Dimension

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This article is about . For , see Dimension (disambiguation).

Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space.

In common usage, the dimensions of an object are the measurements that define its shape and size. That usage is related to, but different from, what this article is about. Also, in science fiction, a "dimension" can also refer to an alternate universe or plane of existence, though this meaning is not discussed in this article.

Physical 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.

Time is often referred to as the 'fourth dimension'. It is, in essence, one way to measure physical change. It is different from the three spatial dimensions in that there is only one of it, and movement seems to be possible in only one direction. On the macroscopic scale that we perceive, physical processes are not symmetric with respect to time. However, at the subatomic Planck scale, almost all physical processes are time symmetric (ie. the equations used to describe these processes are the same regardless of the direction of time), although this doesn't imply that subatomic particles can move backwards in time.

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. Of course we cannot describe these dimensions because we cannot imagine them.

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.

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 article 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 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.

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.

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.

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 (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

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". The concept is used to suggest that if one travels into "another dimension", one is therefore traveling beyond the bounds of human understanding. One could conceivably encounter alien beings in their own "natural" habitat.

Anaglyph

This section should be merged into Anaglyph image.

To understand how Anaglyph 3D works, you must understand how Sterioscope 3D works. It's basicly 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 sterio graph image without the need of crossing your eyes the whole time you're looking at the image.

3-D film

This section should be merged into 3-D film.

In the 1950's, 3D movies were very popular, and cool, 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.

Modern 3-D films

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

More dimensions

Further reading

  • Thomas Banchoff, (1996) Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition, Freeman
  • Clifford A. Pickover, (1999) Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford University Press
  • Rudy Rucker (1984), The Fourth Dimension, Houghton-Mifflin
  • Edwin A. Abbott, (1884) Flatland

See also

External links

da:Dimension de:Dimension (Physik) fr:Dimension ko:차원 io:Dimensiono id:Dimensi it:Dimensione he:ממד nl:Dimensie ja:次元 pl:Wymiar (matematyka) simple:Dimension zh:維度