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3D symmetry group

In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation preserving) symmetries and 60 orientation reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type It may be represented by Coxeter notation [5,3] and Coxeter diagram The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters.

Icosahedral symmetry is a mathematical property of objects indicating that an object has the same symmetries as a regular icosahedron.

The full symmetry group is the Coxeter group of type It may be represented by Coxeter notation [5,3] and Coxeter diagram The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters. Schoe.(Orb. )CoxeternotationElementsMirror diagramsOrthogonalStereographic projectionIh(*532) [5,3] Mirrorlines:15 I(532) [5,3]+Gyrationpoints:125203302

Every polyhedron with icosahedral symmetry has 60 rotational (or orientation preserving) symmetries and 60 orientation reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The edges of a spherical compound of five octahedra represent the 15 mirror planes as colored great circles. Each octahedron can represent 3 orthogonal mirror planes by its edges. The pyritohedral symmetry is an index 5 subgroup of icosahedral symmetry, with 3 orthogonal green reflection lines and 8 red order 3 gyration points. There are 5 different orientations of pyritohedral symmetry. The icosahedral rotation group 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 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 of index 2. 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. Ih 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 D3d and 6 versions of D5d (symmetries like antiprisms). I is also isomorphic to PSL2(5), but Ih is not isomorphic to SL2(5).

Each line in the following table represents one class of conjugate (i. e., geometrically equivalent) subgroups. The column "Mult." (multiplicity) gives the number of different subgroups in the conjugacy class. Explanation of colors: green = the groups that are generated by reflections, red = the chiral (orientation preserving) groups, which contain only rotations. The groups are described geometrically in terms of the dodecahedron. The abbreviation "h. t. s.(edge)" means "halfturn swapping this edge with its opposite edge", and similarly for "face" and "vertex". Schön. CoxeterOrb. H MStructureCyc. OrderIndexMult. DescriptionIh[5,3]*5322/mA5×Z2 12011full groupD2h[2,2]*222mmmD4×D2=D23 8155fixing two opposite edges, possibly swapping themC5v [5] *55 5mD10 10 126fixing a faceC3v [3] *33 3mD6=S3 6 2010fixing a vertexC2v [2] *22 2mmD4=D22 4 3015fixing an edgeCs [ ] * or mD2 2 6015reflection swapping two endpoints of an edgeTh[3+,4]3*2mA4×Z2 2455pyritohedral groupD5d [2+,10] 2*5 m2D20=Z2×D10 20 66fixing two opposite faces, possibly swapping themD3d [2+,6] 2*3 mD12=Z2×D612 1010fixing two opposite vertices, possibly swapping them D1d = C2h [2+,2] 2* 2/mD4=Z2×D24 3015halfturn around edge midpoint, plus central inversionS10 [2+,10+] 5× Z10=Z2×Z510 126rotations of a face, plus central inversionS6 [2+,6+] 3× Z6=Z2×Z36 2010rotations about a vertex, plus central inversionS2 [2+,2+] × Z22 601central inversionI[5,3]+532532A5 6021all rotationsT[3,3]+332332A4 12105rotations of a contained tetrahedronD5[2,5]+522522D1010126rotations around the center of a face, and h. t. s.(face)D3[2,3]+322322D6=S362010rotations around a vertex, and h. t. s.(vertex)D2[2,2]+222222D4=Z2243015halfturn around edge midpoint, and h. t. s.(edge)C5[5]+555Z55246rotations around a face centerC3[3]+333Z3=A334010rotations around a vertexC2[2]+222Z226015half turn around edge midpointC1[ ]+111Z111201trivial group

Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.

Class Symbols PictureArchimedean sr{5,3} CatalanV3.3.3.3.5

Platonic solidKepler–Poinsot polyhedraArchimedean solids{5,3} {5/2,5} {5/2,3}t{5,3}t{3,5}r{3,5}rr{3,5}tr{3,5}Platonic solidKepler–Poinsot polyhedraCatalan solids{3,5} = {5,5/2} = {3,5/2} = V3.10.10V5.6.6V3.5.3.5V3.4.5.4V4.6.10

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 ; a modern exposition is given in
Klein's investigations continued with his discovery of order 7 and order 11 symmetries in and (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.