Symmetry (geometry)

A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.

A geometric object has symmetry if there is an "operation" or "transformation" (such as an isometry or affine map) that maps the figure/object onto itself; i.e., it is said that the object has an invariance under the transform. For instance, a circle rotated about its center will have the same shape and size as the original circle—all points before and after the transform would be indistinguishable. A circle is said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure, the figure is said to have reflectional symmetry or line symmetry; moreover, it is possible for a figure/object to have more than one line of symmetry.

The types of symmetries that are possible for a geometric object depend on the set of geometric transforms available, and on what object properties should remain unchanged after a transform. Because the composition of two transforms is also a transform and every transform has an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical group, the symmetry group of the object.