Symmetry ‒ Introduction

**A symmetry operation ** is an operation that can be performed either physically or imaginatively, where a crystal is transformed into a state indistinguishable from the starting state. In crystals, the symmetry is internal, meaning that it is an ordered geometrical arrangement of atoms and molecules in the **crystal lattice**. This internal symmetry is always reflected in the external form of perfect crystals. We can distinguish three types of symmetry operations, i.e. centric symmetry, mirror symmetry and rotational symmetry.

**Centric Symmetry ** is an inversion through a point. In this operation, lines are drawn from all points in the object through a point in the center of the object. The lines each have lengths that are equidistant from the original points. When the ends of the lines are connected, the original object is reproduced inverted from its original appearance.

**A mirror symmetry operation ** is done by imagining that you cut the object in half, then place a mirror next to one of the halves of the object along the cut. If the reflection in the mirror reproduces the other half of the object, then the object is said to have mirror symmetry. The plane of the mirror is an element of symmetry referred to as a **mirror plane**.

**A rotational symmetry ** is done by a rotation of the object about its axis. The axis along which the rotation is performed is an element of symmetry referred to as a rotation axis. The following types of rotation symmetry axis are possible in crystals. **1-Fold Rotation Axis **. An object that requires rotation of a full 360° in order to restore it to its original appearance has no rotational symmetry. Since it repeats itself 1 time every 360° it is said to have a 1-fold axis of rotational symmetry. **2-fold Rotation Axis **. If an object appears identical after a rotation of 180° (that is twice in a 360° rotation), then it is said to have a 2-fold rotation axis. A filled oval shape represents the point where the 2-fold rotation axis intersects the page. **3-Fold Rotation Axis **. Objects that repeat themselves upon rotation of 120° are said to have a 3-fold axis of rotational symmetry, and they will repeat 3 times in a 360° rotation. A filled triangle is used to symbolize the location of the 3-fold rotation axis. **4-Fold Rotation Axis **. If an object repeats itself after 90° of rotation, it will repeat 4 times in a 360° rotation, as illustrated previously. A filled square is used to symbolize the location of the 4-fold axis of rotational symmetry. **6-Fold Rotation Axis **. If rotation of 60° about an axis causes the object to repeat itself, then it has a 6-fold axis of rotational symmetry. A filled hexagon is used as the symbol for a 6-fold rotation axis.