Improper rotation

Last updated October 03, 2023

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 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).

Related Research Articles

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.

In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space $; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n).$

In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in many crystals in the cubic crystal system, a rotation of the unit cell by 90 degrees around an axis that is perpendicular to one of the faces of the cube is a symmetry operation that moves each atom to the location of another atom of the same kind, leaving the overall structure of the crystal unaffected.

The Schoenfliesnotation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead. The naming of full space groups usually follows another common convention, the Hermann–Mauguin notation, also known as the international notation.

In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

A regular octahedron has 24 rotational symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron.

<span class="mw-page-title-main">Tetrahedral symmetry</span> 3D symmetry group

A regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih_n.

In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) that does not change the object.

In a group, the conjugate by g of h is ghg⁻¹.

In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.

In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin. This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.

In chemistry and crystallography, a symmetry element is a point, line, or plane about which symmetry operations can take place. In particular, a symmetry element can be a mirror plane, an axis of rotation, or a center of inversion. For an object such as a molecule or a crystal, a symmetry element corresponds to a set of symmetry operations, which are the rigid transformations employing the symmetry element that leave the object unchanged. The set containing these operations form one of the symmetry groups of the object. The elements of this symmetry group should not be confused with the "symmetry element" itself. Loosely, a symmetry element is the geometric set of fixed points of a symmetry operation. For example, for rotation about an axis, the points on the axis do not move and in a reflection the points that remain unchanged make up a plane of symmetry.

In group theory, geometry, representation theory and molecular geometry, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, as transformations of an object in space, rotations, reflections and inversions are all symmetry operations. Such symmetry operations are performed with respect to symmetry elements. In the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasizes its usefulness.

Physical properties must be invariant with respect to symmetry operations.
Symmetry operations can be collected together in groups which are isomorphic to permutation groups.

In geometry, a point reflection is a transformation of affine space in which every point is reflected across a specific fixed point. When dealing with crystal structures and in the physical sciences the terms inversion symmetry, inversion center or centrosymmetric are more commonly used.

In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

<span class="mw-page-title-main">Symmetry (geometry)</span> Geometrical property

In geometry, an object has symmetry if there is an operation or transformation that maps the figure/object onto itself. Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry.

References

1 2 3 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
1 2 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 .
1 2 3 Bishop, David M. (1993), Group Theory and Chemistry, Courier Dover Publications, p. 13, ISBN 978-0-486-67355-4 .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Morawiec-1] 1 2 3 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] 1 2 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] 1 2 3 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]