Icosahedron

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 symbol	{3,5}
Wythoff symbol	5 \| 2 3
Coxeter-Dynkin
Symmetry	I_h (*532)
References	U₂₂, C₂₅, W₄
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 (Greek: εικοσάεδρον, from eikosi twenty + hedron seat; pronounced /ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/; plural: -drons, -dra /-drə/) is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is 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 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.

1 Dimensions
2 Area and volume
3 Cartesian coordinates
4 Other facts
5 Construction by a system of equiangular lines
6 Symmetry
7 Stellations
8 Geometric relations
9 Uniform colorings and subsymmetries
10 Related polyhedra and polytopes
11 Uses and natural forms
12 See also
13 References
14 External links

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

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

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

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

where $\varphi$ (also called $\tau$ ) is the golden ratio.

Area and volume

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

$A = 5\sqrt{3}a^2 \approx 8.66025404a^2$

$V = \frac{5}{12} (3+\sqrt5)a^3 \approx 2.18169499a^3.$

Cartesian coordinates

Golden rectangles in an icosahedron

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

(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 $\frac{\sqrt{5}-1}{2}$ , 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, as do the two icosahedra that can be defined in this way from any given octahedron.

Other facts

It has 143,380 nets.
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

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 $6\times 6$ 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 $A^2=5 I$ (where $I$ is the 6x6 identity matrix). This implies that $A$ has eigenvalues $-\sqrt{5}$ and $\sqrt{5}$ , both with multiplicity 3 since $A$ is symmetric and of trace zero.

The matrix $A+\sqrt{5}I$ induces thus an 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 $A+\sqrt{5}I$ 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 $\arccos\frac{1}{\sqrt{5}}$ . Orthogonal projection of $\pm v_1,\dots,\pm v_6$ onto the $\sqrt{5}$ -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 $A_5$ 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 $C_2$ 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.^[1]

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.

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 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 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

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.

Name	Regular icosahedron	alternated truncated octahedron	snub tetrahedron	Pentagonal gyroelongated bipyramid
Coxeter-Dynkin
Schläfli symbol	{3,5}	h_0,1{3,4}	s{3,3}
Wythoff symbol	5 \| 3 2		\| 3 3 2
Symmetry	I_h (*532)	T_h (3*2)	T (332)	D_5d (2*5)
Symmetry order	60	24	12	10
Uniform coloring	(11111)	(11212)	(11213)	(11122)&(22222)

Related polyhedra and polytopes

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

Picture	Icosahedron	Truncated icosahedron	Icosidodecahedron	Truncated dodecahedron	Dodecahedron
Coxeter-Dynkin

Name	tetrahedron	rectified tetrahedron (octahedron)	truncated tetrahedron	cantellated tetrahedron (cuboctahedron)	omnitruncated tetrahedron (truncated octahedron)	Snub tetrahedron (icosahedron)
Schläfli	{3,3}	t₁{3,3}	t_0,1{3,3}	t_0,2{3,3}	t_0,1,2{3,3}	s{3,3}
Coxeter-Dynkin
Graph (A₃)
Graph (A₂)
Solid
Tiling

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
Coxeter-Dynkin

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.

Many viruses, e.g. herpes virus, have the shape of an icosahedron. Viral structures are built of repeated identical protein subunits 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.

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 closo-carboranes are chemical compounds with shape very close to isosahedron. Icosahedral twinning also occurs in crystals, especially nanoparticles.

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.

R. Buckminster Fuller, a 20th century architect known for his geodesic domes, created the Fuller projection world map utilizing the icosahedron. The spherical shape of the world is morphed onto an icosahedron, then the figure is unfolded so that it may be laid out on a flat surface with all land masses still connected. This projection, created with Japanese cartographer Shoji Sadao,^[2] is one of the most accurate world map projections, producing a maximum distortion of only 2%. Fuller's namesake carbon allotrope, fullerene, has the structure of a truncated icosahedron (20 hexagons and 12 pentagons).

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 printed on a regular icosahedron.

The icosahedron displayed in a functional form is seen in the Sol de la Flor light shade. The rosette formed by the overlapping pieces show a resemblance to the Frangipani flower.

Many borides and allotropes of boron contain boron B₁₂ icosahedron as a basic structure unit.

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.^[3]

References

↑ Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999), The fifty-nine icosahedra (3rd ed.), Tarquin, MR 676126, ISBN 978-1-899618-32-3 (1st Edn University of Toronto (1938))
↑ "Fuller and Sadao: Partners in Design". September 19, 2006. http://library.stanford.edu/depts/spc/exhibits/fullersadao.html. Retrieved 2010-01-26.
↑ Klein, Douglas J. (2002), "Resistance-Distance Sum Rules" (PDF), Croatica Chemica Acta 75 (2): 633–649, http://public.carnet.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf, retrieved 2006-09-15

Klein, Felix (1888), Lectures on the ikosahedron and the solution of equations of the fifth degree, http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03070001, Dover edition ISBN 978-0-486-49528-6.

External links

Weisstein, Eric W., "Icosahedron" from MathWorld.
Paper models of the icosahedron
The Uniform Polyhedra
K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
Interactive Icosahedron model - works right in your web browser
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Tulane.edu A discussion of viral structure and the icosahedron
Paper Models of Polyhedra Many links
Origami Polyhedra - Models made with Modular Origami
Video of icosahedral mirror sculpture
[1] Principle of virus architecture

Polyhedra

Dihedron • Tetrahedron • Pentahedron • Hexahedron • Heptahedron • Octahedron • Decahedron • Dodecahedron • Tetradecahedron • Icosahedron

Polyhedron navigator

Platonic solids (regular)	tetrahedron · cube · octahedron · dodecahedron · icosahedron

Archimedean solids (Semiregular/Uniform)	truncated tetrahedron · cuboctahedron · truncated cube · truncated octahedron · rhombicuboctahedron · truncated cuboctahedron · snub cube · icosidodecahedron · truncated dodecahedron · truncated icosahedron · rhombicosidodecahedron · truncated icosidodecahedron · snub dodecahedron

Catalan solids (Dual semiregular)	triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedron

Dihedral regular	dihedron · hosohedron

Dihedral uniform	prisms · antiprisms

Duals of dihedral uniform	bipyramids · trapezohedra

Dihedral others	pyramids · truncated trapezohedra · gyroelongated bipyramid · cupola · bicupola · pyramidal frusta

Degenerate polyhedra are in italics.


Family	A_n	BC_n	D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
n-polytopes	n-simplex	n-orthoplex • n-hypercube	n-demicube
Topics: Polytope families • Regular polytope • List of regular polytopes