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 (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 | + | 532 | 60 |
| Ih | *532 | 120 |
Presentations corresponding to the above are:
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).
Read more about this topic: Icosahedral Symmetry
Famous quotes containing the words point and/or group:
“I would rather produce my passions than brood over them at my expense; they grow languid when they have vent and expression. It is better that their point should operate outwardly than be turned against us.”
—Michel de Montaigne (1533–1592)
“Remember that the peer group is important to young adolescents, and there’s nothing wrong with that. Parents are often just as important, however. Don’t give up on the idea that you can make a difference.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, I, ch.5 (1985)