rotoreflections

Rotoreflections are a type of geometric transformation in Euclidean space that combines a rotation about an axis with a reflection in a plane perpendicular to that axis. The order of these operations matters, meaning a rotation followed by a reflection is generally not the same as a reflection followed by a rotation. These transformations are classified as improper isometries, meaning they preserve distances but reverse orientation. In three-dimensional space, a rotoreflection is often denoted as $S_n$, where $n$ is the order of the rotation. For instance, $S_2$ represents an inversion through a point, which is equivalent to a 180-degree rotation followed by a reflection in a plane passing through that point. Rotoreflections are fundamental concepts in crystallography and molecular symmetry, where they are used to describe the symmetry elements of crystals and molecules. Identifying rotoreflectional symmetry can help predict the physical properties of materials. The point group classification system extensively utilizes rotoreflections to categorize all possible symmetries of objects.