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Encyclopedia > Dimension
From left to right, the square has two dimensions, the cube has three and the tesseract has four.

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. 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. In these n-dimensional spaces a point is located by n co-ordinates (x1, x2, … xn). Some theories, such as those used in fractal geometry, make use of non-integer and negative dimensions. An open surface with X-, Y-, and Z-contours shown. ... This article is about the mathematical construct. ... For other uses, see Sphere (disambiguation). ... Look up line in Wiktionary, the free dictionary. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Higher dimension in mathematics refers to any number of dimensions greater than three. ... In mathematics, an n-dimensional space is a vector space or manifold whose dimension is n, or in some cases, a point in, or subset of, such a space. ... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ... Not to be confused with Natural number. ... A negative number is a number that is less than zero, such as âˆ’2. ...

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. In general, E n is n-dimensional. Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

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." For other uses, see Tesseract (disambiguation). ...

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 in mathematics refers to any number of dimensions greater than three. ... Bernhard Riemann. ... Habilitation is a term used within the university system in Germany, Austria, and some other European countries such as the German-speaking part of Switzerland, in Poland, the Czech Republic, Slovakia, Hungary, Slovenia, Russia, and other countries of former Soviet Union, such as Armenia, Azerbaijan, Moldova, Kirgizstan, Kazakhstan, Uzbekistan, etc. ...

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. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

### 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. This article is about mathematics. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In mathematics, something is said to occur locally in the category of topological spaces if it occurs on small enough open sets. ... This word should not be confused with homomorphism. ...

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. In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... In mathematics, the PoincarÃ© conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ...

### 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 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". In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements. ... Not to be confused with Natural number. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X...

### 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 ind(X) or the large inductive dimension Ind(X). ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...

### 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.[4] The box dimension or Minkowski 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. The boundary of the Mandelbrot set is a famous example of a fractal. ... In mathematics, the Hausdorff dimension (also known as the Hausdorffâ€“Besicovitch dimension) is an extended non-negative real number associated to any metric space. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In fractal geometry, the Minkowski-Bouligand dimension or Minkowski dimension is a way of determining the fractal dimension of a set S in a Euclidean space , or more generally of a metric space (X,d). ... In fractal geometry, the Minkowski-Bouligand dimension or Minkowski dimension is a way of determining the fractal dimension of a set S in a Euclidean space , or more generally of a metric space (X,d). ... 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 one zooms down to finer and finer scales. ...

### Hilbert spaces

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 mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... 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. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ...

### 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 number of strict inclusions in a maximal chain of prime ideals. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Wolfgang Krull (1899 - 1971) was a German mathematician, after whom Krull dimension, the Krull topology, and Krulls principal ideal theorem are named. ... 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.[5] Beno t Mandelbrot was the first to use a computer to plot the Mandelbrot set. ...

## In physics

### Spatial dimensions

A three dimensional Cartesian coordinate system.

### Time

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. For other uses of this term, see Spacetime (disambiguation). ... This article or section does not cite its references or sources. ...

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). Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... T-symmetry is the symmetry of physical laws under a time-reversal transformationâ€” The universe is not symmetric under time reversal, although in restricted contexts one may find this symmetry. ... C-symmetry means the symmetry of physical laws over a charge-inversion transformation. ... In physics, a parity transformation (also called parity inversion) is the simultaneous flip in the sign of all spatial coordinates: A 3Ã—3 matrix representation of P would have determinant equal to â€“1, and hence cannot reduce to a rotation. ... The laws of thermodynamics, in principle, describe the specifics for the transport of heat and work in thermodynamic processes. ... For other uses, see: information entropy (in information theory) and entropy (disambiguation). ...

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 PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... â€œEinsteinâ€ redirects here. ... For a generally accessible and less technical introduction to the topic, see Introduction to special relativity. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... For other uses of this term, see Spacetime (disambiguation). ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

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. This box:      String theory is a still developing mathematical approach to theoretical physics, whose original building blocks are one-dimensional extended objects called strings. ... M-theory is a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together. ...

### 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. 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. Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... The former Weights and Measures office in Middlesex, England. ... Miles per hour is a unit of speed, expressing the number of international miles covered per hour. ...

## Science fiction

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. Science fiction is a form of speculative fiction principally dealing with the impact of imagined science and technology, or both, upon society and persons as individuals. ... Parallel universe or alternate reality in science fiction and fantasy is a self-contained separate reality coexisting with our own. ... For other uses of the word plane, see plane. ...

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. 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 various uses of the term Flatlander, see Flatlander (disambiguation) Flatland: A Romance of Many Dimensions is a 1884 novella by Edwin Abbott Abbott, still popular among mathematics and computer science students, and considered useful reading for people studying topics such as the concept of other dimensions. ... For various uses of the term Flatlander, see Flatlander (disambiguation) Flatland: A Romance of Many Dimensions is a 1884 novella by Edwin Abbott Abbott, still popular among mathematics and computer science students, and considered useful reading for people studying topics such as the concept of other dimensions. ...

## More dimensions

In algebraic geometry, the dimension of an algebraic variety V is defined, informally speaking, as the number of independent rational functions that exist on V. So, for example, an algebraic curve has by definition dimension 1. ... In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements. ... In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space). ... If P and Q be posets on the same set X, Q is an extension of P if in P implies in Q, for all . ... In graph theory, the metric dimension of a graph G is the minimum cardinality of a resolving set for G. Finding the metric dimension of a graph is an NP-complete problem. ... In fractal geometry, the generalized Hurst exponent, (named after Harold Edwin Hurst (1880-1978)) Hq = H(q), for a time series g(t) (t = 1, 2,...) is defined by the scaling properties of its structure functions Sq(Ï„): where q > 0, Ï„ is the time lag and averaging is over the time... In fractal geometry, the fractal dimension is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. ... In chaos theory the correlation dimension (denoted by Î½) is a measure of the dimensionality of the space occupied by a set of random points. ...

### Other

The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ... In a data warehouse, a dimension is a data element that categorizes each item in a data set into non-overlapping regions. ... A dimension table is a data warehousing concept. ... In physics, hyperspace is a theoretical entity. ...

This article or section contains information that has not been verified and thus might not be reliable. ... Clifford A. Pickover is an author, editor, and columnist in the fields of science, mathematics, and science fiction. ... Rudy Rucker, Fall 2004, photo by Georgia Rucker. ... Edwin Abbott Abbott (December 20, 1838 - 1926), English schoolmaster and theologian, is best known as the author of the mathematical satire Flatland (1884). ... For various uses of the term Flatlander, see Flatlander (disambiguation) Flatland: A Romance of Many Dimensions is a 1884 novella by Edwin Abbott Abbott, still popular among mathematics and computer science students, and considered useful reading for people studying topics such as the concept of other dimensions. ... Project Gutenberg, abbreviated as PG, is a volunteer effort to digitize, archive and distribute cultural works. ...

## References

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

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