Symmetry ‒ Introduction

**A symmetry operation** is a physical or conceptual action that transforms a crystal into a state indistinguishable from its original form. In crystals, symmetry is an internal property, representing the ordered geometric arrangement of atoms and molecules within the crystal lattice. This internal symmetry is always reflected in the external form of perfect crystals. This internal symmetry is also reflected in the external structure of perfectly formed crystals. Three primary types of symmetry operations are observed: centric symmetry, mirror symmetry and rotational symmetry. **Centric Symmetry** involves inversion through a central point. In this operation, lines are drawn from all points in the object through a central point. Each line is equidistant from the central point, and when the endpoints are connected, the original object is recreated as an inverted version of its initial form. **Mirror symmetry** is observed by imagining the object split in half and placing a mirror along the split. If the reflection in the mirror reproduces the other half of the object, the object has mirror symmetry. The plane of the mirror serves as a mirror plane, an element of symmetry. **Rotational symmetry** occurs when an object can be rotated around an axis to match its original appearance. The axis along which the rotation occurs is known as the rotation axis, with various types present in crystals. A 1-fold rotation axis describes an object requiring a full 360° rotation to appear identical, meaning it lacks rotational symmetry beyond this complete turn. An object that appears identical after a 180° rotation, repeating twice within 360°, is said to have a 2-fold rotation axis, typically represented by a filled oval indicating the axis intersection. Objects that repeat every 120°, thus repeating three times within a full rotation, are defined as having a 3-fold rotation axis and are symbolized by a filled triangle. When an object repeats after a 90° rotation, occurring four times in a complete rotation, it has a 4-fold rotation axis, represented by a filled square. Finally, an object that repeats after each 60° turn, resulting in six repetitions in a 360° rotation, possesses a 6-fold rotation axis, represented by a filled hexagon.