Although I did a lot of my functional-analysis-definitions for 2015 I certainly didn’t let that pass from the stage. My letter-I choice for the End 2016 A-to-Z was Image. This is useful when you think about functions as ways to turn one set (the domain) into another set (the range). A lot of definitions and tests for conditions, such as continuity, become much less technical and fussy when you start looking at images. It’s one of those changes in perspective that makes work easier.
My 2015 A-to-Z was before I asked for topic nominations. So what topics I did cover tended to focus on my own particular interests, and the courses I remember liking. So there was a good bit of functional analysis, or things that go into functional analysis. Into covers one of those topics. And you get some bonus coverage of “onto”. The terms describe ways that a function can use its range. To an extent it’s taxonomy, distinguishing functions that use the whole range versus those that use part. But it gives you vocabulary to discuss what cases you have to cover in proving a thing about a function. So it’s one of those small but useful terms.
Sorry to be late. We discovered baby fish in a tank that we had thought was empty, and that needed some quick attention. The water seems nearly all right and we’re taking measures to get the nitrate back in line. Fish-keeping is a great hobby for someone with tendency towards obsessions and who likes numbers because there is no end of tests you can start running and charts you can start to keep.
So I’ve seen two baby fish, one about the width of a fingernail and one about half that. We’re figuring to keep them inside until they’re large enough not to be accidentally eaten by the bigger goldfish, which means they might just be in there until we move fish inside for the winter. We’ll see.
Back to my archives, though. The hypersphere is a piece from the first A-to-Z I ever did. I could probably write a more complicated essay today. But the hypersphere is a good example of taking a concept familiar, as circles and as spheres, and generalizing it. Looking at what’s particularly interesting in a concept and how it might apply in different contexts. So it’s a good introduction to a useful bit of geometry, yes, but also to a kind of thinking mathematicians do all the time.
While looking through my past H essays I noticed a typo in Hamiltonian, an essay from the 2019 A-to-Z. Every time I look at an old essay I find a typo, even ones I’ve done this for before. Still, I choose to take it as a sign that this is an auspicious choice.
The Hamiltonian is one of the big important functions of mathematical physics. For all that, I remember being introduced to it, in a Classical Mechanics class, very casually, as though this were just a slightly different Lagrangian. Hamiltonians are very like Lagrangians. Both are rewritings of Newtonian mechanics. They demand more structure, more setup, to use. But they give fine things in trade. So they are worth knowing a bit about.
I’d like today to share a piece from 2017. Gaussian Primes are a fun topic, as they’re one of those things that steps into group theory without being too abstract. And they show how we can abstract a familiar enough idea — here, prime numbers — into something that applies in new contexts. In this case, in complex numbers, which are looking likely to be the running theme for this year’s A-to-Z.
Later in 2017 I talked talk about prime numbers in general, and how “prime” isn’t an idea that exists in the number itself. It exists in the number and the kind of number and how multiplication works for that kind of number.
If you looked at my appeal for A-to-Z topics for the letter G, when I posted it a couple weeks back, you maybe looked over a bunch of essays I quite liked. I still do; G has been a pretty good letter for me. So one of the archive pieces I’d like to bring back to attention is Grammar, from the Leap Day 2016 A-to-Z. It’s about how we study how to make mathematical systems. That you can form theorems about the mechanism for forming theorems is a wild discovery, and the subject can be hard to understand. At least some of its basic principles are accessible, I hope.
And if you’d like me to discuss more topics in mathematical logic, or other fields of mathematics that start with J, K, or L, please leave a comment at this link. Thank you.
There are important pieces of mathematics. Anyone claiming that differential equations are a niche interest is lying to you. And then there are niche interests. These are worthwhile fields. It’s just you can get a good well-rounded mathematical education while being only a little aware of them. And things can move from being important to niche, or back again.
Continued fractions are one of those things I had understood to have fallen from importance. They had a vogue, in Western mathematics, where they do some problems pretty neatly and cleverly. But they’re discussed more rarely these days. The speculation I’ve seen is that they don’t quite have a logical place, as being a little too hard when you’re learning fractions but seeming too easy when you’re learning infinite series, that sort of thing. My experience, it turns out, was not universal, and that’s an exciting thing to learn in the comments.
My impression, not checked against evidence, is that my recaps here feature the 2019 series more than any other. Well, I really liked the 2019 series. I don’t think that’s just recentism. On rereading them, I often feel little pleasant surprises along the way. That’s a good feeling.
So here was my ‘F’ entry for 2019: Fourier series. They’re important. They’re built out of easy pieces, though. And they’re full of weird bits. You can understand why someone would spend a career studying them. And I almost give enough information to actually use the things, if you have enough background to understand how to use them. I like hitting that sweet spot.
If it does turn out that P equals NP we would, at least in principle, have wrecked encryption as we know it. So let me take this chance to mention my essay on Encryption Schemes, from last year’s A-to-Z. And that discusses some of what we look for in encryption, which includes both secrecy and error-free transmission.
There’s one past A-to-Z essay for the letter e that’s compelling after I looked at the exponential function on Thursday. That would be the number that’s the base of the natural logarithm. It’s a number that was barely mentioned in that piece, because I ended up not needing it.
But a couple years ago I wrote a piece that was all e, including points like how curious a number it is. I hope that you enjoy that piece too.
There are some words that mathematicians use a lot, and to suggest things that are similar but not identical. “Normal” is one of them. Distribution is another, and in the End 2016 A-to-Z I discussed statistical distributions. To look at how a process affects a distribution, rather than a particular value, is one of the great breakthroughs of 19th century mathematical physics. This has implications about what it means to understand and to predict the behavior of a system.
I’d like today to share a less-old essay. This one is from the 2019 A-to-Z, and it’s about one of those fundamental topics. Differential equations permeate much of mathematics. Someone might mistake them for being all of advanced mathematics, or at least the kind of mathematics that professionals do. The confusion is reasonable. So I talk a bit here about why they seem to be part of everything.
I’ve learned a couple tricks to these essays. One is that I can always make them longer, and do. I’m trying not to. Another is in how to pick fruitful subjects. There’s always some chance to this and I’m still being surprised. But if I’ve found anything to be a sure bet, it’s picking the candidate topic that seems to be about the most obvious property. For example, everyone interested in mathematics knows about commutative things. But these are great because they make me focus on some “obvious” bit of mathematics lore. The “commutative” essay, in 2018, was an example of this. I didn’t achieve an insight to match that of “asymptote”, in which I finally understood a thing I had known for decades. But I improved my understanding at least.
There are some topics that only the most confident pop-mathematics bloggers can avoid writing. The topics are well-covered already. But they are fascinating, and they are accessible, and that is a powerful combination. One of them is the cardinality of infinitely large sets. That we can say some infinitely large sets are the same size, and others are larger, and have something that seems to make coherent sense.
To date I haven’t written that, not exactly. I have come close, though. One of them is from the End 2016 A-to-Z and its essay on Cantor’s Middle Third. It is a scattering of dust along a line segment. It is a set of points which, altogether, cover no length. But there are as many points in this set as there are in the entire real number line. It’s neat to discover. Please consider it.
The one topic in my 2017 series that I picked myself, without a nomination, was the second, on Benford’s Law. It’s one that seems to defy the notion that numbers are independent of human construction. It’s a mathematical principle discovered in the modern day by experimentation. It’s one that likely would not have been found, in the form we know, if electronic computers were abundant and cheap two centuries ago.
For a small point I wanted to mention how (United States) street addresses can serve as a rough proxy for position. So I needed some house numbers. One that was small and one larger. It’s hard for a human to pick random numbers. We tend to pick odd numbers more than even. We tend to shy away from ‘edge’ numbers, taking (say) 1 to be somehow less random than 3 or, better, 7. When I have to pick an arbitrary number then I try to pick even numbers, and try to run toward the edges. I know this makes me no more random than anyone else. But it means at least my numbers look different.
A person important important to me lives on the 400 block of their street. So I picked a 400 number, and changed the last two digits away from their actual address. Then I needed a larger number. If 400 is a plausible enough ordinary number, how about the 1400 block? So I wrote into my essay the ideas of a house at 418 and another at 1418. I also wanted an even higher street number, and if -18 is a good plausible low number, why not -88 as a good plausible high number? And so I went with that and put it to press.
And, I swear to you, I did not think about it past that.
The trouble is that white supremacists have adopted the number 1-4-8-8 as a dogwhistle. It’s used as reference to the Hitlerian agenda of boundless evil.
When I realized this I thought about what to do. One tempting option was to leave it as is. As a set of digits this is as good as any other. A symbol has power if it is taken to represent that thing; why give one more inch or jot to evil people? And anyone who knows me would know better than to think —
And there’s the rub. Anyone who knows me would know my longing for a just and decent world. More than seven billion people do not know me, and never will. How much time do I demand they spend studying my politics to know that I did not make this completely arbitrary choice to deniably signal cruelty? One point of a dogwhistle is to make something that the perpetrators understand, and the targets understand. And skeptical onlookers will think a meaningless or coincidental choice. This because the overt action of the dogwhistle is something that looks arbitrary or insignificant. That it is about something that seems trivial is important. It lets the aggressors paint their targets as paranoid and thus ridiculous, finding dire patterns in randomness. Who am I to make other people study me to know whether I intended something or whether I thought “I know someone who lives on the 400 block of their street. What’s a number that also has a 4, but is not too much bigger”?
This decided me. Were the number something relevant to the essay, I could justify keeping it in. If I were writing about James IV’s ascension to the throne of Scotland I couldn’t skip naming the year it happened. But for this? And so my imaginary houses moved to 419 and 1419 and 1489.
Mathematicians like to present the field as a universal thing, free of the human culture and concerns and thoughts that create it. It’s not, and can’t be. This example turned up, with thematic unity, in an essay about a thing that turns up studying things that seem to be independent of human culture. It’s a lesson I shall remember.
I’d like to bring back to attention another piece from my 2015 A-to-Z. Bijection is another term from analysis, which you end up doing a lot of as a mathematics major, and start to slowly understand in grad school. I’m almost to the point of understanding the basics myself these days.
The essay also shows how my style has changed since I started. That essay’s rather technical, giving more of the sort of definition you could use to see whether something was a bijection. A bunch of things, particularly a string of topics whose precise definition was heaps of technical terms, got me writing more about the context and history and culture of terms. There’s Mathworld if someone needs to know the four conditions a concept must satisfy. But there is something to say for essays that lay out in clear language what a thing is and how to know whether you have one. I may do some more like that.
I’d like to use today’s publication slot to highlight another older post. This one, less old; it’s from the 2019 sequence.
Abacus led off a series of essays that, generally, I quite liked. Normally right after publication I feel like I just barely had enough words to be presentable and maybe they make any sense. This essay started out with some talk about how to use an abacus. But then I realized the real question was why use an abacus, and that opened the essay in good ways. There’s a writing lesson for me in there somewhere, and I won’t learn it well enough.
I want to cut down on the amount of hard new writing I’m doing, at least until I feel at ease with the All 2020 A-to-Z and maybe anything in 2020 at all. But I also want to not disappear into the void except for a weekly appearance. This may be strange since some of the blogs I like best publish once a month or even more rarely, but I don’t have their confidence. Please bear with me as I bring some older posts to attention, then.
The first I want to highlight is the first A-to-Z post I ever did, a million years ago. The summer 2015 sequence started with “ansatz”, and it was the start of my learning a lot. The most important thing was learning that a term I’d picked up in grad school was, at least, idiosyncratic. I don’t know how small a mathematical dialect it is even has the word. If I knew what of other dialect words — not terms that are only used by a specific field, but that are only used by a particular group of people independent of their field — I’d share them. (If you know any please share! Language is fascinating.)
That discovery’s important. One thing I have learned in the past half-decade is to better appreciate the culture and the history that go into mathematics. Catching a word shows some of how human a subject mathematics is.
And to beat the drum once more, I’m eager to hear of topics starting with the letters D, E, and F that I might write about this year. (Or even revisit from an earlier essay sequence.) Thank you.
So the first bit of news: I’m hosting the Playful Math Education Blog Carnival later this month. This is a roaming blog link party, sharing blogs that delight or educate, or ideally both, about mathematics. As mentioned the other day Iva Sallay of Find the Factors hosted the 135th of these. My entry, the 136th, I plan to post sometime the last week of March.
And I’ll need help! If you’ve run across a web site, YouTube video, blog post, or essay that discusses something mathematical in a way that makes you grin, please let me know, and let me share it with the carnival audience.
This Saturday is March 14th, which we’ve been celebrating as Pi Day. I remain skeptical that it makes a big difference in people’s view of mathematics or in their education. But an afternoon spent talking about mathematics with everyone agreeing that, for today, we won’t complain about how hard it always was or how impossible we always found it, is pleasant. And that’s a good thing. I don’t know how much activity there’ll be for it, since the 14th is a weekend day this year. And the Covid-19 problem has got all the schools in my state closed through to April, so any calendar relevance is shattered.
But I have some things in the archive anyway. Last year I gathered Six Or Arguably Four Things For Pi Day, a collection of short essays about ways to calculate π well or poorly, and about some of the properties we’re pretty sure that π has, even if we can’t prove it. Also this fascinating physics problem that yields the digits of π.
And the middle of March often brings out Comic Strip Master Command. It looks like I’ve had at least five straight Pi Day editions of Reading the Comics, although most of them cover strips from more than just the 14th of March. From the past:
What will 2020 offer? There’s no guessing about anything in 2020 anymore, really. But when I get to look at the Pi Day comic strips for 2020 my essay on them should appear at this link. Thanks ever for reading. And for letting me know about sites that would be good for this month’s Carnival.
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”.