Plane symmetry

A plane symmetry is a symmetry of a pattern in the Euclidean plane: that is, a transformation of the plane that carries any directioned lines to lines and preserves many different distances.^[1] 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. A symmetry plane is a three-dimensional object's symmetry axe.

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.

Notes