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Icosahedral symmetry

 
Icosahedral Symmetry

 
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Icosahedral symmetry
 
 
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with full icosahedral symmetry. This commonplace object is not, however, a regular icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
; it is a spherical
Spherical polyhedron

In mathematics, the surface of a sphere may be divided by line segments into bounded regions, to form a spherical tessellation or spherical polyhedron....
 truncated icosahedron
Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
. ]] A regular icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
 has 60 rotational (or orientation-preserving) symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is the dual of the icosahedron.

The set of orientation-preserving symmetries forms a group referred to as A5 (the alternating group on 5 letters), and the full symmetry group (including reflections) is the product A5 × C2 of A5 with a cyclic group of order 2.

R> 		Schönflies
Arthur Moritz Schönflies

Arthur Moritz Sch?nflies was a Germany mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology....
 crystallographic notation		 Coxeter notation Orbifold notation Order
I [3,5]+ 532 60
Ih [3,5] *532 120
Presentations
Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generating set of a group so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators....
 corresponding to the above are:

I:

Ih:

The first presentation was given by William Rowan Hamilton
William Rowan Hamilton

Sir William Rowan Hamilton was an Ireland physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra....
 in 1856, in his paper on Icosian Calculus
Icosian Calculus

The Icosian Calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.In modern terms, he gave a group presentation of the icosahedral group by Generating set of a group and relations....
.

Note that other presentations are possible, for instance as an alternating group
Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt....
 (for I).

icosahedral rotation group I is of order 60.






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with full icosahedral symmetry. This commonplace object is not, however, a regular icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
; it is a spherical
Spherical polyhedron

In mathematics, the surface of a sphere may be divided by line segments into bounded regions, to form a spherical tessellation or spherical polyhedron....
 truncated icosahedron
Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
. ]] A regular icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
 has 60 rotational (or orientation-preserving) symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is the dual of the icosahedron.

The set of orientation-preserving symmetries forms a group referred to as A5 (the alternating group on 5 letters), and the full symmetry group (including reflections) is the product A5 × C2 of A5 with a cyclic group of order 2.

As point group


]]

Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries
Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere....
 (or equivalently, symmetries on the sphere
List of spherical symmetry groups

List of symmetry groups on the sphere Spherical symmetry groups are also called point groups in three dimensions. This article is about Point_groups_in_three_dimensions#Finite_isometry_groups....
) with the largest symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
s.

Icosahedral symmetry is not compatible with translational symmetry
Translational symmetry

In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation....
, so there are no associated crystallographic point groups
Crystal system

A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of Point groups in three dimensions....
 or space group
Space group

The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. The word 'group' in the name comes from the group , which is used to build the set of space groups....
s.

Schönflies
Arthur Moritz Schönflies

Arthur Moritz Sch?nflies was a Germany mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology....
 crystallographic notation		 Coxeter notation Orbifold notation Order
I [3,5]+ 532 60
Ih [3,5] *532 120
Presentations
Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generating set of a group so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators....
 corresponding to the above are:

I:

Ih:

The first presentation was given by William Rowan Hamilton
William Rowan Hamilton

Sir William Rowan Hamilton was an Ireland physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra....
 in 1856, in his paper on Icosian Calculus
Icosian Calculus

The Icosian Calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.In modern terms, he gave a group presentation of the icosahedral group by Generating set of a group and relations....
.

Note that other presentations are possible, for instance as an alternating group
Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt....
 (for I).

Group structure

The icosahedral rotation group I is of order 60. The group I is isomorphic to A5, the alternating group
Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt....
 of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes
Compound of five cubes

This polyhedral compound is a symmetric arrangement of five cubes. This compound was first described by Edmund Hess in 1876.It is one of five Polyhedral_compound#Regular_compounds, and dual to the compound of five octahedra....
 (which inscribe in the dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
), the compound of five octahedra
Compound of five octahedra

This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....
, or either of the two compounds of five tetrahedra
Compound of five tetrahedra

This Polyhedron compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876....
 (which are enantiomorphs, and inscribe in the dodecahedron).

The group contains 5 versions of Th with 20 versions of D3 (10 axes, 2 per axis), and 6 versions of D5.

The full icosahedral group Ih has order 120. It has I as normal subgroup
Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
 of index 2. The group Ih is isomorphic to I × C2, or A5 × C2, with the inversion in the center
Inversion in a point

In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*....
  corresponding to element (identity,-1), where C2 is written multiplicatively.

Ih acts on the compound of five cubes
Compound of five cubes

This polyhedral compound is a symmetric arrangement of five cubes. This compound was first described by Edmund Hess in 1876.It is one of five Polyhedral_compound#Regular_compounds, and dual to the compound of five octahedra....
 and the compound of five octahedra
Compound of five octahedra

This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....
, but -1 acts as the identity (as cubes and octahedra are centrally symmetric). It acts on the compound of ten tetrahedra
Compound of ten tetrahedra

This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....
: I acts on the two chiral halves (compounds of five tetrahedra
Compound of five tetrahedra

This Polyhedron compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876....
), and -1 interchanges the two halves. Notably, it does not act as S5, and these groups are not isomorphic; see below for details.

The group contains 10 versions of D3d and 6 versions of D5d (symmetries like antiprisms).

I is also isomorphic to PSL2(5), and Ih is isomorphic to SL2(5) (they are both central extensions by C2).

Commonly confused groups

The following groups all have order 120, but are not isomorphic:
  • S5, the symmetric group
    Symmetric group

    In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
     on 5 elements
  • Ih, the full icosahderal group (subject of this article)
  • 2I, the binary icosahedral group
    Binary icosahedral group

    In mathematics, the binary icosahedral group is an group extension of the icosahedral group I of order 60 by a cyclic group of order 2. It can be defined as the preimage of the icosahedral group under the 2:1 covering homomorphism...
They correspond to the following short exact sequences (which do not split) and product In words,
  • is a normal subgroup
    Normal subgroup

    In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
     of
  • is a factor of , which is a direct product
    Direct product

    In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
  • is a quotient group
    Quotient group

    In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element....
     of


Conjugacy classes

The conjugacy class
Conjugacy class

In mathematics, especially group theory, the elements of any group may be partition of a set into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure....
es of I are:
  • identity
  • 12 × rotation by 72°, order 5
  • 12 × rotation by 144°, order 5
  • 20 × rotation by 120°, order 3
  • 15 × rotation by 180°, order 2


Those of Ih include also each with inversion:
  • inversion
  • 12 × rotoreflection by 108°, order 10
  • 12 × rotoreflection by 36°, order 10
  • 20 × rotoreflection by 60°, order 6
  • 15 × reflection, order 2


Subgroups

All of these classes of subgroups are conjugate (i.e., all vertex stabilizers are conjugate), and admit geometric interpretations.

Note that the stabilizer
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 of a vertex/edge/face/polyhedron and its opposite are equal, since is central.

Vertex stabilizers
Stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate.
  • vertex stabilizers in I give cyclic group
    Cyclic group

    In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
    s C3
  • vertex stabilizers in Ih give dihedral groups
    Dihedral symmetry in three dimensions

    This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ....
     D3
  • stabilizers of an opposite pair of vertices in I give dihedral groups D3
  • stabilizers of an opposite pair of vertices in Ih give


Edge stabilizers
Stabilizers of an opposite pair of edges can be interpreted as stabilizers of the rectangle they generate.
  • edges stabilizers in I give cyclic groups C2
  • edges stabilizers in Ih give Klein four-group
    Klein four-group

    In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of Order 2 ....
    s
  • stabilizers of a pair of edges in I give Klein four-group
    Klein four-group

    In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of Order 2 ....
    s ; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
  • stabilizers of a pair of edges in Ih give ; these are 5 of these, given by reflections in 3 perpendicular axes.


Face stabilizers
Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate.
  • face stabilizers in I give cyclic groups C5
  • face stabilizers in Ih give dihedral groups D5
  • stabilizers of an opposite pair of faces in I give dihedral groups D5
  • stabilizers of an opposite pair of faces in Ih give


Polyhedron stabilizers
For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, .
  • stabilizers of the inscribed tetrahedra in I are a copy of T
  • stabilizers of the inscribed tetrahedra in Ih are a copy of Th
  • stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedrons) in I are a copy of O
  • stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedrons) in Ih are a copy of Oh


Fundamental Domain

Fundamental domains for the icosahedral rotation group and the full icosahedral group are given by:


]]]]


]] In the disdyakis triacontahedron
Disdyakis triacontahedron

A disdyakis triacontahedron, or hexakis icosahedron is a Catalan solid and the dual to the Archimedean solid truncated icosidodecahedron. As such it is face uniform but with irregular face polygons....
 one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface.

Solids with icosahedral symmetry


Full icosahedral symmetry

Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s - regular polyhedra (all faces of the same type)




Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s - polyhedra with more than one polygon face type.

3.10.10
4.6.10
5.6.6
3.4.5.4
3.5.3.5


Catalan solid
Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
s - duals of the Archimedean solids.

V3.10.10
V4.6.10
V5.6.6
V3.4.5.4
V3.5.3.5


See also


  • tetrahedral symmetry
    Tetrahedral symmetry

    A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation....
  • octahedral symmetry
    Octahedral symmetry

    A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
  • binary icosahedral group
    Binary icosahedral group

    In mathematics, the binary icosahedral group is an group extension of the icosahedral group I of order 60 by a cyclic group of order 2. It can be defined as the preimage of the icosahedral group under the 2:1 covering homomorphism...
  • Icosian Calculus
    Icosian Calculus

    The Icosian Calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.In modern terms, he gave a group presentation of the icosahedral group by Generating set of a group and relations....