Editing John Horton Conway (section)

==Major areas of research==

===Recreational mathematics===
{{Main|Conway's Game of Life}}
[[File:Gospers glider gun.gif|thumb|right|A single [[Bill Gosper|Gosper]]'s [[Gun (cellular automaton)|Glider Gun]] creating "[[Glider (Conway's Life)|gliders]]" in [[Conway's Game of Life]]]]
Conway invented the Game of Life, one of the early examples of a [[cellular automaton]]. His initial experiments in that field were done with pen and paper, long before personal computers existed. Since Conway's game was popularized by Martin Gardner in ''[[Scientific American]]'' in 1970,<ref>{{Cite magazine|title=Mathematical Games: The fantastic combinations of John Conway's new solitaire game "Life"|first=Martin|last=Gardner|magazine=Scientific American|volume=223|date=October 1970|pages=120–123|jstor=24927642|url=https://web.stanford.edu/class/sts145/Library/life.pdf}}</ref> it has spawned hundreds of computer programs, web sites, and articles.<ref>{{Cite web |url=https://www.dmoz.org/Computers/Artificial_Life/Cellular_Automata/Conway%27s_Game_of_Life |title=DMOZ: Conway's Game of Life: Sites |access-date=11 January 2017 |archive-url=https://web.archive.org/web/20170317103511/http://www.dmoz.org/Computers/Artificial_Life/Cellular_Automata/Conway%27s_Game_of_Life/ |archive-date=17 March 2017 |url-status=dead }}</ref> It is a staple of recreational mathematics. The [[LifeWiki]] is devoted to curating and cataloging the various aspects of the game.<ref>{{Cite web|url=https://www.conwaylife.com/wiki/Main_Page|title=LifeWiki|website=www.conwaylife.com}}</ref> From the earliest days, it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. Conway came to dislike how discussions of him heavily focused on his Game of Life, feeling that it overshadowed deeper and more important things he had done, although he remained proud of his work on it.<ref>{{Cite AV media |url=https://www.youtube.com/watch?v=E8kUJL04ELA |title=Does John Conway hate his Game of Life? |date=2014-03-03 |last=Numberphile |access-date=2024-11-05 |via=YouTube}}</ref> The game helped to launch a new branch of mathematics, the field of [[cellular automata]].<ref>MacTutor History: The game made Conway instantly famous, but it also opened up a whole new field of mathematical research, the field of cellular automata.</ref>
The Game of Life is known to be [[Turing completeness|Turing complete]].<ref>{{cite book |last=Rendell |first=Paul |date=July 2015 |title=Turing Machine Universality of the Game of Life |series=Emergence, Complexity and Computation |volume=18 |publisher=Springer |isbn=978-3319198415 |doi=10.1007/978-3-319-19842-2 |url=https://books.google.com/books?id=w92moAEACAAJ}}</ref><ref>{{cite news |last=Case |first=James |date=1 April 2014 |title=Martin Gardner's Mathematical Grapevine |website=[[Society for Industrial and Applied Mathematics|SIAM]] NEWS |url=https://sinews.siam.org/Details-Page/martin-gardners-mathematical-grapevine |at=Book reviews of Gardner, Martin, 2013 ''Undiluted Hocus-Pocus: The Autobiography of Martin Gardner''. Princeton University Press and Henle, Michael; Hopkins, Brian (edts.) 2012 ''Martin Gardner in the Twenty-First Century''. MAA Publications }}</ref>

===Combinatorial game theory===
Conway contributed to [[combinatorial game theory]] (CGT), a theory of [[partisan game]]s. He developed the theory with [[Elwyn Berlekamp]] and [[Richard K. Guy|Richard Guy]], and also co-authored the book ''[[Winning Ways for your Mathematical Plays]]'' with them. He also wrote ''[[On Numbers and Games]]'' (''ONAG'') which lays out the mathematical foundations of CGT.

He was also one of the inventors of the game [[Sprouts (game)|sprouts]], as well as [[phutball|philosopher's football]]. He developed detailed analyses of many other games and puzzles, such as the [[Soma cube]], [[peg solitaire]], and [[Conway's soldiers]]. He came up with the [[angel problem]], which was solved in 2006.

He invented a new system of numbers, the [[surreal numbers]], which are closely related to certain games and have been the subject of a mathematical novelette by [[Donald Knuth]].<ref>[http://discovermagazine.com/1995/dec/infinityplusonea599 Infinity Plus One, and Other Surreal Numbers] by Polly Shulman, [[Discover Magazine]], 1 December 1995</ref> He also invented a nomenclature for exceedingly [[large number]]s, the [[Conway chained arrow notation]]. Much of this is discussed in the 0th part of ''ONAG''.

===Geometry===
In the mid-1960s with [[Michael Guy (computer scientist)|Michael Guy]], Conway established that there are sixty-four [[uniform polychoron|convex uniform polychora]] excluding two infinite sets of prismatic forms. They discovered the [[grand antiprism]] in the process, the only [[non-Wythoffian]] uniform [[polychoron]].<ref>{{cite journal|author=Conway, J. H. |title=Four-dimensional Archimedean polytopes|journal=Proc. Colloquium on Convexity, Copenhagen |year=1967|publisher= Kobenhavns Univ. Mat. Institut|pages= 38–39}}</ref> Conway also suggested a system of notation dedicated to describing [[polyhedra]] called [[Conway polyhedron notation]].

In the theory of tessellations, he devised the [[Conway criterion]] which is a fast way to identify many prototiles that tile the plane.<ref name=rhoads>{{cite journal| doi=10.1016/j.cam.2004.05.002 | volume=174 | issue=2 | title=Planar tilings by polyominoes, polyhexes, and polyiamonds | year=2005 | journal=Journal of Computational and Applied Mathematics | pages=329–353 | last1 = Rhoads | first1 = Glenn C.| bibcode=2005JCoAM.174..329R | doi-access=free }}</ref>

He investigated lattices in higher dimensions and was the first to determine the symmetry group of the [[Leech lattice]].

===Geometric topology===
In knot theory, Conway formulated a new variation of the [[Alexander polynomial]] and produced a new invariant now called the Conway polynomial.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Conway Polynomial |url=https://mathworld.wolfram.com/ConwayPolynomial.html |access-date=2024-11-05 |website=mathworld.wolfram.com |language=en}}</ref> After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel [[knot polynomial]]s.<ref>Livingston, Charles (1993) ''Knot Theory''. MAA Textbooks. {{ISBN|0883850273}}</ref> Conway further developed [[tangle theory]] and invented a system of notation for tabulating knots, now known as [[Conway notation (knot theory)|Conway notation]], while correcting a number of errors in the 19th-century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings.<ref>{{cite journal|journal=Topology Proceedings |volume=7 |year=1982|pages= 109–118|author=Perko, Ken |url=http://topology.nipissingu.ca/tp/reprints/v07/tp07110.pdf|title=Primality of certain knots}}</ref> The [[Conway knot]] is named after him.

Conway's conjecture that, in any [[thrackle]], the number of edges is at most equal to the number of vertices, is still open.

===Group theory===
He was the primary author of the ''[[ATLAS of Finite Groups]]'' giving properties of many [[finite simple group]]s. Working with his colleagues Robert Curtis and [[Simon P. Norton]] he constructed the first concrete representations of some of the [[sporadic group]]s. More specifically, he discovered three sporadic groups based on the symmetry of the [[Leech lattice]], which have been designated the [[Conway groups]].<ref>{{cite journal |last=Harris |first=Michael |author-link=Michael Harris (mathematician) |date=2015 |title=Mathematics: The mercurial mathematician |journal=Nature |volume=523 |pages=406–7 |doi=10.1038/523406a |others=Review of ''Genius At Play: The Curious Mind of John Horton Conway''
|issue=7561 |bibcode=2015Natur.523..406H |doi-access=free }}</ref> This work made him a key player in the successful [[classification of the finite simple groups]].

Based on a 1978 observation by mathematician [[John McKay (mathematician)|John McKay]], Conway and Norton formulated the complex of conjectures known as [[monstrous moonshine]]. This subject, named by Conway, relates the [[monster group]] with [[elliptic modular function]]s, thus bridging two previously distinct areas of mathematics—[[finite group]]s and [[complex function theory]]. Monstrous moonshine theory has now been revealed to also have deep connections to [[string theory]].<ref>{{Cite web |last=Darling |first=David |title=Monstrous Moonshine conjecture |url=https://www.daviddarling.info/encyclopedia/M/Monstrous_Moonshine_conjecture.html |access-date=2024-11-05 |website=www.daviddarling.info}}</ref>

Conway introduced the [[Mathieu groupoid]], an extension of the [[Mathieu group M12|Mathieu group M<sub>12</sub>]] to 13 points.

===Number theory===
As a graduate student, he proved one case of a [[Waring's problem|conjecture]] by [[Edward Waring]], that every integer could be written as the sum of 37 numbers each raised to the fifth power, though [[Chen Jingrun]] solved the problem independently before Conway's work could be published.<ref>{{cite journal|url=http://www.ems-ph.org/journals/newsletter/pdf/2005-09-57.pdf#page=34 |title=Breakfast with John Horton Conway|pages=32–34|journal=EMS Newsletter |date=September 2005|author=[[Jorge Nuno Silva]]|volume=57}}</ref> In 1972, Conway proved that a natural generalization of the [[Collatz conjecture|Collatz problem]] is algorithmically [[Undecidable problem|undecidable]]. Related to that, he developed the esoteric programming language [[FRACTRAN]]. While lecturing on the Collatz conjecture, [[Terence Tao]] (who was taught by him in graduate school) mentioned Conway's result and said that he was "always very good at making extremely weird connections in mathematics".<ref>{{Citation |title=Day 2 - The notorious Collatz conjecture - Terence Tao | date=30 October 2021 |url=https://www.youtube.com/watch?v=X2p5eMWyaFs |access-date=2023-03-23 |language=en}}</ref>

===Algebra===
Conway wrote a textbook on [[Stephen Kleene]]'s theory of state machines, and published original work on [[algebraic structure]]s, focusing particularly on [[quaternion]]s and [[octonion]]s.<ref name="Bulletin of the AMS Baez reviews Conway and Smith Quaternians">{{cite journal |last=Baez |first=John C. |date=2005 |title=Book Review: ''On quaternions and octonions: Their geometry, arithmetic, and symmetry'' |journal=Bulletin of the American Mathematical Society |volume=42 |issue=2 |pages=229–243 |doi=10.1090/S0273-0979-05-01043-8 |doi-access=free }}</ref>  Together with [[Neil Sloane]], he invented the [[icosian]]s.<ref>{{cite web| url=http://math.ucr.edu/home/baez/week20.html| title=This Week's Finds in Mathematical Physics (Week 20)| author=Baez, John | date=2 October 1993}}</ref>

===Analysis===
He invented a [[Conway base 13 function|base 13 function]] as a counterexample to the [[converse (logic)|converse]] of the [[intermediate value theorem]]: the function takes on every real value in each interval on the real line, so it has a [[Darboux property]] but is ''not'' [[continuous function|continuous]].

===Algorithmics===
For calculating the day of the week, he invented the [[Doomsday algorithm]]. The algorithm is simple enough for anyone with basic arithmetic ability to do the calculations mentally. Conway could usually give the correct answer in under two seconds. To improve his speed, he practised his calendrical calculations on his computer, which was programmed to quiz him with random dates every time he logged on. One of his early books was on [[finite-state machine]]s.

===Theoretical physics===
In 2004, Conway and [[Simon B. Kochen]], another Princeton mathematician, proved the [[free will theorem]], a version of the "[[Hidden-variable theory|no hidden variables]]" principle of [[quantum mechanics]]. It states that given certain conditions, if an experimenter can freely decide what quantities to measure in a particular experiment, then elementary particles must be free to choose their spins to make the measurements consistent with physical law. Conway said that "if experimenters have [[free will]], then so do elementary particles."<ref>''[http://www.cs.auckland.ac.nz/~jas/one/freewill-theorem.html Conway's Proof Of The Free Will Theorem] {{Webarchive|url=https://web.archive.org/web/20171125154530/https://www.cs.auckland.ac.nz/~jas/one/freewill-theorem.html |date=25 November 2017 }}'' by Jasvir Nagra</ref>