RIP John Conway

[lpetrich]

 John Horton Conway 26 December 1937 – 11 April 2020)

COVID-19 Kills Renowned Princeton Mathematician, 'Game Of Life' Inventor John Conway In 3 Days | Mercer Daily Voice

He was a British mathematician who did work in a variety of arcane fields, and he also worked in recreational mathematics, inventing his cellular-automaton Game of Life. For instance, in group theory:

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.[39] This work made him a key player in the successful classification of the finite simple groups.

I'll describe a tiny bit of his professional mathematics.

A group is a set S with binary operation S*S -> S where that operation is associative, has an identity, and has an inverse for every element. The operation need not be commutative, but if it is, that makes a commutative or abelian group. Addition of integers is a group, for instance. Addition of nonnegative integers has only the first two properties, making it a "monoid" that is not a group. Addition of positive integers has only the first property, making it a "semigroup" that is not a monoid or a group.

Let us consider what groups are known to exist. All the finite abelian groups are known: products of cyclic groups with power-of-prime order. Cyclic group Z is addition of integers modulo n. Infinite groups and finite nonabelian (noncommutative) groups are much more difficult, but the problem has been solved for finite "simple" groups, those that cannot be broken down in a certain way. It took a massive effort spread over some 10,000 journal pages, but it has been done. Prime-order cyclic groups are the only simple abelian ones, and there are several infinite families of nonabelian ones, with 26 additional "sporadic groups". John Conway had discovered three of those 26 groups.

 Conway's Game of Life is a cellular-automaton system with lots of interesting properties.

Its world is a rectangular grid that is stepped forward in time. Each grid cell can be alive or dead. Any dead cell with exactly three neighbors becomes a live cell, and otherwise stays dead, and any live cell with two or three neighbors stays alive, and otherwise dies.

John Conway arrived at these rules by trial and error, hoping to find rules that gave interesting behavior. He succeeded.

For a finite board, every possible configuration will eventually give a fixed point or a limit cycle. A trivial fixed point is all dead, but there are several nontrivial ones that are known to exist. Several limit cycles are also known to exist, like oscillations in place and motion of the pattern. Though period 2 is the most common one to emerge from random initial conditions, several others have been observed, like 3, 4, 8, 14, 15, 30.

Among them is a "glider gun", a pattern that periodically emits traveling patterns. There is even a sort of pattern that emits glider guns.

Patterns can interest with other patterns, and one can make logic gates with them, doing operations like "not", "and", and "or". It's possible to make a finite-state machine that is connected to two counters, and that construction is Turing-complete. Several different programmable architectures have been simulated, including a version of Tetris.

There is a lot more that I could say, and many kinds of  Life-like cellular automaton have also been studied.