It has been a few days since I last posted anything, but I have actually been writing like crazy. It has just been math this time. As I already mentioned, after writing a number of papers on some topic, I tend to get bored of proving the 73rd version of Lemma 3.7. Incidentally, did you know that apparently if you ask somebody to mention a random number between 0 and 100, a surprising amount of people say 37, followed by 73? Anyway, I used to work a lot on hyperbolic 3-manifolds, and then at some point I moved on. But a few months ago, I started discussing with some people about a problem—constructing interesting examples of hyperbolic mapping tori for mapping classes of surfaces with infinite topology—and I evidently decided about a week ago that it was time to write it up. It was not a conscious decision, but rather something that just happened. Or at least, that’s how it seems, because I have been writing like a maniac and had definitely not scheduled any specific time to do that.

In the process, I have been reminded of how much I like 3-manifolds. Okay, it might be that the 73rd proof of Lemma 3.7 seems less boring after 10 years of proving other Lemmas 3.7. Or it might be that I enjoyed the feeling of rust peeling off that specific part of the machinery. But I genuinely like 3-manifolds. I love that I can imagine them. That I can “see” them. Evidently, I can’t actually see them. Nobody can see a 3-dimensional torus, let alone a more complicated manifold. But one can still see them in one’s brain. And this brings me to the title of this rant.

When I was in high school, I wanted to build boats. Or bridges. Then chemistry got in the way, but this was only in the later spring of my last year. Before that, the idea was to become an engineer. At the time, to study engineering in Spain, one needed to know technical drawing—with paper and such. I guess, or hope, or assume, that now they do it all digitally. But actually, I’m not so sure if “hope” is the right verb, because I have always maintained that learning (or practicing) some technical drawing is one of the things that has helped me most in math.

Math drawings

As a kid, I used to draw what kids draw, but I was always totally unable to draw a human face, a figure, a hand, or a cat that didn’t look like a cartoon. What I could draw were buildings, squares, trees, windows, rooms, maps, floor plans of houses, and things like that. This largely relies on perspective and shading to give depth to the picture, and these are the kinds of things that come easily to me. Technical drawing, where you had to do things like drawing a 3D picture from the side views of some object, came really easily to me. I have always maintained that it trained my brain to “see” 3-manifolds. Although I shouldn’t exaggerate. At the end of the day, knots play a big role in 3-manifolds, but if I draw a knot, it’s pretty likely to be just a messed-up unknot.

Indeed, the pictures I draw when I do math are invariably rubbish. It’s always a surface, mostly of genus 2, but sometimes I go fancy and it has genus 3, with a bunch of curves on it. I doubt anybody else would see anything there unless I’m talking at the same time. In fact, I have sometimes wondered how some stranger would react if they went through the endless stack of identical pictures I leave on the table after working for a while. I assume they’d think I have some sort of mental disease, almost as if I had been copying the same sentence over and over:

Call me Ishmael. Some years ago - never mind how long precisely - having little or no money...
Call me Ishmael. Some years ago - never mind how long precisely - having little or no money...
Call me Ishmael. Some years ago - never mind how long precisely - having little or no money...
Call me Ishmael. Some years ago - never mind how long precisely - having little or no money...
Call me Ishmael. Some years...

But these pictures, always the same, help me see new things. Those “probably unknots” help me visualize whatever funky knot I’m fighting at the moment. The same picture can have lots of meanings. The curves can be on the top, layered, knotted, short, long, etc. They always look the same, but every time they are different. Probably because drawing them is a dynamic process. The picture is not a still life. If it were, I could save lots of paper: just print it on a titanium plate and stare at it every time I’m working. It doesn’t work like that.

Anyway, in my high school, we didn’t learn technical drawing, so I went for a while to a place where a guy gave classes to a small group of people. He explained, for example, how to use the compass and the ruler to draw a regular heptagon—something famously impossible. But when you did it, it looked really regular. I don’t remember how he did it. In any case, I can be sure that I was the only one who went there voluntarily. Everybody else was a few years older and had evidently failed the technical drawing exams at engineering school. Honestly, why else would they be there? I assume that they were actually wondering why the fuck was I there. Anyway, I had enormous fun drawing those 3D models. I’m going to see if I can find some of those exercises online.



P.S.: I guess it’s human nature that everybody wants to see themselves in their kids, and that’s why it made me really happy when the being got totally excited when she was 5 or 6 and I drew a dodecahedron for her. After I explained the picture to her, she could see it.