# 15.7: Symmetry

- Page ID
- 31577

We have already met the concept symmetry in relation to crystal structures: the lattice generates the translational symmetry—the motif is repeated on every lattice point.

Other types of symmetry exist, including:

- rotation axes
- mirror planes
- centre of symmetry
- inversion axes (combination of rotation and centre of symmetry operations)

An n fold rotational symmetry operation rotates an object by 360°/n. Only n = 1, 2, 3, 4, and 6 are permitted in a periodic lattice

An object has mirror symmetry if reflection of the object in a plane brings it into coincidence with itself:

Some objects have special symmetry about an origin such that, for any point at position *x*, *y*, *z*, there is an exactly similar point at –*x*, –*y*, –*z*. The origin is called a centre of symmetry ( “inversion centre”). Such an object is said to be centrosymmetric:

An n-fold inversion axis is a combination of a rotation by 360/n followed by a centre of symmetry operation. An example of a 4-fold inversion axis is show in the following animation: