Improper rotation

In geometry, an improper rotation (also called rotation-reflection, rotoreflection, 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 a 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.

There is a distinction between rotary reflection and rotary inversion symmetry operations and their associated symmetry elements. Rotary reflections are generally used to describe the symmetry of individual molecules and are defined as a 360°/n rotation about an n-fold rotation axis followed by a reflection over a mirror plane perpendicular to the n-fold rotation axis. Rotoinversions are generally used to describe the symmetry of crystals and are defined as a 360°/n rotation about an n-fold rotation axis followed by an inversion through the origin. Although rotary reflection operations have a rotoinversion analogue and vice versa, rotoreflections and rotoinversions of the same order need not be identical. For example, a 6-fold rotoinversion axis and its associated with symmetry operations are distinct from those resulting from a 6-fold reflection axis.

{| class=wikitable align=center width=400

|+ Example polyhedra with rotoreflection symmetry

!Group

! S₄

! S₆

! S₈

! S₁₀

! S₁₂

|- align=center

!Subgroups

| C₂

| C₃, S₂ = C_i

| C₄, C₂

| C₅, S₂ = C_i

| C₆, S₄, C₃, C₂

|- align=center

!Example

|80px<BR>beveled digonal antiprism

| 80px<BR>triangular antiprism

| 80px<BR>square antiprism

| 100px<BR>pentagonal antiprism

| 100px<BR>hexagonal antiprism

|- align=center

| colspan=6 | Antiprisms with directed edges have rotoreflection symmetry.<BR>p-antiprisms for odd p contain inversion symmetry, C_i.

|}

Three dimensions

In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and inversion in a point on the 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).