A Tribute to John Conway and the Game of Life

Updated: May 31, 2020

Hi everybody! I hope we all had a good spring break despite everything that's going on. I'm back with a new article, and today, we're paying tribute to a legend in the math world and some of his incredible mathematical achievements.

Last Saturday, April 11, John Conway, a professor at Princeton and one of the most creative mathematicians ever in the field, died of the coronavirus. His ideas were so revolutionary—and so new—and his method of thinking was so unique that Conway's good friend and mathematician Simon Kochen often called him a "magical genius." He was 82 years old.

Dr. Conway earned many prizes throughout his life for his large body of work ranging from number theory to game theory to topology to combinatorics and more. We'll talk a lot more about his achievements (and what some of those words mean) throughout this article, but for now, just know that Conway was an incredibly versatile mathematician and a character at that.

There are so many stories of Conway's eccentricity. He delighted in the idea that math should always be fun and playful, and his lectures were always engaging and exciting; in fact, he often didn't choose a lecture topic until the day of, going with whatever his students were most excited about. He loved fun demonstrations and once took a knife to a turnip to show students how to slice an icosahedron, and of course, he ate all the scraps as he went.

An icosahedron (a 20-sided regular solid with triangular faces) and some turnips. Photos courtesy of ResearchGate and FloriCulture Greens.

As for Conway's career, his most famous discoveries are a game called the Game of Life, a new type of number called "surreal numbers", and an object known as the Conway group, a structure in 24-dimensional space! He, like his friend Martin Gardner who made his career writing recreational math columns for the Scientific American, loved to bring his ideas about math to the masses, and the Game of Life was a major success! One estimate says that at the game's peak of popularity, one-quarter of the world's computers were playing it.

So what is the Game of Life? No, it's not the brightly colored Milton Bradley board game, but instead a set of rules that regenerate a starting position into complex patterns. This isn't a game that you play but that you watch, and it's the excitement of programming it and discovering how it works that made it so popular.

Start with a grid and enter some kind of shape into it, like so:

Then, follow the following ruleset. Consider the 8 neighbor cells to any cell (up, down, right, left or diagonal). You consider the cells in your shape itself to be "alive" and all the remaining cells to be "dead" as the game continues.

1. If an alive cell has zero or only one neighbor, it dies of loneliness.

2. If an alive cell has four or more neighbors, it dies of overcrowding.

3. If an alive cell has two or three neighbors, it is stable and it survives.

4. If a dead cell has exactly three neighbors, it is born and becomes an alive cell.

Let's see what happens to our starting setup. The two cells on the ends are lonely, so they die. The cell in the center is stable, so it survives. There are also two cells directly above and below our setup that have exactly three neighbors, so they are born. This gives:

This setup switches (or if you want to be fancy, oscillates) between these two positions, back and forth and back and forth. In fact, it's often called the blinker:

Blinker. Courtesy of Wikipedia.

Can you use this idea to explain what happens when we put in a rectangle of length 1 but of any width? What if we start with a block of 4 instead of a block of 3? A block of 5? What if the width is no longer 1? See if you answer some of these questions (and some of the ones to come!) in the forum.

Let's see what else we can discover about our ruleset. Are there any shapes that don't change at all? Here's two:

What conditions do these setups have to meet? Well, for all the alive cells to remain living, they got to have exactly two or three neighbors. For all the dead cells to remain dead, none of them must have exactly three neighbors. We see that these two setups meet those conditions. Can you come up with any more? (Hint: what does a beehive look like?)

The thing that makes the Game of Life so fascinating is most shapes (even basic shapes) that you input become fascinatingly complex patterns. Here's one:

It takes 12 steps to get from our starting position to the final two positions, which end up alternating back and forth forever (notice the final two positions are simply a variation of our first block of 3). At this point, you should have some kind of idea of the complexity and beauty of the Game of Life.

Let's look at some animations to see if we can get an idea of how incredible the game is. Here's one example animation:

Game of Life man. Gif courtesy of xkcd and New York Times.

And another:

Large animation. Courtesy of Lucy Yu.

There's even one particular well-loved example of the beauty of the Game of Life in this simple figure, called a glider. This figure travels forever down and to the right: