![Translation of a set](https://upload.wikimedia.org/wikipedia/commons/9/91/Translation_of_a_set.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, to
translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
mathematics, continuous translational symmetry is the
invariance of a system of equations under any translation. Discrete translational
symmetry is invariant under
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
translation.
Analogously an
operator on functions is said to be translationally invariant with respect to a
translation operator if the result after applying doesn't change if the argument function is translated.
More precisely it must hold that
Laws of physics
Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
are translationally invariant under a spatial translation if they do not distinguish different points in space. According to
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, space translational symmetry of a physical system is equivalent to the
momentum conservation law.
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.
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 . Fundamental domains are e.g. 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.
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
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
. Different bases of translation vectors generate the same lattice
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
one is transformed into the other by a matrix of integer coefficients of which the
absolute value of the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
is 1. The
absolute value of the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
of the symmetry: any pattern on or in it is possible, and this defines the whole object.
See also
lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...
.
E.g. in 2D, instead of a and b we can also take a and , etc. In general in 2D, we can take and for integers ''p'', ''q'', ''r'', and ''s'' such that 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)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...
.
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 ''p''1 (the same applies without shift). With rotational symmetry of order two of the pattern on the tile we have ''p''2 (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.
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.
Examples
![Translation invariance of less-than-relation](https://upload.wikimedia.org/wikipedia/commons/1/1b/Translation_invariance_of_less-than-relation.svg)
*
Frieze pattern
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 symmetri ...
s all have translational symmetries, and sometimes other kinds.
* The
Fourier transform with subsequent computation of absolute values is a translation-invariant operator.
* The mapping from a
polynomial function to the polynomial degree is a translation-invariant functional.
* The
Lebesgue measure is a
complete translation-invariant
measure.
See also
*
Glide reflection
In 2-dimensional geometry, a glide reflection (or transflection) 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 ...
*
Displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
* Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
*
Periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
*
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...
*
Translation operator (quantum mechanics)
*
Rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
*
Lorentz symmetry
In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
*
Tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety o ...
*{{slink, List of cycles#Mathematics of waves and cycles
References
*Stenger, Victor J. (2000) and MahouShiroUSA (2007). ''Timeless Reality''. Prometheus Books. Especially chpt. 12. Nontechnical.
Classical mechanics
Symmetry
Conservation laws