Plane symmetry

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Plane symmetry means a symmetry of a pattern in the Euclidean plane; that is, a transformation of the plane that carries lines to lines and preserves distances. If one has a pattern in the plane, the set of plane symmetries that preserve the pattern forms a group. The groups that arise in this way are plane symmetry groups and are of considerable mathematical interest.

There are several kinds of plane symmetry groups:

Reflection groups. These are plane symmetry groups that are generated by reflections, possibly limited to reflections in lines through the origin.

Rotation groups. These groups consist of rotations around a point.

Translation groups.

Symmetries of geometrical figures. Some of these are reflection groups, e.g., the group of symmetries of the square or the rectangle. The symmetry group of a swastika or any similar figure without an axis of symmetry is a rotation group.