I was happy to stumble upon the new book on John Conway’s life and work Genius at Play while I was visiting the Museum of Mathematics in New York City. By the way, the museum is highly recommended. As I like to travel light, I did not buy the hard covered book, but instead bought it here in Toronto. The book was in a highly visible place near the check out at Indigo. How refreshing to see a book about a mathematician upfront at a big chain bookseller!
Conway, who’s academic home is Princeton University, is an expert on many mathematical things. Most mathematicians work on small, isolated part of the giant cliff face of mathematics. A true polymath, Conway is one of the few whose work spans the gamut of mathematics. Conway has made major breakthroughs in number theory, group theory, game theory, combinatorics, and cellular automata.
Two of his most famous accomplishments are the Game of Life, and the discovery of the Monster simple group. I explain both briefly in simple terms.
The Game of Life is a certain kind of cellular automata. Think of an infinite grid whose cells are alive or dead. There are simple rules governing how cells live or die. The rules are:
- If the cell is alive, then it stays alive if it has either 2 or 3 live neighbors. Too few neighbours (isolation) or too many (overcrowding) results in a dead cell.
- If the cell is dead, then it becomes alive only in the case that it has 3 live neighbors.
The incredible thing about the Game of Life is that such simple rules lead to incredibly complex behaviour. In fact, the Game of Life can simulate a universal Turing machine, so essentially can be used to simulate any algorithm. Here is an example of how Life looks like:
Groups are algebraic objects which measure symmetry. A major undertaking in 20th century mathematics was to classify the finite simple groups. A group is simple if it has no normal subgroups. Think of simple groups as the building blocks of all groups like primes are for integers, or like the periodic table of elements making up all matter. While there are infinitely many non-isomorphic simple groups, mathematicians found a classification of all of them. The proof of this fact took decades and thousands of pages of works by many mathematicians. Conway’s contribution was to the discovery of one of the so-called sporadic simple groups, now called the Monster. The Monster has monstrous size! It contains
elements! One of its representation is in 196,883-dimensional space. Although it is the not the largest finite simple group (remember, there are infinitely many of these), it is the largest sporadic group.
My own personal experience with Conway was back in 2000 at the Southeastern Combinatorics Conference in Boca Raton, Florida. Back then, I felt more like a graduate student than a professor (I graduated with my doctorate in 1998). I didn’t really know who Conway was at the time, but I was lucky to have attended the lecture. His lecturing style was one of the most memorable I have seen (and I have seen hundreds of talks by mathematicians). He had no prepared slides for his keynote talk, but just the roll-able transparencies… remember this was 2000, and most lecturers were not using their laptops or Beamer or Powerpoint.
Conway began by defining lexicodes, which are certain greedily defined lexicographically ordered codes over finite fields, and gave some examples. For one hour, he told us the story of what lexicodes are and how they fit into the grand scheme of things. It was a remarkable lecture. Without notes, without prepared slides, he effortlessly glided through the material as if he were chatting to a small group of friends.
I look forward to reading Robert’s biography. By all accounts, her writing is top-notch. Maybe a movie deal about Conway’s life is not far off?
Here is a wonderful interview with Conway. He espouses the view that you should follow your passion and do what you are interested in. I love the discussion at the end where he muses about how no one really knows why the Monster exists.