User:Patrick/symmetry

The set of all symmetry operations considered on all objects in a set X can be modeled 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 symmetrical 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. The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon asymmetry.

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 {…, 1, 2, 5, 6, 9, 10, 13, 14, …} acts transitively on all these points, while {…, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, …} 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.

Simplified
The set of all symmetry operations considered on all objects in a set X can be modeled 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 symmetrical 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, while X is, for some set W, the set of functions x: V → W and the group action is defined by (gx)(v) = x[g−1(v)] (or X is 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. A bijection g acts on an object x, giving an object gx.

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.

Isometries
For a common notion of symmetry in Euclidean space, V is the set of points of an object, or Rn, where n is the dimension of the object, and G is the isometry group of V. An object can be modeled as a function x on V, of which a value may represent a selection of properties such as color, density, chemical composition, etc., and if applicable a special value for "outside the object".

An isometry acting on an object x performs a translation, rotation and/or reflection on the object (at least for n<=3). The symmetry group of an object x consists of all isometries that do not affect it. It is a subgroup of G. It is the intersection over all values of x of the isometry group (with each isometry taken with the same domain V) of the set of points where x has this value.

If V is R3 and x is constant outside a plane it has a larger symmetry group than its restriction to the plane. The restrictions of the isometries to the plane have duplicates. After removing these we get the symmetry group of the restriction of x to the plane. If V does not have all symmetry of x, the symmetry group is smaller. For the symmetry of an object one can choose V to be the smallest Rn, or a subset that does not spoil any symmetry of the object.

If, for some g, g·x = y then x and y are said to be symmetrical to each other.

Example
The object is a rectangle with an image on it. V is either the rectangle or R2. G consists of the symmetry group of the blank version of the rectangle (algebraically the Klein four-group), or the Euclidean group E(2), respectively. The symmetry group of the rectangle with image is a subgroup of G. It has 1, 2 or 4 elements (isometries in G) and is dependent on the choice of V only what regards the domains V of the isometries.