Folks who’ve been with me a long while know one of my happy Christmastime traditions is watching the Aardman Animation film Arthur Christmas. The film also gave me a great mathematical-physics question. You should watch the movie, but you might also consider the questions it raises.
First: Could `Arthur Christmas’ Happen In Real Life? There’s a spot in the movie when Arthur and Grand-Santa are stranded on a Caribbean island while the reindeer and sleigh, without them, go flying off in a straight line. What does a straight line on the surface of the Earth mean?
Second: Returning To Arthur Christmas. From here spoilers creep in and I have to discuss, among other things, what kind of straight line the reindeer might move in. There is no one “right” answer.
Third: Arthur Christmas And The Least Common Multiple. If we suppose the reindeer move in a straight line the way satellites move in a straight line, we can calculate how long Arthur and Grand-Santa would need to wait before the reindeer and sled are back if they’re lucky enough to be waiting on the equator.
Fourth: Six Minutes Off. Waiting for the reindeer to get back becomes much harder if Arthur and Grand-Santa are not on the equator. This has potential dangers for saving the day.
Fifth and last: Arthur Christmas and the End of Time. We get to the thing that every mathematical physics blogger really really wants to get into. This is the paradox that conservation of energy and the fact of entropy seem to force us into some weird conclusions, if the universe can get old enough. Maybe; there’s some extra considerations, though, that can change the conclusion.
I have several times taught a class in a subject I did not already know well. This is always exciting, and is even sometimes fun. It depends on how well you cope with discovering all your notes for the coming week are gibberish to yourself and will need a complete rewriting. One of the courses I taught in those conditions was on digital signal processing. This was a delight, and I’m sorry to not have more excuses to write about it. In the Summer 2015 A-to-Z I wrote about the z-transform, something we get to know really well in signal processing. The z-transform is also related to the Fourier transform, which is related to Fourier series, which do a lot to turn differential equations into polynomials. (And I am surprised I don’t yet have an essay about the Fourier transform specifically. Maybe sometime later.) The z-transform is a good place to finish off the spotlights shone on these older A-to-Z essays.
For a while there in grad school I thought I would do a thesis in knot theory. I didn’t, ultimately. I do better in problems that I can set a computer to, and then start thinking about once it has teased some interesting phenomenon out of simulations. But the affection, at least from me towards knot theory, remains. In the Fall 2018 A-to-Z sequence I got to share several subjects from this field. One of them is the Yamada Polynomial, a polynomial-like construct that lets us describe knots. I don’t know how anyone might not find that a fascinating prospect, even if they aren’t good at making the polynomials themselves.
And for today’s revival I offer something from the 2017 A-to-Z. This is about maybe the most mathematical of possible subjects: x. This was a fun one as I got to get into the cultural import of a mathematical thing, which is right up my alley. This and other of the 2017 A-to-Z essays are at this link.
I’d like today to bring up something from the Fall 2019 A-to-Z. It’s a term which may seem unexciting, but it turns up all over the place. Wlog, short for without-loss-of-generality, is one of those phrases that turns up all over mathematical proofs. It’s usually difficult solving abstract problems. It’s usually less hard solving specific ones. Sometimes, we can find a specific problem that solves all of an abstract problem. Isn’t it wonderful when that happens? That and the other Leap Day 2016 A-to-Z essays are at this link.
There’ve been a few A-to-Z essays which felt like breakthroughs to me. One of them was in the Leap Day 2016 Essay. The essay posted in the middle of April, by the way; it got the title because the sequence started at the end of February. Vector was one of the breakthroughs. The obvious course is to talk about vectors as magnitude and direction, maybe as ordered sets of numbers. Then generalize to all the kinds of things mathematicians might describe as vectors. This time, I realized no: I could start from the general idea of vectors, and then mention how this covers things that would be familiar to people who read pop mathematics. It’s an expository style I’ve relied on since then and I’ve generally liked how it’s served me.
A lot of the fun of an A-to-Z is surprise. People will suggest topics that I wouldn’t have considered. Sometimes I’ll do a bit of preliminary research and find that a topic is more interesting than I guessed. Unit Fractions, from the 2018 A-to-Z, is one of those. A unit fraction looks, from the definition, to be too dull to bother with: what’s interesting about one divided by a whole number? A great deal, it turns out, and as I started writing I threatened to keep on writing.
Today I’d like to share one of the essays from the Leap Day 2016 A-to-Z. (I did two A-to-Z sequences in 2016, which was exhilarating. It seems like too much work, but only in retrospect, especially as they were spaced by more than half a year.) The one to share is Transcendental Number, which I like because I go to share cool stuff about transcendental numbers. Particularly the paradox that fascinates me so. Basically every real number is transcendental. The exceptions are the numbers we ever do anything with. There are a handful of numbers that are interesting and that we know to be transcendental numbers. Not many, though. If that doesn’t fire your imagination, well, maybe try my essay. It might make the case more fully.
One of the topics from the 2018 A-to-Z was suggested by a philosopher. There are many mathematical topics shared between the philosophy and the mathematics departments, including all of logic. The Sorites, or Heap, paradox is among them. There are traits that are true only of an accumulation of things, and that are not true of any of the parts of that thing. Where, then, does the trait come from?
So you know how sometimes you’ll do something as a quick little slightly snarky joke and then it haunts the rest of your life? So I mentioned the Ricci Tensor as just a thing, and then Elke Stangl asked me to write an essay about it. And then another friend, who makes YouTube videos about non-Euclidean geometry, mentioned he wished he understood the Ricci Tensor better since he needed a good way to explain what it represents and why it’s a thing worth knowing and he hadn’t found the way yet. And what does my essay feature? Except for the part about having a clear explanation for what it represents and why it’s a thing worth knowing. I tried, yes, but I felt shaky on the physical significance of the thing. And so does everyone else I can find who’s written any kind of pop mathematics, or pop physics, about the thing. So if you’ve found a good lay explanation for why we want to know about Ricci tensors please let me know. My essay could use a follow-up.
I was surprised, looking them over, to realize I had several Q essays I liked a good bit. There was only one that let me natter on about amusement parks, though, and particularly carousels. So please let me bring to your attention Quasirandom numbers, the rare essay with photographs of my own, and featuring the rare merry-go-rounds with a random element.
If there’s one A-to-Z essay I keep referring people to it’s the one about polynomials. So there’s no need to bring that up again. Instead I’ll pick up one from last year’s A-to-Z, about the Pigeonhole Principle. It’s one of those ideas so simple it hardly seems like anything. But it keeps implying weird and surprising and even counter-intuitive things. So that’s worth highlighting, particularly since some of the implications of the Pigeonhole Principle surprised me when I reread my own essay.
To pick out an essay to revive I looked through past A-to-Zs. There was one I realized I didn’t have any memory at all of writing. Which is odd as when I read it, I liked it. So here’s my pointer to it. The Osculating Circle may seem too simple a thing. It turned out to be more than I realized. Glad to see that sort of thing happening. Also to read an old essay that I’m not thinking of ways it should have been better.
The End 2016 A-To-Z feels like the one where I figured out how to do these things. Normal Numbers, this entry, is a piece that felt like I was making the breakthrough I wanted. Some of it is about the technical definition of normal numbers. But I also got into why normal numbers are interesting. Mysterious, even, in the medieval-theologist sense of of mystery. Almost every number is normal, but we only know a handful of normal numbers. So far as I’m aware, though, we don’t know of any number that’s interesting in its own right that’s also normal. It seems like a paradox.
My first A-to-Z sequence was in the summer of 2015. It’s the most primitive of my sequences and I really see how my writing style has changed from then. Well, there’s not much point to writing a lot if your style doesn’t change. Today, though, I’d like to bring up a piece that still holds up. It’s about Measure, another of those concepts that weaves through so many sections of mathematics. It’s also got a charming little anecdote of maybe dubious relevance as I bring spackle into things. That’s quite me.
I don’t use pictures enough for any of my essays. Even ones that would really benefit from them, like the Julia Set. So let me share one of the rare times I did. I got to use some pictures of my first visit to Niagara Falls, and tell about stepping into the river above the Falls, all as part of discussing what mathematicians mean by “local”.
For a while in grad school it looked like I was going to end up in graph theory. A seminar course in knots caught my imagination and while I never found a problem I could make any progress in, it left an affection that’s never faded. So in my first A-to-Z sequence I introduced knots, as mathematicians mean them, and I still like reminding people about that.
Iva Sallay, of the Find The Factors blog/recreational puzzle, had an outside-the-box suggestion for my Fall 2018 A-to-Z essays. It was some great thinking, though. There are a lot of mathematical jokes out there. They represent different aspects of humor writing. I had a great time exploring some of these different kinds. Is it truly mathematical? I don’t know. I loved writing the essay, though, and I keep enjoying re-reading it.
I’ve written some good essays for i-words. There’s one that stands out from the pack. If I did not refer people to the Infinite Monkey Theorem I would be giving bad advice. This is the one about monkeys typing Shakespeare. Along the way of researching it I discovered that Bob Newhart seems to have played a role in turning a thought-experiment about probability into something any comic strip can make a quick joke about. I still would like to know more about the history of monkeys-at-typewriters jokes. If somebody knows how to get a research grant for this sort of thing, please, hook me up.
That and the other Fall 2018 A-to-Z essays are at this link.
I do love mathematics. Much of what I love, though, is about its history and its culture. Occasionally I get to write about mathematical conventions and notation. Doing that lets me explore both interests. Hat, from the End 2016 A-to-Z, was one such exercise. The rest of the End 2016 A-to-Z essays are at this link.