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Dimension

In common usage, a dimension is a parameter Parameter

In mathematics [i], statistics [i], and the mathematical science [i]s, parameters are quantities that d ... 

 or measurement required to define the characteristics of an object—i.e. length, width, and height or size and shape Shape

In geometry [i], two sets have the same shape if one can be transformed to another by a combination of translations [i] ... 

. In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

, dimensions are the parameters required to describe the position and relevant characteristics of any object within a conceptual space —where the dimensions of a space are the total number of different parameters used for all possible objects considered in the model. Generalizations of this concept are possible and different fields of study will define their spaces by their own relevant dimensions, and use these spaces as frameworks upon which all other study is based.

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In common usage, a dimension is a parameter Parameter

In mathematics [i], statistics [i], and the mathematical science [i]s, parameters are quantities that d ... 

 or measurement required to define the characteristics of an object—i.e. length, width, and height or size and shape Shape

In geometry [i], two sets have the same shape if one can be transformed to another by a combination of translations [i] ... 

.
In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

, dimensions are the parameters required to describe the position and relevant characteristics of any object within a conceptual space —where the dimensions of a space are the total number of different parameters used for all possible objects considered in the model.
Generalizations of this concept are possible and different fields of study will define their spaces by their own relevant dimensions, and use these spaces as frameworks upon which all other study is based. In specialized contexts, units of measurement Units of measurement

The definition, agreement and practical use of units of measurement [i] have played a crucial role in hu ... 

 may sometimes be "dimensions"—meter Metre

The metre, or meter , is a measure of length [i]. ... 

s
or feet Foot

The foot is a biological structure found in many animal [i]s that is used for locomotion [i]. ... 

in geographical Geography

Geography is the study of the Earth's features and of the distribution of life on the earth, including ... 

 space models, or cost and price in models of a local economy Economics

In the social science [i]s, economics is the study of the production [i], ... 

.

For example, locating a point on a plane  requires two parameters — latitude Latitude

Latitude, usually denoted symbolically by the Greek letter f [i] , gives the location of a place on ... 

 and longitude
. The corresponding space has therefore two dimensions, its dimension is two, and this space is said to be 2-dimensional .
Locating the exact position of an aircraft in flight requires another dimension , hence the position of the aircraft can be rendered in a three-dimensional space .

If time Time

Two distinct views exist on the meaning of time.... 

is added as a 3rd or 4th dimension , then the aircraft's estimated "speed" may be calculated from a comparison between the times associated with any two positions. For common uses, simply using "speed" is a useful way of condensing Condensation

[i] to a [[liquid]... 

  the more abstract time Time

Two distinct views exist on the meaning of time.... 

 dimension, even if "speed" is not a dimension, but rather a calculation based on two dimensions. Adding the three Euler angles Euler angles

Euler angles are the classical way of representing rotation [i]s in 3-dimensional [i] Euclidean space [i] ... 

, for a total 6 dimensions, allows the current degrees of freedom —orientation and trajectory —of the aircraft to be known.

Theoretical physics often experiments with dimensions - adding more, or changing their properties - in order to describe unusual conceptual models of space, in order to help better describe concept of quantum mechanics Quantum mechanics

Quantum mechanics is a first quantized [i] quantum theory [i] that supersedes classical mechanics [i] ... 

 —ie. the 'physics beneath the visible physical world.' This concept has been borrowed in science fiction Science fiction

Science fiction is a popular genre of fiction in which the narrative world differs from our own present... 

 as a metaphorical device, where an "alternate dimension" describes extraterrestrial Extraterrestrial life

Extraterrestrial life is life [i] that may exist and originate outside the planet Earth [i], the only pl ... 

 places, species, and cultures which function in various different and unusual ways from human culture.

The physical dimensions are the parameters required to answer to the question where and when happened or will happen some event; for instance: When did Napoleon Napoleon I of France

Napoleon I Bonaparte, Emperor of the French, King of Italy, Mediator of the Swiss Confederation and Prot... 

 die? — On the 5 May 1821 at Saint Helena Saint Helena

Saint Helena is an island of volcanic origin and an overseas territory [i] of ... 

 . They play a fundamental role in our perception of the world around us. According to Immanuel Kant Immanuel Kant

Immanuel Kant , was a German [i] philosopher [i] from Knigsberg in East Prussia [i] . ... 

, we actually do not perceive them but they form the frame in which we perceive events; they form the a priori background in which events are perceived.

Spatial dimensions

Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative amount. Moving diagonally upward and forward is just as the name of the direction implies; i.e., moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions.

Time

Time Time

Two distinct views exist on the meaning of time.... 

 is often referred to as the "fourth dimension Fourth dimension

The concept of a fourth dimension is one that is often described in considering its physical implication... 

." It is, in essence, one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that movement seems to occur at a fixed rate and in one direction.

The equations used by physics to model reality often do not treat time in the same way that humans perceive it. In particular, the equations of classical mechanics Classical mechanics

Classical mechanics is used to describe the motion of macroscopic objects, from projectiles [i] to parts ... 

 are symmetric with respect to time T-symmetry

T-symmetry is the symmetry of physical laws [i] under a time [i] reversal transformation [i] ... 

, and equations of quantum mechanics Quantum mechanics

Quantum mechanics is a first quantized [i] quantum theory [i] that supersedes classical mechanics [i] ... 

 are typically symmetric if both time and other quantities are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics .

The best-known treatment of time as a dimension is Poincaré Henri Poincaré

Jules Henri Poincar , generally known as Henri Poincar, was one of France [i]'s greatest mathematician [i]... 

 and Einstein Albert Einstein

Albert Einstein was a German [i]-born theoretical physicist [i]. ... 

's special relativity Special relativity

The special theory of relativity was proposed in 1905 [i] by Albert Einstein [i] in his article "On the Electrodynamics of Moving Bodies [i] ... 

 , which treats perceived space and time as parts of a four-dimensional manifold Manifold

A manifold is an abstract mathematical space [i] in which every point has a neighborho ... 

.

Additional dimensions

Theories such as string theory String theory

String theory is a model [i] of fundamental physics [i] whose building blocks are on ... 

 predict that the space we live in has in fact many more dimensions , but that the universe measured along these additional dimensions is subatomic in size. As a result, we perceive only the three spatial dimensions that have macroscopic size.

Units

In the physical sciences and in engineering, the dimension of a physical quantity is the expression of the class of physical unit Units of measurement

The definition, agreement and practical use of units of measurement [i] have played a crucial role in hu ... 

 that such a quantity is measured against. The dimension of speed, for example, is length divided by time. In the SI system, the dimension is given by the seven exponents of the fundamental quantities. See Dimensional analysis.

Mathematical dimensions

In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

, no definition of dimension adequately captures the concept in all situations where we would like to make use of it. Consequently, mathematician Mathematician

A mathematician is a person whose primary area of study and research is the field of mathematics [i]. ... 

s have devised numerous definitions of dimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Euclidean n-space E n. The point E 0 is 0-dimensional. The line E 1 is 1-dimensional. The plane E 2 is 2-dimensional. And in general E n is n-dimensional.

A tesseract Tesseract

In geometry [i], the tesseract is the 4-dimensional [i] analog [i] of the cube [i], wh ... 

 is an example of a four-dimensional object.

In the rest of this section we examine some of the more important mathematical definitions of dimension.

Hamel dimension

For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis.
See Hamel dimension for details.

Manifolds

A connected topological manifold Manifold

A manifold is an abstract mathematical space [i] in which every point has a neighborho ... 

 is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold.

The theory of manifolds, in the field of geometric topology, is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to 'work'; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied.

Lebesgue covering dimension

For any topological space, the Lebesgue covering dimension is defined to be n if n is the smallest integer for which the following holds: any open cover has a refinement such that no point is included in more than n + 1 elements. For manifolds, this coincides with the dimension mentioned above. If no such n exists, then the dimension is infinite.

Inductive dimension

The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that n+1-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.

Hausdorff dimension

For sets which are of a complicated structure, especially fractal Fractal

In colloquial usage, a fractal is a shape that is recursively constructed or self-similar [i],... 

s, the Hausdorff dimension Hausdorff dimension

In mathematics [i], the Hausdorff dimension is an extended [i] non-negative real number [i] ... 

 is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values . The upper and lower box dimensions are a variant of the same idea.

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.

Krull dimension of commutative rings

The Krull dimension of a commutative ring, named after Wolfgang Krull , is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

Science fiction

Science fiction Science fiction

Science fiction is a popular genre of fiction in which the narrative world differs from our own present... 

 texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence. This usage is derived from the idea that in order to travel to parallel/alternate universes/planes of existence one must travel in a spatial direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth spatial dimension, not the standard ones.

Penrose's singularity theorem

In his book , scientist Scientist

A scientist is an expert [i] in at least one area of science [i] who uses the scientific method [i] to d ... 

 Sir Roger Penrose Roger Penrose

Sir Roger Penrose, OM [i], FRS [i] is an English [i] mathematical physicist [i] ... 

 explained his singularity theorem. It asserts that all theories that attribute more than three spatial dimensions and one temporal Time

Two distinct views exist on the meaning of time.... 

 dimension to the world of experience are unstable. The instabilities that exist in systems of such extra dimensions would result in their rapid collapse into a singularity. For that reason, Penrose wrote, the unification of gravitation Gravitation

In physics [i], gravitation or gravity is the tendency of objects with mass [i] to accelerate [i] ... 

 with other forces through extra dimensions cannot occur.

More dimensions

  • Dimension of an algebraic variety
  • Topological dimension
  • Isoperimetric dimension
  • Poset dimension
  • Metric dimension
  • Pointwise dimension
  • Lyapunov dimension
  • Kaplan-Yorke dimension
  • Exterior dimension
  • Hurst exponent
  • q-dimension; especially:
    • Information dimension
    • Correlation dimension

See also


Degrees of freedom

  • Zero dimensions
    • Point
    • Zero-dimensional space
  • One dimension
    • Line
  • Two dimensions
    • 2D geometric models
    • 2D computer graphics 2D computer graphics

      2D computer graphics is the computer [i]-based generation of digital image [i]s—mostly from two-di ... 

  • Three dimensions
    • 3D computer graphics 3D computer graphics

      3D computer graphics are works of graphic art [i] that were created with the aid of digital [i] computer [i] ... 

    • 3-D film 3-D film

      In film, the term 3-D is used to describe any visual presentation system that attempts to maintain o... 

      s and video
    • Stereoscopy Stereoscopy

      Stereoscopy, stereoscopic imaging or 3-D imaging is any technique capable of recording three... 

  • Four dimensions
    • Time Time

      Two distinct views exist on the meaning of time.... 

    • Fourth spatial dimension Fourth dimension

      The concept of a fourth dimension is one that is often described in considering its physical implication... 

    • Tesseract Tesseract

      In geometry [i], the tesseract is the 4-dimensional [i] analog [i] of the cube [i], wh ... 

  • Five dimensions
    • Kaluza-Klein theory
    • Fifth dimension Fifth dimension

      In physics [i] and mathematics [i], a sequence of N numbers can be understood to represent a location in a... 

  • Ten, eleven or twenty-six dimensions
    • String theory String theory

      String theory is a model [i] of fundamental physics [i] whose building blocks are on ... 

    • M-theory M-theory

      In physics [i], M-theory is put forward as the master theory that unifies the five superstring theories [i] ... 

    • Why 10 dimensions? Why 10 dimensions?

      Although the human [i] mind comprehends the universe [i] with three spatial dimension [i]s, some theories in physics [i] ... 

    • Calabi-Yau spaces Calabi-Yau manifold

      Calabi-Yau manifolds form a special class of surfaces [i] used in some branches of mathematics [i] ... 

  • Infinitely many dimensions
    • Hilbert space
  • Special relativity Special relativity

    The special theory of relativity was proposed in 1905 [i] by Albert Einstein [i] in his article "On the Electrodynamics of Moving Bodies [i] ... 

  • General relativity General relativity

    General relativity is the geometrical [i] theory [i] of gravitation [i] published by Albert Einstein [i] ... 



Other

  • Dimension Dimension

    In common usage, a dimension is a parameter [i] or measurement [i] required to define the characteristi ... 

     and dimension tables
  • Dimensional analysis
  • Hyperspace

Further reading

  • Thomas Banchoff, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition, Freeman
  • Clifford A. Pickover Clifford A. Pickover

    Clifford A. Pickover is an author, editor, and columnist in the fields of science [i], mathematics [i], ... 

    , Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford University Press
  • Rudy Rucker Rudy Rucker

    Rudolf von Bitter Rucker is an American computer scientist [i] and science fiction [i] author, and is on... 

     , The Fourth Dimension, Houghton-Mifflin
  • Edwin A. Abbott,
  • Imagining the Tenth Dimension - Animated and easy explanation: http://www.tenthdimension.com/