Citizendia
Your Ad Here

For other uses, see Dimension (disambiguation).
From left to right, the square has two dimensions, the cube has three and the tesseract has four.
From left to right, the square has two dimensions, the cube has three and the tesseract has four. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. Geometry The tesseract can be constructed in a number of different ways

In mathematics the dimension of a space is roughly defined as the mimimum number of coordinates needed to specify every point within it[1][2]. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume Dimensions can be thought of as the axes in a Cartesian coordinate system, which in a three-dimensional system run left-right, up-down and forward-backward. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane A set of three co-ordinates on these axes, or any other three-dimensional coordinate system, specifies the position of a particular point in space[3]. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another In the physical world, according to the theory of relativity the fourth dimension is time, which runs before-after. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. In Physics and Mathematics, a sequence of n numbers can be understood as a location in an n -dimensional space For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of An event’s position in space and time is therefore specified if four co-ordinates are given. In relativity, a four-vector is a vector in a four-dimensional real Vector space, called Minkowski space.

On surfaces such as a plane or the surface of a sphere, a point can be specified using just two numbers and so this space is said to be two-dimensional. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Similarly a line is one-dimensional because only one co-ordinate is needed, whereas a point has no dimensions. In mathematics, spaces with more than three dimensions are used to describe other manifolds. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Higher dimension as a term in Mathematics most commonly refers to any number of spatial Dimensions greater than three In these n-dimensional spaces a point is located by n co-ordinates (x1, x2, … xn). In Mathematics, an n -dimensional space is a Topological space whose Dimension is n (where n is a fixed Natural Some theories, such as those used in fractal geometry, make use of non-integer and negative dimensions. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French A negative number is a Number that is less than zero, such as −2

Another 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. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand An example of a dimension is length, abbreviated L, which is the dimension for measurements expressed in units of length, be they meters, nautical miles, or lightyears. Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International A nautical mile or sea mile is a unit of Length. It corresponds approximately to one minute of Latitude along any meridian. A light-year or light year (symbol ly) is a unit of Length, equal to just under ten trillion Kilometres As defined by Another example is time, abbreviated T, whether the measurement is expressed in seconds or in hours. For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units The hour (symbol h) is a unit of Time. It is not an SI unit but is accepted for use with the SI 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.

Contents

In mathematics

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. A mathematician is a person whose primary area of study and research is the field of Mathematics. 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. In general, E n is n-dimensional.

A tesseract is an example of a four-dimensional object. Geometry The tesseract can be constructed in a number of different ways 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 (x_1,\dots,x_n), abstractly, without any geometric picture needed nor implied. Higher dimension as a term in Mathematics most commonly refers to any number of spatial Dimensions greater than three Habilitation is the highest academic qualification a person can achieve by their own pursuit in certain European and Asian countries

The rest of this section examines some of the more important mathematical definitions of dimension.

Hamel dimension

Main article: Hamel dimension

For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis. In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

Manifolds

A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, a phenomenon is sometimes said to occur locally if roughly speaking it occurs on sufficiently small or arbitrarily small Neighborhoods Topological equivalence redirects here see also Topological equivalence (dynamical systems. 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. In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied. In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among

Lebesgue covering dimension

For any normal topological space X, the Lebesgue covering dimension of X is defined to be n if n is the smallest integer for which the following holds: any open cover has an open refinement (a second open 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. In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Mathematics, the Lebesgue covering dimension or topological dimension of a Topological space is defined to be the minimum value of n, such The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection In this case we write dim X = n. For X a manifold, this coincides with the dimension mentioned above. If no such integer n exists, then the dimension of X is said to be infinite, and we write dim X = ∞. Note also that we say X has dimension -1, i. e. dim X = -1 if and only if X is empty. This definition of covering dimension can be extended from the class of normal spaces to all Tychonoff spaces merely by replacing the term "open" in the definition by the term "functionally open".

Inductive dimension

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. In the mathematical field of Topology, the inductive dimension of a Topological space X is either of two values the small inductive dimension For a different notion of boundary related to Manifolds see that article

Hausdorff dimension

For sets which are of a complicated structure, especially fractals, the Hausdorff dimension is useful. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" In Mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative Real number associated The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined [4] The box dimension or Minkowski dimension is a variant of the same idea. In Fractal geometry, the Minkowski-Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the Fractal In Fractal geometry, the Minkowski-Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the Fractal In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values. In Fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a Fractal appears to fill space as

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, an orthonormal basis of an Inner product space V (i In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" 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. In Commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull ( 1899 - 1971) is defined to be the In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers

Negative dimension

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. Benoît B Mandelbrot (born 20 November 1924 is a French mathematician, best known as the father of fractal geometry. [5]

In physics

Spatial dimensions

A three dimensional Cartesian coordinate system.
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. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another 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 Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics 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. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane )

Time

Time is often referred to as the "fourth dimension". In Physics and Mathematics, a sequence of n numbers can be understood as a location in an n -dimensional space 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. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects T Symmetry is the symmetry of physical laws under a Time reversal transformation &mdash T t \mapsto -t In Physics, C-symmetry means the symmetry of physical laws under a charge -conjugation transformation. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate. 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 laws of thermodynamics, in principle describe the specifics for the transport of Heat and work in Thermodynamic processes. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy

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. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity

Additional dimensions

Theories such as string theory and M-theory predict that physical space in general has in fact 10 and 11 dimensions, respectively. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings In Theoretical physics, M-theory is a new limit of String theory in which 11 dimensions of Spacetime may be identified 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.

Penrose's singularity theorem

In his book The Road to Reality: A Complete Guide to the Laws of the Universe, scientist Sir Roger Penrose explained his singularity theorem. The Road to Reality is a book on modern Physics by the British mathematical physicist Roger Penrose, published in 2004 A scientist, in the broadest sense refers to any person that engages in a systematic activity to acquire Knowledge or an individual that engages in such practices Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus The Penrose-Hawking singularity theorems are a set of results in General relativity which attempt to answer the question of whether gravity is necessarily singular It asserts that all theories that attribute more than three spatial dimensions and one temporal dimension to the world of experience are unstable. For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of 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. Gravitation is a natural Phenomenon by which objects with Mass attract one another In Physics, a force is whatever can cause an object with Mass to Accelerate.

Dimensionful quantities

Main article: Dimensional analysis

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. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand The dimension of speed, for example, is LT−1, that is, length divided by time. The units in which the quantity is expressed, such as ms−1 (meters per second) or mph (miles per hour), has to conform to the dimension.

Science fiction

Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence. Parallel universe or alternative reality is a self-contained separate reality coexisting with one's own In Metaphysics and Esoteric cosmology, a plane, other than the Physical plane, is conceived as a subtle state of Consciousness that transcends 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.

One of the most heralded science fiction novellas regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novel Flatland by Edwin A. For other uses see Flatland (disambiguation Flatland A Romance of Many Dimensions is an 1884 Science fiction Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as "The best introduction one can find into the manner of perceiving dimensions. For other uses see Flatland (disambiguation Flatland A Romance of Many Dimensions is an 1884 Science fiction "

More dimensions

See also

By number of dimensions

Other

Further reading

References

  1. ^ Curious About Astronomy
  2. ^ MathWorld: Dimension]
  3. ^ Oxford Illustrated Encyclopedia: The Physical World
  4. ^ Fractal Dimension, Boston University Department of Mathematics and Statistics
  5. ^ Benoit B. Mandelbrot, Negative Fractal Dimension, Yale Mathematics Department

Dictionary

dimension

-noun

  1. A single aspect of a given thing.
  2. A measure of spatial extent in a particular direction, such as height, width or breadth, or depth.
  3. A construct whereby objects or individuals can be distinguished.
  4. (geometry) Any of the independent coordinates used to specify uniquely the location of a point in a space.
  5. (linear algebra) The number of elements of any basis of a vector space.
  6. (physics) One of the physical property regarded as a fundamental measure of a physical quantity, such as mass, length and time.
  7. (computing) Any of the independent ranges of indices in a multidimensional array.
  8. (science fiction) An alternative universe or plane of existence.

-verb

  1. (transitive) To mark, cut or shape something to specified dimensions.
© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic