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 might consider some 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.
And for the last of this year’s (planned) exhumations from my archives? It’s a piece from summer 2017: Zeta Function. As will happen in mathematics, there are many zeta functions. But there’s also one special one that people find endlessly interesting, and that’s what we mean if we say “the zeta function”. It, of course, goes back to Bernhard Riemann.
Also a cute note I saw going around. If you cut off the century years then the date today — the 16th day of the 12th month of the 20th year of the century — you get a rare Pythagorean triplet. and after a moment we notice that’s the famous 3-4-5 Pythagorean triplet all over again. If you miss it, well, that’s all right. There’ll be another along in July of 2025, and one after that in October of 2026.
To dig something out of my archives today, I offer the Zermelo-Fraenkel Axioms. This wrapped up the End 2016 A-to-Z. On the last day of 2016, I see; I didn’t realize I was cutting things that close that year. These are fundamentals of set theory, which is the study of what you can include and what you exclude from a set of things. For a while in the 20th century this looked likely to be the foundation of mathematics, from which everything else could be derived. We’ve moved on now to thinking that category theory is more likely the core. But set theory remains a really good foundation. You can understand a lot of what’s interesting about it without needing more than a child’s ability to make marks on paper and draw circles around some of them. Or, like my essays insist on doing, without even doing the drawings that would make it all easier to follow.
And let me share an essay from the Leap Day 2016 A-to-Z. I am still amazed that I had the energy to write two A-to-Z’s, and at three a week, in 2016. The Yukawa Potential is another mathematical/physics thing named for a person, here Hideki Yukawa, the first Japanese person to win a Nobel Prize. It’s a potential energy. It’s also an essay that inspired me to start the Why Stuff Can Orbit series, although not to quite finish it. (It reached a decent enough conclusion, but I had meant to do more. Maybe for 2021.)
I have many flaws as a pop mathematics blogger. Less depth of knowledge than I should have, for example. A tendency to start a series before I have a clear ending, so that projects will peter out rather than resolve. A-to-Z’s are different, as they have a clear direction and ending. And a frightful cultural bias, too. I’m terribly weak on mathematics outside the western tradition. Yang Hui’s Triangle, an essay I wrote about in the End 2020 A-to-Z, is a slight correction to that. I grew up learning this under a different name, that of a Western mathematician who studied the thing centuries after Yang Hui did. But then Yang Hui credited an earlier-yet mathematician, Jia Xian, for the insight. It’s difficult to get anything in mathematics named for the “correct” person.
As mentioned, ‘X’ is a difficult letter for a glossary project. There aren’t many mathematical terms that start with the letter, as much as it is the default variable name. Making things better is that many of the terms that do are important ones. Xor, from my 2015 A-to-Z, is an example of this. It’s one of the major pieces of propositional logic, and anyone working in logic gets familiar with it really fast.
The letter ‘X’ is a problem for this sort of glossary project. At least around the fourth time you do one, as you exhaust the good terms that start with the letter X. In 2018, I went to the Extreme Value Theorem, using the 1990s Rule that x- and ex- were pretty much the same thing. The Extreme Value Theorem is one of those little utility theorems. On a quick look it seems too obvious to tell us anything useful. It serves a role in proofs that do tell us interesting, surprising things.
And let me tease other W-words I won’t be repeating for my essay this week with the Well-Ordering Principle, discussed in the summer of 2017. This is one of those little properties that some sets of numbers, like whole numbers, have and that others, like the rationals, don’t. It doesn’t seem like anything much, which is often a warning that the concept sneaks into a lot of interesting work. On re-reading my own work, I got surprised, which I hope speaks better of the essay than it does of me.
No reason not to keep showing off old posts while I prepare new ones. A Summer 2015 Mathematics A To Z: well-posed problem shows off one of the set of things mathematicians describe as “well”. Well-posedness is one of those things mathematicians learn to look for in problems, and to recast problems so that they have it. The essay also shows off how much I haven’t been able to settle on rules about how to capitalize subject lines.
I have accepted that this week, at least, I do not have it in me to write an A-to-Z essay. I’ll be back to it next week, I think. I don’t know whether I’ll publish my usual I-meant-this-to-be-800-words-and-it’s-three-times-that piece on Monday or on Wednesday, but it’ll be sometime next week. And, events personal and public allowing, I’ll continue weekly from there. Should still finish the essay series before 2020 finishes. I say this assuming that 2020 will in fact finish.
But now let me look back on a time when I could produce essays with an almost machine-like reliability, except for when I forgot to post them. My 2019 Mathematics A To Z: Versine is such an essay. The versine is a function that had a respectably long life in a niche of computational computing. Cheap electronic computers wiped out that niche. The reasons that niche ever existed, though, still apply, just to different problems. Knowing of past experiences can help us handle future problems.
I am not writing another duplicate essay. I intend to have an A-to-Z essay for the week. I just haven’t had the time or energy to write anything so complicated as an A-to-Z since the month began. Things are looking up, though, and I hope to have something presentable for Friday.
So let me just swap my publication slots around, and share an older essay, as I would have on Friday. My 2018 Mathematics A To Z: Volume was suggested by Ray Kassinger, of the popular web comic Housepets!, albeit as a Mystery Science Theater 3000 reference. It’s a great topic, though. It’s one of those things everyone instinctively understands. But making that instinct precise demands we accept some things that seem absurd. It’s a great example of what mathematics can do, given a chance.
And, then, many of the U- entries in an A-to-Z are negations. Unbounded, from the summer 2015 sequence, is a good example of that. It’s also a concept worth knowing, since a lot of properties of analysis depend on whether you have an unbounded set or not. Or an unbounded function.
In looking over past A-to-Z’s I notice a lot of my U- entries are the negation of something. Unknots, for example. Or unbounded. English makes this construction hard to avoid. Any interesting property is also interesting when it’s absent. But there are also mathematical terms that start with a U on their own terms. The Summer 2017 Mathematics A To Z: Ulam’s Spiral shows off one of them. Stanislaw Ulam’s spiral is one of those things we find as a curious graphical adjunct to prime numbers. The essay also features one of my many pieces in praise of boredom.
I know, it’s strange for me to not post another piece about tiling. But My 2019 Mathematics A To Z: Taylor Series is going to be a good utility essay, useful for a long while to come. Taylor Series represent one of the standard mathematician tricks. This is to rewrite a thing we want to do as a sum of things it’s easy to do. This can make our problem into a long series of little problems. But the advantage is we know what to do with all those little problems. It’s often a worthwhile trade.
Well, this is just embarrassing.
I’ve always held out the option that I would revisit a topic sometime. I thought it would most likely be taking some essay from one of my earliest A-to-Z’s where, with a half-decade’s more experience in pop mathematics writing, I could do much better. And at the request of someone who felt that, like, my piece on duals was foggy. It is, but nobody’s ever cared enough about duals to say anything.
So I went looking at what previous T topics I’d written about here. Usually I pick them the Sunday or Monday of a week, since that’s easy to do. This week, I didn’t have the time until Thursday when I looked and found I wrote up “Tiling” for the 2018 A-to-Z. In about November of that year, too. And after casting aside a suggestion from Mr Wu of the Singapore Maths Tuition blog, although that time at least I was responding to a specific topic suggestion. 2020, you know?
Well, now that the deed is done, I can see what I learned from it anyway. First is picking out the archive pieces before I write the week’s essay. Second is how my approach differed in the 2020 essay. The broad picture is similar enough. The most interesting differences are that in the 2020 essay I look at more specifics. Like, just when Robert Berger found his aperiodic tiling of the plane. And what the Wang Tiles are that he found them with. Or, a very brief sketch of how to show Penrose (rhomboid) tiling is aperiodic. This changes the shape of the essay. Also it makes the essay longer, but that might also might reflect that in 2018 I was publishing two essays a week. This year I’m doing one, and somehow still putting out as many words per week.
I like the greater focus on specifics, although that might just reflect that I’m usually happiest with something I just wrote. As I get distance from it, I come to feel the whole thing’s so bad as to be humiliating. When it’s far enough in the past, usually, I come around again and feel it’s pretty good, and maybe that I don’t know how to write like that anymore. The 2018 essay is, to me, only embarrassing in stuff that I glossed over that in 2020 I made specific. Not to worry, though. I still get foggy and elliptical about important topics anyway.
It feels to me like I did a lot of functional analysis terms in the Leap Day 2016 series. Its essay for the letter ‘S’, Surjective Map, is one of them. We have many ways of dividing up the kinds of functions we have. One of them is in how they use their range. A function has a set called the domain, and a set called the range, and they might be the same set, yes. The function pairs things in the domain with things in the range. Everything in the domain has to pair with something in the range. But we allow having things in the range that aren’t paired to anything in the domain. So we have jargon that tells us, quickly, whether there are unmatched pieces in the range.
Sometimes I write an essay and know it’s something I’m going to refer back to a lot. Sometimes I know it’s just going to sink without trace. Often these deserve it; the subject is something particular and not well-connected to other topics. Sometimes, one sinks without a trace and for not much good reason. Smooth is one of those curiously sunk pieces. It’s about a concept important to analysis. And also a piece that shows my obsession with pointing out cultural factors in mathematics: we care about ‘smooth’ because we’ve found it a useful thing to highlight. And yet it’s gotten no comments, only an average number of likes, and I don’t seem to have linked back to it in any essays where it might be useful. I may have forgotten I wrote the thing. So here’s a referral that maybe will help me remember I have it on hand, ready for future use.
Bernard Riemann is one of those figures you can’t be a mathematics major without learning about. His name attaches to an enormous amount of analysis. One Riemann-named thing every mathematician learns very well is the Riemann Sum. It’s the first analysis model we use to explain why integration works. And we can put together a version of this for numerical integration. Its greatest use, though, is that we can use it to justify other ways to integrate that are easier to actually use. Great little utility.
Part of why I write these essays is to save future time. If I have an essay explaining some complex idea, then in the future, I can use a link and a short recap of the central idea. There’s some essays that have been perennials. I think I’ve linked to polynomials more than anything else on this site. And then some disappear, even though they seem to be about good useful subjects. Riemann sphere, from the Leap Day 2016 sequence, is one of those disappeared topics. This is one of the ways to convert between “shapes on the plane” and “shapes on the sphere”. There’s no way to perfectly move something from the plane to the sphere, or vice-versa. The Riemann Sphere is an approach which preserves the interior angles. If two lines on the plane intersect at a 25 degree angle, their representation on the sphere will intersect at a 25 degree angle. But everything else may get strange.
I feel like I talk group theory a lot in these A-to-Z sequences. Some of that’s deserved. Group theory underlies a lot of modern mathematics. Part of it is surely that it made the deepest impression on me, as a mathematics major, even though my work ended up not touching groups often. Quotient Groups are at that nice intersection of being important yet having a misleading name. You’re introduced to them after learning about groups, which have an operation that works like addition/subtraction; and then rings, which have addition/subtraction plus multiplication. Surely a quotient group is just a ring with division, right? No, it is not. But, lucky thing, there’s one quotient group you certainly know and feel familiar with. You’ll see.
In summer 2015 I picked all the topics for my A-to-Z; I didn’t work up the courage to ask for topics until the next time around. Some, I remember why I chose. I’m not sure why I picked Quintile, as a statistics term, rather than quartile. Both are legitimate terms, and circle around a similar idea. That is that we need to know how data is distributed: what range of numbers are common, what ones are rare. I wonder if I wasn’t saving ‘quartile’ for some later A-to-Z, for fear of running out of Q terms. Or if I felt that quartiles were familiar enough that quintiles would seem a touch strange. That is the sort of thing I’d likely do.
And in last year’s A-to-Z I published one of those essays already becoming a favorite. I haven’t had much chance to link back to it. So let me fix that. My 2019 Mathematics A To Z: Platonic focuses on the Platonic Solids, and questions like why we might find them interesting. Also, what Platonic solids look like in spaces of other than three dimensions. Three-dimensional space has five Platonic solids. There are six Platonic Solids in four dimensions. How many would you expect in a five-dimensional space? Or a ten-dimensional one? The answer may surprise you!
As I did the 2015 A-to-Z I learned how to do them in a way that feels me. In writing about the meaning of Proper, I found an important part of my voice. That’s the part which began with a corny mathematician’s joke. It also shows something I have forgotten how to do: it explains the whole thing, even with a joke to warm things up, in maybe 500 words. Well, I was publishing three A-to-Z essays a week back then; something had to go.
I’m going to take one more day, I think, preparing the Playful Math Education Blog Carnival. It’s hard work. But while you wait let me please share an older piece. In 2017 I wrote about Open Sets. These are important things, born of topology and offering us many useful tools. One of the best is that it lets us define “neighborhoods” and, along the way, “limits” and from that, “continuity”.
It was also a chance for me to finally think about one of those obvious nagging questions. There are open sets and there are closed sets. But it’s not the case that a set is either open or closed. A set can be not-open without being closed, and not-closed without being open. A set can even be both open and closed simultaneously. How can that turn out? And I learned that while “open” and “closed” are an obvious matched pair of words, they’re about describing very different traits of sets.
Occasionally an A-to-Z gives me the chance to naturally revisit an earlier piece. Orthonormal, from the Leap Day 2016 series, was one of those. It builds heavily on orthogonal, discussed the year before. When you know what the terms mean, of course it would. But getting to what the terms mean is part of the point of these essays.
Also, I hope to publish the 141th installment of the Playful Math Education Blog Carnival this weekend. If you’ve found a mathematics page, video, game, anything that delights or teaches or both, please mention in the comments. I’m eager to share it with more people.
For the 2018 A-to-Z I spent some time talking about a big piece of thermodynamics. Anyone taking a statistical mechanics course learns about the Nearest Neighbor Model. It’s a way of handling big systems of things that all interact. This is really hard to do. But if you make the assumption that the nearest pairs are the most important ones, and everything else is sort of a correction or meaningless noise? You get … a problem that’s easier to simulate on a computer. It’s not necessarily easier to solve. But it’s a good starting point for a lot of systems.
The restaurant I was thinking of, when I wrote this, was Woody’s Oasis, which had been kicked out of East Lansing as part of the stage in gentrification where all the good stuff gets the rent raised out from under it, and you get chain restaurants instead. They had a really good vegetarian … thing … called smead, that we guess was some kind of cracked-wheat sandwich filling. No idea what it was. There are other Woody’s Oasises in the area, somehow all different and before the pandemic we kept figuring we’d go and see if they had smead, sometime.
The Summer 2015 A-to-Z was the first I’d done. Its essays tended to be shorter and narrower in focus than what I write these days. But another feature is that they tended to be more practical, like, something that you could use to read a mathematics paper with better understanding. N-tuple is an example this. N-tuples are ordered bunches of numbers, and turn up in many places. They’re not quite vectors and matrices. But the ordinary use of vectors and matrices we represent with n-tuples.
One of the pieces I wrote for the Leap Day 2016 A-to-Z was to explain the Matrix. Matrices turn up a lot in mathematics. They’re nice things. They organize a lot of information economically. They’re vector spaces, so that a lot of really nice tools come along with their existence. They don’t have much to do with what I’ve been writing about this year, but, so what? I can bring back from obscurity pieces I just liked, too.
All my A-to-Z pieces, from every year, should be at this link. And all of the 2020 A-to-Z pieces should be at this link. Also please let me know if you have ideas for the letters P, Q, and R. I’m also still eagerly looking for Playful Math Education Blog Carnival-ready pieces. Thank you.
I avoided most of the technical talk when I discussed the Möbius strip the other day. You — well, a mathematician — could describe the strip as a non-orientable compact manifold with boundary. The boundary part is obvious. The non-orientable bit is easy enough to understand, when you remember that thing about it being a one-sided surface. Compact is an idea worth its own essay sometime. In this context it amounts to “there aren’t any gaps inside it”. Manifold, too, is worth an essay, and I wrote one in 2018 about it. Thanks for reading.
I did not mean my archive pieces this week to all be patching up stuff omitted from my Leibniz essay. But that essay touched a bunch of mathematical points, some of which I already had essays about. One of them was published in the 2018 A-to-Z. We really, really want to use the idea of an infinitesimally tiny change to understand how calculus works. But we can’t do that with logical rigor. (Unless we redefine what “real numbers” are, and some mathematicians go in for that.) Ah, but: what if we could get all the things these infinitesimals give us without having to do anything weird? And that is the modern idea of the limit, which we sorted out about 150 years ago and are pretty satisfied with.
While talking about Leibniz, who isn’t the inventor of calculus — but is the person I’d credit most with showing us how calculus could be — I made some speculations unsupported by evidence about whether he looked into optimization problems. This because of the philosophical work that he’s famous for among lay audiences, the proposition that God’s will implies this must be the best possible universe. (I don’t know what he’s most famous for among professional philosophers.)
I don’t have an essay specifically on optimization theory, as mathematicians see it. Not exactly. But last year I did write about linear programming, which is a particular type of optimization problem. It’s a kind that’s often the best we can do in a complex circumstance. And it lets me introduce you to the word “simplex”, which is fun to say.
If complex numbers aren’t the dominant theme of this year’s A-to-Z, then biographies are. I’ve written biographies for past series, though. Here, from 2018, is a tiny slice about William Thompson, Lord Kelvin, and one of those bits of his work that is mathematical and important and had a huge effect on the world. But it’s also become invisible. So please consider a couple hundred words about that.
In writing about K-theory I mentioned the “kernel”. I didn’t have space to describe what that was, and I failed to link to the essay I wrote in late 2016 about what kernels were and why we care. Let me fix that now. We’re introduced to kernels in group theory, where, for the simple groups, they seem like a complicated way to talk about “zero”. But we also see them in linear algebra. And between those, we get them in analysis. That then leads into not quite all of modern mathematics. But a lot of it.
In my first A-to-Z I wrote a good number of pieces about the kinds of functions there are. For example, jump, a particular kind of discontinuity in functions. This is useful because there are a lot of pieces of functional analysis where we know things are true for continuous functions. And if a function has a jump discontinuity? Usually we know the thing is true except at the discontinuity. There’s more rules, of course. And, like, Fourier series will get strange around jump discontinuities.
I’d have written the essay a bit different today, but I am in awe of a time I could wrap up the point within six hundred words. That never happens anymore.
I can’t guess where this year’s A-to-Z series will lead. Often a theme develops. Complex numbers look like they’re trying to be it. So let me share something from last year’s A-to-Z, and which relies on complex numbers. Julia sets, which are some of the best-known fractals, are calculated by working out functions on complex numbers. By iteration, particularly. That is, start with some number. Evaluate a function where the independent variable has that number. This gets you some (probably) different number. Evaluate the same function again, but using this as the independent variable. This gets you (usually) another number. Evaluate the same function again, with this third number as the independent variable’s value.
You’ve done this sort of iteration when playing with a calculator and hitting the square root or the square or the sine or whatever other function key over and over. These usually end up pretty boring, at 0 or 1 or the calculator reading INF. Put in a slightly different function? You get something beautiful.