Improper rotation

In geometry, an improper rotation^[1] (also called rotation-reflection,^[2] rotoreflection,^[1] rotary reflection,^[3] or rotoinversion^[4]) 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.^[5] 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.

Example polyhedra with rotoreflection symmetry
Group	S₄	S₆	S₈	S₁₀	S₁₂
Subgroups	C₂	C₃, S₂ = C_i	C₄, C₂	C₅, S₂ = C_i	C₆, S₄, C₃, C₂
Example	beveled digonal antiprism	triangular antiprism	square antiprism	pentagonal antiprism	hexagonal antiprism
Antiprisms with directed edges have rotoreflection symmetry. 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.^[1] 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.^[3]

An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis.^[6] 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).^[6] There are several different systems for naming individual improper rotations:

In the Schoenflies notation the symbol S_n (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 be confused with the same notation for symmetric groups).^[6]
In Hermann–Mauguin notation the symbol n is used for an n-fold rotoinversion; i.e., rotation by an angle of rotation of 360°/n with inversion. If n is even it must be divisible by 4. (Note that 2 would be simply a reflection, and is normally denoted "m", for "mirror".) When n is odd this corresponds to a 2n-fold improper rotation (or rotary reflexion).
The Coxeter notation for S_2n is [2n⁺,2⁺] and , as an index 4 subgroup of [2n,2], , generated as the product of 3 reflections.
The Orbifold notation is n×, order 2n.
Subgroups for S₂ to S₂₀.
C₁ is the identity group.
S₂ is the central inversion.
C_n are cyclic groups.

Subgroups

The direct subgroup of S_2n is C_n, order n, index 2, being the rotoreflection generator applied twice.
For odd n, S_2n contains an inversion, denoted C_i or S₂. S_2n is the direct product: S_2n = C_n × S₂, if n is odd.
For any n, if odd p is a divisor of n, then S_2n/p is a subgroup of S_2n, index p. For example S₄ is a subgroup of S₁₂, index 3.

As an indirect isometry

In a wider sense, an improper rotation may be defined as any indirect isometry; i.e., an element of E(3)\E⁺(3): thus it can also be a pure reflection in a plane, or have a glide plane. An indirect isometry is an affine transformation with an orthogonal matrix that has a determinant of −1.

A proper rotation is an ordinary rotation. In the wider sense, a proper rotation is defined as a direct isometry; i.e., an element of E⁺(3): it can also be the identity, a rotation with a translation along the axis, or a pure translation. A direct isometry is an affine transformation with an orthogonal matrix that has a determinant of 1.

In either the narrower or the wider senses, the composition of two improper rotations is a proper rotation, and the composition of an improper and a proper rotation is an improper rotation.

Physical systems

When studying the symmetry of a physical system under an improper rotation (e.g., if a system has a mirror symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general between tensors and pseudotensors), since the latter transform differently under proper and improper rotations (in 3 dimensions, pseudovectors are invariant under inversion).

References

^ ^a ^b ^c Morawiec, Adam (2004), Orientations and Rotations: Computations in Crystallographic Textures, Springer, p. 7, ISBN 978-3-540-40734-8.
^ Miessler, Gary; Fischer, Paul; Tarr, Donald (2014), Inorganic Chemistry (5 ed.), Pearson, p. 78
^ ^a ^b Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 267, ISBN 978-1-930190-09-2.
^ Klein, Philpotts (2013). Earth Materials. Cambridge University Press. pp. 89–90. ISBN 978-0-521-14521-3.
^ Salomon, David (1999), Computer Graphics and Geometric Modeling, Springer, p. 84, ISBN 978-0-387-98682-1.
^ ^a ^b ^c Bishop, David M. (1993), Group Theory and Chemistry, Courier Dover Publications, p. 13, ISBN 978-0-486-67355-4.

[Morawiec-1] Morawiec, Adam (2004), Orientations and Rotations: Computations in Crystallographic Textures, Springer, p. 7, ISBN 978-3-540-40734-8.

[2] Miessler, Gary; Fischer, Paul; Tarr, Donald (2014), Inorganic Chemistry (5 ed.), Pearson, p. 78

[sss-3] Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 267, ISBN 978-1-930190-09-2.

[4] Klein, Philpotts (2013). Earth Materials. Cambridge University Press. pp. 89–90. ISBN 978-0-521-14521-3.

[5] Salomon, David (1999), Computer Graphics and Geometric Modeling, Springer, p. 84, ISBN 978-0-387-98682-1.

[Bishop-6] Bishop, David M. (1993), Group Theory and Chemistry, Courier Dover Publications, p. 13, ISBN 978-0-486-67355-4.

[1]

[2]

[3]

[4]

[5]

[6]