In geometry, a translation "slides" an object by a vector a: T_a(p) = p + a. For other uses, see Geometry (disambiguation). ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... This article is about vectors that have a particular relation to the spatial coordinates. ...

In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariance under discrete (quantized) translation. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Sphere symmetry group o. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ... Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ...

Analogously an operator A on functions is said to be translation invariant with respect to a translation operator T_δ if the result after applying A doesn't change if the argument function is translated. More precisely it must hold that In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

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Laws of physics are translationally invariant if they do not distinguish different points in space. According to Noether's theorem, translational symmetry of a physical system is equivalent to the momentum conservation law. A physical law or a law of nature is a scientific generalization based on empirical observations. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... This article is about momentum in physics. ...

Translational symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group. The symmetry group of an object (e. ...

Geometry

Translational invariance implies that, at least in one direction, the object is infinite: for any given point p, the set of points with the same properties due to the translational symmetry form the infinite discrete set {p+na|n in Z} = p + Z a. Fundamental domains are e.g. H + [0,1] a for any hyperplane H for which a has an independent direction. This is in 1D a line segment, in 2D an infinite strip, and in 3D a slab, such that the vector starting at one side ends at the other side. Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector. A hyperplane is a concept in geometry. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...

In spaces with dimension higher than 1, there may be multiple translational symmetry. For each set of k independent translation vectors the symmetry group is isomorphic with Z^k. In particular the multiplicity may be equal to the dimension. This implies that the object is infinite in all directions. In this case the set of all translations forms a lattice. Different bases of translation vectors generate the same lattice iff one is transformed into the other by a matrix of integer coefficients of which the absolute value of the determinant is 1. The absolute value of the determinant of the matrix formed by a set of translation vectors is the hypervolume of the n-dimensional parallelepiped the set subtends (also called the covolume of the lattice). This parallelepiped is a fundamental region of the symmetry: any pattern on or in it is possible, and this fully defines the whole object. See also lattice (group). In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ... In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. ...

E.g. in 2D, instead of a and b we can also take a and a-b, etc. In general in 2D, we can take pa + qb and ra + sb for integers p, q, r, and s such that ps-qr is 1 or -1. This ensures that a and b themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair. Each pair a, b defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental domain. The vectors a and b can be represented by complex numbers. For two given lattice points, equivalence of choices of a third point to generate a lattice shape is represented by the modular group, see lattice (group). For the cross product in algebraic topology, see Künneth theorem. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. ...

Alternatively, e.g. a rectangle may define the whole object, even if the translation vectors are not perpendicular, if it has two sides parallel to one translation vector, while the other translation vector starting at one side of the rectangle ends at the opposite side.

For example, consider a tiling with equal rectangular tiles with an asymmetric pattern on them, all oriented the same, in rows, with for each row a shift of a fraction, not one half, of a tile, always the same, then we have only translational symmetry, wallpaper group p1 (the same applies without shift). With rotational symmetry of order two of the pattern on the tile we have p2 (more symmetry of the pattern on the tile does not change that, because of the arrangement of the tiles). The rectangle is a more convenient unit to consider as fundamental domain (or set of two of them) than a parallelogram consisting of part of a tile and part of another one. Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ...

In 2D there may be translational symmetry in one direction for vectors of any length. One line, not in the same direction, fully defines the whole object. Similarly, in 3D there may be translational symmetry in one or two directions for vectors of any length. One plane (cross-section) or line, respectively, fully defines the whole object. A 3-D view of a beverage-can stove with a cross section in yellow. ...

Examples

Text

An example of translational symmetry in one direction in 2D nr. 1) is:

Note: The example is not an example of rotational symmetry.

 example example example example example example example example 

(get the same by moving one line down and two positions to the right), and of translational symmetry in two directions in 2D (wallpaper group p1):

 * |* |* |* | |* |* |* |* |* |* |* |* * |* |* |* | |* |* |* |* |* |* |* |* 

(get the same by moving three positions to the right, or one line down and two positions to the right; consequently get also the same moving three lines down).

In both cases there is neither mirror-image symmetry nor rotational symmetry.

For a given translation of space we can consider the corresponding translation of objects. The objects with at least the corresponding translational symmetry are the fixed points of the latter, not to be confused with fixed points of the translation of space, which are non-existent. In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...

Calculus

• The Fourier transform with subsequent computation of absolute values is a translational invariant operator.
• The mapping from a polynomial function to the polynomial degree is a translation invariant functional.

In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

See also

• periodic function
• lattice (group)
• rotational symmetry
• Lorentz symmetry
• tessellation
• mathematics of waves and cycles

In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. ... The triskelion appearing on the Isle of Man flag. ... In physics, Lorentz symmetry is the invariance of physical laws under the Lorentz transformations. ... A tessellated plane seen in street pavement. ... Below is a listing of cycles. ...

References

• Stenger, Victor J. (2000) and MahouShiroUSA (2007). Timeless Reality. Prometheus Books. Especially chpt. 12. Nontechnical.