Dimension: Encyclopedia - Dimension

Dimension

In common usage, the dimensions (from Latin "measured out") of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. In mathematics, the dimensions of a space are the parameters required to describe a particular object in this space. The dimension of a space is the number of these parameters. For example, locating a city on the Earth requires two parameters: longitude and latitude; the corresponding space has therefore two dimensions and its dimension is two. This space is said to be 2-dimensional (for short 2D). Locating an airplane might require a 3D space by addition of the altitude parameter, or even a 6D space, if one adds the three angles required for defining the orientation of the airplane. The airplane is then considered to have six degrees of freedom. Other dimensions can be supplemented like speed, temperature or even colour or price of the plane. Generalisations of the concept are possible and a number of alternative definitions may be introduced. Units are sometimes associated with each dimension, for instance, meters or feet with altitude or dollars with price. In science fiction, a "dimension" can also refer to an alternate universe or plane of existence. This usage is derived from the fact that to get to the alternate universe/plane of existence requires movement in a dimension beyond the normal 3 space + time.

Dimension - Physical dimensions

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 die? — On the 5 May 1821 at Saint Helena (15°56′ S 5°42′ W). They play a fundamental role in our perception of the world around us. According to Immanuel Kant, 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.

Dimension - 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. (See Cartesian coordinate system)

Dimension - Time

Time is often referred to as the "fourth dimension". 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 are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy).

The most well-known treatment of time as a dimension is Einstein's theory of general relativity, which treats perceived space and time as parts of a four-dimensional manifold. Some scientists consider that the fact that Einstein's theory allows time to move at different rates indicates a 2nd time dimension exists.

Dimension - Additional dimensions

Theories such as string theory predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26), 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.

Dimension - Units

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

Dimension - Mathematical dimensions

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

Dimension - Hamel dimension

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

Dimension - Manifolds

A connected topological manifold 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 characterised 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.

Dimension - 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 (a second cover where each element is a subset of an element in the first cover) 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.

Dimension - 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.

Dimension - Hausdorff dimension

For sets which are of a complicated structure, especially fractals, the Hausdorff dimension 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.

Dimension - 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.

Dimension - Krull dimension of commutative rings

The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

Dimension - Science fiction

Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence, or concepts that are beyond the reader. The word gives a sense of authority to a film, and inspires imagination and awe in the minds of the reader, that one could travel to "another dimension". This concept is derived from the idea that in order to travel to 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 (or higher) spatial dimension, not the standard ones.

Dimension - Anaglyph

This section should be merged into Anaglyph image.

To understand how Anaglyph 3D works, you must understand how Sterioscope 3D works. It's basically blending the two images taken eye width apart. The eye thinks it's seeing one normal image by blending together one image we see with both eyes. You blend two slightly different pictures together and look at it. It looks 3D! Anaglyph is just taking a single framed 3D image and making one eye only see one image... the red sees the blue (because the red of the glasses blends in with the red) and the blue sees the red (the blue of the glasses blends in with the blue). This creates a normal stereo graph image without the need of crossing your eyes the whole time you're looking at the image.

Dimension - 3-D film

This section should be merged into 3-D film.

In the 1950's, 3D movies were very popular, but since they didn't have color film then they had an entirely different method. They would take two black and white cameras, line them up eye-with apart and take the film from both at the same time. They took two projectors in the projection booth, both aimed at the screen. The left one with the left film with a blue filter over the lense, and the right with the right film and a red filter over the lense, rather than just having one 3D image blended together.

Dimension - Modern 3-D films

Spy Kids 3D has developed a new craze of 3D, mostly in amusement parks with motion simulators. Spongebob 3D in Paramount's Great America, California has a Polarization 3D movie along with a motion simulator. Shark Boy and Lava Girl is another modern 3D movie, by the same producer of Spy Kids.

Dimension - More dimensions

• Dimension of an algebraic variety
• Topological dimension
• Isoperimetric dimension
• Poset dimension
• Pointwise dimension
• Lyapunov dimension
• Kaplan-Yorke dimension
• Exterior dimension
• Hurst exponent
• q-dimension; especially:
  • Information dimension (corresponding to q=1)
  • Correlation dimension (corresponding to q=2)

See also

Dimension - Degrees of freedom

• Zero dimensions
  • Point
  • Zero-dimensional space
• One dimension
  • Line
• Two dimensions
  • 2D geometric models
  • 2D computer graphics
• Three dimensions
  • 3D computer graphics
  • 3-D films and video
  • Stereoscopy (3-D imaging)
• Four dimensions
  • Time (4th dimension)
  • Fourth spatial dimension
  • Tesseract (four dimensional shapes)
• Five dimensions
  • Kaluza-Klein theory
  • Fifth dimension
• Ten, eleven or twenty-six dimensions
  • String theory
  • M-theory
  • Why 10 dimensions?
  • Calabi-Yau spaces
• Infinitely many dimensions
  • Banach space (only some have infinitely many dimensions)

• Special relativity
• General relativity

Dimension - Other

• Dimension (data warehouse) and dimension tables
• Dimensional analysis

Adapted from the Wikipedia article "Dimension", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki