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Home >Science >News >Opinion | John Conway: Life in the gliding lane, with cats

John Conway seems to have had fun with right-angled triangles, or at least one right-angled triangle. Such triangles, as you perhaps know, are fodder for Pythagoras’ famous Theorem, which says that if you add the squares of the two sides forming the right angle, you get the square of the hypotenuse, the side opposite the right angle. Or, if we call the three sides a, b and c (the hypotenuse):

a2 + b2= c2

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So anyway, Conway was once playing with the right angle triangle whose sides are 1, 2 and the square root of 5. (Check: 12 + 22 = 1 + 4 = 5 = (Ö5)2.) He discovered that he could cut the triangle into 5 “similar" triangles—meaning that each one has sides in the same 1:2:Ö5 ratio. Which of course meant he had to subdivide each of those 5, and each of the 25 that resulted, and then each of the 125 little wedges … in this way, he ended up tiling the plane with this particular triangle.

Tiling the plane, perhaps you will remember from an earlier column (Tiling the plane, published on 24 September 2015), is an area of serious mathematical inquiry. The idea is to find a shape, a polygon preferably, identical copies of which can be laid out so as to cover any large flat surface without leaving any holes. In Conway’s little discovery with this right triangle, there’s nothing intrinsically remarkable. After all, mathematicians already knew that any triangles at all can be used to tile the plane. But what was intriguing about this particular tiling was that these triangles appear in every possible orientation. That is, if you think of the triangle as a warped arrowhead, you’ll find that in the plane it tiles, triangles point in every direction on the compass. The result is an arresting but strangely pleasing image. You think you see patterns, and your eye tries to make sense of them, but they are abruptly interrupted. Is that a larger version of the triangle, you wonder. But one corner is twisted at an odd angle, so no. Does that line stretch across the whole surface, you ask. But over there it suddenly disappears. On and on in that vein.

Not that this tiling is Conway’s best-known work, nor even particularly representative of work. But it does offer a sense of Conway’s spirit, his approach to mathematics. From all accounts, he was constantly playing with shapes, objects, mathematical structures both theoretical and tangible, coaxing mathematical treasures out of them bit by bit. Don’t take my word for it, either. Consider instead some lines from a blurb about his biography, Genius at Play. Referring to when he moved to Princeton University, it says “he deployed cards, ropes, dice, coat hangers, and even the odd Slinky as props to extend his winning imagination and share his mathy obsessions with signature contagion."

So if you had to ask a group of mathematicians and computer scientists who in their community was likely to be playing with slicing and dicing the Ö5 right triangle, I think it’s likely a number of them would say “John Conway". That’s the man he was.

“Was", because on April 11 John Conway died, three days after being infected with the coronavirus. He was 82. He will be mourned and missed. For they don’t make mathematicians like him much anymore.

There is plenty of mathematics and science that Conway took an interest in and made substantial contributions to. There are surreal numbers, for example: they include what we call the real numbers, but also both infinites and infinitesimals, numbers greater or smaller, respectively, than all the positive real numbers. (How is that possible? Mathematically, it is, but we’ll have to leave it there). He did a lot of work with the “monster group", called that because it has no less than 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 members. Just by the way, it also has 196,883 dimensions—compare to the three you are familiar with and give some thought to what it must take to grasp even the idea of nearly 200,000 dimensions. In particular, he discovered a connection between this group and what mathematicians call “modular groups". He promptly labelled that link “monstrous moonshine". In 2006, he and a colleague proposed their “Free Will Theorem". If you and I and other random human beings have a free will, it asserts, then so do the fundamental particles of physics: fermions and the Higgs boson and others.

Intriguing and tantalizing, but too much, too complex, to think of explaining in a column like this.

But Conway is certainly best known for his invention of something much more approachable, that became a cult favourite among computer scientists: his Game of Life. Despite the name, this is not a game in the sense of tennis or chess. Let me paraphrase here my explanation of Life from a 2016 column:

Life happens on a grid of squares that stretch in every direction (imagine an infinite chessboard). Each square, or cell, is either alive or dead (or call it black or white, filled or empty). To start, you choose a certain number of cells to be live —at random, or in a pattern, whatever—and fill those in, maybe with a pencil. Each cell now evolves according to a set of rules that considers its eight neighbours (left, right, above, below, and on the four corners).

* If it’s a dead cell with exactly three live neighbours, it springs to life. Think of this as reproduction (never mind the three parents). Other dead cells stay dead.

* If it’s a live cell that has less than two or more than three live neighbours, it dies. Think of starving if you have too few neighbours, or suffocating if you have too many.

* If it’s a live cell that has two or three live neighbours, it lives on.

Subject each cell to these rules—that’s one iteration—and obviously you change the state of the game. Then apply the rules again, and yet again, on and on, iteration after iteration. You will see the patterns on the board change in different ways. Run it as an animation—and as you watch, as the patterns roam across the board, as they mutate or even disappear, you might swear they are actually alive. There are even some patterns that seem to morph into copies of themselves; one, the famous “glider", replicates itself one square over after four steps, so if you run the iterations rapidly, you’ll see it shoot across the board. Conway’s colleague and friend Richard Guy discovered the glider, and he died on 9 March. That’s two stalwarts of Life, dead just weeks apart. (The 2016 column I mentioned above was actually a doff of the hat to Guy on his 100th birthday— Gliding past a century, published 21 October 2016).

Gliders and mutation, movement and death, from just a few squares you might shade with a pencil. Not for nothing did Conway call his invention “Life".

It’s fun to watch and play with, certainly. But this being a Conway artifact, it’s also much more than a game. It’s an example of a cellular automaton (CA), a concept familiar to computer scientists. Think of a CA as a basic but complete computer. All you need to do, with Life as a CA, is appropriately define and interpret the patterns that appear on the board as the game progresses. Given time, then, it will do everything that computers you know of—your cellular phone, for one—can do.

There’s so much more to explore here, really. Life’s like that, to coin a phrase. But mathematicians will find irony in Conway’s life being taken by a virus. Fundamentally self-replicating in its behaviour, fundamentally mathematical in its spread: corona comes with—for anybody who’s seen the game and played with it—inescapable reminders of Conway’s Life. It’s sad to lose this remarkable mind, no doubt. But there’s a hint of what he meant to mathematicians in a joke that’s making the rounds: Seems John died because he had either less than two or more than three mathematicians as neighbours.

I have no doubt he’d have laughed at that joke. After all, this is a man who once said: “You know, people think mathematics is complicated. Mathematics is the simple bit. It’s stuff we can understand. It’s cats that are complicated."

Go well, John Conway. Thank you for Life, for the mathematics, and, yes, for how I now think of my cat.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun