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Icosahedral symmetry fundamental domains

A Soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry.

A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 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 × Z₂. The latter group is also known as the Coxeter group H₃, and is also represented by Coxeter notation, [5,3] and Coxeter diagram Template:CDD.

As point group

The icosahedral rotation group I with fundamental domain

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 (or equivalently, symmetries on the sphere) with the largest symmetry groups.

Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups.

Schönflies crystallographic notation	Coxeter notation	Orbifold notation	Order
I	[3,5]+	532	60
Ih	[3,5]	*532	120

Presentations corresponding to the above are:

$I:⟨s,t\mid s^2,t^3,(st{)}^5⟩$

$I_h:⟨s,t\mid s^3(st{)}^{-2},t^5(st{)}^{-2}⟩.$

These correspond to the icosahedral groups (rotational and full) being the (2,3,5) triangle groups.

The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus.

Note that other presentations are possible, for instance as an alternating group (for I).

Group structure

The Template:Visible anchor I is of order 60. The group I is isomorphic to A₅, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes (which inscribe in the dodecahedron), the compound of five octahedra, or either of the two compounds of five tetrahedra (which are enantiomorphs, and inscribe in the dodecahedron).

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

The Template:Visible anchor I_h has order 120. It has I as normal subgroup of index 2. The group I_h is isomorphic to I × Z₂, or A₅ × Z₂, with the inversion in the center corresponding to element (identity,-1), where Z₂ is written multiplicatively.

I_h acts on the compound of five cubes and the compound of five octahedra, but -1 acts as the identity (as cubes and octahedra are centrally symmetric). It acts on the compound of ten tetrahedra: I acts on the two chiral halves (compounds of five tetrahedra), 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 D_3d and 6 versions of D_5d (symmetries like antiprisms).

I is also isomorphic to PSL₂(5), but I_h is not isomorphic to SL₂(5).

Commonly confused groups

The following groups all have order 120, but are not isomorphic:

• S₅, the symmetric group on 5 elements
• I_h, the full icosahedral group (subject of this article, also known as H₃)
• 2I, the binary icosahedral group

They correspond to the following short exact sequences (which do not split) and product

$1\to A_5\to S_5\to Z_2\to 1$

$I_h=A_5\times Z_2$

$1\to Z_2\to 2I\to A_5\to 1$

In words,

• $A_5$ is a normal subgroup of $S_5$
• $A_5$ is a factor of $I_h$, which is a direct product
• $A_5$ is a quotient group of $2I$

Note that $A_5$ has an exceptional irreducible 3-dimensional representation (as the icosahedral rotation group), but $S_5$ does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the symmetric group.

These can also be related to linear groups over the finite field with five elements, which exhibit the subgroups and covering groups directly; none of these are the full icosahedral group:

• $A_5\cong \mathrm{PSL}(2,5),$ the projective special linear group, see here for a proof;
• $S_5\cong \mathrm{PGL}(2,5),$ the projective general linear group;
• $2I\cong \mathrm{SL}(2,5),$ the special linear group.

Conjugacy classes

The conjugacy classes 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 I_h 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 of chiral icosahedral symmetry

Schoenflies notation	Coxeter notation	Orbifold notation	Hermann Mauguin notation	Order	Index	Subgroup relations
I	[5,3]+	532	532	60	1
T	[3,3]+	332	332	12	5
D5	[5,2]+	522	522	10	6
D3	[3,2]+	322	322	6	10
C5	[5]+	55	5	5	12
D2	[2,2]+	222	222	4	15
C3	[3]+	33	3	3	20
C2	[2]+	22	2	2	30
C1	[ ]+	11	1	1	60

Subgroups of full icosahedral symmetry

Schoenflies notation	Coxeter notation	Orbifold notation	Hermann Mauguin notation	Order	Index	Subgroup relations
Ih	[5,3]	*532	Template:Overline2/m	120	1
I	[5,3]+	532	532	60	2
D5d	[2+,10]	2*5	Template:Overlinem2	20	6
Th	[4,3+]	3*2	mTemplate:Overline	12	10
T	[3,3]+	332	332	12	10
D3d	[2+,6]	2*3	Template:Overlinem	12	10
D5	[5,2]+	522	522	10	12
C5v	[5]	*55	5m	10	12
S10	[2+,10+]	5×	Template:Overline	10	12
D2h	[2,2]	*222	mmm	8	15
D3	[3,2]+	322	322	6	20
C3v	[3]	*33	3m	6	20
S6	[2+,6+]	3×	Template:Overline	6	20
C5	[5]+	55	5	5	24
D2	[2,2]+	222	222	4	30
C2h	[2,2+]	2*	2/m	4	30
C2v	[2]	*22	mm2	4	30
C3	[3]+	33	3	3	40
Cs	[ ]	*	Template:Overline or m	2	60
C2	[2]+	22	2	2	60
S2	[2+,2+]	×	Template:Overline	2	60
C1	[ ]+	11	1	1	120

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

Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, since $-1$ 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 groups C₃
• vertex stabilizers in I_h give dihedral groups D₃
• stabilizers of an opposite pair of vertices in I give dihedral groups D₃
• stabilizers of an opposite pair of vertices in I_h give $D_3\times \pm 1$

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 Z₂
• edges stabilizers in I_h give Klein four-groups $Z_2\times Z_2$
• stabilizers of a pair of edges in I give Klein four-groups $Z_2\times Z_2$; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
• stabilizers of a pair of edges in I_h give $Z_2\times Z_2\times Z_2$; there 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 I_h 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 I_h give $D_5\times \pm 1$

Polyhedron stabilizers

For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, $I\stackrel{\sim }{\to }{A}_5<S_5$.

• stabilizers of the inscribed tetrahedra in I are a copy of T
• stabilizers of the inscribed tetrahedra in I_h are a copy of T_h
• 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 I_h are a copy of O_h

Fundamental domain

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

Icosahedral rotation group I

Full icosahedral group Ih

Faces of disdyakis triacontahedron are the fundamental domain

In the disdyakis triacontahedron 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

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Chiral solids

Class	Name	picture	Faces	Edges	Vertices
Archimedean solid	snub dodecahedron		92	150	60
Catalan solid	pentagonal hexecontahedron		60	50	92

Full icosahedral symmetry

Platonic solids
{5,3}	{3,5}
Archimedean solids
3.10.10	4.6.10	5.6.6	3.4.5.4	3.5.3.5
Catalan solids
V3.10.10	V4.6.10	V5.6.6	V3.4.5.4	V3.5.3.5

Other objects with icosahedral symmetry

• Barth surfaces
• Virus structure, and Capsid

Liquid crystals with icosahedral symmetry

For the intermediate material phase called liquid qrystals the existence of icosahedral symmetry was proposed by H. Kleinert and K. Maki and its structure was first analyzed in detail in that paper. See the review article here. In aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011.

Related geometries

Icosahedral symmetry is equivalently the projective special linear group PSL(2,5), and is the symmetry group of the modular curve X(5), and more generally PSL(2,p) is the symmetry group of the modular curve X(p). The modular curve X(5) is geometrically a dodecahedron with a cusp at the center of each polygonal face, which demonstrates the symmetry group.

This geometry, and associated symmetry group, was studied by Felix Klein as the monodromy groups of a Belyi surface – a Riemann surface with a holomorphic map to the Riemann sphere, ramified only at 0, 1, and infinity (a Belyi function) – the cusps are the points lying over infinity, while the vertices and the centers of each edge lie over 0 and 1; the degree of the covering (number of sheets) equals 5.

This arose from his efforts to give a geometric setting for why icosahedral symmetry arose in the solution of the quintic equation, with the theory given in the famous Template:Harv; a modern exposition is given in Template:Harv.

Klein's investigations continued with his discovery of order 7 and order 11 symmetries in Template:Harv and Template:Harv (and associated coverings of degree 7 and 11) and dessins d'enfants, the first yielding the Klein quartic, whose associated geometry has a tiling by 24 heptagons (with a cusp at the center of each).

Similar geometries occur for PSL(2,n) and more general groups for other modular curves.

More exotically, there are special connections between the groups PSL(2,5) (order 60), PSL(2,7) (order 168) and PSL(2,11) (order 660), which also admit geometric interpretations – PSL(2,5) is the symmetries of the icosahedron (genus 0), PSL(2,7) of the Klein quartic (genus 3), and PSL(2,11) the buckyball surface (genus 70). These groups form a "trinity" in the sense of Vladimir Arnold, which gives a framework for the various relationships; see trinities for details.

There is a close relationship to other Platonic solids.

See also

• tetrahedral symmetry
• octahedral symmetry
• binary icosahedral group
• Icosian calculus

References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Peter R. Cromwell, Polyhedra (1997), p.296
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 11: Finite symmetry groups

External links

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