Editing Four-dimensional space (section)

==History==
The idea of making [[time]] the fourth dimension began with [[Jean le Rond d'Alembert]]  "Dimensions" published 1754 in the ''[[Encyclopédie, ou dictionnaire raisonné des sciences, des arts et des métiers]]''. <ref name=VanOss>{{Cite journal|last1=Van Oss|first1=Rosine G |location=State University of New York at Buffalo|date=November 1983|title=D'Alembert and the fourth dimension|journal=[[Historia Mathematica]]|volume=10 |issue=4|doi=10.1016/0315-0860(83)90007-1|publication-date=29 June 2004|via=''attr''. Cajori (1926) doi:10.1080/00029890.1926.11986607}}</ref><ref name=Archibald>{{Cite journal|last1=Archibald|first1=R. C. |author-link=Raymond_C._Archibald#Bibliography|url=https://www.ams.org/journals/bull/1914-20-08/S0002-9904-1914-02511-X/S0002-9904-1914-02511-X.pdf|date=1 May 1914|title=<small><small>TIME AS A FOURTH DIMENSION</small></small>|journal=[[Bulletin of the American Mathematical Society]]|volume=20|page=410 (2)|doi=10.1090/S0002-9904-1914-02511-X}}
</ref> That [[mechanics]] can be viewed as occurring also in time was found by [[Joseph-Louis Lagrange]] c.1755 published 1788 in {{lang|fr|[[s:fr:Mécanique analytique|Mécanique analytique]]}}. <ref>{{cite book|last1=Bell|first1=E.T.|title=Men of Mathematics|date=1965|publisher=[[Simon and Schuster]]|location=New York|isbn=978-0-671-62818-5|page=154|edition=1st}}</ref> 

Mathematics of 4D commenced in the nineteenth century. <ref>{{Cite journal|last1=Lawrence|first1=Snezana|location=[[Bath Spa University]]|date=8 January 2015|url= https://link.springer.com/article/10.1007/s00004-014-0221-9#citeas|title=Life, Architecture, Mathematics, and the Fourth Dimension|journal=[[Nexus Network Journal]]|volume=17|page=Abstract|doi=10.1007/s00004-014-0221-9|archive-url=https://archive.today/20250501104604/https://link.springer.com/article/10.1007/s00004-014-0221-9%23citeas|archive-date=1 May 2025}}</ref> 3D form rotation onto its mirror-image is possible in 4D space was realized by [[August Ferdinand Möbius]] {{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence.}}  in  ''[[:s:de:August_Ferdinand_Möbius|Der barycentrische Calcul]]'' published 1827. <ref>{{Cite book|last1=Gray|first1=Jeremy|date=2007|author-link=Jeremy Gray|url=https://link.springer.com/chapter/10.1007/978-1-84628-633-9_13|title=Across the Rhine — Möbius’s Algebraic Version of Projective Geometry. In: Worlds Out of Nothing|doi=10.1007/978-1-84628-633-9_13|publisher=Springer, London|isbn=978-1-84628-633-9}}</ref> An arithmetic of four spatial dimensions, called [[quaternion]]s, was defined by [[William Rowan Hamilton]] in 1843.  Soon after, [[tessarine]]s and [[coquaternion]]s were introduced as other four-dimensional [[algebra over a field|algebras over '''R''']]. Higher dimensional non-Euclidean spaces were put on a firm footing by [[Bernhard Riemann]]'s 1854 [[Habilitationsschrift|thesis]], {{lang|de|Über die Hypothesen welche der Geometrie zu Grunde liegen}}, in which he considered a "point" to be any sequence of coordinates {{math|(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')}}. 

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 [[Regular 4-polytope|six found in 4-dimensional space]].{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli's work was only published posthumously in 1901, and remained largely unknown until publication of [[H.S.M. Coxeter]]'s ''[[Regular Polytopes (book)|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 "[[s: What is the Fourth Dimension?|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. {{efn|The simplest form of Hinton's method is to draw two ordinary 3D cubes in 2D space, one encompassing the other, separated by an "unseen" distance, and then draw lines between their equivalent vertices. This can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube. The eight lines connecting the vertices of the two cubes in this case represent a ''single direction'' in the "unseen" fourth dimension.}} <ref>{{cite book|last1=Hinton|first1=Charles Howard|author-link1=Charles Howard Hinton|title=Speculations on the Fourth Dimension: Selected writings of Charles H. Hinton|date=1980|editor-first=Rudolf v. B.|editor-last=Rucker|publisher=[[Dover Publishing]]|location=New York|isbn=978-0-486-23916-3|page=vii}}</ref>  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''.<ref name="Hinton">{{cite book|last1=Hinton|first1=Charles Howard|title=The Fourth Dimension|orig-year=1904|date=1993|publisher=Health Research|location=Pomeroy, Washington|isbn=978-0-7873-0410-2|page=14|url=https://books.google.com/books?id=_ZG3MA1wvjIC&pg=PA14|access-date=17 February 2017|language=en}}</ref><ref>{{cite book|last1=Gardner|first1=Martin|title=Mathematical Carnival: From Penny Puzzles. Card Shuffles and Tricks of Lightning Calculators to Roller Coaster Rides into the Fourth Dimension|date=1975|publisher=[[Knopf Publishing|Knopf]]|location=New York|isbn=978-0-394-49406-7|pages=42, 52–53|edition=1st}}</ref> 1886, [[Victor Schlegel]] described<ref>{{Cite book |last1=Schlegel |first1=Victor |title=Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper |last2=Waren |year=1886 |language=de |trans-title=On projection models of regular four-dimensional bodies |author-link=Victor Schlegel}}</ref> his method of visualizing [[4-polytope|four-dimensional objects]] with [[Schlegel diagram]]s.

Minkowski's 1908 paper<ref>{{Cite journal |last=Minkowski |first=Hermann |author-link=Hermann Minkowski |year=1909 |title=Raum und Zeit |trans-title=Space and Time |url=https://en.wikisource.org/wiki/de:Raum_und_Zeit_(Minkowski) |journal=Physikalische Zeitschrift |language=de |volume=10 |pages=75–88 |access-date=October 27, 2022 |via=Wikisource}}</ref> consolidating the role of time as the fourth dimension of [[spacetime]] provided the geometric basis for Einstein's theories of special and general relativity.<ref name="Møller">{{cite book|last1=Møller|first1=C.|title=The Theory of Relativity|url=https://archive.org/details/theoryofrelativi0000mlle|url-access=registration|date=1972|publisher=Clarendon Press|location=Oxford|isbn=978-0-19-851256-1|page=[https://archive.org/details/theoryofrelativi0000mlle/page/93 93]|edition=2nd}}</ref> 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:
{{Quote|Little, if anything, is gained by representing the fourth Euclidean dimension as ''time''. In fact, this idea, so attractively developed by [[The Time Machine|H. G. Wells in ''The Time Machine'']], has led such authors as [[An Experiment with Time|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''{{Sfn|Coxeter|1973|p=119}}}}