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Symmetry Groups In One Dimension

Symmetry groups in one dimension

A pattern in 1D can be represented as a function f(x) for, say, the color at position x. The 1D isometries map x to x+a and to a−x. Isometries which leave the function unchanged are translations x+a with a such that f(x+a)=f(x) and reflections a−x with a such that f(a−x)=f(x)

Translational symmetry

Consider all patterns in 1D which have translational symmetry, i.e. functions f(x) such that for some a>0, f(x+a)=f(x) for all x. For these patterns, the values of a for which this property holds form a group.

Discrete symmetry groups

We first consider patterns for which the group is discrete, i.e. for which the positive values in the group have a minimum. By rescaling we make this minimum value 1. Such patterns fall in two categories, the two 1D space groups. In the simpler case the only isometries of R which map the pattern to itself are translations; this applies e.g. for the pattern − −−− − −−− − −−− − −−− Each isometry can be characterized by an integer, namely plus or minus the translation distance. Therefore the symmetry group is Z. In the other case, among the isometries of R which map the pattern to itself there are also reflections; this applies e.g. for the pattern − −−− − − −−− − − −−− − We choose the origin for x at one of the points of reflection. Now all reflections which map the pattern to itself are of the form a−x with a an integer (the increments of a are 1 again, because we can combine a reflection and a translation to get another reflection, and we can combine two reflections to get a translation). Therefore all isometries can be characterized by an integer and a code, say 0 or 1, for translation or reflection. Thus:
-

(a,0):x \mapsto x+a

(a,1):x\mapsto a-x

The latter is a reflection with respect to the point a/2 (an integer or an integer plus 1/2). Group operations (function composition, the one on the right first) are, for integers a and b:
- (a,0)o(b,0)=(a+b,0)
- (a,0)o(b,1)=(a+b,1)
- (a,1)o(b,0)=(a−b,1)
- (a,1)o(b,1)=(a−b,0) E.g in the 3rd case: translation by an amount b changes x into x+b, reflection with respect to 0 gives −x−b, and a translation a gives a−b−x. This group is called the generalized dihedral group of Z, Dih(Z), and also D_∞. It is a semidirect product of Z and C₂. It has a normal subgroup of index 2 isomorphic to Z: the translations. Also it contains an element f of order 2 such that, for all n in Z, n f = f n⁻¹: the reflection with respect to the reference point, (0,1). The two groups are called lattice groups. The lattice is Z. As translation cell we can take the interval 0 ≤ x < 1. In the first case the fundamental domain can be taken the same; topologically it is a circle (1-torus); in the second case we can take 0 ≤ x ≤ 0.5. The actual discrete symmetry group of a translationally symmetric pattern can be:
- of group 1 type, for any positive value of the smallest translation distance
- of group 2 type, for any positive value of the smallest translation distance, and any positioning of the lattice of points of reflection (which is twice as dense as the translation lattice) The set of translationally symmetric patterns can thus be classified by actual symmetry group, while actual symmetry groups, in turn, can be classified as type 1 or type 2. These space group types are the symmetry groups "up to conjugacy with respect to affine transformations": the affine transformation changes the translation distance to the standard one (above: 1), and the position of one of the points of reflections, if applicable, to the origin. Thus the actual symmetry group contains elements of the form gag⁻¹ = b, which is a conjugate of a.

Non-discrete symmetry groups

For a homogeneous "pattern" the symmetry group contains all translations, and reflection in all points. The symmetry group is isomorphic to Dih(R). There are also less trivial patterns/functions with translational symmetry for arbitrarily small translations, e.g. the group of translations by rational distances. Even apart from scaling and shifting, there are infinitely many cases, e.g. by considering rational numbers of which the denominators are powers of a given prime number. The translations form a group of isometries. However, there is no pattern with this group as symmetry group.

Patterns without translational symmetry

For a pattern without translational symmetry there are the following possibilities (1D point groups):
- the symmetry group is the trivial group (no symmetry)
- the symmetry group is one of the groups each consisting of the identity and reflection in a point (isomorphic to C₂)

1D-symmetry of a function vs. 2D-symmetry of its graph

Symmetries of a function (in the sense of this article) imply corresponding symmetries of its graph. However, 2-fold rotational symmetry of the graph does not imply any symmetry (in the sense of this article) of the function: function values (in a pattern representing colors, grey shades, etc.) are nominal data, i.e. grey is not between black and white, the three colors are simply all different. Even with nominal colors there can be a special kind of symmetry, as in: −−−−−−− -- − −−− − − − (reflection gives the negative image). This is also not included in the classification.

Group action

Group actions of the symmetry group that can be considered in this connection are:
- on R
- on the set of real functions of a real variable (each representing a pattern) This section illustrates group action concepts for these cases. The action of G on X is called
- transitive if for any two x, y in X there exists an g in G such that g·x = y; - for neither of the two group actions this is the case for any discrete symmetry group
- faithful (or effective) if for any two different g, h in G there exists an x in X such that g·x ≠ h·x; - for both group actions this is the case for any discrete symmetry group (because, except for the identity, symmetry groups do not contain elements that "do nothing")
- free if for any two different g, h in G and all x in X we have g·x ≠ h·x; - this is the case if there are no reflections
- regular (or simply transitive) if it is both transitive and free; this is equivalent to saying that for any two x, y in X there exists precisely one g in G such that g·x = y.

Orbits and stabilizers

Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx: :

Gx = \left\

Case that the group action is on R:
- For the trivial group, all orbits contain only one element; for a group of translations, an orbit is e.g. , for a reflection e.g. , and for the symmetry group with translations and reflections e.g. (translation distance is 10, points of reflection are ..,-7,-2,3,8,13,18,23,..). The points within an orbit are "equivalent". If a symmetry group applies for a pattern, then within each orbit the color is the same. Case that the group action is on patterns:
- The orbits are sets of patterns, containing translated and/or reflected versions, "equivalent patterns". A translation of a pattern is only equivalent if the translation distance is one of those included in the symmetry group considered, and similarly for a mirror image. The set of all orbits of X under the action of G is written as X/G. If Y is a subset of X, we write GY for the set . We call the subset Y invariant under G if GY = Y (which is equivalent to GY ⊆ Y). In that case, G also operates on Y. The subset Y is called fixed under G if g·y = y for all g in G and all y in Y. In the example of the orbit , is invariant under G, but not fixed. For every x in X, we define the stabilizer subgroup of x (also called the isotropy group or little group) as the set of all elements in G that fix x: :

G_x = \

If x is a reflection point, its stabilizer is the group of order two containing the identity and the reflection in x. In other cases the stabilizer is the trivial group. For a fixed x in X, consider the map from G to X given by g |-> g·x. The image of this map is the orbit of x and the coimage is the set of all left cosets of G_x. The standard quotient theorem of set theory then gives a natural bijection between G/G_x and Gx. Specifically, the bijection is given by hG_x |-> h·x. This result is known as the orbit-stabilizer theorem. If, in the example, we take x=3, the orbit is , and the two groups are isomorphic with Z. If two elements x and y belong to the same orbit, then their stabilizer subgroups, G_x and G_y, are isomorphic. More precisely: if y = g·x, then G_y = gG_x g⁻¹. In the example this applies e.g. for 3 and 23, both reflection points. Reflection about 23 corresponds to a translation of -20, reflection about 3, and translation of 20.

Isometry

:For the mechanical engineering and architecture usage, see isometric projection. In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Geometric figures which can be related by an isometry are called congruent. Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.

Definitions

The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and one should guess from context which one is intended. Let

X

and

Y

be metric spaces with metrics

d_X

and

d_Y

. A map

f:X\to Y

is called distance preserving if for any

x,y\in X

one has

d_Y(f(x),f(y))=d_X(x,y).

A distance preserving map is automatically injective. A global isometry is a bijective distance preserving map. A path isometry or arcwise isometry is a map which preserves the lengths of curves (not necessarily bijective). Two metric spaces X and Y are called isometric if there is an isometry from X to Y. The set of isometries from a metric space to itself forms a group with respect to function composition, called the isometry group.

Examples

- Any reflection, translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group.
- The map R

\to

R defined by

x\mapsto |x|

is a path isometry but not a global isometry.
- The isometric linear maps from Cⁿ to itself are the unitary matrices.

Generalizations

- Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map

f:X\to Y

between metric spaces such that
- # for

x,x'\in X

one has

|d_Y(f(x),f(x'))-d_X(x,x')|<\epsilon

, and
- # for any point

y\in Y

there exists a point

x\in X

with

d_Y(y,f(x))<\epsilon.

:That is, an ε-isometry preserves distances to within ε and leaves no element of the codomain further than ε away from the image of an element of the domain. Note that ε-isometries are not assumed to be continuous.
- Quasi-isometry is yet an other useful generalization.

Reflection (mathematics)

:This article is about reflection in geometry. For reflexivity of binary relations, see reflexive relation. In mathematics, a reflection (also spelt reflexion) is a map that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q. In order to reflect a planar figure one needs the "mirror" to be a line ("axis of reflection"), while for reflections in the three-dimensional space one would use a plane for a mirror. Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle. Geometrically, to find the reflection of a point one drops a perpendicular from the point onto the line (plane) used for reflection, and continues the same distance on the other side. To find the reflection of a figure, one reflects each point in the figure. A reflection done twice brings us back where we started. A reflection preserves the distance between points. A reflection does not move the points which are on the mirror, and the dimension of the mirror is by one smaller than the dimension of the space in which the reflection takes places. These observations allow one to formalize the definition of reflection: a reflection is an involutive isometry of an Euclidean space whose set of fixed points is an affine subspace of codimension 1. A figure which does not change upon undergoing a certain reflection is said to have reflection symmetry. Closely related to reflections are oblique reflections and circle inversions. These transformations are still involutions with the set of fixed points having codimension 1, but they are no longer isometries. On a somewhat unrelated note, in LAPACK the term reflector with the types block reflector and elementary reflector is used to describe the functionality of the routines that implement the Householder transformation.

Formulas

Given a vector a in Euclidean space Rⁿ, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by :

\mathrm_a(v) = v - 2\fraca

where v·a denotes the dot product of v with a. Note that the second term in the above equation is just twice the projection of v onto a. One can easily check that
- Ref_a(v) = − v, if v is parallel to a, and
- Ref_a(v) = v, if v is perpendicular to a. Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix whose entries are :

R_ = \delta_ - 2\frac

where δ_ij is the Kronecker delta. The formula for the reflection in the affine hyperplane

v\cdot a = c

is given by :

\mathrm_(v) = v - 2\fraca.

External link

- [http://www.cut-the-knot.org/Curriculum/Geometry/Reflection.shtml Reflection in Line] at cut-the-knot Category:Euclidean symmetries

Symmetry

Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by one of the operations. In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometries: translations, rotations, reflections, and glide reflections. glide reflection

Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetric to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of G. If the symmetry group of x is the trivial group then x is said to be asymmetric, otherwise symmetric. A general example is that G is a group of bijections g: V → V acting on the set of functions x: V → W by (gx)(v)=x(g−1(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of x consists of all g for which x(v)=x(g(v)) for all v. G is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of G may not be the symmetry group of any object. For example, if the group contains for every v and w in V a g such that g(v)=w, then only the symmetry groups of constant functions x contain that group. However, the symmetry group of constant functions is G itself. In a modified version for vector fields, we have (gx)(v)=h(g,x(g−1(v))) where h rotates any vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors) according to rotation and inversion in g, see symmetry in physics. The symmetry group of x consists of all g for which x(v)=h(g,x(g(v))) for all v. In this case the symmetry group of a constant function may be a proper subgroup of G: a constant vector has only rotational symmetry with respect to rotation about an axis if that axis is in the direction of the vector, and only inversion symmetry if it is zero. For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space. The rotation group of an object is the symmetry group if G is restricted to E+(n), the group of direct isometries. (For generalizations, see the next subsection.) Objects can be modeled as functions x, of which a value may represent a selection of properties such as color, density, chemical composition, etc. Depending on the selection we consider just symmetries of sets of points (x is just a boolean function of position v), or, at the other extreme, e.g. symmetry of right and left hand with all their structure. For a given symmetry group, the properties of part of the object, fully define the whole object. Considering points equivalent which, due to the symmetry, have the same properties, the equivalence classes are the orbits of the group action on the space itself. We need the value of x at one point in every orbit to define the full object. A set of such representatives forms a fundamental domain. An object with a desired symmetry can be produced by choosing for every orbit a single function value. Starting from a given object x we can e.g.:
- take the values in a fundamental domain (i.e., add copies of the object)
- take for each orbit some kind of average or sum of the values of x at the points of the orbit (ditto, where the copies may overlap) If it is desired to have no more symmetry than that in the symmetry group, then the object to be copied should be asymmetric. As pointed out above, some groups of isometries are not the symmetry group of any object, except in the modified model for vector fields. For example, this applies in 1D for the group of all translations. The fundamental domain is only one point, so we can not make it asymmetric, so any "pattern" invariant under translation is also invariant under reflection (these are the uniform "patterns"). In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry. If there is also reflectional symmetry, the constant function value contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D example is an infinite cylinder with a current perpendicular to the axis; the magnetic field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero. For vectors (in particular the current density) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. This cylindrical symmetry without mirror planes through the axis is also only possible in the vector field version of the symmetry concept. A similar example is a cylinder rotating about its axis, where magnetic field and current density are replaced by angular momentum and velocity, respectively. A symmetry group is said to act transitively on a repeated feature of an object if, for every pair of occurrences of the feature there is a symmetry operation mapping the first to the second. For example, in 1D, the symmetry group of acts transitively on all these points, while does not act transitively on all points. Equivalently, the first set is only one conjugacy class with respect to isometries, while the second has two classes.

Non-isometric symmetry

As mentioned above, G (the symmetry group of the space itself) may differ from the Euclidean group, the group of isometries. Examples:
- G is the group of similarity transformations, i.e. affine transformations with a matrix A that is a scalar times an orthogonal matrix. Thus dilations are added, self-similarity is considered a symmetry
- G is the group of affine transformations with a matrix A with determinant 1 or -1, i.e. the transformation which preserve area; this adds e.g. oblique reflection symmetry.
- G is the group of all bijective affine transformations
- In inversive geometry, G includes circle reflections, etc.
- More generally, an involution defines a symmetry with respect to that involution.

Reflection symmetry

See reflection symmetry.

Rotational symmetry

See rotational symmetry.

Translational symmetry

See main article translational symmetry. Translational symmetry leaves an object invariant under a discrete or continuous group of translations Ta(p) = p + a

Glide reflection symmetry

A glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. It implies translational symmetry with twice the translation vector. The symmetry group is isomorphic with Z.

Rotoreflection symmetry

In 3D, rotoreflection or improper rotation in the strict sense is rotation about an axis, combined with reflection in a plane perpendicular to that axis. As symmetry groups with regard to a roto-reflection we can distinguish:
- the angle has no common divisor with 360°, the symmetry group is not discrete
- 2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); a special case is n=1, inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
- Cnh (angle of 360°/n); for odd n this is generated by a single symmetry, and the abstract group is C2n, for even n this is not a basic symmetry but a combination. See also point groups in three dimensions.

Screw axis symmetry

In 3D, screw axis symmetry is invariance under a rotation about an axis combined with translation along that axis . We can distinguish:
- there is invariance for every angle and a proportional translation distance, this applies e.g. for an infinite helix and double helix;
- the angle has no common divisor with 360°; the symmetry group is discrete, although the set of angles is not; it does not contain pure translations
- n-fold screw axis (angle of 360°/n) See also space group.

Symmetry combinations

See symmetry combinations.

Color

symmetry combinations With a color image one can associate a greyshade or black-and-white image. One way is to associate with each color a greyshade or either black or white. Alternatively, boundaries may be represented in black, and interior areas in white. When considering symmetry "ignoring colors" this tends to mean that dark colors become black and light colors white, or that boundaries become black. Sometimes there is only one meaningful conversion, in other cases the conversion has to be specified to avoid ambiguity (see e.g. the tetrakis square tiling). The new image may have more symmetry. Also colors may provide a special kind of symmetry, e.g. with corresponding points having opposite colors (including black and white), such as in the yin and yang symbol. Compare the modified symmetry model for vector fields, above.

Similarity vs. sameness

Although two objects with great similarity appear the same, they must logically be different. For example, if one rotates an equilateral triangle around its center 120 degrees, it will appear the same as it was before the rotation to an observer. In theoretical euclidean geometry, such a rotation would be unrecognizable from its previous form. In reality however, each corner of any equilateral triangle composed of matter must be composed of separate molecules in separate locations. Therefore, symmetry in real physical objects is a matter of similarity instead of sameness. The difficulty for an intelligence to differentiate such a seemingly exact similarity is understandable.

More on symmetry in geometry

The German geometer Felix Klein enunciated a very influential Erlangen programme in 1872, suggesting symmetry as unifying and organising principle in geometry (at a time when that was read 'geometries'). This is a broad rather than deep principle. Initially it led to interest in the groups attached to geometries, and the slogan transformation geometry (an aspect of the New Math, but hardly controversial in modern mathematical practice). By now it has been applied in numerous forms, as kind of standard attack on problems. A fractal, as conceived by Mandelbrot, has symmetry involving scaling. For example an equilateral triangle can be shrunk so that each of its sides are one third the length of the original's sides. These smaller triangles can be rotated and translated until they are adjacent and in the center of each of the larger triangle's lines. The smaller triangles can repeat the process, resulting in even smaller triangles on their sides. Fascinating intricate structures can be created by repeating such scaling symmetrical operations many times. If a structure has a symmetry plane then for every part of the structure there are two possibilities:
- the part has itself a symmetry plane (the same plane)
- it has a mirror image counterpart

Symmetry in mathematics

:(main article: symmetry in mathematics) An example of a mathematical expression exhibiting symmetry is a2c + 3ab + b2c. If a and b are exchanged, the expression remains unchanged due to the commutativity of addition and multiplication. Like in geometry, for the terms there are two possibilities:
- it is itself symmetric
- it has one or more other terms symmetric with it, in accordance with the symmetry kind See also symmetric function, duality (mathematics).

Symmetry in logic

A dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba. Thus, “is the same age as” is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul. Symmetric binary logical connectives are "and" (&and;, <math>\land</math>, or &), "or" (&or;), "biconditional" (iff) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or").

Generalization of symmetry

If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of symmetry group to that of a groupoid. Physicists have come up with other directions of generalization, such as supersymmetry and quantum groups.

Symmetry in physics

(see main article: symmetry in physics) Symmetry in physics has been generalized to mean invariance under any kind of transformation. This has become one of the most powerful tools of theoretical physics. See Noether's theorem (which, as a gross oversimplification, states that for every symmetry law, there is a conservation law).

Symmetry in biology

See symmetry in nature, bilateral symmetry, facial symmetry, pentamerism.

Symmetry in chemistry

See Spectroscopy, Molecular orbital

Symmetry in the arts and crafts

You can find the use of symmetry across a wide variety of arts and crafts.

Architecture

Architecture Image:Monticello.PNG Symmetry has long been a predominant design element in architecture; prominent examples include the Leaning Tower of Pisa, Monticello, the Astrodome, the Sydney Opera House, Gothic church windows, and the Pantheon. Symmetry is used in the design of the overall floor plan of buildings as well as the design of individual building elements such as doors, windows, floors, frieze work, and ornamentation; many facades adhere to bilateral symmetry. Links:
- [http://members.tripod.com/vismath/kim/ Williams: Symmetry in Architecture]
- [http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml Aslaksen: Mathematics in Art and Architecture]

Pottery

Image:Persian_Pottery.jpg The ancient Chinese used symmetrical patterns in their bronze castings since the 17th century B.C. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design. Persian pottery dating from 6000 B.C. used symmetric zigzags, squares, and cross-hatchings. Links:
- [http://www.chinavoc.com/arts/handicraft/bronze.htm Chinavoc: The Art of Chinese Bronzes]
- [http://www-oi.uchicago.edu/OI/MUS/VOL/NN_SUM94/NN_Sum94.html Grant: Iranian Pottery in the Oriental Institute]
- [http://www.metmuseum.org/collections/department.asp?dep=14 The Metropolitan Museum of Art - Islamic Art]

Quilts

Image:kitchen_kaleid.PNG As quilts are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry. Links:
- [http://its.guilford.k12.nc.us/webquests/quilts/quilts.htm Quate: Exploring Geometry Through Quilts]
- [http://log24.com/log04/0809.htm Quilt Geometry]

Carpets, rugs

Image:orientalrug.JPG A long tradition of the use of symmetry in rug patterns spans a variety of cultures. American Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly most rugs use quadrilateral symmetry -- a motif reflected across both the horizontal and vertical axes. Links:
- [http://www.marlamallett.com/default.htm Mallet: Tribal Oriental Rugs]
- [http://navajocentral.org/rugs.htm Dilucchio: Navajo Rugs]

Music

Form

Symmetry has been used as a formal constraint by many composers, such as the arch form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney (or swell). In classical music, Bach used the symmetry concepts of permutation and invariance; see (external link "Fugue No. 21," [http://jan.ucc.nau.edu/~tas3/wtc/ii21s.pdf pdf] or [http://jan.ucc.nau.edu/~tas3/wtc/ii21.html Shockwave]).

Pitch structures

Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chord. Symmetrical scales or chords, such as the whole tone scale, augmented chord, or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal center, and have a less specific diatonic functionality. However, composers such as Alban Berg, Béla Bartók, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non-tonal tonal centers. Perle (1992) explains "C-E, D-F#, [and] Eb-G, are different instances of the same interval...the other kind of identity...has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows:" Thus in addition to being part of the interval-4 family, C-E is also a part of the sum-4 family (with C equal to 0). Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are enharmonic with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of Romantic composers such as Gustav Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander Scriabin, Edgard Varese, and the Vienna school. At the same time, these progressions signal the end of tonality. The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op. 3 (1910). (Perle, 1990)

Equivalency

Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. See also Asymmetric rhythm.

Other arts and crafts

Chair      Bracelet      Celtic knotwork The concept of symmetry is applied to the design of objects of all shapes and sizes -- you can find it in the design of beadwork, furniture, sand paintings, knotwork, masks, and musical instruments (to name just a handful of examples).

Aesthetics

Symmetry does not by itself confer beauty to an object — many symmetrical designs are boring or overly challenging, and on the other hand preference for, or dislike of, exact symmetry is apparently dependent on cultural background. Along with texture, color, proportion, and other factors, symmetry does however play an important role in determining the aesthetic appeal of an object. See also M. C. Escher, wallpaper group, tiling.

Symmetry in games and puzzles

- See also symmetric games.
- See sudoku. Puzzles
- [http://m759.freeservers.com/puzzle.html The Diamond 16 Puzzle] Board Games
- [http://www.symmetryperfect.com/shots The Symmetrical Chess Collection]

Symmetry in literature

See palindrome.

Symmetry in telecommunications

Some telecommunications services (specifically data products) may be referred to as symmetrical or asymmetrical. This refers to the bandwidth allocated for data sent and received. Most internet services used by residential customers are asymmetrical: the data sent to the server normally is far less than that returned by the server.

Moral symmetry

- Tit for tat
- Reciprocity
- Golden Rule
- Empathy & Sympathy
- Reflective equilibrium

External links

- [http://0waldo.com (A)symmetrical wallpaper tiles] by Walter Muncaster (user:0waldo)
- [http://home.earthlink.net/~akuster/music/bartok/quartet4.htm An Analysis of the first movement of the Fourth String Quartet (1928)] by Andrew Kuster
- [http://www.usask.ca/education/coursework/skaalid/theory/theory.htm Skaalid: Design Theory]
- [http://mathforum.org/library/topics/sym_tess/ Mathforum: Symmetry/Tesselations]
- [http://www.teachersnetwork.org/teachnet/westchester/symmetry.htm Calotta: A World of Symmetry]
- [http://www.uwgb.edu/dutchs/SYMMETRY/2DPTGRP.HTM Dutch: Symmetry Around a Point in the Plane]
- [http://www.punahou.edu/acad/sanders/MathArt/MACch2sym.html Sanders: Transformations and Symmetry]
- [http://daphne.palomar.edu/design/conclude.html Saw: Design Notes]
- [http://home.earthlink.net/~jdc24/symmetry.htm Chapman: Aesthetics of Symmetry]
- [http://www.bangor.ac.uk/~mas009/psym.htm Abas: The Wonder Of Symmetry]
- [http://www.mi.sanu.ac.yu/~jablans/isis0.htm ISIS Symmetry]
- [http://etext.lib.virginia.edu/cgi-local/DHI/dhi.cgi?id=dv4-46 Symmetry and Asymmetry at The Dictionary of the History of Ideas]

References

- Perle, George (1990). The Listening Composer, p. 112. California: University of California Press. ISBN 0520069919.
- Perle, George (1992). Symmetry, the Twelve-Tone Scale, and Tonality. Contemporary Music Review 6 (2), pp. 81-96
- Weyl, Hermann (1952). Symmetry. Princeton University Press. ISBN 0-691-02374-3.
- Category:Arts ja:対称性

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one. For example, the integers, Z, form a discrete subgroup of the reals, R, but the rational numbers, Q, do not. Since topological groups are homogeneous, one need only look at a single point to determine if the group is discrete. In particular, a topological group is discrete if and only if the singleton containing the identity is a clopen set. Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms of a discrete group are exactly the group homomorphisms of the underlying group. Hence, there is an isomorphism between the categories of groups and of discrete groups and indeed, discrete groups can generally be identified with the underlying (non-topological) groups. With this in mind, the term discrete group theory is used to refer to the study of groups without topological structure, in contradistinction to topological or Lie group theory. It is divided, logically but also technically, into finite group theory, and infinite group theory. If G is a finite or countably infinite group, then the discrete topology suffices to make it a zero-dimensional Lie group. Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete. There are some occasions when a topological group or Lie group is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the Bohr compactification, and in group cohomology theory of Lie groups. A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G.

Examples

- Frieze groups and wallpaper groups are discrete subgroups of the isometry group of the Euclidean plane. Wallpaper groups are cocompact, but Frieze groups are not.
- A space group is a discrete subgroup of the isometry group of Euclidean space of some dimension.
- A crystallographic group usually means a cocompact, discrete subgroup of the isometries of some Euclidean space. Sometimes, however, a crystallographic group can be a cocompact discrete subgroup of a nilpotent or solvable Lie group.
- Every triangle group T is a discrete subgroup of the isometry group of the sphere (when T is finite), the Euclidean plane (when T has a Z + Z subgroup of finite index), or the hyperbolic plane.
- Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane.
- A Fuchsian group that preserves orientation and acts on the upper half-plane model of the hyperbolic plane is a discrete subgroup of the Lie group PSL(2,R), the group of orientation preserving isometries of the upper half-plane model of the hyperbolic plane.
- A Fuchsian group is sometimes considered as a special case of a Kleinian group, by embedding the hyperbolic plane isometrically into three dimensional hyperbolic space and extending the group action on the plane to the whole space.
- The modular group is PSL(2,Z), thought of as a discrete subgroup of PSL(2,R). The modular group is a lattice in PSL(2,R), but it is not cocompact.
- Kleinian groups are, by definition, discrete subgroups of the isometry group of hyperbolic 3-space. These include quasi-Fuchsian groups.
- A Kleinian group that preserves orientation and acts on the upper half space model of hyperbolic 3-space is a discrete subgroup of the Lie group PSL(2,C), the group of orientation preserving isometries of the upper half-space model of hyperbolic 3-space.
- A lattice in a Lie group is a discrete subgroup such that the Haar measure of the quotient space is finite.

Links to more examples

- crystallographic point group
- congruence subgroup
- arithmetic group

Symmetry group

:(This page currently does not yet describe various aspects of symmetry groups in theoretical physics, especially in (quantum and classical) field theory.) ---- The symmetry group of an object (image, signal, etc., e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant with composition as the operation. It is a subgroup of the isometry group of the space concerned. (If not stated otherwise, we consider symmetry groups in Euclidean geometry here, but the concept may also be studied in wider contexts, see below.) The "objects" may be geometric figures, images and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g. a function of position with values in a set of colors. For symmetry of e.g. 3D bodies one may also want to take physical composition into account. The group of isometries of space induces a group action on objects in it. The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations) under which the figure is invariant. The subgroup of orientation-preserving isometries (i.e. translations, rotations and compositions of these) which leave the figure invariant is called its proper symmetry group. The proper symmetry group of an object is equal to its full symmetry group iff the object is chiral. Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is a subgroup of the special orthogonal group SO(n) then, and therefore also called rotation group of the figure. Discrete symmetry groups come in two types: finite point groups, which include only rotations, reflections, and combinations - they are in fact just the finite subgroups of O(n), and infinite lattice groups, which also include translations and possibly glide reflections. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups. Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E(n) (the isometry group of Rⁿ), where two subgroups H₁, H₂ of a group G are conjugate, if there exists g ∈ G such that H₁=g^-1H₂g ). For example:
- two 3D figures have mirror symmetry, but with respect to a different mirror plane
- two 3D figures have 3-fold rotational symmetry, but with respect to a different axis
- two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction Sometimes a broader concept of "same symmetry type" is used, resulting in e.g. 17 wallpaper groups. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This excludes for example in 1D the group of translations by a rational number. A "figure" with this symmetry group is non-drawable and up to arbitrarily fine detail homogeneous, without being really homogeneous.

One dimension

The isometry groups in 1D where for all points the set of images under the isometries is topologically closed are:
- the trivial group C₁
- the groups of two elements generated by a reflection in a point; they are isomorphic with C₂
- the infinite discrete groups generated by a translation; they are isomorphic with Z
- the infinite discrete groups generated by a translation and a reflection in a point; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D_∞ (which is a semidirect product of Z and C₂).
- the group generated by all translations (isomorphic with R); this group cannot be the symmetry group of a "pattern": it would be homogeneous, hence could also be reflected. However, a uniform 1D vector field has this symmetry group.
- the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group of R, Dih(R). See also symmetry groups in one dimension.

Two dimensions

Up to conjugacy the discrete point groups in 2 dimensional space are the following classes:
- cyclic groups C₁, C₂, C₃, C₄,... where C_n consists of all rotations about a fixed point by multiples of the angle 360°/n
- dihedral groups D₁, D₂, D₃, D₄,... where D_n (of order 2n) consists of the rotations in C_n together with reflections in n axes that pass through the fixed point. C₁ is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter F. C₂ is the symmetry group of the letter Z, C₃ that of a triskelion, C₄ of a swastika, and C₅, C₆ etc. are the symmetry groups of similar swastika-like figures with five, six etc. arms instead of four. D₁ is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter A. D₂, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle, and D₃, D₄ etc. are the symmetry groups of the regular polygons. The actual symmetry groups in each of these cases have two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors. The remaining isometry groups in 2D with a fixed point, where for all points the set of images under the isometries is topologically closed are:
- the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the circle group S¹, the multiplicative group of complex numbers of absolute value 1. It is the proper symmetry group of a circle and the continuous equivalent of C_n. There is no figure which has as full symmetry group the circle group, but for a vector field it may apply (see the 3D case below).
- the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. It is also called Dih(S¹) as it is the generalized dihedral group of S¹. For non-bounded figures, the additional isometry groups can include translations; the closed ones are:
- the 7 frieze groups
- the 17 wallpaper groups
- for each of the symmetry groups in 1D, the combination of all symmetries in that group in one direction, and the group of all translations in the perpendicular direction
- ditto with also reflections in a line in the first direction

Three dimensions

Up to conjugacy the set of 3D point groups consists of 7 infinite series, and 7 separate ones. In crystallography they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 from the 7 infinite series, and 5 of the 7 others). See point groups in three dimensions. The continuous symmetry groups with a fixed point include those of:
- cylindrical symmetry without a symmetry plane perpendicular to the axis, this applies for example often for a bottle
- cylindrical symmetry with a symmetry plane perpendicular to the axis
- spherical symmetry For objects and scalar fields the cylindrical symmetry implies vertical planes of reflection. However, for vector fields it does not: in cylindrical coordinates with respect to some axis,

\mathbf = A_\rho\boldsymbol + A_\phi\boldsymbol + A_z\boldsymbol

has cylindrical symmetry with respect to the axis iff

A_\rho, A_\phi,

and

A_z

have this symmetry, i.e., they do not depend on φ. Additionally there is reflectional symmetry iff

A_\phi = 0

. For spherical symmetry there is no such distinction, it implies planes of reflection. The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. See also subgroups of the Euclidean group.

Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it; this is one way of looking at the Erlangen programme. For example, automorphism groups of certain models of finite geometries are not "symmetry groups" in the usual sense, although they preserve symmetry. They do this by preserving families of point-sets rather than point-sets (or "objects") themselves. See [http://log24.com/theory/patt.html pattern groups]. Like above, the group of automorphisms of space induces a group action on objects in it. For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent iff the two images of the figure are the same (here "the same" does not mean something like e.g. "the same up to translation and rotation", but it means "exactly the same"). Then the equivalence class of the identity is the symmetry group of the figure, and every equivalence class corresponds to one isomorphic version of the figure. There is a bijection between every pair of equivalence classes: the inverse of a representative of the first equivalence class, composed with a representative of the second. In the case of a finite automorphism group of the whole space, its order is the order of the symmetry group of the figure multiplied by the number of isomorphic versions of the figure. Examples:
- Isometries of the Euclidean plane, the figure is a rectangle: there are infinitely many equivalence classes; each contains 4 isometries.
- The space is a cube with Euclidean metric; the figures include cubes of the same size as the space, with colors or patterns on the faces; the automorphisms of the space are the 48 isometries; the figure is a cube of which one face has a different color; the figure has a symmetry group of 8 isometries, there are 6 equivalence classes of 8 isometries, for 6 isomorphic versions of the figure. Compare Lagrange's theorem and its proof.

External link

- [http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Overview of the 32 crystallographic point groups] - form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7 separate 3D point groups Category:Geometry Category:Symmetry Category:Group theory

Semidirect product

In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups.

Some equivalent definitions

Let G be a group, N a normal subgroup of G (i.e.,

N\triangleleft G

) and H a subgroup of G. The following statements are equivalent:
- G = NH and N ∩ H = (with e being the identity element of G)
- G = HN and N ∩ H =
- Every element of G can be written in one and only one way as a product of an element of N and an element of H
- Every element of G can be written in one and only one way as a product of an element of H and an element of N
- The natural embedding H → G, composed with the natural projection G → G / N, yields an isomorphism between H and G / N
- There exists a homomorphism G → H which is the identity on H and whose kernel is N If one (and therefore all) of these statements hold, we say that G is a semidirect product of N and H, or that G splits over N.

Elementary facts and caveats

If G is the semidirect product of the normal subgroup N and the subgroup H, and both N and H are finite, then the order of G equals the product of the orders of N and H. Note that, as opposed to the case with the direct product, a semidirect product is not, in general, unique; if G and G' are two groups which both contain N as a normal subgroup and H as a subgroup, and both are a semidirect product of N and H, then it does not follow that G and G' are isomorphic. This remark leads to an extension problem, of describing the possibilities.

Outer semidirect products

If G is a semidirect product of N and H, then the map φ : H → Aut(N) (where Aut(N) denotes the group of all automorphisms of N) defined by φ(h)(n) = hnh^–1 for all h in H and n in N is a group homomorphism. Together N, H and φ determine G up to isomorphism, as we now show. Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism φ : H → Aut(N), define a new group N ×_φ H, the semidirect product of N and H with respect to φ, as follows: the underlying set is the cartesian product N × H, and the group operation
- is given by :(n₁, h₁)
- (n₂, h₂) = (n₁ φ(h₁)(n₂), h₁ h₂) for all n₁, n₂ in N and h₁, h₂ in H. This is a group in which the identity element is (e_N, e_H) and the inverse of the element (n, h) is (φ(h^–1)(n^–1), h^–1). Pairs N × form a normal subgroup isomorphic to N, while pairs × H form a subgroup isomorphic to H. The full group is a semidirect product of those two subgroups in the sense given above. Conversely, suppose that we are given an internal semidirect product as defined above, i.e. a group G with a normal subgroup N, a subgroup H, and such that every element g of G may be written uniquely in the form g=nh where n lies in N and h lies in H. Let φ : H→Aut(N) be the homomorphism :φ(h)(n)=hnh^–1. Then G is isomorphic to the outer semidirect product N ×_φ H; the isomorphism sends the product nh to the tuple (n,h). In G, we have the rule :(n₁h₁)(n₂h₂) = n₁(h₁n₂h₁^–1)(h₁h₂) and this is the deeper reason for the above definition of the outer semidirect product, and an easy way to memorize it. A version of the splitting lemma for groups states that a group G is isomorphic to a semidirect product of the two groups N and H if and only if there exists a short exact sequence :

0\longrightarrow N \longrightarrow^\ \, G \longrightarrow^\ \,  H \longrightarrow 0

and a group homomorphism γ : H → G such that α ○ γ = id_H, the identity map on H. In this case, φ : H → Aut(N) is given by :φ(h)(n) = β⁻¹(γ(h) β(n)γ(h⁻¹)).

Examples

The dihedral group D_n with 2n elements is isomorphic to a semidirect product of the cyclic groups C_n and C₂. Here, the non-identity element of C₂ acts on C_n by inverting elements; this is an automorphisms since C_n is abelian. The Euclidean group O(2) of all rigid motions (isometries) of the plane (maps f : R² → R² such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in R²) is isomorphic to a semidirect product of the abelian group R² (which describes translations) and the group O(2) of orthogonal 2×2 matrices (which describes rotations and reflections which keep the origin fixed). n is a translation, h a rotation or reflection. Applying a translation and then a rotation or reflection corresponds to applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the conjugate of the original translation). Every orthogonal matrix acts as an automorphism on R² by matrix multiplication. The orthogonal group O(n) of all orthogonal real n×n matrices (intuitively the set of all rotations and reflections of n-dimensional space which keep the origin fixed) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of n-dimensional space) and C₂. If we represent C₂ as the multiplicative group of matrices , where R is a reflection of n dimensional space which keeps the origin fixed (i.e. an orthogonal matrix with determinant –1 representing an involution), then φ : C₂ → Aut(SO(n)) is given by φ(H)(N) = H N H^–1 for all H in C₂ and N in SO(n). In the non-trivial case ( H is not the identity) this means that φ(H) is conjugation of operations by the reflection (a rotation axis and the direction of rotation are replaced by their "mirror image").

Relation to direct products

Suppose G is a semidirect product of the normal subgroup N and the subgroup H. If H is also normal in G, or equivalently, if there exists a homomorphism G → N which is the identity on N, then G is the direct product of N and H. The direct product of two groups N and H can be thought of as the outer semidirect product of N and H with respect to φ(h) = id_N for all h in H. Note that in a direct product, the order of the factors is not important, since N × H is isomorphic to H × N. This is not the case for semidirect products, as the two factors play different roles.

Generalizations

The construction of semidirect products can be pushed much further. There is a version in ring theory, the crossed product of rings. This is seen naturally as soon as one constructs a group ring for a semidirect product of groups. There is also the semidirect sum of Lie algebras. Given a group action on a topological space, there is a corresponding crossed product which will in general be non-commutative even if the group is abelian. This kind of ring (see crossed product for a related construction) can play the role of the space of orbits of the group action, in cases where that space cannot be approached by conventional topological techniques - for example in the work of Alain Connes (cf. noncommutative geometry). There are also far-reaching generalisations in category theory. They show how to construct fibred categories from indexed categories. This is an abstract form of the outer semidirect product construction.

Notation

Sources differ in their notation for the semidirect product. Some texts discuss it with no explicit notation. Others use the subscripted "times" symbol (×_φ) as above to modify the direct product by inclusion of a homomorphism, writing the normal group on the left. Other notation reshapes the times symbol; Unicode [http://www.unicode.org/charts/symbols.html] lists four variants: : Although the Unicode description of the rtimes symbol says "right normal factor", a number of authors use it with a left normal factor. Therefore the usual caution for mathematical notation applies: When reading, be careful to notice the conventions adopted by the author, and when writing, explain notation choices for the reader. The choice of symbol may vary, but putting the normal factor on the left seems fairly consistent.

Index of a subgroup

In mathematics, if G is a group, H a subgroup of G, and g an element of G, then :gH = is a left coset of H in G, and :Hg = is a right coset of H in G.

Some properties

We have gH = H if and only if g is an element of H. Any two left cosets are either identical or disjoint. The left cosets form a partition of G: every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and H itself is the only coset that is a subgroup. The left cosets of H in G are the equivalence classes under the equivalence relation on G given by x ~ y if and only if x^-1y ∈ H. Similar statements are also true for right cosets. A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here. Some books on very applied group theory erroneously identify the conjugacy class as 'the' equivalence class as opposed to a particular type of equivalence class. All left cosets and all right cosets have the same number of elements (or cardinality in the case of an infinite H). Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H]. Lagrange's theorem allows us to compute the index in the case where G and H are finite, as per the formula: :|G| = [G : H] · |H| This equation also holds in the case where the groups are infinite (but is somewhat less useful). The subgroup H is normal if and only if gH = Hg for all g in G. In this case one can turn the set of all cosets into a group, the factor group of G by H.

Fundamental domain

In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern. The set of orbits of the symmetry group define a partitioning of space. Each partition consists of points which, based on the symmetry, have equal properties, e.g., for a 2D color pattern, have the same color. A fundamental domain is a set of representatives of these orbits. This is not unique, but typically a convenient connected part of space is chosen. Examples in 3D:
- for n-fold rotation: an orbit is either a set of n points around the axis, or a single point on the axis; the fundamental domain is a sector
- for reflection in a plane: an orbit is either a set of 2 points, one on each side of the plane, or a single point in the plane; the fundamental domain is a half-space bounded by that plane
- for inversion in a point: an orbit is a set of 2 points, one on each side of the center, except for one orbit, consisting of the center only; the fundamental domain is a half-space bounded by any plane through the center
- for 180° rotation about a line: an orbit is either a set of 2 points opposite to each other with respect to the axis, or a single point on the axis; the fundamental domain is a half-space bounded by any plane through the line
- for discrete translational symmetry in one direction: the orbits are translates of a 1D lattice in the direction of the translation vector; the fundamental domain is an infinite slab
- for discrete translational symmetry in two directions: the orbits are translates of a 2D lattice in the plane through the translation vectors; the fundamental domain is an infinite bar with parallelogrammatic cross section
- for discrete translational symmetry in three directions: the orbits are translates of the lattice; the fundamental domain is a primitive cell which is e.g. a parallelepiped, or a Wigner-Seitz cell, also called Voronoi cell. In the case of translational symmetry combined with other symmetries, the fundamental domain is part of the primitive cell. For example, for wallpaper groups the fundamental domain is a factor 1, 2, 3, 4, 6, 8, or 12 smaller than the primitive cell. More generally, in mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets of Γ in G, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. A fundamental domain always contains a free regular set U, an open set moved around by G into disjoint copies, and nearly as good as D in representing the cosets. One typical condition is that D is almost an open set, in the sense that D is the symmetric difference of an open set in G with a set of measure zero, for the Haar measure on G. For example, when G is Euclidean space of dimension n, and Γ is Zⁿ, the quotient G/Γ is the n-torus. A fundamental domain (also called fundamental region) here can be taken to be [0,1)ⁿ, which is the open set (0,1)ⁿ up to a set of measure zero. In practice the main use of a fundamental domain may be to compute integrals on G/Γ, in which case the set of measure zero is mentioned only to keep straight the pedantic assertion that D is exactly a set of coset representatives, and may quickly be forgotten. Other uses, for example in ergodic theory, are similarly based on having a reasonable set D up to sets of measure zero.

Example

up to The existence and description of a fundamental domain is in general something requiring painstaking work to establish. The diagram to the right shows part of the construction of the fundamental domain for the action of the modular group Γ on the upper half-plane H. This famous diagram appears in all classical books on elliptic modular functions. (It was probably well known to C. F. Gauss, who dealt with fundamental domains in the guise of the reduction theory of quadratic forms.) Here, each triangular region (bounded by the blue lines) is a free regular set of the action of Γ on H. The boundaries (the blue lines) are not a part of the free regular sets. To construct a fundamental domain of H/Γ, one must also consider how to assign points on the boundary, being careful not to double-count such points. Thus, the free regular set in this example is :

U = \left\.

The fundamental domain is built by adding the boundary on the left plus half the arc on the bottom: :

D=U\cup\left\ \cup \left\.

The choice of which points of the boundary to include as a part of the fundamental domain is arbitrary, and varies from author to author. The core difficulty of defining the fundamental domain lies not so much with the definition of the set per se, but rather with how to treat integrals over the fundamental domain, when integrating functions with poles and zeros on the boundary of the domain.

Torus

Geometry

In geometry, a torus (pl. tori) is a doughnut-shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle. The sphere is a special case of the torus obtained when the axis of rotation is a diameter of the circle. If the axis of rotation does not intersect the circle, the torus has a hole in the middle and resembles a ring doughnut, a hula hoop or an inflated tire. The other case, when the axis of rotation is a chord of the circle, produces a sort of squashed sphere resembling a round cushion. Torus was the Latin word for a cushion of this shape. A torus can be defined parametrically by: :

x(u, v) =  (R + r\cos) \cos \,

y(u, v) =  (R + r \cos) \sin \,

z(u, v) = r \sin \,

where :u, v ∈ [0, 2π], :R is the distance from the center of the tube to the center of the torus, :r is the radius of the tube. The equation in Cartesian coordinates for a torus azimuthally symmetric about the z-axis is :

\left(R - \sqrt\right)^2 + z^2 = r^2

The surface area and interior volume of this torus are given by :

A = 4\pi^2 Rr = \left( 2\pi r \right) \left( 2 \pi R \right) \,

V = 2\pi^2R r^2 = \left( \pi r^2 \right) \left( 2\pi R \right). \,

According to a broader definition, the generator of a torus need not be a circle but could also be an ellipse or any other conic section.

Topology

conic section Topologically, a torus is a closed surface defined as product of two circles: S¹ × S¹. The surface described above, given the relative topology from R³, is homeomorphic to a topological torus as long as it does not intersect its own axis. The torus can also be described as a quotient of the Euclidean plane under the identifications :(x,y) ~ (x+1,y) ~ (x,y+1) Or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon

ABA^B^

. The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: :

\pi_1(\mathbb^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathbb \times \mathbb

Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. The first homology group of the torus is isomorphic to the fundamental group (since the fundamental group is abelian).

The n-torus

One can easily generalize the torus to arbitrary dimensions. An n-torus is defined as a product of n circles: :

\mathbb^n = S^1 \times S^1 \times \cdots \times S^1

The torus discussed above is the 2-torus. The 1-torus is just the circle. The 3-torus is rather difficult to visualize. Just as for the 2-torus, the n-torus can be described as a quotient of Rⁿ under integral shifts in any coordinate. That is, the n-torus is Rⁿ modulo the action of the integer lattice Zⁿ (with the action being taken as vector addition). Equivalently, the n-torus is obtained from the n-cube by gluing the opposite faces together. An n-torus is an example of an n-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication. Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n. The cohomology ring H^•(Tⁿ,Z) can be identified with the exterior algebra over the Z-module Zⁿ whose generators are the duals of the n nontrivial cycles.

External links

- [http://www.cut-the-knot.org/shortcut.shtml#torus Creation of a torus] at cut-the-knot
- [http://www.mathsisfun.com/geometry/torus.html More Torus Images] (from [http://www.mathsisfun.com/ Math is Fun])
- Eric W. Weisstein. "Torus." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Torus.html
- [http://www.bubblerings.com/bubblerings/media.cfm Images and movies of bubble rings] from David Whiteis' [http://www.bubblerings.com BubbleRings.com] Category:Surfaces ja:トーラス

Point group

In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. Point groups can exist in a Euclidean space of any dimension. In 2D, a discrete point group is sometimes called a rosette group, and is used to describe the symmetries of an ornament. The 3D discrete point groups are heavily used in chemistry, especially to describe the symmetries of a molecule and of orbitals forming covalent bonds, and in this context they are also called molecular point groups. See point groups in three dimensions. There are infinitely many discrete point groups in each number of dimensions. However, only a finite number is compatible with translational symmetry. This is stated in the crystallographic restriction theorem. In 1D there are 2, in 2D 10, and in 3D 32. They are called crystallographic point groups. crystallographic point group flag has C₅ symmetry; the star on each petal has D₅ symmetry.]] Point groups in 2D fall into two distinct families, according to whether they consist of rotations only, or include reflections. The cyclic groups, C_n (abstract group type Z_n), consist of rotations by 360°/n, and all integer multiples. For example, a swastika has symmetry group C₄, consisting of rotations by 0°, 90°, 180°, and 270°. The symmetry group of a square belongs to the family of dihedral groups, D_n (abstract group type Dih_n), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S₁ is distinct from Dih(S₁) because it explicitly includes the reflections. Note that an infinite group need not be continuous; for example, we have a group of all integer multiples of rotation by 360°/√2, which does not include rotation by 180°. Depending on application, homogeneity up to arbitrarily fine detail in transverse direction may be considered equivalent to full homogeneity in that direction, in which case these symmetry groups can be ignored. See also point groups in two dimensions. C_n and D_n for n = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups. More complex symmetries arise in 3D, see point groups in three dimensions. In any dimension, d, the continuous group of all possible fixed point isometries is the orthogonal group, denoted by O(d); and its continuous subgroup of all possible rotations is the special orthogonal group, denoted by SO(d). This is not Schönflies notation, but the conventional names from Lie group theory.

External links

- [http://www.chemistry.emory.edu/pointgrp/ Downloadable point group tutorial] (Mac and Windows only)
- [http://www.uniovi.es/qcg/d-MolSym/ Molecular symmetry examples]
- [http://www.reciprocalnet.org/edumodules/symmetry/index.html Web-based point group tutorial] (needs Java and Flash) Category:Crystallography Category:Symmetry Category:Group theory

Group action

This article is about the mathematical concept. For the sociology term, see group action (sociology). In mathematics, a symmetry group describes all symmetries of objects. This is formalized by the notion of a group action: every element of the group "acts" like a bijective map (or "symmetry") on some set. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices, and is usually considered in the finite-dimensional case—it is the same as a group action of G on an ordered basis of a vector space.

Definition

If G is a group and X is a set, then a (left) group action of G on X is a binary function g : G × X → X (where the image of g in G and x in X is written as g·x) which satisfies the following two axioms:
- g·(h·x) = (gh)·x for all g, h

\in

G and x

\in

X.
- e·x = x for every x

\in

X; here e denotes the identity element of G. From these two axioms, it follows that for every g in G, the function which maps x in X to g·x is a bijective map from X to X. Therefore, one may alternatively and equivalently define a group action of G on X as a group homomorphism G → Sym(X), where Sym(X) denotes the group of all bijective maps from X to X. If a group action G × X → X is given, we also say that G acts on the set X or X is a G-set. In complete analogy, one can define a right group action of G on X as a function X × G → X by the two axioms
- (x·g)·h = x·(gh)
- x·e = x. Note that the difference between left and right actions is the order in which gh acts on x. For left actions h acts first followed by g, while for right actions g acts first followed by h. In the sequel, we consider only left group actions.

Examples

- Every group G acts on G in two natural but essentially different ways: g·x = (gx) for all x in G, or g·x = (gxg⁻¹) for all x in G.
- The symmetric group S_n and its subgroups act on the set by permuting its elements.
- The symmetry group of a polyhedron acts on the set of vertices of that polyhedron.
- The symmetry group of any geometrical object acts on the set of points of that object: for example, [http://log24.com/theory/plane.html the eightfold cube] and [http://log24.com/theory/dtheorem.html the diamond theorem].
- The automorphism group of a vector space (or graph, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).
- The general linear group GL(n,R), special linear group SL(n,R), orthogonal group O(n,R), and special orthogonal group SO(n,R) are Lie groups which act on Rⁿ.
- The Galois group of a field extension E/F acts on the bigger field E. So does every subgroup of the Galois group.
- The additive group of the real numbers (R, +) acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems): if t is in R and x is in the phase space, then x describes a state of the system, and t.x is defined to be the state of the system t seconds later if t is positive or -t seconds ago if t is negative.
- The additive group of the real numbers (R, +) acts on the set of real functions of a real variable with (g · f)(x) equal to e.g. f(x+g), f(x) + g,

f(x e^g)

f(x) e^g

f(x+g) e^g

, or

f(x e^g)+g

, but not

f(x e^g+g)

- The quaternions with modulus 1, as a multiplicative group, act on R³: for any such quaternion

z = \cos\frac + \sin\frac\,

Symmetry groups in one dimension

Translational symmetry

Discrete symmetry groups

Non-discrete symmetry groups

Patterns without translational symmetry

1D-symmetry of a function vs. 2D-symmetry of its graph

Group action

Orbits and stabilizers

See also

Isometry

Definitions

Examples

Generalizations

See also

Reflection (mathematics)

Formulas

See also

External link

Symmetry

Mathematical model for symmetry

Non-isometric symmetry

Reflection symmetry

Rotational symmetry

Translational symmetry

Glide reflection symmetry

Rotoreflection symmetry

Screw axis symmetry

Symmetry combinations

Color

Similarity vs. sameness

More on symmetry in geometry

Symmetry in mathematics

Symmetry in logic

Generalization of symmetry

Symmetry in physics

Symmetry in biology

Symmetry in chemistry

Symmetry in the arts and crafts

Architecture

Pottery

Quilts

Carpets, rugs

Music

Form

Pitch structures

Equivalency

Other arts and crafts

Aesthetics

Symmetry in games and puzzles

Symmetry in literature

Symmetry in telecommunications

Moral symmetry

Related topics

External links

References

Discrete group

Examples

Links to more examples

See also

Symmetry group

One dimension

Two dimensions

Three dimensions

Symmetry groups in general

See also

External link

Semidirect product

Some equivalent definitions

Elementary facts and caveats

Outer semidirect products

Examples

Relation to direct products

Generalizations

Notation

See also

Index of a subgroup

Some properties

See also

Fundamental domain

Example