Regular icosahedron

Net folding into icosahedron

If the edge length of a regular icosahedron is ${\displaystyle a}$, the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is

$${\displaystyle r_{u}={\frac {a}{2}}{\sqrt {\varphi {\sqrt {5}}}}={\frac {a}{4}}{\sqrt {10+2{\sqrt {5}}}}=a\sin {\frac {2\pi }{5}}\approx 0.951\,056\,5163\cdot a}$$

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

$${\displaystyle r_{i}={\frac {\varphi ^{2}a}{2{\sqrt {3}}}}={\frac {\sqrt {3}}{12}}\left(3+{\sqrt {5}}\right)a\approx 0.755\,761\,3141\cdot a}$$

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

$${\displaystyle r_{m}={\frac {a\varphi }{2}}={\frac {1}{4}}\left(1+{\sqrt {5}}\right)a=a\cos {\frac {\pi }{5}}\approx 0.809\,016\,99\cdot a}$$

where ${\displaystyle \varphi }$ is the golden ratio.

The surface area ${\displaystyle A}$ and the volume ${\displaystyle V}$ of a regular icosahedron of edge length ${\displaystyle a}$ are:

$${\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660\,254\,04a^{2}}$$

$${\displaystyle V={\frac {5}{12}}\left(3+{\sqrt {5}}\right)a^{3}\approx 2.181\,694\,99a^{3}}$$

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 ${\displaystyle {\frac {\sqrt {3}}{4}}a^{2}}$ times its height ${\displaystyle r_{i}}$.

The volume filling factor of the circumscribed sphere is:

$${\displaystyle f={\frac {V}{{\frac {4}{3}}\pi r_{u}^{3}}}={\frac {20\left(3+{\sqrt {5}}\right)}{\left(2{\sqrt {5}}+10\right)^{\frac {3}{2}}\pi }}\approx 0.605\,461\,3829,}$$

compared to 66.49% for a dodecahedron. A sphere inscribed in an icosahedron will enclose 89.635% of its volume, compared to only 75.47% for a dodecahedron.

The midsphere of an icosahedron will have a volume 1.01664 times the volume of the icosahedron, which is by far the closest similarity in volume of any platonic solid with its midsphere. This arguably makes the icosahedron the "roundest" of the platonic solids.

Icosahedron vertices form three orthogonal golden rectangles

The vertices of an icosahedron centered at the origin with an edge length of 2 and a circumradius of ${\textstyle {\sqrt {\varphi ^{2}+1}}\approx 1.902}$ are[2]

$${\displaystyle (0,\pm 1,\pm \varphi )}$$

$${\displaystyle (\pm 1,\pm \varphi ,0)}$$

$${\displaystyle (\pm \varphi ,0,\pm 1)}$$

where ${\textstyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$ is the golden ratio. Taking all permutations of these coordinates (not just cyclic permutations) results in the Compound of two icosahedra.

The vertices of the icosahedron 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 ${\textstyle {\frac {1}{\varphi }}=\varphi -1={\frac {{\sqrt {5}}-1}{2}}}$.

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.

Regular icosahedron and its circumscribed sphere. Vertices of the regular icosahedron lie in four parallel planes, forming in them four equilateral triangles; this was proved by Pappus of Alexandria

The vertices of an icosahedron centred at the origin with an edge length of ${\textstyle {\sqrt {2\cdot \left(1-{\frac {1}{\sqrt {5}}}\right)}}\approx 1.051}$ and a circumradius of 1 (with four points lying in the x-y plane) are

$${\displaystyle \pm (1,0,0)}$$

$${\displaystyle \pm \left({\frac {1}{\sqrt {5}}},{\frac {2}{\sqrt {5}}},0\right)}$$

$${\displaystyle \pm \left({\frac {1}{\sqrt {5}}},{\frac {1-{\frac {1}{\sqrt {5}}}}{2}},\pm {\sqrt {\frac {1+{\frac {1}{\sqrt {5}}}}{2}}}\right)}$$

$${\displaystyle \pm \left({\frac {1}{\sqrt {5}}},{\frac {-1-{\frac {1}{\sqrt {5}}}}{2}},\pm {\sqrt {\frac {1-{\frac {1}{\sqrt {5}}}}{2}}}\right)}$$

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

Orthogonal projections

Centered by	Face	Edge	Vertex
Coxeter plane	A2	A3	H3
Graph
Projective symmetry	[6]	[2]	[10]
Graph	Face normal	Edge normal	Vertex normal

This configuration matrix represents the icosahedron. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole icosahedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[3][4]

${\displaystyle {\begin{bmatrix}{\begin{matrix}12&5&5\\2&30&2\\3&3&20\end{matrix}}\end{bmatrix}}}$

Here is the configuration expanded with k-face elements and k-figures. The diagonal element counts are the ratio of the full Coxeter group H₃, order 120, divided by the order of the subgroup with mirror removal.

H3		k-face	fk	f0	f1	f2	k-fig	Notes
A2		( )	f0	12	5	5	{5}	H3/H2 = 120/10 = 12
A1A1		{ }	f1	2	30	2	{ }	H3/A1A1 = 120/4 = 30
H2		{3}	f2	3	3	20	( )	H3/A2 = 120/6 = 20

The icosahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Orthographic projection	Stereographic projection

• 

  arcctg 1/2 ≈ 63,434949°
  An icosahedron has 43,380 distinct nets.[5]
• To color the icosahedron, such that no two adjacent faces have the same color, requires at least 3 colors.[lower-alpha 1]
• A problem dating back to the ancient Greeks is to determine which of two shapes has larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by Hero, Pappus, and Fibonacci, among others.[6] Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas.[7] Both volumes have formulas involving the golden ratio, but taken to different powers.[8] As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).[9]
• Icosahedral angle - the angle between the closest vertices of the icosahedron, relative to the center of the body of the icosahedron (3D), is equal to the diagonal angle of a double and / or half square (= arcctg 1/2 ≈ 63.434949°)[10]

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 ${\displaystyle \mathbb {Q} [{\sqrt {5}}]}$ necessary in more elementary approaches.

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

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

A straightforward computation yields ${\displaystyle A^{2}=5I}$ (where ${\displaystyle I}$ is the 6 × 6 identity matrix). This implies that ${\displaystyle A}$ has eigenvalues ${\displaystyle -{\sqrt {5}}}$ and ${\displaystyle {\sqrt {5}}}$, both with multiplicity 3 since ${\displaystyle A}$ is symmetric and of trace zero.

The matrix ${\displaystyle A+{\sqrt {5}}I}$ induces thus a Euclidean structure on the quotient space ${\displaystyle \mathbb {R} ^{6}/\operatorname {ker} (A+{\sqrt {5}}I)}$, which is isomorphic to ${\displaystyle \mathbb {R} ^{3}}$ since the kernel ${\displaystyle \operatorname {ker} (A+{\sqrt {5}}I)}$ of ${\displaystyle A+{\sqrt {5}}I}$ has dimension 3. The image under the projection ${\displaystyle \pi$ :\mathbb {R} ^{6}\to \mathbb {R} ^{6}/\operatorname {ker} (A+{\sqrt {5}}I)} of the six coordinate axes in ${\displaystyle \mathbb {R} ^{6}}$ forms a system of six equiangular lines in ${\displaystyle \mathbb {R} ^{3}}$ intersecting pairwise at a common acute angle of ${\displaystyle \arccos 1/{\sqrt {5}}}$. Orthogonal projection of the positive and negative basis vectors of ${\displaystyle \mathbb {R} ^{6}}$ onto the ${\displaystyle {\sqrt {5}}}$-eigenspace of ${\displaystyle A}$ yields thus the twelve vertices of the icosahedron.

A second straightforward construction of the icosahedron uses representation theory of the alternating group ${\displaystyle A_{5}}$ acting by direct isometries on the icosahedron.

Full Icosahedral symmetry has 15 mirror planes (seen as cyan great circles on this sphere) meeting at order ${\displaystyle \pi /5,\pi /3,\pi /2}$ angles, dividing a sphere into 120 triangle fundamental domains. There are 6 5-fold axes (blue), 10 3-fold axes (red), and 15 2-fold axes (magenta). The vertices of the regular icosahedron exist at the 5-fold rotation axis points.

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 1884). 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 ${\displaystyle C_{2}}$ of size two, which is generated by the reflection through the center of the icosahedron.

The icosahedron has a large number of stellations. According to specific rules defined in the book The Fifty-Nine Icosahedra,[11] 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.

21 of 59 stellations

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.

The small stellated dodecahedron, great dodecahedron, and great icosahedron are three facetings of the regular icosahedron. They share the same vertex arrangement. They all have 30 edges. The regular icosahedron and great dodecahedron share the same edge arrangement but differ in faces (triangles vs pentagons), as do the small stellated dodecahedron and great icosahedron (pentagrams vs triangles).

Convex	Regular stars
icosahedron	great dodecahedron	small stellated dodecahedron	great icosahedron

Icosahedral symmetry subgroups

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.

	Regular	Uniform		2-uniform			3-uniform
Name	Regular icosahedron	Snub octahedron	Snub tetratetrahedron	Gyroelongated pentagonal bipyramid	Triangular gyrobianticupola	Snub triangular antiprism[14]	Snub square bipyramid
Image
Face coloring	(11111)	(11212)	(11213)	(11122) (22222)	(12332) (23333)	(11213) (11212)	(11424) (22434) (33414)
Coxeter diagram
Schläfli symbol	{3,5}	s{3,4}	sr{3,3}	() || {n} || r{n} || ()		ss{2,6}	sdt{2,4}
Conway	I	HtO	sT	k5A5		sY3 = HtA3	HtdP4
Symmetry	Ih [5,3] (*532)	Th [3+,4] (3*2)	T [3,3]+ (332)	D5d [2+,10] (2*5)	D3d [2+,6] (2*3)	D3 [3,2]+ (322)	D2h [2,2] (*222)
Symmetry order	120	24	12	20	12	6	8

Regular icosahedron graph
3-fold symmetry
Vertices	12
Edges	30
Radius	3
Diameter	3
Girth	3
Automorphisms	120 (A5 × Z2)
Chromatic number	4
Properties	Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph
Table of graphs and parameters

The skeleton of the icosahedron (the vertices and edges) forms a graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.

The high degree of symmetry of the polygon is replicated in the properties of this graph, which is distance-transitive 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.[22]

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

Orthogonal projection

There are 4 related Johnson solids, including pentagonal faces with a subset of the 12 vertices. The similar dissected regular icosahedron has 2 adjacent vertices diminished, leaving two trapezoidal faces, and a bifastigium has 2 opposite sets of vertices removed and 4 trapezoidal faces. The pentagonal antiprism is formed by removing two opposite vertices.

Form	J2	Bifastigium	J63	J62	Dissected icosahedron	s{2,10}	J11
Vertices	6 of 12	8 of 12	9 of 12	10 of 12			11 of 12
Symmetry	C5v, [5], (*55) order 10	D2h, [2,2], *222 order 8	C3v, [3], (*33) order 6	C2v, [2], (*22) order 4		D5d, [2+,10], (2*5) order 20	C5v, [5], (*55) order 10
Image

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}	t{5,3}	r{5,3}	t{3,5}	{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:

Family of uniform tetrahedral polyhedra
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

Uniform octahedral polyhedra
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{31,1}	t{3,4} t{31,1}	{3,4} {31,1}	rr{4,3} s2{3,4}	tr{4,3}	sr{4,3}	h{4,3} {3,3}	h2{4,3} t{3,3}	s{3,4} s{31,1}
		=	=	=				= or	= or	=
Duals to uniform polyhedra
V43	V3.82	V(3.4)2	V4.62	V34	V3.43	V4.6.8	V34.4	V33	V3.62	V35

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

*n32 symmetry mutation of regular tilings: {3,n}
Spherical				Euclid.	Compact hyper.		Paraco.	Noncompact hyperbolic
3.3	33	34	35	36	37	38	3∞	312i	39i	36i	33i

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 ${\displaystyle n=6}$, and hyperbolic plane for any higher ${\displaystyle n}$. The series can be considered to begin with ${\displaystyle n=2}$, with one set of faces degenerated into digons.

n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry n32	Spherical				Euclidean	Compact hyperbolic		Paracomp.
	232	332	432	532	632	732	832	∞32
Snub figures
Config.	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.∞
Gyro figures
Config.	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.∞

Spherical		Hyperbolic tilings
{2,5}	{3,5}	{4,5}	{5,5}	{6,5}	{7,5}	{8,5}	...	{∞,5}

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.

• Great icosahedron
• Geodesic grids use an iteratively bisected icosahedron to generate grids on a sphere
• Icosahedral twins
• Infinite skew polyhedron
• Jessen's icosahedron
• Regular polyhedron
• Truncated icosahedron

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.

• Klitzing, Richard. "3D convex uniform polyhedra x3o5o – ike".
• Hartley, Michael. "Dr Mike's Math Games for Kids".
• K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Tulane.edu A discussion of viral structure and the icosahedron
• Origami Polyhedra – Models made with Modular Origami
• Video of icosahedral mirror sculpture
• Principle of virus architecture

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.