If you like geometric group theory or amazing pictures (but especially geometric group theory), you might want to start reading Geometry and the Imagination, written by University of Chicago mathematician Danny Calegari. I’ve been following it for a while, but I got inspired to write about it here by a recent post on some new software he wrote, kleinian. Logically enough, it is a tool for visualizing Kleinian groups. And isn’t this visualization beautiful?
The universal cover of a genus 3 handlebody, visualized using the program kleinian. Image: Danny Calegari.
If I had to sum up the blog in a sentence, I’d say that Geometry and the Imagination contains expository posts about hyperbolic geometry and geometric group theory written for a mathematically sophisticated audience, with a few flights of fancy (like this post on solving the Rubik’s cube) thrown in. Like many research-oriented blogs, it delves into many of the technical details of theorems and proofs but with a lot more helpful “big picture” signposts than most research articles have. For me, at least, as a grad student and early-career researcher, seeing the forest for the trees has been the most difficult part of starting to do research, so I am a fan of blogs with signposts.
A few years ago, Calegari’s student Alden Walker wrote a series of posts containing notes from Calegari’s hyperbolic geometry course. (Calegari and Walker were both at Caltech at the time. Walker is now a postdoc at the University of Chicago.) As I mentioned in my last post, I find notes like this very helpful because even when I know a subject, I haven’t always thought deeply about what order to present it in, what examples to use, and so on.
One fun post is on Kenyon’s squarespirals:
“The other day by chance I happened to look at Richard Kenyon’s web page, and was struck by a very beautiful animated image there. The image is of a region tiled by colored squares, which are slowly rotating. As the squares rotate, they change size in such a way that the new (skewed, resized) squares still tile the same region. I thought it might be fun to try to guess how the image was constructed, and to produce my own version of his image.”
In the rest of the post, Calegari moves from square tilings of rectangles to square tilings of a torus to arrive at an image that is quite similar to Kenyon’s original. Aesthetically, I prefer this intermediate gif, of rotating squares that do not spiral, so I’m including it instead.
Squares that rotate but do not spiral. Image: Danny Calegari.
If heavier math is your thing, Calegari has a three–part series on Ian Agol’s proof of the virtual Haken conjecture and a post on his and Walker’s result that random groups contain surface subgroups.
But my favorite post is Random Turtles in the Hyperbolic Plane. With a title like that, how could it not be? In it, he riffs on his daughter’s Logo programming project. She programmed a turtle to take a random walk in the Euclidean plane, so he looks at what happens when turtles take random walks in the hyperbolic plane. Hyperbolic turtles display some very interesting behavior, with a “phase transition” from a beeline to the boundary to something that really “looks like” a random walk, depending on the how size of each step compares to the size of the turn the turtle makes each time. It’s a surprisingly rich line of inquiry!
The walk of a random turtle in the hyperbolic plane. Image: Danny Calegari.