Dimension
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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.
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Physical dimensions
The spacetime in which we live appears to be 4-dimensional. It is conventional (and for most practical purposes entirely sensible) to consider this as three spatial dimensions and one of time. We can move up-or-down, north-or-south, or east-or-west, and 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 northwest is merely a combination of moving north and moving west.
Time is often referred to as the 'fourth dimension'. It is somewhat different to 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.
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 dimensions
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 infer that if one travels into 'another dimension' they are traveling beyond the bounds of human understanding.
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)
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 (http://sources.wikipedia.org/wiki/Flatland)
See also
- 2D geometric models
- Stereoscopy (3-D imaging)
- 2D computer graphics
- 3D computer graphics
- 3-D films and video
- Data warehousing and dimension tables
- Dimensional analysis
- Special relativity
- General relativityda:Dimension
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