Involutional symmetry C_{s}, (*) [ ] = |
Cyclic symmetry C_{nv}, (*nn) [n] = |
Dihedral symmetry D_{nh}, (*n22) [n,2] = | |
Polyhedral group, [n,3], (*n32) | |||
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Tetrahedral symmetry T_{d}, (*332) [3,3] = |
Octahedral symmetry O_{h}, (*432) [4,3] = |
Icosahedral symmetry I_{h}, (*532) [5,3] = |
In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) that does not change the object.
They are the finite symmetry groups on a cone. For n = ∞ they correspond to four frieze groups. Schönflies notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses, orbifold notation.
C_{2h}, [2,2^{+}] (2*) and C_{2v}, [2], (*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. C_{2v} applies e.g. for a rectangular tile with its top side different from its bottom side.
In the limit these four groups represent Euclidean plane frieze groups as C_{∞}, C_{∞h}, C_{∞v}, and S_{∞}. Rotations become translations in the limit. Portions of the infinite plane can also be cut and connected into an infinite cylinder.
Notations | Examples | ||||
---|---|---|---|---|---|
IUC | Orbifold | Coxeter | Schönflies^{*} | Euclidean plane | Cylindrical (n=6) |
p1 | ∞∞ | [∞]^{+} | C_{∞} | ||
p1m1 | *∞∞ | [∞] | C_{∞v} | ||
p11m | ∞* | [∞^{+},2] | C_{∞h} | ||
p11g | ∞× | [∞^{+},2^{+}] | S_{∞} |
S_{2}/C_{i} (1x): | C_{4v} (*44): | C_{5v} (*55): | |
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Parallelepiped |
Square pyramid |
Elongated square pyramid |
Pentagonal pyramid |