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dimension, in mathematics
dimension,in mathematics, number of parameters or coordinates required locally to describe points in a mathematical object (usually geometric in character). For example, the space we inhabit is three-dimensional, a plane or surface is two-dimensional, a line or curve is one-dimensional, and a point is zero-dimensional. By means of a coordinate system one can specify any point with respect to a chosen origin (and coordinate axes through the origin, in the case of two or more dimensions). Thus, a point on a line is specified by a number x giving its distance from the origin, with one direction chosen as positive and the other as negative; a point on a plane is specified by an ordered pair of numbers (x,y) giving its distances from the two coordinate axes; a point in space is specified by an ordered triple of numbers (x,y,z) giving its distances from three coordinate axes. Mathematicians are thus led by analogy to define an ordered set of four, five, or more numbers as representing a point in what they define as a space of four, five, or more dimensions. Although such spaces cannot be visualized, they may nevertheless by physically significant. For example, the quadruple of numbers (x,y,z,t), where t represents time, is sometimes interpreted as a point in four-dimensional space-time (see relativityrelativity,
physical theory, introduced by Albert Einstein, that discards the concept of absolute motion and instead treats only relative motion between two systems or frames of reference.
..... Click the link for more information. ). The state of the weather or the economy, in current models, is a point in a many-dimensional space. Many features of plane and solid Euclidean geometry have mathematical analogues in higher dimensional spaces.
dimension, in physics
dimension,in physics, an expression of the character of a derived quantity in relation to fundamental quantities, without regard for its numerical value. In any system of measurement, such as the metric system, certain quantities are considered fundamental, and all others are considered to be derived from them. Systems in which length (L), time (T), and mass (M) are taken as fundamental quantities are called absolute systems. In an absolute system force is a derived quantity whose dimensions are defined by Newton's second law of motionmotion,
the change of position of one body with respect to another. The rate of change is the speed of the body. If the direction of motion is also given, then the velocity of the body is determined; velocity is a vector quantity, having both magnitude and direction, while speed
..... Click the link for more information. as ML/T2, in terms of the fundamental quantities. Pressure (force per unit area) then has dimensions M/LT2; work or energy (force times distance) has dimensions ML2/T2; and power (energy per unit time) has dimensions ML2/T3. Additional fundamental quantities are also defined, such as electric charge and luminous intensity. The expression of any particular quantity in terms of fundamental quantities is known as dimensional analysis and often provides physical insight into the results of a mathematical calculation.
The dimension of a geometric figure is equal to one if the figure is a curve, two if the figure is a surface, and three if the figure is a solid. From the standpoint of analytic geometry, the dimension of a figure is equal to the number of coordinates needed to determine the position of a point on the figure. For example, the position of a point on a curve can be determined by a single coordinate, that of a point on a surface by two coordinates, and that of a point in three-dimensional space by three coordinates.
Until the mid-19th century, geometry dealt only with figures of three or less dimensions. With, however, the development of the concept of a multidimensional space, geometry began studying figures of any dimension. The simplest figures of dimension m are m-dimensional manifolds. An m-dimensional manifold in n-dimensional space is determined by n - m equations. For example, a curve, or one-dimensional manifold, is defined in three-dimensional space by 3 – 1 = 2 equations. The position of a point on an m-dimensional manifold is determined by what are called curvilinear coordinates. Thus, the position of a point on a sphere is determined by the point’s “geographic coordinates,” or latitude and longitude. The position of a point on a torus is defined in an analogous manner.
The above statements are valid only under certain restrictive assumptions. A truly general definition of the dimension of any closed bounded set lying in n-dimensional Euclidean space was given by P. S. Urysohn: the dimension of such a set is less than or equal to m if and only if the set admits of an ε-cover, for any ε > 0, by closed sets of multiplicity at most n + 1. This general definition of dimension can be extended in a natural way to extremely broad classes of topological spaces. In 1921, Urysohn constructed dimension theory, one of the most profound theories of modern topology. The further development of dimension theory has been due primarily to such Soviet mathematicians as P. S. Aleksandrov and L. S. Pontriagin.
REFERENCEAleksandrov, P. S., and B. A. Pasynkov. Vvedenie v teoriiu razmernosti. Moscow, 1973.
dimension(1) See dimension table.
(2) One axis in an array. In programming, a dimension statement defines the array and sets up the number of elements within the dimensions.