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:
“We should have learnt by now that laws and court decisions can only point the way. They can establish criteria of right and wrong. And they can provide a basis for rooting out the evils of bigotry and racism. But they cannot wipe away centuries of oppression and injustice—however much we might desire it.”
—Hubert H. Humphrey (1911–1978)
“Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
—Simone Weil (1910–1943)