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.
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:
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:
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:
In fact, we can even create a setup that generates gliders over and over again, forever:
I think the fascinating success of this "Glider Gun" speaks for itself.
For even more about spaceships and looping patterns and other aspects of the Game of Life, visit this article.
So what does this all mean? From our perspective, this seems like a super recreational bit of math. It's fun but it's hardly practical. However, this game may be more applicable than you realize. The Game of Life is part of a larger class of systems called cellular automata systems that consist of a grid of cells that cycle through states (like the "alive" or "dead" here). These systems all have different rules and some even cycle through more than two states (perhaps three or four different colors).
You may have realized the rules for the Game of Life are somewhat arbitrary. After all, who is born from three parents? But the idea of death by too few neighbors (it's difficult to get food or water if you're isolated) and the idea of death by too many neighbors (overcrowding makes resources scarce) are real concepts that carry over into the real world. The cellular automata systems as a whole can model aspects of our lives—everything from the patterns of seashells to plants' intake of gases to computer processors can modeled by sets of rules in cellular automata. If you want to learn more about the applications of cellular automata systems, feel free to visit this Wikipedia article.
I think the reason the Game of Life is so appealing is it shows just how complex and beautiful an incredibly simplified version of life is—and in turn, reveals just how much more complex and beautiful life itself may be.
So join the popular nerds of the 1980s! Try the Game of Life yourself. You can very rudimentarily test it by making a grid in Excel (like I did to make these diagrams) or visit https://playgameoflife.com/ to try it on a larger scale or—if you're even more ambitious—program it yourself! There are way more cool ideas and aspects that come from this game that I could possibly hope to explain in one article, so see what you can discover and share it with me in the forums!
Now, I know John Conway would not want me to focus an entire article about him on the Game of Life. It was, after all, something fun he created on the side that he didn't expect to gain quite as much popularity as it did. His work encompasses way more than this game (however fun and exciting it may be), and I definitely want to mention a couple more things—though even given a lifetime, I could never even hope to understand all John Conway achieved throughout his life in so many different fields.
Conway himself was most proud of his surreal numbers, a system of numbers that actually went beyond the real numbers to include concepts like infinities and infinitesimals as concrete numbers. As abstract as they sound, surreal numbers actually have applications in game theory, a field of math that investigates strategies to win, lose or maximize your odds in all sorts of different games from checkers to dominoes to dice to any game you can think of! I often find that game theory is one of the most fun and practical fields to explore, and we even used some game theory principles when we discussed the Prisoner's Dilemma in one of my previous articles.
Surreal numbers themselves don't apply to every part of game theory (it's a very large field after all!) but they certainly made their dent, and they're a super fascinating concept to explore. I'd love to explain them in another article sometime, but it takes a lot, so for now, I'm just going to give you some resources:
John Conway's 24-dimensional symmetry object, the Conway group, is probably one of his most impactful discoveries:
John Conway even released the Monstrous Moonshine conjecture, a paper about a symmetry group with 196,883 dimensions, early in his career. Symmetry groups are a very abstract area of math, but they're definitely one of the areas where research is cutting-edge right now. Later in his career, John Conway wrote a book with four other mathematicians called "The ATLAS of Finite Groups" which became one of the most important books in group theory.
Some other amazing achievements of Conway's include the Free Will Conjecture (it turns out not just humans but also elementary particles have free will!), the design of a water-powered computer Conway named Winnie (Water Initiated Nonchalantly Numerical Integrating Engine), and the creation of a game called Phutball (a checker-like game called Philosopher's Football), and that's not even mentioning hundreds of groundbreaking theorems!
To wrap up this article, all I want to say is that John Conway will be dearly missed. He was a blessing to the math community and such an influence on all of us. He's gone too soon, and it's truly tragic that his death was caused by the current state of events. I learned quite a bit about him at Georgia's Governors Honor's Programs (GHP) last summer, and it's been so fascinating to see his work and watch videos about him online. He's made an immense impact on the math world, and his creative genius will be felt for many years to come.
Read the New York Time's obituary: https://www.nytimes.com/2020/04/15/technology/john-horton-conway-dead-coronavirus.html
If you watch one video about John Conway, watch this one:
Here's another great video. (It turns out Conway was a tongue gymnast!)
I'd also recommend these two videos about the Game of Life:
And this memorial podcast to Conway's life:
There's also a biography of John Conway out now called Genius at Play by Siobhan Roberts.
I hope you learned something interesting today, and I encourage you to do more research! Learn about the Game of Life, learn about the surreal numbers, learn about Conway's life... I'm always such a proponent of learning about the math and the mathematician, and one way to keep Conway's memory alive is by learning a bit about the fun ideas he loved to explore himself. Have a great week and see you next Sunday!
Cover Image Courtesy of Dith Pran from the New York Times