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.