ndimensional space
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An ndimensional space is a topological space whose dimension is n, given by the cardinality of a maximal set of linearly independent elements. An example is ndimensional Euclidean space (E^{n}), which describes Euclidean geometry in n dimensions. ndimensional spaces with large values of n are sometimes called highdimensional spaces. In linear algebra an ndimensional space is called a vector space.
Many familiar geometric objects can be generalized to any number of dimensions. The twodimensional triangle and the threedimensional tetrahedron are specific instances of the ndimensional simplex; the circle and the sphere are specific instances of the ndimensional hypersphere. An ndimensional manifold is a space that locally is an ndimensional Euclidean space, but whose global structure may be nonEuclidean. Elliptical (S^{n}) and hyperbolic spaces (H^{n}) are ndimensional spaces with constant positive and negative curvature respectively.
The state of an object with n degrees of freedom can be described as a point in some ndimensional space; for example, classical mechanics describes the threedimensional position and momentum of a point particle as a point in 6dimensional phase space. The numbers that describe a point may represent anything; for this, multivariate statistics and machine learning theory make extensive use of highdimensional spaces. Fractional and negative dimensions can apply to more general spaces, such as the Hausdorff dimension in topology and Kodaira dimension in algebraic geometry. Infinitedimensional spaces can be formulated in a meaningful way; an example is the Hilbert space, a concept in quantum mechanics.
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[edit] History
The abstract notion of coordinates was preceded by the homogeneous coordinates of August Ferdinand Möbius in 1827. The introduction of Cartesian coordinates reduced the three spatial dimensions to a list of three real numbers. Since this list did not have a restriction on its number of members, there existed the possibility of higherdimensional geometry. Bernhard Riemann, in his 1854 habilitation Über die Hypothesen welche der Geometrie zu Grunde liegen, considered a point to be represented by a list of n numbers without any geometric picture implied. He explained the value of this abstraction thus:^{[1]}
 "Solche Untersuchungen, welche, wie hier ausgeführt, von allgemeinen Begriffen ausgehen, können nur dazu dienen, dass diese Arbeit nicht durch die Beschränktheit der Begriffe gehindert und der Fortschritt im Erkennen des Zusammenhangs der Dinge nicht durch überlieferte Vorurteile gehemmt wird."
Loosely translated:
 "Abstract studies such as these allow one to observe relationships without being limited by narrow terms, and prevent traditional prejudices from inhibiting one's progress."
[edit] Applications
Pure and applied mathematics include many abstract sets and applied models with arbitrarily high and infinite dimensions, due to the simple constructions required to construct the necessary spaces. For instance, the configuration space of a rigid body in Euclidean 3space is the 6dimensional group of rigid motions E^{+}(3), with 3 dimensions for position (translation) and 3 for orientation (rotation); a blackandwhite image with 1000 pixels can be viewed as a point in 1000dimensional space, whose components indicate the darkness of each pixel. This approach allows for generalization of many analysis techniques, and is important in disciplines such as machine learning. Sometimes, dimension reduction techniques are used to deal with very high dimensional data, particularly when the number of dimensions is higher than the number of data points available. In geometric topology, 5 or moredimensional spaces are considered high due to the nature of the difficulties in the subject that imply dimensions 3 and 4 being the most resistant and therefore most studied.
[edit] Properties
Spaces with a dimension higher than 3 have properties that differ greatly from the intuitions generated by the 3dimensional space of reality. In particular, as the dimension increases, the volume of an object increases faster. The implications of this include:
 almost all of the volume within a highdimensional hypersphere lies in a thin shell near its outside. This is important to gaining an understanding of how the chisquared test works.
 the volume within a highdimensional hypersphere, relative to a hypercube of the same width, tends to zero as dimensionality tends to infinity, and almost all of the volume of the hypercube is concentrated in its "corners".
 the curse of dimensionality, the intractability of problems in highdimensional spaces.
An understanding of the derivation of the Shannon limit in coding theory depends on the properties of highdimensional spaces.
[edit] See also
 Coordinate space
 ndimensional calculus
 Facet (mathematics)
 Fourvector
 Kissing number problem
 Vector decomposition
 Combinatorial explosion
 Curse of dimensionality
 Dimension theory
 Highdimensional statistics
 Hilbert space
 Minkowski spacetime
 Supporting hyperplane
[edit] References
 ^ Werke, p. 268, edition of 1876, cited in Pierpont, NonEuclidean Geometry, A Retrospect
[edit] External links
 Highdimensional data analysis
 Highdimensional spaces are counterintuitive (in five parts)
 Visualizing Multidimensional Geometry with Parallel Coordinates by Alfred Inselberg
 Other Dimensions of the Universe
