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] }^{ [3] }^{ [4] }^{ [5] }^{ [6] }^{ [7] } On 11 April 2020, at age 82, he died of complications from COVID-19.^{ [8] }
Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce.^{ [9] }^{ [7] } He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician.^{ [10] }^{ [11] } After leaving sixth form, he studied mathematics at Gonville and Caius College, Cambridge.^{ [9] } 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".^{ [12] }^{ [13] }
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.^{ [11] } 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. He was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at Sidney Sussex College, Cambridge.^{ [14] } After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University.^{ [14] }
Conway was especially known for the invention of 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 the game was introduced by Martin Gardner in Scientific American in 1970,^{ [15] } it has spawned hundreds of computer programs, web sites, and articles.^{ [16] } It is a staple of recreational mathematics. There is an extensive wiki devoted to curating and cataloging the various aspects of the game.^{ [17] } 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 used to hate the Game of Life—largely because it had come to overshadow some of the other deeper and more important things he has done.^{ [18] } Nevertheless, the game did help launch a new branch of mathematics, the field of cellular automata.^{ [19] }
The Game of Life is known to be Turing complete.^{ [20] }^{ [21] }
Conway's career was intertwined with that of mathematics popularizer and Scientific American columnist 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.^{ [22] }^{ [23] } 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.^{ [24] } For instance, he discussed Conway's game of Sprouts (Jul 1967), Hackenbush (Jan 1972), and his angel and devil problem (Feb 1974). In the September 1976 column, he reviewed Conway's book On Numbers and Games and even managed to explain Conway's surreal numbers.^{ [25] }
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.^{ [26] } Gardner used these results when he introduced the world to Penrose tiles in his January 1977 column.^{ [27] } The cover of that issue of Scientific American features the Penrose tiles and is based on a sketch by Conway.^{ [23] }
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.^{ [28] }^{ [29] }
Conway was widely known for his contributions to combinatorial game theory (CGT), a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays . He also wrote the book On Numbers and Games (ONAG) which lays out the mathematical foundations of CGT.
He was also one of the inventors of 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 mid-1960s with Michael Guy, 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.^{ [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, nowadays 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.^{ [35] }(Some might say "all but 3½ of the non-alternating primes with 11 crossings." The typographical duplication in the published version of his 1970 table seems to be an effort to include one of the two missing knots that was included in the draft of the table that he sent to Fox [Compare D. Lombardero's 1968 Princeton Senior Thesis, which distinguished this one, but not the other, from all others, based on its Alexander polynomial].) In knot theory the Conway knot is named after him.
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] }
Conway has written textbooks and done original work in algebra, focusing particularly on quaternions and octonions.^{ [39] } Together with Neil Sloane, he invented the icosians.^{ [40] }
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 finite-state machines.
In 2004, Conway and Simon B. Kochen, another Princeton mathematician, proved the free will theorem, a startling 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. In Conway's provocative wording: "if experimenters have free will, then so do elementary particles."^{ [41] }
Conway received the Berwick Prize (1971),^{ [42] } was elected a Fellow of the Royal Society (1981),^{ [43] } became a fellow of the American Academy of Arts and Sciences in 1992, was the first recipient of the Pólya Prize (LMS) (1987),^{ [42] } 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,^{ [44] } and in 2014 one from Alexandru Ioan Cuza University.^{ [45] }
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 "off-beat" 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).^{ [43] }
In 2017 Conway was given honorary membership of the British Mathematical Association.^{ [46] }
On 8 April 2020, Conway developed symptoms of COVID-19.^{ [47] } On 11 April, he died in New Brunswick, New Jersey at the age of 82.^{ [47] }^{ [48] }^{ [49] }^{ [50] }^{ [51] }
Martin Gardner was an American popular mathematics and popular science writer, with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literature—especially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton. He was also 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 was regarded as one of the most important 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. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.
A pentomino is a polyomino of order 5, that is, a polygon in the plane made of 5 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 12 different free pentominoes. When reflections are considered distinct, there are 18 one-sided pentominoes. When rotations are also considered distinct, there are 63 fixed pentominoes.
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject.
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.
On Numbers and Games is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpretentious manner and many chapters are accessible to non-mathematicians. Martin Gardner discussed the book at length, particularly Conway's construction of surreal numbers, in his Mathematical Games column in Scientific American in September 1976.
Combinatorial game theory (CGT) is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition. CGT has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field.
A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.
Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that satisfies the given conditions. Mathematical puzzles require mathematics to solve them. Logic puzzles are a common type of mathematical puzzle.
In geometry and group theory, a lattice in is a subgroup of the additive group which is isomorphic to the additive group , and which spans the real vector space . In other words, for any basis of , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell.
Elwyn Ralph Berlekamp was an American mathematician known for his work in computer science, coding theory and combinatorial game theory. He was a professor emeritus of mathematics and EECS at the University of California, Berkeley.
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 co-authorship 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 tongue-in-cheek "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.
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.
Solomon Wolf Golomb was an American mathematician, engineer, and professor of electrical engineering at the University of Southern California, best known for his works on mathematical games. Most notably, he invented Cheskers in 1948 and coined the name. He also fully described polyominoes and pentominoes in 1953. He specialized in problems of combinatorial analysis, number theory, coding theory, and communications. His game of pentomino inspired Tetris.
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.
Michael J. T. Guy 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, computer algebra, and the theory of polyhedra in higher dimensions. He worked closely with John Horton Conway, and is the son of Conway's collaborator Richard K. Guy.
In mathematics, tiny and miny are operators that yield infinitesimal values when applied to numbers in combinatorial game theory. Given a positive number G, tiny G is equal to {0||0|-G} for any game G, whereas miny G is tiny G's negative, or {G|0||0}.
David Anthony Klarner was an American mathematician, author, and educator. He is known for his work in combinatorial enumeration, polyominoes, and box-packing.
Chaim Goodman-Strauss is an American mathematician who works in convex geometry, especially aperiodic tiling. He is on the faculty of the University of Arkansas and is a co-author with John H. Conway of The Symmetries of Things, a comprehensive book surveying the mathematical theory of patterns.