Happy birthday, John Conway!

Today marks the 77th birthday of one of the world’s most eminent mathematicians: John Horton Conway, currently Professor of Mathematics and John Von Neumann Professor in Applied and Computational Mathematics at Princeton University. Conway earns my admiration for countless cool contributions in many branches of mathematics.

Conway became interested in mathematics at a very early age: as a four year old kid, he could already recite the powers of two, and at the age of eleven his ambition was to become a mathematician.

A selection of the topics Conway has touched:

- Conway invented a number system called the surreal numbers, which form the largest possible ordered field (in some sense). Study of this system was motivated by mathematical games, which could be solved using the surreal numbers. Conway wrote the delightful book
*On Numbers and Games*about it. - During his research into mathematical games, he invented a whole bunch of new ones: Sprouts, Hackenbush, Phutball, Conway’s Soldiers, the Angel problem…
- One of the early and still celebrated examples of a cellular automaton, the Game of Life, is a creation of Conway, whose early experiments were done with pen and paper, long before personal computers existed.

- Conway’s 15 theorem states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers.
- Together with Michael Guy he established the classification of convex uniform polychora (4-dimensional analogues of polyhedra), discovering the grand antiprism in the process.
- Conway extensively investigated lattices in higher dimensions, and determined the symmetry group of the Leech lattice.

- In knot theory, there is a variant of the classical Alexander polynomial named the Alexander–Conway polynomial which is an invariant for knots. He also developed the beautiful tangle theory, which built a bridge between knot-like structures and fraction arithmetic.
- Conway played a major role in the classification of finite simple groups. He discovered the three sporadic Conway groups, based on the symmetry of the Leech lattice, and was the primary author of the ATLAS of Finite Groups.
- Without hesitation, the most famous finite group is the Monster group. Together with Simon Norton, Conway formulated a conjecture relating this gargantuan group with modular forms, and labeled it “monstrous moonshine”.

- He extended the Mathieu group to the Mathieu
*groupoid*and presented it as a sliding tile puzzle played on a projective plane. - Conway proposed the Turing-complete esoteric programming language FRACTRAN, in which a program is an ordered list of positive fractions together with an initial integer input value.
- Ever heard of Conway’s icosian numbers? They’re a specific set of quaternions and exhibit lots of symmetry.
- Conway’s doomsday algorithm can be used to calculate days of the week. The story goes Conway’s computer isn’t protected by passwords, but by a quiz of random dates, in order to improve his mental arithmetic speed.
- Together with Simon Kochen he proved the free will theorem. In Conway’s own wording, the theorem states that “if experimenters have free will, then so do elementary particles”.

- The LUX method is an algorithm to generating magic squares.
- Conway introduced and analyzed the look-and-say sequence and proved the Cosmological Theorem: every sequence eventually splits into a sequence of “atomic elements”, finite subsequences that never again interact with their neighbors.
- As a spectacular counterexample to the converse of the intermediate value theorem, Conway defined the monstrous discontinuous base 13 function, which takes on every real value in each interval on the real line.

- Conway’s criterion gives a simple but powerful sufficient criterion for a prototile to tile the plane.
- Pinwheel tilings are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations, and were based on a construction due to Conway.
- The thrackle conjecture is an open problem in graph theory.