Icosahedral symmetry
Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.
The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A5 on 5 letters.
The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus.
The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A5 on 5 letters.
The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the center corresponding to element (identity,-1), where Z2 is written multiplicatively.
If Pk is the product of taking the permutation Pi and applying Pj to it, then for the same values of i, j and k, it is also true that Qk is the product of taking Qi and applying Qj, and also that premultiplying a vector by Mk is the same as premultiplying that vector by Mi and then premultiplying that result with Mj, that is Mk = Mj × Mi.
The following groups all have order 120, but are not isomorphic: They correspond to the following short exact sequences (the latter of which does not split) and product In words, Note that
Each line in the following table represents one class of conjugate (i.e., geometrically equivalent) subgroups.
The full icosahedral symmetry group [5,3] () of order 120 has generators represented by the reflection matrices R0, R1, R2 below, with relations R02 = R12 = R22 = (R0×R1)5 = (R1×R2)3 = (R0×R2)2 = Identity.
Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.
For the intermediate material phase called liquid crystals the existence of icosahedral symmetry was proposed by H. Kleinert and K. Maki[2] and its structure was first analyzed in detail in that paper.
In aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011.
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 (Klein 1888); a modern exposition is given in (Tóth 2002, Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, p. 66).
Similar geometries occur for PSL(2,n) and more general groups for other modular curves.
