Improper rotation
In
Group | S4 | S6 | S8 | S10 | S12 |
---|---|---|---|---|---|
Subgroups | C2 | C3, S2 = Ci | C4, C2 | C5, S2 = Ci | C6, S4, C3, C2 |
Example | ![]() beveled digonal antiprism |
triangular antiprism
|
![]() square antiprism |
![]() pentagonal antiprism |
![]() hexagonal antiprism |
inversion symmetry , Ci.
|
Three dimensions
In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and
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 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 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 S2n is [2n+,2+] and
, as an index 4 subgroup of [2n,2],
, generated as the product of 3 reflections.
- The central inversion..
Cn are cyclic groups
Subgroups
- The direct subgroup of S2n is Cn, order n, index 2, being the rotoreflection generator applied twice.
- For odd n, S2n contains an inversion, denoted Ci or S2. S2n is the direct product: S2n = Cn × S2, if n is odd.
- For any n, if odd p is a divisor of n, then S2n/p is a subgroup of S2n, 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
See also
References
- ^ ISBN 978-3-540-40734-8.
- ^ Miessler, Gary; Fischer, Paul; Tarr, Donald (2014), Inorganic Chemistry (5 ed.), Pearson, p. 78
- ^ ISBN 978-1-930190-09-2.
- ISBN 978-0-521-14521-3.
- ISBN 978-0-387-98682-1.
- ^ ISBN 978-0-486-67355-4.