John Horton Conway is one of the greatest living mathematicians. Born in England, he has lived in the USA since 1986. He has contributed a huge number of new and innovative ideas to many different branches of mathematics, including knot theory, finite group theory, number theory, coding and the geometrical study of crystalline lattices.

**Biographical Sketch**

Conway was born in Liverpool in 1937^{1}. From an early age, he showed a penchant for mathematics; by the age of four, he could already recite the powers of two, and when interviewed at age 11 for entry into secondary school, he said he wanted to be a mathematician when he grew up. He studied mathematics at Gonville and Caius^{2} College, Cambridge, receiving a BA in 1959 and a doctorate in 1964.

From 1964 onwards, Conway worked as a lecturer in Pure Mathematics in the University of Cambridge, and continued his research into the subject. Conway's first major published work was in 1968: a study of the symmetry group G of the Leech lattice, a 24-dimensional close packing of spheres. He found a way of simplifying the system and tidying up the untidy world of group theory. Conway's work was new and unusual, so he was immediately recognised as an important name in the mathematical world.

Conway's fame spread to the amateur mathematical and computer scientist communities in 1970 when Martin Gardner published an account in *Scientific American* of Conway's newest invention, the *Game of Life*, a type of cellular automaton that appears to mimic real life itself.

As Conway progressed through his career, he received promotion within the University of Cambridge: first as Reader in Pure Mathematics and Mathematical Statistics (1973) and then Professor of Mathematics (1983). He also received fellowships from two Cambridge colleges - Sidney Sussex (1964) and Gonville and Caius (1970) - as well as from the Royal Society (1981).

In 1986, Conway took up the post of John von Neumann Chair of Mathematics in Princeton, New Jersey, USA, and has lived in Princeton since then. He has been married a number of times and has a large number of children.

**Description**

Conway's most important mathematical work involved simplifying the world of group theory. His new groups brought together many previously unclassifiable groups. This extreme neatness in the mathematical world is a sharp constrast to Conway's untidiness in real life - he's a hairy person with an enormous tangled beard. He has a reputation for surrounding himself with clutter and for losing stuff.

Conway is a brilliant speaker and a showman. He loves to play games - he's a mean backgammon player. But games are not just for relaxation; Conway's thoughts on games have led him on to many mathematical discoveries. His Surreal Number system was developed from a study of the Oriental game of Go. He also creates new games with a mathematical flavour - Sprouts is one he created with student Michael Paterson.

**Some of Conway's Work**

**Mrs Perkins's Quilt**

One of Conway's first published works was an analyis of a problem presented by the great puzzler of the 19th Century, Henry Dudeney. This concerns how to arrange a number of squares of different sizes to make a larger square. While this sounds trivial, Conway published it under that title in a respected British mathematical journal.

**Group Theory**

Conway's advances in group theory are what made him famous in the world of mathematicians, but they are beyond the scope of this Guide to explain. Simply, a group is a collection of objects and rule for operating on them which obeys certain conditions. Groups are very important in the study of number systems. They are one of those things in mathematics which appear so blindingly obvious that they are hardly worth stating, but some very complex mathematics is built on top of this plain foundation.

**Sprouts**

Invented by Conway and a student, Michael Paterson, in 1967, Sprouts is a pen and paper game in which two players take turns drawing lines and spots on a sheet. Each player tries to block the other from being able to play.

The rules are simply stated. Before the game starts, a few spots (typically four) are drawn on the paper. Now each player takes turn. The player must join two spots with a curvy line, or join one spot to itself using a loop. The line is not allowed to go through or touch any other spot, or to cross any other line. There can't be more than three lines coming out of any spot. After drawing the line, the player must draw a new spot on the line. Then it is the other player's turn. When one player can't move, the other player is the winner.

**The Game of Life**

Conway devised the Game of Life, which is really more of a simulation than a game, based on an idea by von Neumann. It was published by Martin Gardner in his Mathematical Games column in *Scientific American* magazine in 1970. It took the mathematical world by storm; here was something easy to understand for the amateurs, which nevertheless warranted a serious study by the professionals. Every computer programmer tried to automate the game and competitions were held to find the most interesting patterns. The theory of cellular automata is a development of the basic ideas of the game.

The Game of Life is too big a subject to be treated fully in this Entry. Simply stated, it attempts to simulate a primitive form of life on a grid of squares similar to a chessboard, but without the black-and-white colouring and with no limit on the number of squares. Each square may be occupied by a live 'cell' or may be empty. Simple rules are used to determine which cells will die and which empty squares will give birth to new cells; all births and deaths take place together, then there is a pause before the next cycle of life and death takes place. As the game progresses, the pattern of cells changes, sometimes growing, sometimes dying away. Some 'organisms' known as 'gliders' are mobile and move across the grid. Others stay still but shoot out a continuous stream of gliders.

As well as providing endless fascination for the amateurs, Life has a serious side. Conway showed that certain simple patterns can be used to perform basic arithmetic functions such as adding and subtracting. On top of this he was able to build all the components necessary to make a Universal Computing Machine^{3}.

**Surreal Numbers**

Conway invented a new way of defining numbers based purely on the concept of greater than and less than, which produced a superset of the real numbers. Author and computer scientist, Donald Knuth, coined the name 'surreal numbers' meaning 'on top of the real numbers' for these and the name stuck.

A surreal number is defined as two sets of numbers; every element in the first set is less than the number, while every element in the second set is greater than the number. By providing appropriate rules for addition, multiplication, negation, etc, Conway was able to show that certain surreal numbers are exact analogues to the real numbers we know so well. Others behave like infinitely large numbers (similar to the Cantor ordinals); infinitely small non-zero numbers (the infinitesimals so beloved of Isaac Newton as he tried to put his invention of 'the Calculus' on a sound mathematical footing); or numbers infinitely close to existing real numbers without actually being the real number. In this way, the surreal numbers are a superset of the reals: every real number is also surreal, but there are surreals which correspond to no real number, yet can be manipulated by the rules of addition, multiplication and so on. For the first time ever, we can ask whether infinity is an even or odd number and at least expect an answer^{4}.

**Knot Theory**

A normal knot is a twist of one or more ropes. A mathematical knot is not that different - it is a curved line which is tangled around itself and then twisted so that the ends meet to form a loop. Like the continuous loops of Celtic knotwork, a mathematical knot has no ends. It is a topological entity: you can bend it, stretch it, any way you want to, as long you don't cut it. It is the way in which it is tangled around itself that matters. Two apparently different tangles may turn out to be the same knot after manipulation by tucking a loop in here, putting a twist there and so on. Mathematicians would like some procedure which can be applied to two knots which will prove either that they are the same or different. This is known as the 'comparison problem' and it is the most important one in knot theory.

One approach to this problem is to define a 'knot invariant': a mathematical description of the knot which does not change when simple topological transformations are done to the knot. If two knots have different invariants, then they must be different knots, although they're not necessarily the same knot if they have identical invariants. In 1923, a mathematician called JW Alexander came up with such an invariant, the Alexander polynomial, but it was very difficult to calculate.

Conway made a major step forward when he invented a very simple way of coming up with a knot invariant. The Conway polynomial is formed by changing a knot into two simpler knots by a process of cutting and re-attaching at a particular crossing. This process is repeated at different crossings until the knots are simple enough that their polynomials are known: the plain loop has a polynomial of 1 and all others can be got by combining and multiplying by x at appropriate points. Conway's polynomial is simple enough that non-mathematicians can easily understand it, although the mathematical proof that it all works the way it is supposed to is beyond the amateur.

The ultimate goal is an invariant that is always different for two different knots. Conway's polynomial is a long way from this, and was pushed out of the limelight in 1984 when Vaughan Jones produced a much better invariant. Nevertheless, Conway's polynomial was a simple but effective step in the right direction. There is still much work to be done on the comparison problem.

**Work on Tessellation**

Conway did important work with Roger Penrose on his kite and dart tessellations^{5}. Penrose invented two tiles, in the shape of a kite and a sort of boomerang; armed with endless supplies of just these two tiles, a tiler can completely cover the infinite plane in a pattern which is irregular and never repeats itself. Conway and Penrose together managed to prove that this can be done in an infinite number of ways.

**The Doomsday Algorithm**

No, it's not anything to do with the end of the world, it is a method for calculating the day of the week. Conway noticed that the five dates 4/4, 6/6, 8/8, 10/10 and 12/12 (4-Apr, 6-Jun, 8-Aug etc) are all on the same day of the week. He developed a simple algorithm based on this which will tell you what day of the week any particular date is.

OK, computers can do this for you in a flash, but Conway's Doomsday algorithm is easy enough that you can do it in your head; he even includes a mnemonic based on the fingers of your left hand.

**Conclusion**

Here we have a mathematician who appears to be interested in everything. Acknowledged by amateur and professional alike as a world-class mathematician, Conway continues to produce novel approaches to problems and to see more clearly than those around him. He worries sometimes that his work on games might be considered trivial and might overshadow his more serious work, such as his contributions to group theory. Considering how many different fields that he has contributed to, this seems unlikely.