One-dimensional symmetry group

Last updated January 19, 2022

A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D).

Affine symmetry groups represent translation. 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). The reflections can be represented by the affine Coxeter group [∞], or Coxeter-Dynkin diagram representing two reflections, and the translational symmetry as [∞]⁺, or Coxeter-Dynkin diagram as the composite of two reflections.

Point group

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 Z₂)

Group	Coxeter		Description
C₁	[ ]⁺		Identity, Trivial group Z₁
D₁	[ ]		Reflection. Abstract groups Z₂ or Dih₁.

Discrete symmetry groups

These affine symmetries can be considered limiting cases of the 2D dihedral and cyclic groups:

Group	Coxeter		Description
C_∞	[∞]⁺		Cyclic: ∞-fold rotations become translations. Abstract group Z_∞, the infinite cyclic group.
D_∞	[∞]		Dihedral: ∞-fold reflections. Abstract group Dih_∞, the infinite dihedral group.

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.

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 or line 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 where the constant "a" is 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:

$(a,0)\circ (b,0)=(a+b,0)$
$(a,0)\circ (b,1)=(a+b,1)$
$(a,1)\circ (b,0)=(a-b,1)$
$(a,1)\circ (b,1)=(a-b,0)$

E.g., in the third 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.

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 a 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\{g\cdot x\mid g\in G\right\}.

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. {..,−9,1,11,21,..}, for a reflection e.g. {2,4}, and for the symmetry group with translations and reflections, e.g., {−8,−6,2,4,12,14,22,24,..} (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 {g · y : y $\in$ Y and g $\in$ G}. We call the subset Yinvariant 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 = yfor all g in G and all y in Y. In the example of the orbit {−8,−6,2,4,12,14,22,24,..}, {−9,−8,−6,−5,1,2,4,5,11,12,14,15,21,22,24,25,..} 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}=\{g\in G\mid g\cdot x=x\}.

If x is a reflection point, its stabilizer is the group of order two containing the identity and the reflection inx. In other cases the stabilizer is the trivial group.

For a fixed x in X, consider the map from G to X given by $g\mid \rightarrow g\cdot x$ . 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 $hG_{x}\mid \rightarrow h\cdot x$ . This result is known as the orbit-stabilizer theorem. If, in the example, we take $x=3$ , the orbit is {−7,3,13,23,..}, 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 $y=g\cdot x$ , 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.

Related Research Articles

Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.

In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.

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, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetries. The set of symmetries of a frieze pattern is called a frieze group.

Here a wallpaper is a drawing that covers a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper corresponds a group of such congruent transformations, in which the function composition operates. The present article classifies such groups.

In 2-dimensional geometry, a glide reflection is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection and translation can look different from the starting configuration, so objects with glide symmetry are in general, not symmetrical under reflection alone. In group theory, the glide plane is classified as a type of opposite isometry of the Euclidean plane

In geometry and group theory, a lattice in is a subgroup of the additive group $which is isomorphic to the additive group, and which spans the real vector space . In other words, for any basis of, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell.$

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

In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by $a : T a (p) = p + a$ .

In mathematics, D₃ (sometimes alternatively denoted by D₆) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S₃ of degree 3. It is also the smallest possible non-abelian group.

A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. Both the regular dodecahedron and the rhombic triacontahedron have the same set of symmetries.

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.

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 mathematics, the infinite dihedral group Dih_∞ is an infinite group with properties analogous to those of the finite dihedral groups.

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.

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.

The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. Each one is an infinite extension of a finite symmetric group, the group of permutations (rearrangements) of a finite set. In addition to their geometric description, the affine symmetric groups may be defined as collections of permutations of the integers that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied as part of the fields of combinatorics and representation theory.

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.