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In mathematics, four-dimensional space ("4D") is a geometric space with four dimensions. It typically is more specifically four-dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space. It has been studied by mathematicians and philosophers for over two centuries, both for its own interest and for the insights it offered into mathematics and related fields. Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular a vector with four elements (a 4-tuple) can be used to represent a position in four-dimensional space. The space is a Euclidean space, so has a metric and norm, and so all directions are treated as the same: the additional dimension is indistinguishable from the other three. In modern physics, space and time are unified in a four-dimensional Minkowski continuum called spacetime, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing). Spacetime is ''not'' a Euclidean space. ==History== Lagrange wrote in his ''Mécanique analytique'' (published 1788, based on work done around 1755) that mechanics can be viewed as operating in a four-dimensional space — three of dimensions of space, and one of time.〔Bell, E.T. (1937). ''Men of Mathematics'', Simon and Schuster, p. 154.〕 In 1827 Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image,〔Coxeter, H. S. M. (1973). ''Regular Polytopes'', Dover Publications, Inc., p. 141.〕 and by 1853 Ludwig Schläfli had discovered many polytopes in higher dimensions, although his work was not published until after his death.〔Coxeter, H. S. M. (1973). ''Regular Polytopes'', Dover Publications, Inc., pp. 142–143.〕 Higher dimensions were soon put on firm footing by Bernhard Riemann's 1854 Habilitationsschrift, ''Über die Hypothesen welche der Geometrie zu Grunde liegen'', in which he considered a "point" to be any sequence of coordinates (''x''1, ..., ''x''''n''). The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in ''A History of Vector Analysis''. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R. One of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay ''What is the Fourth Dimension?''; published in the Dublin University magazine.〔Rudolf v.B. Rucker, editor ''Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton'', p. vii, Dover Publications Inc., 1980 ISBN 0-486-23916-0〕 He coined the terms ''tesseract'', ''ana'' and ''kata'' in his book ''A New Era of Thought'', and introduced a method for visualising the fourth dimension using cubes in the book ''Fourth Dimension''. In 1886 Victor Schlegel described〔Victor Schlegel (1886) ''Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper'', Waren〕 his method of visualizing four-dimensional objects with Schlegel diagrams. In 1908, Hermann Minkowski presented a paper〔 *Various English translations on Wikisource: Space and Time 〕 consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity.〔 〕 But the geometry of spacetime, being non-Euclidean, is profoundly different from that popularised by Hinton. The study of Minkowski space required new mathematics quite different from that of four-dimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy blurring the distinction, so in 1973 H. S. M. Coxeter felt compelled to write: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Four-dimensional space」の詳細全文を読む スポンサード リンク
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