In geometry, an improper rotation[.] (also called rotation-reflection,[.] or rotoinversion[.]
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
!
! S6
! S8
! S10
! S12 |
|
|
with directed edges have rotoreflection symmetry.
p-antiprisms for odd p contain inversion symmetry, Ci. |
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.
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 mirror image. 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).
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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 be confused with the same notation for ).
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In Hermann–Mauguin notation the symbol 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 would be simply a reflection, and is normally denoted "m", for "mirror".) When n is odd this corresponds to a 2 n-fold improper rotation (or rotary reflexion).
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The Coxeter notation for S2 n is 2 n+,2+ and , as an index 4 subgroup of 2 n,2, , generated as the product of 3 reflections.
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The Orbifold notation is n×, order 2 n..
S2 is the central inversion.
Cn are .]]
Subgroups
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The direct subgroup of S2 n is C n, order n, index 2, being the rotoreflection generator applied twice.
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For odd n, S2 n contains an inversion, denoted Ci or S2. S2 n is the direct product: S2 n = C n × S2, if n is odd.
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For any n, if odd p is a divisor of n, then S2 n/ p is a subgroup of S2 n, 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
Euclidean group(3)\E
+(3): thus it can also be a pure reflection in a plane, or have a
glide reflection. 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
(as well as scalars and
, and in general between
and
), since the latter transform differently under proper and improper rotations (in 3 dimensions, pseudovectors are invariant under inversion).
See also
