dimension

—dimensional, adj. —dimensionality, n. —dimensionally, adv. —dimensionless, adj.

/di men"sheuhn, duy-/, n.

1. Math.

a. a property of space; extension in a given direction: A straight line has one dimension, a parallelogram has two dimensions, and a parallelepiped has three dimensions.

b. the generalization of this property to spaces with curvilinear extension, as the surface of a sphere.

c. the generalization of this property to vector spaces and to Hilbert space.

d. the generalization of this property to fractals, which can have dimensions that are noninteger real numbers.

e. extension in time: Space-time has three dimensions of space and one of time.

2. Usually, dimensions.

a. measurement in length, width, and thickness.

b. scope; importance: the dimensions of a problem.

3. unit (def. 6).

4. magnitude; size: Matter has dimension.

5. Topology.

a. a magnitude that, independently or in conjunction with other such magnitudes, serves to define the location of an element within a given set, as of a point on a line, an object in a space, or an event in space-time.

b. the number of elements in a finite basis of a given vector space.

6. Physics. any of a set of basic kinds of quantity, as mass, length, and time, in terms of which all other kinds of quantity can be expressed; usually denoted by capital letters, with appropriate exponents, placed in brackets: The dimensions of velocity are [LT^-1]. Cf. dimensional analysis.

7. dimensions, Informal. the measurements of a woman's bust, waist, and hips, in that order: The chorus girl's dimensions were 38-24-36.

8. See dimension lumber.

v.t.

9. to shape or fashion to the desired dimensions: Dimension the shelves so that they fit securely into the cabinet.

10. to indicate the dimensions of an item, area, etc., on (a sketch or drawing).

[1375-1425; late ME dimensioun ( < AF) < L dimension- (s. of dimensio) a measuring, equiv. to dimens(us) measured out (ptp. of dimetiri, equiv. to di- DI-² + metiri to measure) + -ion- -ION]

Syn. 2b. range, extent, magnitude.

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In mathematics, a number indicating the fewest coordinates necessary to identify a point in a geometric space; more generally, a number indicating a measurement of length (see length, area, and volume).

One-dimensional space can be represented by a numbered line, on which a single number identifies a point. In two-dimensional space, a coordinate system may be superimposed, requiring only two numbers to identify a point. Three numbers suffice in three-dimensional space, and so on.

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▪ geometry

      in common parlance, the measure of the size of an object, such as a box, usually given as length, width, and height. In mathematics, the notion of dimension is an extension of the idea that a line is one-dimensional, a plane is two-dimensional, and space is three-dimensional. In mathematics and physics one also considers higher-dimensional spaces, such as four-dimensional space-time, where four numbers are needed to characterize a point: three to fix a point in space and one to fix the time. Infinite-dimensional spaces, first studied early in the 20th century, have played an increasingly important role both in mathematics and in parts of physics such as quantum field theory, where they represent the space of possible states of a quantum mechanical (quantum mechanics) system.

      In differential geometry one considers curves as one-dimensional, since a single number, or parameter, determines a point on a curve—for example, the distance, plus or minus, from a fixed point on the curve. A surface, such as the surface of the Earth, has two dimensions, since each point can be located by a pair of numbers—usually latitude and longitude. Higher-dimensional curved spaces were introduced by the German mathematician Bernhard Riemann (Riemann, Bernhard) in 1854 and have become both a major subject of study within mathematics and a basic component of modern physics, from Albert Einstein (Einstein, Albert)'s theory of general relativity and the subsequent development of cosmological models of the universe to late-20th-century superstring theory (string theory).

      In 1918 the German mathematician Felix Hausdorff introduced the notion of fractional dimension. This concept has proved extremely fruitful, especially in the hands of the Polish-French mathematician Benoit Mandelbrot, who coined the word fractal and showed how fractional dimensions could be useful in many parts of applied mathematics.

Robert Osserman

 

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