Four-dimensional Space - Wikipedia

The idea of making time the fourth dimension began with Jean le Rond d'Alembert's "Dimensions," published in 1754 in the Encyclopédie, ou dictionnaire raisonné des sciences, des arts et des métiers.[1][2] That physical mechanics can be viewed as occurring also in time was posited by Joseph-Louis Lagrange in 1755 and published in 1788 in his Mécanique analytique.[3]

Mathematics of 4D commenced in the nineteenth century.[4] 3D form rotation onto its mirror-image is possible in 4D space was realized by August Ferdinand Möbius [5] in Der barycentrische Calcul published 1827.[6] An arithmetic of four spatial dimensions, called quaternions, was defined by William Rowan Hamilton in 1843. Soon after, tessarines and coquaternions were introduced as other four-dimensional algebras over R. Higher-dimensional non-Euclidean spaces were put on a firm footing by Bernhard Riemann's 1854 thesis, Über die Hypothesen welche der Geometrie zu Grunde liegen, in which he considered a "point" to be any sequence of coordinates (x1, ..., xn).

Euclidean spaces of more than three dimensions were first described in 1852, when Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n, using both synthetic and algebraic methods. He discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension, including six found in 4-dimensional space.[7] Schläfli's work was only published posthumously in 1901, and remained largely unknown until publication of H.S.M. Coxeter's Regular Polytopes in 1947. During that interval many others also discovered higher-dimensional Euclidean space. One of the first popular 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, in which he explained the concept of a "four-dimensional cube" with a step-by-step generalization of the properties of lines, squares, and cubes. [a] [8] He coined the terms tesseract, ana and kata in his book A New Era of Thought and introduced a method for visualizing the fourth dimension using cubes in the book Fourth Dimension.[9][10] 1886, Victor Schlegel described[11] his method of visualizing four-dimensional objects with Schlegel diagrams.

Hermann Minkowski's 1908 paper[12] consolidating the role of time as the fourth dimension of spacetime provided the geometric basis for Einstein's theories of special and general relativity.[13] The geometry of spacetime, being non-Euclidean, is profoundly different from that explored by Schläfli and popularised by Hinton. Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by Martin Gardner in his January 1962 "Mathematical Games column" in Scientific American. 1967, The associative algebra of W R Hamilton was the source of the science of vector analysis in three dimensions as recounted by Michael J. Crowe in A History of Vector Analysis. The study of Minkowski space required Riemann's mathematics which is 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:

Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H. G. Wells in The Time Machine, has led such authors as John William Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.

— H. S. M. Coxeter, Regular Polytopes[14]