Coxeter notation
Classification system for symmetry groups in geometry / From Wikipedia, the free encyclopedia
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In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.
More information , [ ] = [1]C1v, , [2]C2v ...
, [ ] = [1] C1v |
, [2] C2v |
, [3] C3v |
, [4] C4v |
, [5] C5v |
, [6] C6v |
---|---|---|---|---|---|
Order 2 |
Order 4 |
Order 6 |
Order 8 |
Order 10 |
Order 12 |
[2] = [2,1] D1h |
[2,2] D2h |
[2,3] D3h |
[2,4] D4h |
[2,5] D5h |
[2,6] D6h |
Order 4 |
Order 8 |
Order 12 |
Order 16 |
Order 20 |
Order 24 |
, [3,3], Td | , [4,3], Oh | , [5,3], Ih | |||
Order 24 |
Order 48 |
Order 120 | |||
Coxeter notation expresses Coxeter groups as a list of branch orders of a Coxeter diagram, like the polyhedral groups, = [p,q]. Dihedral groups, , can be expressed as a product [ ]×[n] or in a single symbol with an explicit order 2 branch, [2,n]. |
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