John H. Conway
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John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
s,
knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
, 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 ''The Game of Life'', also known as ''Life'', is an 1860 board game by Milton Bradley. Game of Life also often refers to: *Conway's Game of Life, in mathematics, a cellular automaton Game of Life or The Game of Life may also refer to: Games * ' ...
. 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. On 11 April 2020, at age 82, he died of complications from COVID-19.


Early life and education

Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Cambridge. 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". 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. 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. In 1964, Conway was awarded his doctorate and was appointed as College Fellow and Lecturer in Mathematics at
Sidney Sussex College, Cambridge Sidney Sussex College (referred to informally as "Sidney") is a constituent college of the University of Cambridge in England. The College was founded in 1596 under the terms of the will of Frances Sidney, Countess of Sussex (1531–1589), wife ...
. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University. There, he won the school's Pi Day pie-eating contest.


Conway and Martin Gardner

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. 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. For instance, he discussed Conway's game of Sprouts (July 1967),
Hackenbush Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line. Gameplay The game starts with the p ...
(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. 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. Gardner used these results when he introduced the world to Penrose tiles in his January 1977 column. The cover of that issue of ''Scientific American'' features the Penrose tiles and is based on a sketch by Conway.


Personal life and death

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 in 2001. He had three grandchildren and two great-grandchildren. On 8 April 2020, Conway developed symptoms of COVID-19. On 11 April, he died in New Brunswick, New Jersey, at the age of 82.


Major areas of research


Recreational mathematics

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, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics. There is an extensive wiki devoted to curating and cataloging the various aspects of the game. 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 the Game of Life, feeling that it overshadowed deeper and more important things he had done. Nevertheless, the game helped to launch a new branch of mathematics, the field of cellular automata. The Game of Life is known to be Turing complete.


Combinatorial game theory

Conway contributed to combinatorial game theory (CGT), a theory of partisan games. He developed the theory with Elwyn Berlekamp and 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, 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. 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''.


Geometry

In the mid-1960s with
Michael Guy Michael J. T. Guy (born 1 April 1943) is a British computer scientist and mathematician. He is known for early work on computer systems, such as the Phoenix system at the University of Cambridge, and for contributions to number theory, comput ...
, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms. They discovered the grand antiprism in the process, the only non-Wythoffian uniform polychoron. 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 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:Will It Tile? Try ...
which is a fast way to identify many prototiles that tile the plane. 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. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials. Conway further developed
tangle theory In mathematics, a tangle is generally one of two related concepts: * In John Conway's definition, an ''n''-tangle is a proper embedding of the disjoint union of ''n'' arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2''n ...
and invented a system of notation for tabulating knots, now known as 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. The
Conway knot In mathematics, in particular in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway. It is related by mutation to the Kinoshita–Terasaka knot, with which it shares the same ...
is named after him. Conway's conjecture that, in any
thrackle A thrackle is an embedding of a graph in the plane, such that each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. ...
, 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
s. Working with his colleagues Robert Curtis and
Simon P. Norton Simon Phillips Norton (28 February 1952 – 14 February 2019)
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 In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by . The largest of the Conway groups, Co0, is the group of auto ...
. 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 group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
s and complex function theory. Monstrous moonshine theory has now been revealed to also have deep connections to
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
. Conway introduced the
Mathieu groupoid In mathematics, the Mathieu groupoid M13 is a groupoid acting on 13 points such that the stabilizer of each point is the Mathieu group M12. It was introduced by and studied in detail by . Construction The projective plane of order 3 has 13 point ...
, an extension of the Mathieu group M12 to 13 points.


Number theory

As a graduate student, he proved one case of a
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
by Edward Waring, that every integer could be written as the sum of 37 numbers each raised to the fifth power, though
Chen Jingrun Chen Jingrun (; 22 May 1933 – 19 March 1996), also known as Jing-Run Chen, was a Chinese mathematician who made significant contributions to number theory, including Chen's theorem and the Chen prime. Life and career Chen was the third son in ...
solved the problem independently before Conway's work could be published.


Algebra

Conway wrote a textbook on Stephen Kleene's theory of state machines and published original work on
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s, focusing particularly on
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s and octonions. Together with
Neil Sloane __NOTOC__ Neil James Alexander Sloane (born October 10, 1939) is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing. Sloane is best known for being the creator ...
, he invented the icosians.


Analysis

He invented a base 13 function as a counterexample to the converse of the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two import ...
: the function takes on every real value in each interval on the real line, so it has a Darboux property but is ''not''
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
.


Algorithmics

For calculating the day of the week, he invented the
Doomsday algorithm The Doomsday rule, Doomsday algorithm or Doomsday method is an algorithm of determination of the day of the week for a given date. It provides a perpetual calendar because the Gregorian calendar moves in cycles of 400 years. The algorithm for men ...
. 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 machines.


Theoretical physics

In 2004, Conway and
Simon B. Kochen Simon Bernhard Kochen (; born 14 August 1934, Antwerp) is a Canadian mathematician, working in the fields of model theory, number theory and quantum mechanics. Biography Kochen received his Ph.D. (''Ultrafiltered Products and Arithmetical Extens ...
, another Princeton mathematician, proved the
free will theorem The free will theorem of John H. Conway and Simon B. Kochen states that if we have a free will in the sense that our choices are not a function of the past, then, subject to certain assumptions, so must some elementary particles. Conway and Kochen's ...
, 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."


Awards and honours

Conway received the
Berwick Prize The Berwick Prize and Senior Berwick Prize are two prizes of the London Mathematical Society awarded in alternating years in memory of William Edward Hodgson Berwick, a previous Vice-President of the LMS. Berwick left some money to be given to the ...
(1971), was elected a Fellow of the Royal Society (1981), became a fellow of the American Academy of Arts and Sciences in 1992, was the first recipient of the
Pólya Prize (LMS) The Pólya Prize is a prize in mathematics, awarded by the London Mathematical Society. Second only to the triennial De Morgan Medal in prestige among the society's awards, it is awarded in the years that are not divisible by three – those in wh ...
(1987), 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, and in 2014 one from
Alexandru Ioan Cuza University The Alexandru Ioan Cuza University (Romanian: ''Universitatea „Alexandru Ioan Cuza"''; acronym: UAIC) is a public university located in Iași, Romania. Founded by an 1860 decree of Prince Alexandru Ioan Cuza, under whom the former Academia Mih ...
. His FRS nomination, in 1981, reads: In 2017 Conway was given honorary membership of the British Mathematical Association. Conferences called
Gathering 4 Gardner Gathering 4 Gardner (G4G) is an educational foundation and non-profit corporation (Gathering 4 Gardner, Inc.) devoted to preserving the legacy and spirit of prolific writer Martin Gardner. G4G organizes conferences where people who have been inspi ...
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.Bellos, Alex (2008)
The science of fun
''The Guardian'', 30 May 2008


Select publications

* 1971 – ''Regular algebra and finite machines''. Chapman and Hall, London, 1971, Series: Chapman and Hall mathematics series, . * 1976 – '' On numbers and games''. Academic Press, New York, 1976, Series: L.M.S. monographs, 6, . * 1979 – ''On the Distribution of Values of Angles Determined by Coplanar Points'' (with
Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
,
Michael Guy Michael J. T. Guy (born 1 April 1943) is a British computer scientist and mathematician. He is known for early work on computer systems, such as the Phoenix system at the University of Cambridge, and for contributions to number theory, comput ...
, and H. T. Croft). Journal of the London Mathematical Society, vol. II, series 19, pp. 137–143. * 1979 – ''Monstrous Moonshine'' (with
Simon P. Norton Simon Phillips Norton (28 February 1952 – 14 February 2019)
). Bulletin of the London Mathematical Society, vol. 11, issue 2, pp. 308–339. * 1982 – '' Winning Ways for your Mathematical Plays'' (with
Richard K. Guy Richard Kenneth Guy (30 September 1916 – 9 March 2020) 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 mathemati ...
and Elwyn Berlekamp). Academic Press, . * 1985 – '' Atlas of finite groups'' (with Robert Turner Curtis,
Simon Phillips Norton Simon Phillips Norton (28 February 1952 – 14 February 2019)
,
Richard A. Parker Richard A. Parker (born 29 January 1953, in Surrey) is a mathematician and freelance computer programmer in Cambridge, England. He invented many of the algorithms for computing the modular character tables of finite simple groups. He discovered ...
, and Robert Arnott Wilson). Clarendon Press, New York, Oxford University Press, 1985, . * 1988 – ''Sphere Packings, Lattices, and Groups'' (with
Neil Sloane __NOTOC__ Neil James Alexander Sloane (born October 10, 1939) is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing. Sloane is best known for being the creator ...
). Springer-Verlag, New York, Series: Grundlehren der mathematischen Wissenschaften, 290, . * 1995 – ''Minimal-Energy Clusters of Hard Spheres'' (with
Neil Sloane __NOTOC__ Neil James Alexander Sloane (born October 10, 1939) is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing. Sloane is best known for being the creator ...
, R. H. Hardin, and
Tom Duff Tom or TOM may refer to: * Tom (given name), a diminutive of Thomas or Tomás or an independent Aramaic given name (and a list of people with the name) Characters * Tom Anderson, a character in ''Beavis and Butt-Head'' * Tom Beck, a character ...
). Discrete & Computational Geometry, vol. 14, no. 3, pp. 237–259. * 1996 – ''The Book of Numbers'' (with
Richard K. Guy Richard Kenneth Guy (30 September 1916 – 9 March 2020) 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 mathemati ...
). Copernicus, New York, 1996, . * 1997 – ''The Sensual (quadratic) Form'' (with Francis Yein Chei Fung).
Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
, Washington, DC, 1997, Series: Carus mathematical monographs, no. 26, . * 2002 – ''On Quaternions and Octonions'' (with Derek A. Smith).
A. K. Peters A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science. They published the journals ''Experimental Mathematics'' and the '' Journal ...
, Natick, MA, 2002, . * 2008 – ''
The Symmetries of Things ''The Symmetries of Things'' is a book on mathematical symmetry and the symmetries of geometric objects, aimed at audiences of multiple levels. It was written over the course of many years by John Horton Conway, Heidi Burgiel, and Chaim Goodman- ...
'' (with Heidi Burgiel and Chaim Goodman-Strauss).
A. K. Peters A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science. They published the journals ''Experimental Mathematics'' and the '' Journal ...
, Wellesley, MA, 2008, .


See also

*
List of things named after John Horton Conway This is a list of things named after the English mathematician John Horton Conway (1937–2020). * Conway algebra – an algebraic structure introduced by Paweł Traczyk and Józef H. Przytycki * Conway base 13 function – a function used as ...


References


Sources

* Alpert, Mark (1999).
Not Just Fun and Games
' ''Scientific American'', April 1999 * Boden, Margaret (2006). ''Mind As Machine'', Oxford University Press, 2006, p. 1271 * du Sautoy, Marcus (2008). ''Symmetry'', HarperCollins, p. 308 * Guy, Richard K (1983).
Conway's Prime Producing Machine
' Mathematics Magazine, Vol. 56, No. 1 (Jan. 1983), pp. 26–33 * * * * Princeton University (2009)
Bibliography of John H. Conway
Mathematics Department * Seife, Charles (1994).
Impressions of Conway
' The Sciences * Schleicher, Dierk (2011)
Interview with John Conway
Notices of the AMS


External links

* * * ** ** * Conway leading a tour of brickwork patterns in Princeton, lecturing on the ordinals and on sums of powers and the Bernoulli numbers
necrology by Keith Hartnett in Quanta Magazine, April 20, 2020
{{DEFAULTSORT:Conway, John Horton 1937 births 2020 deaths 20th-century English mathematicians 21st-century English mathematicians Algebraists Group theorists Combinatorial game theorists Cellular automatists Mathematics popularizers Recreational mathematicians Alumni of Gonville and Caius College, Cambridge Fellows of Sidney Sussex College, Cambridge Fellows of the Royal Society Princeton University faculty Scientists from Liverpool British expatriate academics in the United States Researchers of artificial life Deaths from the COVID-19 pandemic in New Jersey