| 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 |
Ih, H3, [5,3], (*532) |
| Rotation group |
I, [5,3]+, (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 (
/ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/) 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.
The name comes from the Greek: εικοσάεδρον, from είκοσι (eíkosi) "twenty" and ἕδρα (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
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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
A019881
and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is
- Failed to parse (Missing texvc executable; please see math/README to configure.): r_i = \frac{\varphi^2 a}{2 \sqrt{3}} = \frac{\sqrt{3}}{12} \left(3+ \sqrt{5} \right) a \approx 0.7557613141\cdot a
A179294
while the midradius, which touches the middle of each edge, is
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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
A019863
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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): A = 5\sqrt{3}a^2 \approx 8.66025404a^2,
A010527
- Failed to parse (Missing texvc executable; please see math/README to configure.): V = \frac{5}{12} (3+\sqrt5)a^3 \approx 2.18169499a^3.
A102208
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
- Failed to parse (Missing texvc executable; please see math/README to configure.): f=V/(4 \pi r_u^3/3) = \frac{20(3+\surd 5)}{(2\surd 5+10)^{3/2}\pi}\approx 0.6054613829.
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
Seen as a pentagonal gyroelongated bipyramid, with D5d, dihedral symmetry, the icosahedron vertices can be positioned in spherical coordinates, with two vertices are placed on the poles of a sphere, the remaining vertices are located at latitude ±arctan(1/2). The longitudes can be found using the antiprismatic symmetry of the remaining ten vertices, thus fixing a point then rotating 36° about the polar axis and reflecting about the equator until we return to the original point.
The icosahedron has three special orthogonal projections, centered, on a face, edge, and vertex:
Orthogonal projections
| Centered by |
Face |
Edge |
Vertex |
| Coxeter plane |
A2 |
A3 |
H3 |
| Graph |
 |
 |
 |
Projective
symmetry |
[6] |
[4] |
[10] |
| Graph |
Face normal |
|
Vertex normal |
- 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%).
The following construction of the icosahedron avoids tedious computations in the number field Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Q}[\sqrt{5}]
necessary in more elementary approaches.
The existence of the icosahedron amounts to the existence of six equiangular lines in Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle -\sqrt{5}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \sqrt{5}
, both with multiplicity 3 since A is symmetric and of trace zero.
The matrix Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle A+\sqrt{5}I
induces thus a Euclidean structure on the quotient space Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R^6/\ker(A+\sqrt{5}I)
which is isomorphic to Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R^3
since the kernel Failed to parse (Missing texvc executable; please see math/README to configure.): \ker(A+\sqrt{5}I)
of Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle {A+\sqrt{5}I}
has dimension 3. The image under the projection Failed to parse (Missing texvc executable; please see math/README to configure.): \pi:\mathbb R^6 \longrightarrow \mathbb R^6/\ker(A+\sqrt{5}I)
of the six coordinate axes Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R v_1,\dots,\mathbb R v_6
in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R^6
forms thus a system of six equiangular lines in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R^3
intersecting pairwise at a common acute angle of Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\arccos}\tfrac{1}{\sqrt{5}}
. Orthogonal projection of ±v1, ..., ±v6 onto the Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
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.
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]
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.
The icosahedron can be constructed from the tetrahedron (4 sided pyramid) 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 icosahedron can be transformed by a truncation sequence into its dual, the dodecahedron:
This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
| Name |
tetrahedron |
rectified tetrahedron
(octahedron) |
truncated tetrahedron |
cantellated tetrahedron
(cuboctahedron) |
omnitruncated tetrahedron
(truncated octahedron) |
Snub tetrahedron
(icosahedron) |
| Schläfli |
{3,3} |
t1{3,3} |
t0,1{3,3} |
t0,2{3,3} |
t0,1,2{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.
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. |
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 |
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]+ |
Symmetry
order |
6 |
12 |
24 |
60 |
∞ |
Coxeter
Schläfli |
s{2,3} |
s{3,3} |
s{4,3} |
s{5,3} |
s{6,3} |
s{7,3} |
s{8,3} |
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 |
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 |
Gold nanoparticle viewed in electron microscope.
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.
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 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%.
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.
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.
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.[7]
The company logo of the TDK Corporation contains a geometric figure which is based on the stellation diagram of the icosahedron.
- ^ Weisstein, Eric W., "Icosahedral group" from MathWorld.
- ^ Weisstein, Eric W., "Icosahedron" from MathWorld.
- ^ Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H.T.; Petrie, J.F. (1999), The Fifty-Nine Icosahedra (3rd ed.), Tarquin, ISBN 978-1-899618-32-3, MR676126 (1st Edn University of Toronto (1938))
- ^ C. Michael Hogan. 2010. Virus. Encyclopedia of Earth. National Council for Science and the Environment. eds. S. Draggan and C. Cleveland
- ^ Bobik, T.A. (2007), "Bacterial Microcompartments", Microbe (Am. Soc. Microbiol.) 2: 25–31, http://www.microbemagazine.org/index.php/01-2007-home/2308-bacterial-microcompartments
- ^ "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
