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Concept
Improper rotation
Summary
In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.
It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.
In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and inversion in a point on the axis. For this reason it is also called a rotoinversion or rotary inversion. The two definitions are equivalent because rotation by an angle θ followed by reflection is the same transformation as rotation by θ + 180° followed by inversion (taking the point of inversion to be in the plane of reflection). In both definitions, the operations commute.
A three-dimensional symmetry that has only one fixed point is necessarily an improper rotation.
An improper rotation of an object thus produces a rotation of its . The axis is called the rotation-reflection axis. This is called an n-fold improper rotation if the angle of rotation, before or after reflexion, is 360°/n (where n must be even). There are several different systems for naming individual improper rotations:
In the Schoenflies notation the symbol Sn (German, Spiegel, for mirror), where n must be even, denotes the symmetry group generated by an n-fold improper rotation. For example, the symmetry operation S6 is the combination of a rotation of (360°/6)=60° and a mirror plane reflection. (This should not to be confused with the same notation for symmetric groups).
In Hermann–Mauguin notation the symbol is used for an n-fold rotoinversion; i.e.
Official source
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