This has taken some time, but I’m pleased to announce that the New York Times Numberplay blog, written by Gary Antonick, is going to feature a problem based on chapter 15 of my book, The Universe in Zero Words. My latest information is that his post will go up on Monday, October 8. It will include excerpts from the book (and possibly the complete chapter — I’m not really certain).
Chapter 15 is one of my favorite chapters, because it’s about non-Euclidean geometry — a kind of geometry that was discovered in the early 1800s, which is absolutely fundamental to three-dimensional topology (or the study of “alternate universes” with three dimensions, just like ours), and yet remains relatively little known to people who aren’t experts in mathematics. In some ways that is completely understandable. It is mathematically a little bit more difficult than high-school geometry. To figure out the distance between points, you not only need the “square root” button on your calculator, you also need the “log” or “ln” button. Wow! A lot harder!
Okay, so I’m being facetious. The thing about non-Euclidean geometry is that it really forces you to re-examine your conceptions of what geometry is all about, and how you know the things you think that you know. It’s a question that really goes beyond mathematics and into philosophy, because Immanuel Kant (remember him from Philosophy 101?) argued that Euclidean geometry was the example par excellence of “synthetic a priori” knowledge — in other words, absolutely sure knowledge about the universe that is arrived at by pure reason.
And guess what? Kant was wrong. Non-Euclidean geometry burst his bubble. (Of course, he was already dead by then.) In fact, in Chapter 15 of my book, I explain why Euclidean geometry is not the only “natural” geometry of the universe. If whales had invented geometry, they almost certainly would consider non-Euclidean geometry to be “natural.” And there would probably be a cetacean Immanuel Kant telling us that non-Euclidean geometry was synthetic a priori knowledge! (Or maybe the whales would be smarter than us from the beginning?)
Gary’s blog post and my book will give you a tiny introduction to this wonderful world of whale geometry (or to use the “correct” word, hyperbolic geometry). If you want more, go to my book, and if you want more than that… Well, I’ll send you in two possible directions.
One book you can read is Euclidean and Non-Euclidean Geometries: Development and History, by Marvin J. Greenberg, which is kind of textbook-y but still very accessible to college math students. The other is a truly unique book, The Shape of Space, written by Jeff Weeks, a MacArthur Fellow, a three-dimensional topologist, and (by the way) a graduate school classmate of mine. He took about two extra years to get his Ph.D., which was unheard of at Princeton, because he was writing this book instead of his dissertation. It’s intended to be an explanation at the high-school level of the main ideas of three-dimensional topology and why it’s such an incredibly cool subject. Here non-Euclidean geometry is only a tool, but an important one. Although it may be a bit dated (written in the early 1980s), the book is a one-of-a-kind labor of love and definitely worth reading.
P.S. For any visitors who have already read my book — please do take a look at Gary Antonick’s column anyway, because he puts a nice twist on the example in the book. I was not at all certain that it would be possible to turn this into a “recreational math” problem, but I think he has done a good job.
