- Higher dimension
"Higher dimension" as a term in
mathematics most commonly refers to any number of spatialdimension s greater than three.The three standard dimensions are
length ,width , andbreadth (orheight ). The first higher dimension required is oftentime , andspace-time is the most common example of a four-dimensional space.In physics and chemistry, the dimensions of a system are referred to as its "degrees of freedom".
History
The introduction of
Cartesian coordinates reduced the three spatial dimensions to three real numbers. The possibility of "geometry of higher dimensions" was thereby opened up: the list of numbers could in principle be longer than three. Applications togeometry awaited the needs of mathematicians.Historically, the notion of higher dimensions was introduced by
Bernhard Riemann , in his 1854Habilitationsschrift , "Über die Hypothesen welche der Geometrie zu Grunde liegen", where he considered a point to be any "n" numbers , abstractly, without any geometric picture needed nor implied. He explained the value of this abstraction thus: [Werke, p. 268, edition of 1876, cited in [http://projecteuclid.org/euclid.bams/1183493815 Pierpont, Non-Euclidean Geometry, A Retrospect] ] : "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."The abstract notion of coordinates was preceded by the
homogeneous coordinates ofAugust Ferdinand Möbius , of 1827.Application
It is commonplace in advanced pure and
applied mathematics to study abstract sets and applied models with many dimensions. For instance, theconfiguration space of arigid body in Euclidean 3-space is the 6-dimensional
group of rigid motions "E"+(3), with 3 dimensions for position (translation ) and 3 for orientation (rotation ).Fairly simple constructions yield spaces with arbitrarily high positive integer dimension, and only slightly more sophistication is required to construct spaces of infinite dimension.
In
geometric topology , the nature of the difficulties in the subject has turned out to be such that dimensions 3 and 4 are the most resistant (see for exampleWhitney disc ). Therefore in that context "higher dimension" usually means dimension ≥ 5.ee also
*
Fourth dimension
*Fifth dimension
*Minkowski spacetime References
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