In geometry, an icosahedron 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 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.
A regular icosahedron is a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism
in any of six orientations.
Icosahedrite is a mineral that has the crystal shape of an icosahedron.
The name comes from Greek είκοσι, meaning "twenty", and εδρα, meaning "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 is
A019881 and the radius of an inscribed sphere is
A179294 while the midradius, which touches the middle
of each edge, is A019863
where φ is the golden ratio.
Area and volume The surface area A and the volume V of a regular
icosahedron of edge length a are: A010527
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
Cartesian coordinates
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= is the golden ratio.
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.
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, then the other ten vertices
are at latitude ±arctan(1/2) ≈ ±26.57°.
These ten vertices are at evenly spaced longitudes, 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:
Spherical tiling 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.
Other facts An icosahedron has 43,380 distinct nets.
To color the icosahedron, such that no two adjacent faces have the same color, requires
at least 3 colors.
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.
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.
Both volumes have formulas involving the golden ratio, but taken to different powers.
As it turns out, the icosahedron occupies less of the sphere's volume than the dodecahedron.
Construction by a system of equiangular lines 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 A2 = 5I.
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 ±v1, ..., ±v6 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 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,.
See icosahedral symmetry: related geometries for further history, and related symmetries
on seven and eleven letters.
The full symmetry group of the icosahedron 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.
Pseudoicosahedron
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.
Pyritohedral symmetry
If the 8 equilateral triangles are geometrically identical, the pseudoicosahedron has pyritohedral
symmetry, [4,3+], with order 24.
A lower tetrahedral symmetry, [3,3]+, exists as well, seen as the 8 triangles marked in
alternating pairs of four, with order 12.
These symmetries offer Coxeter-Dynkin diagrams: and respectfully, each representing the lower
symmetry to the regular icosahedron ,, [5,3] icosahedral symmetry of order 120.
Cartesian coordinates The coordinates of the 12 vertices can be
defined by the vectors defined by all the possible cyclic permutations and sign-flips
of coordinates of the form.
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, where φ is the golden ratio.
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:
Crystal pyrite Iron pyrites have been observed to have formed
crystals in the form of pseudoicosahedra.
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.
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.
However, when combined with suitable cells having smaller dihedral angles, icosahedra
can be used as cells in semi-regular polychora, just as hexagons can be used as faces in semi-regular
polyhedra.
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.
The pyritohedral symmetry version is sometimes called a pseudoicosahedron, and is dual to
the pyritohedron.
Uses and natural forms
Biology Many viruses, e.g. herpes virus, have icosahedral
shells.
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.
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.
Also, icosahedrite has a crystal shape of an icosahedron.
Toys and games
In several roleplaying games, such as Dungeons & Dragons, the twenty-sided die 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, 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.
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 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.
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
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.
The icosahedral graph is Hamiltonian: there is a cycle containing all the vertices.
It is also a planar graph.
Related polyhedra and polytopes There are 4 related Johnson solids, including
pentagonal faces with a subset of the 12 vertices: The icosahedron can be transformed by a truncation
sequence into its dual, the dodecahedron: As a snub tetrahedron, and alternation of
a truncated octahedron it also exists in the tetrahedral and octahedral symmetry families:
This polyhedron is topologically related as a part of sequence of regular polyhedra with
Schläfli symbols {3,n}, continuing into the hyperbolic plane.
The regular icosahedron, seen as a snub tetrahedron, is a member of a sequence of snubbed polyhedra
and tilings with vertex figure and Coxeter–Dynkin diagram . These figures and their duals have
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.
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.
See also 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
References
External links Weisstein, Eric W., "Icosahedron", MathWorld.
Richard Klitzing, 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
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
Origami Polyhedra – Models made with Modular Origami
Video of icosahedral mirror sculpture [1] Principle of virus architecture