle {A+\sqrt{5}I}$ has dimension 3. The image under the projection $\pi:\mathbb R^6 \longrightarrow \mathbb R^6/\ker(A+\sqrt{5}I)$ of the six coordinate axes $\mathbb R v_1,\dots,\mathbb R v_6$ in $\mathbb R^6$ forms thus a system of six equiangular lines in $\mathbb R^3$ intersecting pairwise at a common acute angle of $\scriptstyle{\arccos}\tfrac{1}{\sqrt{5}}$. Orthogonal projection of ±v1, ..., ±v6 onto the $\scriptstyle \sqrt{5}$-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 A5 acting by direct isometries on the icosahedron.

## Symmetry

The rotational symmetry group of the regular icosahedron is isomorphic to the alternating group on five letters. This non-abelian simple group is the only non-trivial 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 non-abelian, 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 C2 of size two, which is generated by the reflection through the center of the icosahedron.

## Stellations

According to specific rules defined in the book The Fifty-Nine 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.[3]

 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.

## Geometric relations

There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. 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 Th-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 n-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the snub 24-cell), just as hexagons can be used as faces in semi-regular polyhedra (for example the truncated icosahedron). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral 120-cell, one of the ten non-convex 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.

## Uniform colorings and subsymmetries

The icosahedron can be constructed from the tetrahedron by a rotation of the triangular faces, inserting pairs of new triangles in place of the original 6 edges.

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 Coxeter-Dynkin Schläfli symbol Wythoff symbol Regular icosahedron alternatedtruncated octahedron snubtetrahedron Pentagonalgyroelongated bipyramid {3,5} h0,1{3,4} s{3,3} 5 | 3 2 | 3 3 2 Ih [5,3] (*532) Th [3+,4] (3*2) T [3,3]+ (332) D5d [2+,10] (2*5) 60 24 12 10 (11111) (11212) (11213) (11122)&(22222)

## Related polyhedra and polytopes

The icosahedron can be transformed by a truncation sequence into its dual, the dodecahedron:

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532)[5,3]+, (532)
{5,3}t0,1{5,3}t1{5,3}t0,1{3,5}{3,5}t0,2{5,3}t0,1,2{5,3}s{5,3}
Duals to uniform polyhedra
V5.5.5V3.10.10V3.5.3.5V5.6.6V3.3.3.3.3V3.4.5.4V4.6.10V3.3.3.3.5

As a snub tetrahedron, and alternation of a truncated octahedron it also exists in the tetrahedral and octahedral symmetry families:

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332)[3,3]+, (332)
{3,3}t0,1{3,3}t1{3,3}t1,2{3,3}t2{3,3}t0,2{3,3}t0,1,2{3,3}s{3,3}
Duals to uniform polyhedra
V3.3.3V3.6.6V3.3.3.3V3.6.6V3.3.3V3.4.3.4V4.6.6V3.3.3.3.3
Uniform octahedral polyhedra
Symmetry: [4,3], (*432)[4,3]+, (432)[1+,4,3], (*332)[4,3+], (3*2)
{4,3}t0,1{4,3}t1{4,3}t1,2{4,3}{3,4}t0,2{4,3}t0,1,2{4,3}s{4,3}h{4,3}h1,2{4,3}
Duals to uniform polyhedra
V4.4.4V3.8.8V3.4.3.4V4.6.6V3.3.3.3V3.4.4.4V4.6.8V3.3.3.3.4V3.3.3V3.3.3.3.3

This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.

 {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.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+
SphericalEuclideanHyperbolic
232
[2,3]+
D3
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

s{2,3}

s{3,3}

s{4,3}

s{5,3}

s{6,3}

s{7,3}

s{8,3}

s{∞,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
V3.3.3.3.8V3.3.3.3.∞
Coxeter
SphericalHyperbolic 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 Coxeter-Dynkin Great dodecahedron Small stellated dodecahedron Great icosahedron

The icosahedron can tessellate hyperbolic space in the order-3 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 3-space.

 It is shown here as an edge framework in a Poincaré disk model, with one icosahedron visible in the center.

## Uses and natural forms

Twenty-sided die
Gold nanoparticle viewed in electron microscope.
Structure of γ-boron.

### Biology

Many viruses, e.g. herpes virus, have icosahedral shells.[4] 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.[5] 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.

### Chemistry

The closo-carboranes 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 B12 icosahedron as a basic structure unit.

### Toys and games

In several roleplaying games, such as Dungeons & Dragons, the twenty-sided 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 ten-sided die, or d10), but most modern versions are labeled from "1" to "20". See d20 System.

An icosahedron is the three-dimensional 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 8-Ball, various answers to yes-no questions are inscribed on a regular icosahedron.

### Others

R. Buckminster Fuller and Japanese cartographer Shoji Sadao[6] 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.[7]

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.

## As a graph

A planar representation of the icosahedral graph.

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 distance-transitive, distance-regular, 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.[8]

The icosahedral graph is Hamiltonian: there is a cycle containing all the vertices. It is also a planar graph.