Editing Symmetry group (section)

==Three dimensions==
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{{see also|Point groups in three dimensions}}

Up to conjugacy the set of three-dimensional point groups consists of 7 infinite series, and 7 other individual groups. In [[crystallography]], only those point groups are considered which preserve some crystal lattice (so their rotations may only have order 1, 2, 3, 4, or 6). This [[crystallographic restriction theorem|crystallographic restriction]] of the infinite families of general point groups results in 32 [[crystallographic point group]]s (27 individual groups from the 7 series, and 5 of the 7 other individuals).

The continuous symmetry groups with a fixed point include those of:
*cylindrical symmetry without a symmetry plane perpendicular to the axis. This applies, for example, to a [[bottle]] or [[cone]].
*cylindrical symmetry with a symmetry plane perpendicular to the axis
*spherical symmetry

For objects with [[scalar field]] patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true for [[vector field]] patterns: for example, in [[cylindrical coordinates]] with respect to some axis, the vector field 
<math>\mathbf{A} = A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}</math> has cylindrical symmetry with respect to the axis whenever <math>A_\rho, A_\phi,</math> and <math> A_z</math> have this symmetry (no dependence on <math>\phi</math>); and it has reflectional symmetry only when <math>A_\phi = 0</math>.

For spherical symmetry, there is no such distinction: any patterned object has planes of reflection symmetry.

The continuous symmetry groups without a fixed point include those with a [[screw axis]], such as an infinite [[helix]]. See also [[Euclidean group#Subgroups|subgroups of the Euclidean group]].