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For other uses of "Dimension" and "Dimensions", see Dimension (disambiguation).
2-dimensional renderings (i.e., flat drawings) of a 0-dimensional point, a 1-dimensional line segment, a 2-dimensional square, a 3-dimensional cube, and a 4-dimensional tesseract
The term dimension has various different, although related, meanings in common usage, in mathematics, and in physics.
In common usage, a dimension (Latin, "measured out") is a measurable aspect of an object. The most commonly used dimensions give the measurements describing the size of a roughly block-shaped object: length, width, and height. However, dimensions can also concern other physical aspects, such as the mass and electric charge of an object, or even, in a context where cost is relevant, an economic aspect such as its price.
In mathematics, the dimension of a mathematical abstract space is the number of parameters required to describe the position of a point within that space. For example, in the Euclidean plane, a point can be located with just two numbers, so the dimension of the plane is 2. A system relating the values of the parameters to the positions in the space is called a coordinate system, and the parameters describing the position of a point are then called coordinates. Generalizations of this concept are possible, and different fields of mathematics use different, specialized definitions.
In physics, the term dimension is used in two rather different senses. The first is essentially the mathematical meaning. Physics makes use of mathematical models that capture physically relevant aspects of reality, and these models may involve mathematical spaces. For example, in general relativity makes use of a concept of spacetime, modelled as a four-dimensional mathematical space.
The second meaning of the term "dimension" in physics relates to the nature of a measurable quantity. In general, physical measurements that must be expressed in units of measurement, and quantities obtained by such measurements are dimensionful. An example if a dimension is length, abbreviated L, which is the dimension for measurements expressed in units of length, be they meters, nautical miles, or lightyears. Another example is time, abbreviated T, whether the measurement is expressed in seconds or in hours. Speed, which is the distance (length) travelled in a certain amount of time, is a dimensionful quantity that has the dimension LT−1 (meaning L/T). Acceleration, the change in speed per time unit, has dimension LT−2.
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A three dimensional Cartesian coordinate system.
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 Space and Cartesian coordinate system.)
Time is often referred to as the "fourth dimension". It is, one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, that movement in time occurs at the fixed rate of one second per second, and that we cannot move freely in time but subjectively move in one direction.
The equations used in physics to model reality do not treat time in the same way that humans perceive it. 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 best-known treatment of time as a dimension is Poincaré and Einstein\'s special relativity (and extended to general relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime, and in the special, flat case as Minkowski space.
Theories such as string theory and M-theory predict that physical space in general has in fact 10 and 11 dimensions, respectively. The extra dimensions are spacelike. We perceive only three spatial dimensions, and no physical experiments have confirmed the reality of additional dimensions. A possible explanation that has been suggested is that space is as it were "curled up" in the extra dimensions on a very small, subatomic scale, possibly at the quark/string level of scale or below.
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 SI, the dimension of a derived quantity is given by the product of powers of the dimensions of the seven base quantities.
See also:
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. Whereas outside of mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions," mathematicians usually express this as: "The tesseract has dimension 4," or: "The dimension of the tesseract is 4."
Historically, the notion of higher dimensions in mathematics was introduced by Bernhard Riemann, in his 1854 Habilitationsschrift, where he considered a point to be any n numbers , abstractly, without any geometric picture needed nor implied.
The rest of this section examines some of the more important mathematical definitions of dimension.
Main article: Hamel dimension
For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis.
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 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.
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.
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.
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.Fractal Dimension, Boston University Department of Mathematics and Statistics The box dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values.
Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space 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.
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.
The negative (fractal) dimension is introduced by Benoit Mandelbrot, in which, when it is positive gives the known definition, and when it is negative measures the degree of "emptiness" of empty sets.Benoit B. Mandelbrot, Negative Fractal Dimension, Yale Mathematics Department
Science fiction 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 (or higher) spatial dimension, not the standard ones.
In his book The Road to Reality: A Complete Guide to the Laws of the Universe, scientist Sir Roger Penrose explained his singularity theorem. It asserts that all theories that attribute more than three spatial dimensions and one temporal 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 with other forces through extra dimensions cannot occur.
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