John Horton Conway  

Born  
Died  11 April 2020 82) New Brunswick, New Jersey, U.S.  (aged
Education  Gonville and Caius College, Cambridge (BA, MA, PhD) 
Known for  
Awards 

Scientific career  
Fields  Mathematics 
Institutions  University of Cambridge Princeton University 
Thesis  Homogeneous ordered sets (1964) 
Doctoral advisor  Harold Davenport ^{ [1] } 
Doctoral students 

Website  Archived version @ web.archive.org 
John Horton Conway FRS (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life.
Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career.^{ [2] } On 11 April 2020, at age 82, he died of complications from COVID19.^{ [3] }
Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce.^{ [2] }^{ [4] } He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician.^{ [5] }^{ [6] } After leaving sixth form, he studied mathematics at Gonville and Caius College, Cambridge.^{ [4] } A "terribly introverted adolescent" in school, he took his admission to Cambridge as an opportunity to transform himself into an extrovert, a change which would later earn him the nickname of "the world's most charismatic mathematician".^{ [7] }^{ [8] }
Conway was awarded a BA in 1959 and, supervised by Harold Davenport, began to undertake research in number theory. Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals.^{ [6] } It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room.^{ [2] }
In 1964, Conway was awarded his doctorate and was appointed as College Fellow and Lecturer in Mathematics at Sidney Sussex College, Cambridge.^{ [9] }
After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University.^{ [9] } There, he won the Princeton University Pi Day pieeating contest.^{ [10] }
Conway's career was intertwined with that of Martin Gardner. When Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity.^{ [11] }^{ [12] } Gardner and Conway had first corresponded in the late 1950s, and over the years Gardner had frequently written about recreational aspects of Conway's work.^{ [13] } For instance, he discussed Conway's game of Sprouts (July 1967), Hackenbush (January 1972), and his angel and devil problem (February 1974). In the September 1976 column, he reviewed Conway's book On Numbers and Games and even managed to explain Conway's surreal numbers.^{ [14] }
Conway was a prominent member of Martin Gardner's Mathematical Grapevine. He regularly visited Gardner and often wrote him long letters summarizing his recreational research. In a 1976 visit, Gardner kept him for a week, pumping him for information on the Penrose tilings which had just been announced. Conway had discovered many (if not most) of the major properties of the tilings.^{ [15] } Gardner used these results when he introduced the world to Penrose tiles in his January 1977 column.^{ [16] } The cover of that issue of Scientific American features the Penrose tiles and is based on a sketch by Conway.^{ [12] }
Conway was married three times. With his first two wives he had two sons and four daughters. He married Diana in 2001 and had another son with her.^{ [17] } He had three grandchildren and two greatgrandchildren.^{ [2] }
On 8 April 2020, Conway developed symptoms of COVID19.^{ [18] } On 11 April, he died in New Brunswick, New Jersey, at the age of 82.^{ [18] }^{ [19] }^{ [20] }^{ [21] }^{ [22] }
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,^{ [23] } it has spawned hundreds of computer programs, web sites, and articles.^{ [24] } It is a staple of recreational mathematics. There is an extensive wiki devoted to curating and cataloging the various aspects of the game.^{ [25] } 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.^{ [26] } The game helped to launch a new branch of mathematics, the field of cellular automata.^{ [27] } The Game of Life is known to be Turing complete.^{ [28] }^{ [29] }
Conway contributed to combinatorial game theory (CGT), a theory of partisan games. He developed the theory with Elwyn Berlekamp and Richard Guy, and also coauthored 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, as well as 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.^{ [30] } He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG.
In the mid1960s with Michael Guy, Conway established that there are sixtyfour convex uniform polychora excluding two infinite sets of prismatic forms. They discovered the grand antiprism in the process, the only nonWythoffian uniform polychoron.^{ [31] } Conway has 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.^{ [32] }
He investigated lattices in higher dimensions and was the first to determine the symmetry group of the Leech lattice.
In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial.^{ [33] } After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials.^{ [34] } Conway further developed tangle theory and invented a system of notation for tabulating knots, now known as Conway notation, while correcting a number of errors in the 19thcentury knot tables and extending them to include all but four of the nonalternating primes with 11 crossings.^{ [35] } 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.
He was the primary author of the ATLAS of Finite Groups giving properties of many finite simple groups. Working with his colleagues Robert Curtis and Simon P. Norton he constructed the first concrete representations of some of the sporadic groups. More specifically, he discovered three sporadic groups based on the symmetry of the Leech lattice, which have been designated the Conway groups.^{ [36] } 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, Conway and Norton formulated the complex of conjectures known as monstrous moonshine. This subject, named by Conway, relates the monster group with elliptic modular functions, thus bridging two previously distinct areas of mathematics—finite groups and complex function theory. Monstrous moonshine theory has now been revealed to also have deep connections to string theory.^{ [37] }
Conway introduced the Mathieu groupoid, an extension of the Mathieu group M_{12} to 13 points.
As a graduate student, he proved one case of a 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.^{ [38] } In 1972, Conway proved that a natural generalization of the Collatz problem is algorithmically 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".^{ [39] }
Conway wrote a textbook on Stephen Kleene's theory of state machines, and published original work on algebraic structures, focusing particularly on quaternions and octonions.^{ [40] } Together with Neil Sloane, he invented the icosians.^{ [41] }
He invented a base 13 function as a counterexample to the 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.
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 finitestate machines.
In 2004, Conway and Simon B. Kochen, another Princeton mathematician, proved the free will theorem, a version of the "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."^{ [42] }
Conway received the Berwick Prize (1971),^{ [43] } was elected a Fellow of the Royal Society (1981),^{ [44] }^{ [45] } became a fellow of the American Academy of Arts and Sciences in 1992, was the first recipient of the Pólya Prize (LMS) (1987),^{ [43] } won the Nemmers Prize in Mathematics (1998) and received the Leroy P. Steele Prize for Mathematical Exposition (2000) of the American Mathematical Society. In 2001 he was awarded an honorary degree from the University of Liverpool,^{ [46] } and in 2014 one from Alexandru Ioan Cuza University.^{ [47] }
His FRS nomination, in 1981, reads:
A versatile mathematician who combines a deep combinatorial insight with algebraic virtuosity, particularly in the construction and manipulation of "offbeat" algebraic structures which illuminate a wide variety of problems in completely unexpected ways. He has made distinguished contributions to the theory of finite groups, to the theory of knots, to mathematical logic (both set theory and automata theory) and to the theory of games (as also to its practice).^{ [44] }
In 2017 Conway was given honorary membership of the British Mathematical Association.^{ [48] }
Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, and Conway himself was often a featured speaker at these events, discussing various aspects of recreational mathematics.^{ [49] }^{ [50] }
Martin Gardner was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literature – especially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton. He was a leading authority on Lewis Carroll; The Annotated Alice, which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies. He had a lifelong interest in magic and illusion and in 1999, MAGIC magazine named him as one of the "100 Most Influential Magicians of the Twentieth Century". He was considered the doyen of American puzzlers. He was a prolific and versatile author, publishing more than 100 books.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of threedimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an ndimensional polytope or npolytope. For example, a twodimensional polygon is a 2polytope and a threedimensional polyhedron is a 3polytope. In this context, "flat sides" means that the sides of a (k + 1)polytope consist of kpolytopes that may have (k – 1)polytopes in common.
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
= 2^{46} · 3^{20} · 5^{9} · 7^{6} · 11^{2} · 13^{3} · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
≈ 8×10^{53}.
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
Ralph William Gosper Jr., known as Bill Gosper, is an American mathematician and programmer. Along with Richard Greenblatt, he may be considered to have founded the hacker community, and he holds a place of pride in the Lisp community. The Gosper curve and the Gosper's algorithm are named after him.
Elwyn Ralph Berlekamp was a professor of mathematics and computer science at the University of California, Berkeley. Berlekamp was widely known for his work in computer science, coding theory and combinatorial game theory.
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979.
Richard Kenneth Guy was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathematics, combinatorics, and graph theory. He is best known for coauthorship of Winning Ways for your Mathematical Plays and authorship of Unsolved Problems in Number Theory. He published more than 300 scholarly articles. Guy proposed the partially tongueincheek "strong law of small numbers", which says there are not enough small integers available for the many tasks assigned to them – thus explaining many coincidences and patterns found among numerous cultures. For this paper he received the MAA Lester R. Ford Award.
In a cellular automaton, a gun is a pattern with a main part that repeats periodically, like an oscillator, and that also periodically emits spaceships. There are then two periods that may be considered: the period of the spaceship output, and the period of the gun itself, which is necessarily a multiple of the spaceship output's period. A gun whose period is larger than the period of the output is a pseudoperiod gun.
Simon Phillips Norton was a mathematician in Cambridge, England, who worked on finite simple groups.
Robert Arnott Wilson is a retired mathematician in London, England, who is best known for his work on classifying the maximal subgroups of finite simple groups and for the work in the Monster group. He is also an accomplished violin, viola and piano player, having played as the principal viola in the Sinfonia of Birmingham. Due to a damaged finger, he now principally plays the kora.
Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series. These are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others.
A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by nonoverlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.
Daniel T. Wise is an American mathematician who specializes in geometric group theory and 3manifolds. He is a professor of mathematics at McGill University.
In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements: The tile must be a closed topological disk with six consecutive points A, B, C, D, E, and F on the boundary such that:
Miranda ChihNing Cheng is a Taiwaneseborn and Dutcheducated mathematician and theoretical physicist who works as an associate professor at the University of Amsterdam. She is known for formulating the umbral moonshine conjectures and for her work on the connections between K3 surfaces and string theory.
Chaim GoodmanStrauss is an American mathematician who works in convex geometry, especially aperiodic tiling. He retired from the faculty of the University of Arkansas and currently serves as outreach mathematician for the National Museum of Mathematics. He is coauthor with John H. Conway and Heidi Burgiel of The Symmetries of Things, a comprehensive book surveying the mathematical theory of patterns.