## How Pop-Up Cards and Books Work

Through the Interesting Esoterica postings on Mathstodon I learned of this neat post. Joseph O’Rourke published this year Pop-Up Geometry: The Mathematics Behind Pop-Up Cards. I haven’t got the book (yet), but O’Rourke has a page with animated GIFs showing how basic shapes work. The animations, even without narrative, are eye-opening, revealing how to make complicated and curved motions with a single rotating plane and fixed-length attachments. It isn’t properly origami but the subject is related.

Interesting Esoterica has an abstract to this entry here. An advantage to searching there is the archive of interesting topics, searchable by tags. These sprawl considerably over difficulty range: Under the tag ‘things to make and do’ are this piece on Pop-up Geometry, but also on group theory as it applies to laying model train tracks, or a 1959 essay describing how to build a computer out of paper. Or, if you’re looking for a more advanced project, Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-Style Square-Filling Curves. (This one is closer to the train tracks paper than you imagine, and you can follow its point from looking at the pictures.) You’re likely to find something delightful there.

## Have You Considered Spending Next Month Drawing Mathematics?

While poking around on Mathstodon, the mathematics-themed instance of the Twitter-like Mastodon, I ran across this. It’s “Mathober 2022”, the third of a series of daily doodling prompts, all built on mathematics themes.

The list of topics, and the goal of the exercise, is described here. The idea is to take a chance to do a sketch or a doodle or write a little bit about each of 31 mathematics topics, and share what you do. There’s no obligation to do all of them, no standards on how finished to do things. Or whether you can work ahead, or enter things late. The goal is to encourage creative expression.

Some of the prompts, like ‘cubic’ or ‘Moiré’, seem to have obvious artistic interpretation. Others, like ‘fundamental’ or ‘singularity’, will be more challenging.

This turns out to be the third Mathober and I regret not being aware of earlier ones. Daily prompt projects can be great ways to find motivation to do new creative projects.

## Some Fun Ways to Write Numbers but Complicated

I have a delightful trifle for you today. It is, like a couple of other arithmetic games, from a paper by Inder J Taneja, who has a wonderful eye for this sort of thing. It’s based on the sort of puzzle you might use to soothe your thoughts: how can you represent a whole number, using the string of digits 1 through 9 in order, and the ordinary arithmetic operations? That is, something like 12 x 34 – 56 + 78 ÷ 9? (If that would be a whole number.) Or in reverse order: 987 – 65 x 4 ÷ 32 + 1? (Again, if that’s a whole number.)

Dr Taneja has a set of answers for you, with the numbers from 0 through 11,111 written as strings of increasing and of decreasing digits. Here’s the arXiv link to the paper, for those who’d like to see the answers.

There is one missing number: Dr Taneja could find no way to produce 10,958 using the digits in increasing order. I imagine, given the paper was last updated in 2014, that there’s not a way to do this without adding some new operation such as factorials or roots, into the mix. Still, some time when you need to think of something soothing? Maybe give this a try. You might surprise everyone.

## How August 2022 Treated My Mathematics Blog: Romania Has Tired Of Me

With the start of another month it’s a chance to use my weekly publication slot to review the previous month. Also I’ve somehow settled on publishing one essay a week. That was never a deliberate choice, just an attempt to keep my schedule in line with my energy and enthusiasm during a time that’s drained most of both. August having had five Wednesdays in it, though, I published five things. Here they are, ranked most to least popular:

There are too few data points to do a real test. It does look like this isn’t just chronological order, though. Also, that Kickstarter has closed, but it was very successful. Denise Gaskin’s project collected more than nine times the initial goal and reached all but one of its stretch goals. You can still donate, though, to support an educational-publishing project.

It was a month of decline in my readership, though. There were 1,760 page views during the month, possibly because whatever drove hundreds of views from Romania in July did not repeat. In fact, there were only two page views from Romania in August. This is below the twelve-month running mean of 2,163.2 views per month, and the twelve-month running median of 2,105.5.

WordPress’s estimate of the number of unique visitors decreased too. There were 1,101 unique visitors here in August. The twelve-month running mean was 1,407.2, for the twelve months leading up to August. The running median was 1,409 unique visitors.

There were 17 things liked here in August, the same number as July. That’s below the mean of 30.9 and median of 29.5. There were no comments in August, for the second month in a row; as you might imagine, this is crashing far below the running median of 5.1 and median of 4. The figures look less bad if you pro-rate things by the number of posts. Then at least the views and unique visitors are between the mean and median numbers. Likes and comments are still low, though.

WordPress estimates that I published 2,617 words in August, bringing my total for the year to 48,472. So my average post length dwindled a bit in August, and it’s reduced my average post length this year to 915.

As of the start of September, WordPress says, I’ve gotten 167,896 page views in total, from a recorded 100,719 unique visitors. And, for good measure, a total of 1,728 posts since I began this blog … eleven? … years ago, and 3,321 comments over that time.

## You Could Help Make an Educational Kickstarter More Successful

Many of my readers likely remember Denise Gaskins, who organizes the Playful Math Education Blog Carnival. Gaskins has other projects to help people, particularly parents, with mathematics education. One of them is a Kickstarter. It’s nearly completed — it’s to close the 1st of September at 9 pm Eastern — and it has already made most of its goals. But a bit of help could get it over its highest (listed) stretch goal.

The project is the publication of Word Problems From Literature, as described here. It’s a guide about not just how to calculate, but how to tackle word problems. Everyone learning mathematics hates these, but they’re the heart of so much real mathematics. You need to work out what to calculate, and where to find the pieces needed for that. Gaskins’s project is to publish word problems inspired by the books kids might actually read, and to guide them in setting up calculations and finding what they need to do the calculations.

The project has already met its basic and most of its stretch goals, which is nice to see. If you’d like to see more details, or help it reach its highest stretch goals, though, please check it at this link. Thank you.

## Reading the Comics, August 14, 2022: Not Being Wrong Edition

The handful of comic strips I’ve chosen to write about this week include a couple with characters who want to not be wrong. That’s a common impulse among people learning mathematics, that drive to have the right answer.

Will Henry’s Wallace the Brave for the 8th opens the theme, with Rose excited to go to mathematics camp as a way of learning more ways to be right. I imagine everyone feels this appeal of mathematics, arithmetic particularly. If you follow these knowable rules, and avoid calculation errors, you get results that are correct. Not just coincidentally right, but right for all time. It’s a wonderful sense of security, even when you get past that childhood age where so little is in your control.

A thing that creates a problem, if you love this too closely, is that much of mathematics builds on approximations. Things we know not to be right, but which we know are not too far wrong. You expect this from numerical mathematics, yes. But it happens in analytic mathematics too. I remember struggling in high school physics, in the modeling a pendulum’s swing. To do this you have to approximate the sine of the angle the pendulum bob with the angle itself. This approximation is quite good, if the angle is small, as you can see from comparing the sine of 0.01 radians to the number 0.01. But I wanted to know when that difference was accounted for, and it never was.

(An alternative interpretation is to treat the path swung by the end of the pendulum as though it were part of a parabola, instead of the section of circle that it really is. A small arc of parabola looks much like a small arc of circle. But there is a difference, not accounted for.)

Nor would it be. A regular trick in analytic mathematics is to show that the thing you want is approximated well enough by a thing you can calculate. And then show that if one takes a limit of the thing you can calculate you make the error infinitesimally small. This is all rigorous and you can in time come to accept it. I hope Rose someday handles the discovery that we get to right answers through wrong-but-useful ones well.

Charles Schulz’s Peanuts Begins for the 8th is one that I have featured here before. It’s built on Lucy not accepting that the answer to a multiplication can be zero, even if it is zero times zero. It’s also built on the mixture of meanings between “zero” and “nothing” and “not existent”. Lucy’s right that zero times zero has to be something, as in a thing with some value. But we also so often use zero to mean “nothing that exists” makes zero a struggle to learn and to work with.

Dan Thompson’s Brevity for the 12th is an anthropomorphic numerals joke, built on the ancient playground pun about why six is afraid of seven. And a bit of wordplay about odd and even numbers on top of that. For this I again offer the followup joke that I first heard a couple of years ago. Why was it that 7 ate 9? Because 7 knows to eat 3-squared meals a day!

Lincoln Pierce’s Big Nate for the 14th is a baseball statistics joke. Really a sabermetrics joke. Sabermetrics and other fine-grained sports analysis study at the enormous number of games played, and situations within those games. The goal is to find enough similar situations to make estimates about outcomes. This is through what’s called the “frequentist” interpretation of statistics. That is, if this situation has come up a hundred times before, and it’s led to one particular outcome 85 of those times, then there’s an 85 percent chance of that outcome in this situation.

Baseball is well-posed to set up this sort of analysis. The organized game has always demanded the keeping of box scores, close records of what happened in what order. Other sports can have the same techniques applied, though. It’s not likely that Randy has thrown enough pitches to estimate his chance of giving up a walk-off grand slam. But combine all the little league teams there are, and all the seasons they’ve played? That starts to sound plausible. Doesn’t help the feeling that one was scheduled for a win and then it didn’t happen.

And that’s enough comics for now. All of my Reading the Comics posts should be at this link, and I hope to have another next week. Thanks for reading.

## Reading the Comics, August 5, 2022: Catching Up Edition

I’ve had several weeks since my last Reading the Comics post. They’ve been quiet enough weeks. Let me share some of the recent offerings from Comic Strip Master Command that I enjoyed, though. I enjoy many comic strips but not all of them mention something on-point here.

Isabella Bannerman’s Six Chix for the 18th of July is a wordplay joke, naming a dog Isosceles to respect his front and hindlegs being equal in length. As I say, I’m including these because I like them, not because I have deep thoughts about them.

Bill Amend’s FoxTrot for the 24th of July gets a bit more solidly mathematical, as it’s natural to think of this sort of complicated polyhedron as something mathematicians do. Geometers at least. There’s a comfortable bit of work to be done in these sorts of shapes. They sometimes have appealing properties, for instance balancing weight loads well. Building polyhedrons out of toothpicks and gumdrops, or straws and marshmallows, or some other rigid-and-soft material, is a fine enough activity. I think every mathematics department has some dusty display shelf with a couple of these.

There are many shapes that Paige Fox’s construction might be. To my eye Paige Fox seems to be building a truncated icosahedron, that is to say, the soccer-ball shape. It’s an Archimedean solid, one of the family of thirteen shapes made of nonintersecting regular convex polygons. These are the shapes you discover if you go past Platonic solids. The family is named for that Archimedes, although the work in which he discussed them is now lost.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th of July is a different take on the challenge of motivating students to care about mathematics. The “right approach” argument has its appeal, although it’s obviously thinking of “professor” as the only job a mathematician can hold. And even granting that none of the people who understand your work have the power to fire you, that means people who don’t understand your work do have it. Also, people shut down very hard at your promise that they could understand anything about what you do, even when you know how to express it in common-language terms. This gets old fast. I am also skeptical that women are impressed by men at bars who claim to be employed for their intellect.

Anyway, academic jobs are great and more jobs should work by their rules, which, yes, do include extremely loose set hours and built-in seasons where the amount and kind of work you do varies.

Patrick Roberts’s Todd the Dinosaur for the 5th of August is a mathematics-anxiety dream, represented with the sudden challenge to do mental arithmetic. 47 times 342 is an annoying problem, yes, but one can at least approximate it fairly well quickly. 47 is almost 50, which is half a hundred. So 47 times 342 has to be nearly half of a hundred times 342, that is, half of 34,200. This is an easy number to cut in half, though: 17,100. To get this exactly? 47 is three less than fifty, so, subtract three times 342. 342 is about a third of a thousand, we can make a better estimate by subtracting a thousand: 16,100.

If you’re really good you notice that 342 is nine more than 333, so, three times 342 is three times nine more than three times 333. That is, it’s 27 more than 999. So the 16,100 estimate is 26 more than the correct number, 16,074. And I believe if you check, you will find the card in your hand is the ace of clubs. Am I not right, professor?

And that’s a look at comic strips through to a bit over a week ago. I intend to have more soon. All my Reading the Comics posts should be at this link. Thanks for being with me for these reviews. See you again soon.

## Here’s the 157th Playful Math Blog Carnival

I’d missed the chance to share it last week, as the readership report somehow took priority and the publication slot. Sue VanHattum’s Math Momma Writes … blog is the most recent host of the Playful Math Blog Carnival. Here’s the July 2022 installment, the 157th of the series. VanHattum won me over right away by leading with a nighttime picture of a carnival. That isn’t required for people who host the educational-and-recreational mathematics feature, but it doesn’t hurt.

Besides the expected references to blogs and games and such, VanHattum has a nice section describing present or upcoming books. These are mostly aimed at kids, which I suspect might be younger than my usual audience. But you might know someone who’d like to know.

And if you have a mathematics or education blog, and want to try gathering a bundle of interesting or educational or fun mathematics links together, I recommend contacting Denise Gaskins and hosting for a month. I regret that I haven’t had the time or resources to host myself in a long while, and hope that wil change soon. It’s a fun challenge.

## How July 2022 Treated My Mathematics Blog: Romania liked me

I have not given up on my mathematics blog, though I admit to its commanding less attention than I have sometimes given it. I have had less attention to give everything. In a month of writing that comes pretty close to simple maintenance mode, I expect pretty average readership figures. I did not have them.

WordPress says that I received 3,071 page views in July, which is the biggest total I’ve had since October 2019, and I believe my second most-read month ever. This is because for some reason I got about a thousand page views from Romania, mostly in the second week of the month. I don’t know why. I usually get about a thousand page views from the United States — in July there were 860 — so this is odd. There were also 339 page views from India, which is up from the usual of one to two hundred, but not so much more as to be clearly wrong. So, much as I like having a big month, I can’t believe in it.

Because sure, 3,071 is way above the twelve-month running mean of 2,064.8 views per month, and the running median of 2,080 per month. When I look at the count of unique visitors, though? That’s a less exalted 1,193 for July. THat’s close to what June offered me, and below the running mean of 1,418.1 and running median of 1,409 visitors. The things that measure interactions were even more dire: only 17 likes were given around here in July. That’s the lowest figure in at least two and a half years, and below the running mean of 32.3 and running median of 30.5. And finally there were no comments in July, the third time that’s ever happened. My running mean is 7.3 and median 6.5 right now. (Well, there was one submitted comment, on my announcement that I won’t be doing an A-to-Z this year. But it was just a one-word “Nice”. I imagine that’s a spammer doing that thing rather than an attempt to hurt my feelings.)

I would like to report what the relative popularity of July’s posts were. For some reason WordPress won’t tell me. I can get the similar data for my humor blog, so I don’t know what the issue is. Well, here’s what I published this last month:

For the year to date, WordPress figures I’ve published 45,855 words, not counting this post. That’s coming in at an average 955 words per posting, which is dwindling a little. I blame Comic Strip Master Command for not giving me stuff I can go on about all night.

As of the start of August, WordPress says, I’ve gotten 166,185 page views, from a recorded 99,618 unique visitors.

## About my 2022 Mathematics A-to-Z

I don’t plan to have one.

Some context. Since 2015 I’ve run a series of A-to-Z essays. This is writing a short glossary about various mathematics terms. Most years they were a complete pass through the alphabet, with some fudging to allow for there being fewer terms that start with ‘X’ than you’d think. For 2021, I changed the format up a little, writing instead one for each letter in the phrase “Mathematics A to Z”. You can read all 198 of the completed essays at this link.

Hard as I thought 2021 was, 2022 has been worse, in wearing down my energy and enthusiasm for big projects. Or projects at all. And an A-to-Z is a big, worthwhile but exhausting project. I am already too late in the year to run one essay a week and complete the full alphabet. Given the lead times necessary even a partial alphabet would be hard to fit in. (I don’t want to run these projects across years, although I accepted that for the 2021 project.) In 2015 and 2016 I was able to write three essays a week, a commitment of time and energy I can’t imagine making now.

So rather than sit idly hoping something will turn up, I accept the situation. I am not up for an A-to-Z this year. I hope that 2023 will be a happier year, one in which I can do a project of that magnitude. I suppose we all hope for that.

## Reading the Comics, July 16, 2022: Brevity is the Soul of Wit Edition

Once more Comic Strip Master Command shows great parsimony with its mathematics-themed comic strips. The one with most substance to it is Dan Thompson’s Brevity, and even that’s not one I can talk about at length. Watch me try, anyway.

Dan Thompson’s Brevity for the 11th of July is the anthropomorphic numerals-and-symbols joke for the week. And it correctly identifies + as representing the addition operator. I talked about operators for the 2019 A-to-Z, though my focus there was on functions of functions. Here, we’ve just got a binary operator, taking two integers (or real numbers) and, if we follow its rule, matching this pair with an integer (or real number). Well, we can embed the integers (or real numbers) in the space of functions, if we want.

It happens that “smooth” is also a term of mathematical art. It refers to a function that you can take enough derivatives of. “Enough” here depends on our use. Depending on what our work is, we might need anything from one derivative to infinitely many derivatives. I’m not sure that I would call the ordinary arithmetic-type addition operator “smooth”. But I’m not confident in saying it’s not.

Greg Cravens’s The Buckets for the 12th of July is more wordplay than mathematics play. I’m including it because I like the playful energy of the comic strip and want more people to notice it. I like the idea of calling such an assembled doll a fraction figure, though.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 14th of July is your usual story problem setup. It’s a standard enough insult joke, although it does use the mathematics content. There aren’t a lot of spelling questions that would let you set up an ambiguous “how many are left” joke. Maybe an English class.

This takes us through the middle of July, as these go. All my Reading the Comics posts should be at this link,. Will there be more comics for next week? I have my suspicions.

## Reading the Comics, July 12, 2022: Numerals Edition

The small set of comic strips with some interesting mathematical content for the first third of the month include an anthropomorphic numerals one and one about the representation of infinity. That’s enough to make a title.

Richard Thompson’s Cul de Sac repeat for the 3rd is making its third appearance in my column! I had mentioned it when it first ran, in July 2012, and then again in its July 2017 repeat. But in neither of those past times did I actually include the comic as I felt it likely GoComics would keep the link to it stable. I’m less confident now that they will keep the link up, as Thompson has died and his comic strip — this century’s best, to date — is in perpetual rerun.

I admit not having many thoughts which I haven’t said twice already. It’s a joke about making a character out of the representation of a number. Alice gives it personality and backstory and, as the kids say these days, lore. Anyway, be sure to check out this blog for the comic’s repeat the 4th of July, 2027. I hope to still be reading then.

I think mathematicians do tend to give, if not personality, at least traits to mathematical constructs. Like, 60, a number with abundant divisors, is likely to be seen as a less difficult number to calculate with than 58 is. A mathematician is likely to see $e^{\imath sin(x)}$ as a pleasant, well-behaved function, but is likely to see $\sin(e^{\frac{\imath}{x}})$ as a troublesome one. These examples all tie to how easy they are to do other stuff with. But it is natural to think fondly of things you see a lot that work nicely for you. These don’t always have to be “nice” things. If you want to test an idea about continuous curves, for example, it’s convenient to have handy the Koch curve. It’s this spiky fractal that’s nothing but corners, and you can use it to see if the idea holds up. Do this enough, and you come to see a reliable partner to your work.

Bill Amend’s FoxTrot for the 3rd is the one built on representing infinity. Best that one could hope for given Peter’s ambitious hopes here. I know the characters in this strip and I just little brother Jason wanted a Möbius-strip burger.

Tauhid Bondia’a Crabgrass for the 7th — a strip new to newspaper syndication, by the way — is a cryptography joke. This sort of thing is deeply entwined into mathematics, most deeply probability. This because we know that a three-letter (English) word is more likely to be ‘the’ or ‘and’ or ‘you’ than it is to be ‘qua’ or ‘ado’ or ‘pyx’. And either of those is more likely than ‘pqd’. So, if it’s a simple substitution, a coded word like ‘zpv’ gives a couple likely decipherings. The longer the original message, and the more it’s like regular English, the more likely it is that we can work out the encryption scheme.

But this is simple substitution. What’s a complex substitution? There are many possible schemes here. Their goal is to try to make, in the code text, every set of three-letter combinations to appear about as often as every other pair. That is, so that we don’t see ‘zpv’ happen any more, or less, than ‘sgd’ or ‘zmc’ do, so there’s no telling which word is supposed to be ‘the’ and which is ‘pyx’. Doing that well is a genuinely hard problem, and why cryptographers are paid I assume big money. It demands both excellent design of the code and excellent implementation of it. (One cryptography success for the Allies in World War II came about because some German weather stations needed to transmit their observations using two different cipher schemes. The one which the Allies had already cracked then gave an edge to working out the other.)

It also requires thinking of the costs of implementation. Kevin and Miles could work out a more secure code, but would it be worth it? They just need people to decide their message is too much effort to be worth cracking. Mrs Campbell seems to have reached that conclusion, at a glance. Not sure what Principal Sanders would have decided, were Miles not eager to get out of there. Operational security is always a challenge.

And that’s enough for the start of the month. All my Reading the Comics posts should be at this link,. I hope to have some more of them to discuss next week. We’ll see what happens.

## How June 2022 Treated My Mathematics Blog

The folks who signed up to get my posts delivered by e-mail — it’s a box on the rightmost column of this page — know I used that title last month. A typo, basically; I was thinking of the promise of the new month and did not notice my subject line was a month early. I fixed that old post, and nobody seems to have mentioned it. But I like being open about my mistakes as well as my great moments. Also I would like to have more great moments. In any event, here’s to specifics.

WordPress records me as getting 1,749 page views in June. This is a fair bit below the running averages for the twelve months running up to June 2022. The mean has been 2,128.0 views per month, and the median 2,105.5. The number of unique visitors continued its decline to, to 1,159 visitors, compared to a running mean of 1,467.6 and running median of 1,436. I’m sure there’s something that could be done about these figures but it’s impossible to say what. It is setting up a botnet to send spurious page hits.

I’m staying likeable, at least. 29 posts got liked in June, just about on target for the running mean of 32.9 and running median of 31.5. There were only two comments, but the averages there are a mean of 8.0 and median of 4.5, so it’s not that far off the norm.

Prorated to the scant number of posts made — five this month — the figures look more competitive. There were 349.8 views per posting, on average; the running averages are a mean of 323.5 and median of 302.8. There were 231.8 unique visitors per posting; the mean was 222.2 and median 211.3. 5.8 likes per posting, compared to a mean of 4.6 and median of 4.2. 0.4 comments per posting, the only point where the prorated count was below the average. The running mean per posting was 1.0 and median 0.8.

So here’s the roster of what I posted in June, ranked from most to least popular. Or at least clicked-on; I don’t have the energy to compare how many likes things get.

WordPress figures that I posted 3,256 words in June, an average of 993 words per posting. This is about on par with my recent average, and brings me to 43,706 words for the year to date. That includes a stretch back in January when I was rerunning a lot of material. If you’d like to be a regular reader, I told you up top how to get e-mails sent to you by mail. If you’d rather have them in your WordPress reader, you can use the ‘Follow Nebusresearch button’, in the right column of the page. Or you could set up your RSS reader to use https://nebusresearch.wordpress.com/feed. That’s the best option, really, but the one I won’t see in any of my statistics.

## Reading the Comics, June 26, 2022: First Doldrums of Summer Edition

I have not kept secret that I’ve had little energy lately. I hope that’s changing but can do little more than hope. I find it strange that my lack of energy seems to be matched by Comic Strip Master Command. Last week saw pretty slim pickings for mathematically-themed comics. Here’s what seems worth the sharing from my reading.

Lincoln Peirce’s Big Nate for the 22nd is a Pi Day joke, displaced to the prank day at the end of Nate’s school year. It’s also got a surprising number of people in the comments complaining that 3.1416 is only an approximation to π. It is, certainly, but so is any representation besides π or a similar mathematical expression. And introducing it with 3.1416 gives the reader the hint that this is about a mathematics expression and not an arbitrary symbol. It’s important to the joke that this be communicated clearly, and it’s hard to think of better ways to do that.

Dave Whamond’s Reality Check for the 24th is another in the line of “why teach algebra instead of something useful” strips. There are several responses. One is that certainly one should learn how to do a household budget; this was, at least back in the day, called home economics, and it was a pretty clear use of mathematics. Another is that a good education is about becoming literate in all the great thinking of humanity: you should come out knowing at least something coherent about mathematics and literature and exercise and biology and music and visual arts and more. Schools often fail to do all of this — how could they not? — but that’s not reason to fault them on parts of the education that they do. And anther is that algebra is about getting comfortable working with numbers before you know just what they are. That is, how to work out ways to describe a thing you want to know, and then to find what number (or range of numbers) that is. Still, these responses hardly matter. Mathematics has always lived in a twin space, of being both very practical and very abstract. People have always and will always complain that students don’t learn how to do the practical well enough. There’s not much changing that.

Charles Schulz’s Peanuts Begins for the 26th sees Violet challenge Charlie Brown to say what a non-perfect circle would be. I suppose this makes the comic more suitable for a philosophy of language blog, but I don’t know any. To be a circle requires meeting a particular definition. None of the things we ever point to and call circles meets that. We don’t generally have trouble connecting our imperfect representations of circles to the “perfect” ideal, though. And Charlie Brown said something meaningful in describing his drawing as being “a perfect circle”. It’s tricky pinning down exactly what it is, though.

And that is as much as last week moved me to write. This and my other Reading the Comics posts should be at this link. We’ll see whether the upcoming week picks up any.

## Reading the Comics, June 18, 2022: Pizza Edition

I’m back with my longest-running regular feature here. As I’ve warned I’m trying not to include every time one of the newspaper comics (that is, mostly, ones running on Comics Kingdom or GoComics) mentions the existence of arithmetic. So, for example, both Frank and Ernest and Rhymes with Orange did jokes about the names of the kinds of triangles. You can clip those at your leisure; I’m looking to discuss deeper subjects.

Scott Hilburn’s The Argyle Sweater is … well, it’s just an anthropomorphic-numerals joke. I have a weakness for The Wizard of Oz, that’s all. Also, I don’t know but somewhere in the nine kerspillion authorized books written since Baum’s death there be at least one with a “wizard of odds” plot.

Bill Amend’s FoxTrot reads almost like a word problem’s setup. There’s a difference in cost between pizzas of different sizes. Jason and Marcus make the supposition that they could buy the difference in sizes. They are asking for something physically unreasonable, but in a way that mathematics problems might do. The ring of pizza they’d be buying would be largely crust, after all. (Some people like crust, but I doubt any are ten-year-olds like Jason and Marcus.) The obvious word problem to spin out of this is extrapolating the costs of 20-inch or 8-inch pizzas, and maybe the base cost of making any pizza however tiny.

You can think of a 16-inch-wide circle as a 12-inch-wide circle with an extra ring around it. (An annulus, we’d say in the trades.) This is often a useful way to look at circles. If you get into calculus you’ll see the extra area you get from a slight increase in the diameter (or, more likely, the radius) all over the place. Also, in three dimensions, the difference in volume you get from an increase in diameter. There are also a good number of theorems with names like Green’s and Stokes’s. These are all about what you can know about the interior of a shape, like a pizza, from what you know about the ring around the edge.

Jim Meddick’s Monty sees Sedgwick, spoiled scion of New Jersey money, preparing for a mathematics test. He’s allowed the use of an abacus, one of the oldest and best-recognized computational aides. The abacus works by letting us turn the operations of basic arithmetic into physical operations. This has several benefits. We (generally) understand things in space pretty well. And the beads and wires serve as aides to memory, always a struggle. Sedgwick also brings out a “hyperbolic abacus”, a tool for more abstract operations like square roots and sines and cosines. I don’t know of anything by that name, but you can design mechanical tools to do particular computations. Slide rules, for example, generally have markings to let one calculate square roots and cube roots easily. Aircraft pilots might use a flight computer, a set of plastic discs to do quick estimates of flight time, fuel consumption, ground speed, and such. (There’s even an episode of the original Star Trek where Spock fiddles with one!)

I have heard, but not seen, that specialized curves were made to let people square circles with something approximating a compass-and-straightedge method. A contraption to calculate sines and cosines would not be hard to imagine. It would need to be a post on a hinge, mostly, with a set of lines to read off sine and cosine values over a range of angles. I don’t know of one that existed, as it’s easy enough to print out a table of trig functions, but it wouldn’t be hard to make.

And that’s enough for this week. This and all my other Reading the Comics posts should be at this link. I hope to get this back to a weekly column, but that does depend on Comic Strip Master Command doing what’s convenient for me. We’ll see how it turns out.

## I shouldn’t keep hiding the Playful Math Education Carnivals from you

I have not had the time or energy to host the Playful Math Education Carnival for a while now. I hope that changes but I don’t know when it will. Still, there is no good reason for me not to let you know when Denise Gaskins’ project, of gathering educational or recreational or just delightful mathematics links, has a new edition.

Nature Study Australia, which started as a nature-study and activity blog, hosts the most recent essay. It’s the 156th of the sequence, and so starts with the sorts of fun facts abou the number 156. From there it spreads into calculation tricks and practices, and eventually into the games and activities that highlight these sequences.

If you should write about mathematics even just sometimes, you might consider hosting the blog. It’s a worthwhile challenge, and you can sign up for future months at Denise Gaskins’s blog.

## Reading the Comics, June 3, 2022: Prime Choices Edition

I intended to be more casual about these comics when I resumed reading them for their mathematics content. I feel like Comic Strip Master Command is teasing me, though. There has been an absolute drought of comics with enough mathematics for me to really dig into. You can see that from this essay, which covers nearly a month of the strips I read and has two pieces that amount to “the cartoonist knows what a prime number is”. I must go with what I have, though.

Mark Anderson’s Andertoons for the 12th of May I would have sworn was a repeat. If it is, I don’t seem to have featured it before. It gives us Wavehead — I’ve learned his name is not consistent — learning about division. The first kind of division, at least, with a quotient and a remainder. The novel thing here, with integer division, is that the result is not a single number, but rather an ordered pair. I hadn’t thought about it that way before, I suppose since integer division and ordered pairs are introduced so far apart in one’s education.

We mostly put away this division-with-remainders as soon as we get comfortable with decimals. 19 &div; 4 becoming “4 remainder 3” or “4.75” or “4 $latex\frac{3}{4}$” all impose a roughly equal cognitive load. But this division reappears in (high school) algebra, when we start dividing polynomials. (Almost anything you can do with integers there’s a similar thing you can do with polynomials. This is not just because you can rewrite the integer “4” as the polynomial “f(x) = 0x + 4”.) There may be something easier to understand in turning $\left(x^2 + 3x - 3\right) \div \left(x - 2\right)$ into $\left(x + 1\right)$ remainder $\left(4x - 1\right)$.

A thing happening here is that integer arithmetic is a ring. We study a lot of rings, as it’s not hard to come up with things that look like addition and subtraction and multiplication. Rings we don’t assume to have division that stays in your set. They can turn into pairs, like with integers or with polynomials. Having that division makes the ring into a field, so-called because we don’t have enough things called a “field” already.

Scott Hilburn’s The Argyle Sweater for the 16th of May is one of the prime number strips from this collection. About the only note worth mention is that the indivisibility of 3 depends on supposing we mean the integer 3. If we decided 3 was a real number, we would have every real number other than zero as a divisor. There’s similar results for complex numbers or polynomials. I imagine there’s a good fight one could get going about whether 3-in-integer-arithmetic is the same number as 3-in-real-arithmetic. I’m not ready for that right now, though.

I like the blood bag Dracula’s drinking from. Nice touch.

Dave Coverly’s Speed Bump for the 16th of May names the ways to classify triangles based on common side lengths (or common angles). There is some non-absurdity in the joke’s premise. Not the existence of these particular pennants. But that someone who loves a subject enough to major in it will often be a bit fannish about it? Yes. It’s difficult to imagine going any other way. You need to get to a pretty high leve of mathematics to go seriously into triangles, but the option is there.

Dave Whamond’s Reality Check for the 3rd of June is the other comic strip playing on the definition of “prime”. Here it’s applied to the hassle of package delivery, and the often comical way that items will get boxed in what seems to be no logical pattern. But there is a reason behind that lack of pattern. It is an extremely hard problem to get bunches of things together at once. It gets even harder when those things have to come from many different sources, and get warehoused in many disparate locations. Add to that the shipper’s understandable desire to keep stuff sitting around, waiting, for as little time as possible. So the waste in package and handling and delivery costs seems worth it to send an order in ten boxes than in finding how to send it all in one.

It feels like an obvious offense to reason to use four boxes to send five items. It can be hard to tell whether the cost of organizing things into fewer boxes outweighs the additional cost of transporting, mostly, air. This is not to say that I think the choice is necessarily made correctly. I don’t trust organizations to not decide “I dunno, we always did it this way”. I want instead to note that when you think hard about a question it often becomes harder to say what a good answer would be.

I can give you a good answer, though, if your question is how to read more comic strips alongside me. I try to put all my Reading the Comics posts at this link. You can see something like a decade’s worth of my finding things to write about students not answering word problems. Thank you for reading along with this.

## How May 2022 Treated My Mathematics Blog

The easy way to put this article is, if I don’t read my mathematics blog why should anyone else? There is truth to this. I have mentioned several times that this has been a difficult year for me, and I’ve had to ration where I put my energy. I’ve avoided going a whole week without a post, but it’s only by reposting old material that I’ve managed that. Even the old standby of writing about the mathematics in comic strips has fallen short, as Comic Strip Master Command isn’t sending so many worth my attention these days. These are strange times.

The result is a decline in my readership, although it’s less of one than I had expected. There were no comments at all around here in May, which, have to say, seems fair. There wasn’t much to comment on, especially with just four essays posted. That’s my lowest posting volume in years. It’s also not the first time I had zero comments in a month, which takes some sting off.

So there were 2,057 page views here in May. That’s a bit below the twelve-month running mean of 2,212.3 views per month leading up to May. And below the running median of 2,114.5 views. Per posting, the number looks impressive, though, with 514.3 page views per posting. That beats the running mean of 309.1 and median of 302.8.

There were 1,358 unique visitors recorded in May. That’s again a slight decline from the 1,528.2 running mean and 1,461.5 running median. And, again, per posting the numbers seem impressive. 339.5 unique visitors with each posting, above the mean of 213.2 and median of 211.3. The implication, yes, is if I didn’t post at all I’d have infinitely many readers, a conclusion which hurts my feelings.

There were twenty likes given in May, up from April but still below the mean of 35.3 and median of 33. It’s a per-posting average of 5.0 likes per posting, above the mean of 4.6 and median of 4.2 but there’s no way there’s statistical significance to that. And, of course, no comments, compared to a running mean of 9.7 and median of 7.

With so few essays posted it’s easy to report the order of their popularity. I’m not sure whether their order depends on how interesting the text was or how early in the month they were posted. There’s no way the difference is statistically significant. But here’s the May 2022 pieces ranked most popular to least:

WordPress figures I started the month with a grand total of 1,714 posts. These all together drew 3,319 comments and 161,316 page views from 97,265 recorded unique visitors. It also figures my average post for the month had 876 words in it, bringing my average post for the year 2022 down to 1,037 words per posting. I’ve managed to put together 40,451 words so far this year. This surprises me by being close to half what I’ve managed on my humor blog, where I post every day. There, I have several regular columns, such as story comic plot summaries, that are popular and relatively easy to write.

Having said all that, will this look at May’s figures affect my writing any? I do think I have enough comic strips for a post, that should be next Wednesday, at least. If Comic Strip Master Command works with me, there could be more. But this all will depend on my emotional and energy reserves.

Some of my faithful readers may wonder: am I preparing to say something sad about this year’s A-to-Z? I’m not prepared to say, not yet. What I am is thinking about whether I want to commit to such a big, hard project. I am aware how much it would tax me to do, and while I would like to have it done, there is so much doing to get there. It will depend on how June treats me.

## About Chances of Winning on The Price Is Right, Again

While I continue to wait for time and muse and energy and inspiration to write fresh material, let me share another old piece. This bit from a decade ago examines statistical quirks in The Price Is Right. Game shows offer a lot of material for probability questions. The specific numbers have changed since this was posted, but, the substance hasn’t. I got a bunch of essays out of one odd incident mentioned once on the show, and let me do something useful with that now.

To the serious game show fans: Yes, I am aware that the “Item Up For Bid” is properly called the “One-Bid”. I am writing for a popular audience. (The name “One-Bid” comes from the original, 1950s, run of the show, when the game was entirely about bidding for prizes. A prize might have several rounds of bidding, or might have just the one, and that format is the one used for the Item Up For Bid for the current, 1972-present, show.)

Putting together links to all my essays about trapezoid areas made me realize I also had a string of articles examining that problem of The Price Is Right, with Drew Carey’s claim that only once in the show’s history had all six contestants winning the Item Up For Bids come from the same seat in Contestants’ Row. As with the trapezoid pieces they form a more or less coherent whole, so, let me make it easy for people searching the web for the likelihood of clean sweeps or of perfect games on The Price Is Right to find my thoughts.

## Something Neat About Triangles, Again

I apologize for not having anything fresh to share today. It’s been a difficult week, one of many. So I would like to share something from years ago, and something I still find delightful.

I was reading a biography of Donald Coxeter, one of the most important geometers of the 20th century, and it mentioned in passing something Coxeter referred to as Morley’s Miracle Theorem. The theorem was proved in 1899 by Frank Morley, who taught at Haverford College (if that sounds vaguely familiar that’s because you remember it’s where Dave Barry went) and then Johns Hopkins (which may be familiar on the strength of its lacrosse team), and published this in the first issue of the Transactions of the American Mathematical Society. And, yes, perhaps it isn’t actually important, but the result is so unexpected and surprising that I wanted to share it with you. The biography also includes a proof Coxeter wrote for the theorem, one that’s admirably straightforward, but let me show the result without the proof so you can wonder about it.

First, start by drawing a triangle. It doesn’t have to have any particular interesting properties other than existing. I’ve drawn an example one.

The next step is to cut into three equal pieces each of the interior angles of the triangle, and draw those lines. I’m doing that in separate diagrams for each of the triangle’s three original angles because I want to better suggest the process.

I should point out, this trisection of the angles can be done however you like, which is probably going to be by measuring the angles with a protractor and dividing the angle by three. I made these diagrams just by sketching them out, so they aren’t perfect in their measure, but if you were doing the diagram yourself on a sheet of scratch paper you wouldn’t bother getting the protractor out either. (And, famously, you can’t trisect an angle if you’re using just compass and straightedge to draw things, but you don’t have to restrict yourself to compass and straightedge for this.)

Now the next bit is to take the points where adjacent angle trisectors intersect — that is, for example, where the lower red line crosses the lower green line; where the upper red line crosses the left blue line; and where the right blue line crosses the upper green line. Draw lines connecting these points together and …

This new triangle, drawn in purple on my sketch, is an equilateral triangle!

(It may look a little off, but that’s because I didn’t measure the trisectors when I drew them in and just eyeballed it. If I had measured the angles and drawn the new ones in carefully, it would have been perfect.)

I’ve been thinking back on this and grinning ever since reading it. I certainly didn’t see that punch line coming.

## Reading the Comics, May 7, 2022: Does Comic Strip Master Command Not Do Mathematics Anymore Edition?

I mentioned in my last Reading the Comics post that it seems there are fewer mathematics-themed comic strips than there used to be. I know part of this is I’m trying to be more stringent. You don’t need me to say every time there’s a Roman numerals joke or that blackboards get mathematics symbols put on them. Still, it does feel like there’s fewer candidate strips. Maybe the end of the 2010s was a boom time for comic strips aimed at high school teachers and I only now appreciate that? Only further installments of this feature will let us know.

Jim Benton’s Jim Benton Cartoons for the 18th of April, 2022 suggests an origin for those famous overlapping circle pictures. This did get me curious what’s known about how John Venn came to draw overlapping circles. There’s no reason he couldn’t have used triangles or rectangles or any shape, after all. It looks like the answer is nobody really knows.

Venn, himself, didn’t name the diagrams after himself. Wikipedia credits Charles Dodgson (Lewis Carroll) as describing “Venn’s Method of Diagrams” in 1896. Clarence Irving Lewis, in 1918, seems to be the first person to write “Venn Diagram”. Venn wrote of them as “Eulerian Circles”, referencing the Leonhard Euler who just did everything. Sir William Hamilton — the philosopher, not the quaternions guy — posthumously published the Lectures On Metaphysics and Logic which used circles in these diagrams. Hamilton asserted, correctly, that you could use these to represent logical syllogisms. He wrote that the 1712 logic text Nucleus Logicae Weisianae — predating Euler — used circles, and was right about that. He got the author wrong, crediting Christian Weise instead of the correct author, Johann Christian Lange.

With 1712 the trail seems to end to this lay person doing a short essay’s worth of research. I don’t know what inspired Lange to try circles instead of any other shape. My guess, unburdened by evidence, is that it’s easy to draw circles, especially back in the days when every mathematician had a compass. I assume they weren’t too hard to typeset, at least compared to the many other shapes available. And you don’t need to even think about setting them with a rotation, the way a triangle or a pentagon might demand. But I also would not rule out a notion that circles have some connotation of perfection, in having infinite axes of symmetry and all points on them being equal in distance from the center and such. Might be the reasons fit in the intersection of the ethereal and the mundane.

Daniel Beyer’s Long Story Short for the 29th of April, 2022 puts out a couple of concepts from mathematical physics. These are all about geometry, which we now see as key to understanding physics. Particularly cosmology. The no-boundary proposal is a model constructed by James Hartle and Stephen Hawking. It’s about the first $10^{-43}$ seconds of the universe after the Big Bang. This is an era that was so hot that all our well-tested models of physical law break down. The salient part of the Hartle-Hawking proposal is the idea that in this epoch time becomes indistinguishable from space. If I follow it — do not rely on my understanding for your thesis defense — it’s kind of the way that stepping away from the North Pole first creates the ideas of north and south and east and west. It’s very hard to think of a way to test this which would differentiate it from other hypotheses about the first instances of the universe.

The Weyl Curvature is a less hypothetical construct. It’s a tensor, one of many interesting to physicists. This one represents the tidal forces on a body that’s moving along a geodesic. So, for example, how the moon of a planet gets distorted over its orbit. The Weyl Curvature also offers a way to describe how gravitational waves pass through vacuum. I’m not aware of any serious question of the usefulness or relevance of the thing. But the joke doesn’t work without at least two real physics constructs as setup.

Liniers’ Macanudo for the 5th of May, 2022 has one of the imps who inhabit the comic asserting responsibility for making mathematics work. It’s difficult to imagine what a creature could do to make mathematics work, or to not work. If pressed, we would say mathematics is the set of things we’re confident we could prove according to a small, pretty solid-seeming set of logical laws. And a somewhat larger set of axioms and definitions. (Few of these are proved completely, but that’s because it would involve a lot of fiddly boring steps that nobody doubts we could do if we had to. If this sounds sketchy, consider: do you believe my claim that I could alphabetize the books on the shelf to my right, even though I’ve never done that specific task? Why?) It would be like making a word-search puzzle not work.

The punch line, the blue imp counting seventeen of the orange imp, suggest what this might mean. Mathematics as a set of statements following some rule, is a niche interest. What we like is how so many mathematical things seem to correspond to real-world things. We can imagine mathematics breaking that connection to the real world. The high temperature rising one degree each day this week may tell us something about this weekend, but it’s useless for telling us about November. So I can imagine a magical creature deciding what mathematical models still correspond to the thing they model. Be careful in trying to change their mind.

And that’s as many comic strips from the last several weeks that I think merit discussion. All of my Reading the Comics posts should be at this link, though. And I hope to have a new one again sometime soon. I’ll ask my contacts with the cartoonists. I have about half of a contact.

## How April 2022 Treated My Mathematics Blog

This past month I moved towards the sort of thing that’s normal for my blog here. Mostly, Reading the Comics posts, with another piece that was about a mathematical curiosity. That is a typical selection of posts when I’m not doing something special, such as an A-to-Z sequence. So, with a new month begun, I like to see how it was received. As usual, I check WordPress’s statistics for the past month, and compare it to the running average for the twelve months leading up to that.

WordPress figures there were 2,121 page views here in April. That’s a little below the running mean of 2,286.8 page views. It’s almost exactly at the running median, though, of 2,122 page views in a month. So this suggests April turned out quite average. There were 1,404 recorded unique visitors. This is below the running mean of 1,602.7 unique visitors, and noticeably below the running median of 1,479. This suggests a month a bit below average.

Per posting, though? That suggests an increasing readership. There were 424.2 page views recorded per posting in April, above the running mean of 301.7 and running median of 302.8. There were 280.8 unique visitors per posting, also well above the 211.1 mean and 211.3 median. That’s not to say every post got 281 visitors, since many of the visitors looked at stuff from before April. This is what keeps me from re-blogging even more repeats.

That it was a slow month seems supported by the record of likes and comments, though. There were 19 likes given in April, well below the mean of 39.5 and median of 39. That’s a little less bad considered per posting, but still. That’s 3.8 likes per posting, below the running mean of 5.0 and running median of 4.5. There were an anemic two comments, way below the mean of 11.3 and median of 9.5. That’s just 0.4 comments per posting, compared to an already not-great mean of 1.4 and median of 1.2.

I had thought I posted more in April than a mere five pieces. Not so. Here’s the order of popularity of my posts, which are not quite in chronological order. I too quirk an eye at what the most popular thing of April was:

WordPress figures I posted 3,089 words in April, my fewest since September. And that comes to an average of 617.8 words per posting, again my lowest since September. For the year I’ve published 36,947 words, and have averaged 1,056 words per posting.

I started May with a total of 159,259 recorded page views from a recorded 95,907 unique visitors. But WordPress didn’t start telling us unique visitor counts until my blog here was a couple years old, so don’t take that too literally.

## How to Add Up Powers of Numbers

Do you need to know the formula to tell you what the sum of the first N counting numbers, raised to a power? No, you do not. Not really. It can save a bit of time to know the sum of the numbers raised to the first power. Most mathematicians would know it, or be able to recreate it fast enough:

$\sum_{n = 1}^{N} n = 1 + 2 + 3 + \cdots + N = \frac{1}{2}N\left(N + 1\right)$

But there are similar formulas to add up, say, the counting numbers squared, or cubed, or so. And a toot on Mathstodon, the mathematics-themed instance of social network Mastodon, makes me aware of a cute paper about this. In it Dr Alessandro Mariani describes A simple mnemonic to compute sums of powers.

It’s a neat one. Mariani describes a way to use knowledge of the sum of numbers to the first power to generate a formula for the sum of squares. And then to use the sum of squares formula to generate the sum of cubes. The sum of cubes then lets you get the sub of fourth-powers. And so on. This takes a while to do if you’re interested in the sum of twentieth powers. But do you know how many times you’ll ever need to generate that formula? Anyway, as Mariani notes, this sort of thing is useful if you find yourself at a mathematics competition. Or some other event where you can’t just have the computer calculate this stuff.

Mariani’s process is a great one. Like many mnemonics it doesn’t make literal sense. It expects one to integrate and differentiate polynomials. Anyone likely to be interested in a formula for the sums of twelfth powers knows how to do those in their sleep. But they’re integrating and differentiating polynomials for which, in context, the integrals and derivatives don’t exist. Or at least don’t mean anything. That’s all right. If all you want is the right answer, it’s okay to get there by a wrong method. At least if you verify the answer is right, which the last section of Mariani’s paper does. So, give it a read if you’d like to see a neat mathematical trick to a maybe useful result.

## Reading the Comics, April 17, 2022: Did I Catch Comic Strip Master Command By Surprise Edition

Part of the thrill of Reading the Comics posts is that the underlying material is wholly outside my control. The subjects discussed, yes, although there are some quite common themes. (Students challenging the word problem; lottery jokes; monkeys at typewriters.) But also quantity. Part of what burned me out on Reading the Comics posts back in 2020 was feeling the need to say something about lots of comic strips . Now?

I mentioned last week seeing only three interesting strips, and one of them, Andertoons, was a repeat I’d already discussed. This week there were only two strips that drew a first note and again, Andertoons was a repeat I’d already discussed. Mark Anderson’s comic for the 17th I covered in enough detail back in August of 2019. I don’t know how many new Andertoons are put into the rotation at GoComics. But the implication is Comic Strip Master Command ordered mathematics-comics production cut down, and they haven’t yet responded to my doing these again. I guess we’ll know for sure if things pick up in a couple weeks, as the lead time allows.

So Rick McKee and Kent Sligh’s Mount Pleasant for the 15th of April is all I have to discuss. It’s part of the long series of students resisting the teacher’s question. The teacher is asking a fair enough question, that of how to do a problem that has several parts. She does ask how we “should” solve the problem of finding what 4 + 4 – 2 equals. The catch is there are several ways to do this, all of them as good. We know this if we’ve accepted subtraction as a kind of addition, and if we’ve accepted addition as commutative.

So the order is our choice. We can add 4 and 4 and then subtract 2. Or subtract 2 from the second 4, and then add that to the first 4. If you want, and can tell the difference, you could subtract 2 from the first 4, and then add the second 4 to that.

For this problem it doesn’t make any difference. But one can imagine similar ones where the order you tackle things in can make calculations easier, or harder. 5 + 7 – 2, for example, I find easier if I work it out as 5 + ( 7 – 2), that is, 5 + 5. So it’s worth taking a moment to consider whether rearranging it can make the calculation more reliable. I don’t know whether the teacher meant to challenge the students to see that there are alternatives, and no uniquely “right” answer. It’s possible McKee and Sligh did not have the teaching plan worked out.

That makes for another week’s worth of comic strips to discuss. All of my Reading the Comics posts should be at this link. Thanks for reading this and I will let you know if Comic Strip Master Command increases production of comics with mathematics themes.

## Reading the Comics, April 10, 2022: Quantum Entanglement Edition

I remember part of why I stopped doing Reading the Comics posts regularly was their volume. I read a lot of comics and it felt like everyone wanted to do a word problem joke. Since I started easing back into these posts it’s seemed like they’ve disappeared. When I put together this week’s collection, I only had three interesting ones. And one was Andertoons for the 10th of April. Andertoons is a stalwart here, but this particular strip was one I already talked about, back in 2019.

Another was the Archie repeat for the 10th of April. And that only lists mathematics as a school subject. It would be the same joke if it were English lit. Saying “differential calculus” gives it the advantage of specificity. It also suggests Archie is at least a good enough student to be taking calculus in high school, which isn’t bad. Differential calculus is where calculus usually starts, with the study of instantaneous changes. A person can, and should, ask how a change can be instantaneous. Part of what makes differential calculus is learning how to find something that matches our intuition about what it should be. And that never requires us to do something appalling like divide zero by zero. Our current definition took a couple centuries of wrangling to find a scheme that makes sense. It’s a bit much to expect high school students to pick it up in two months.

Ripley’s Believe It Or Not for the 10th of April, 2022 was the most interesting piece. This referenced a problem I didn’t remember having heard about, the “36 Officers puzzle” of Leonhard Euler. Euler’s name you know as he did foundational work in every field of mathematics ever. This particular puzzle ates to 1779, according to an article in Quanta Magazine which one of the Ripley’s commenters offered. Six army regiments each have six officers of six different ranks. How can you arrange them in a six-by-six square so that no row or column repeats a rank or regiment?

The problem sounds like it shouldn’t be hard. The two-by-two version of this is easy. So is three-by-three and four-by-four and even five-by-five. Oddly, seven-by-seven is, too. It looks like some form of magic square, and seems not far off being a sudoku problem either. So it seems weird that six-by-six should be particularly hard, but sometimes it happens like that. In fact, this happens to be impossible; a paper by Gaston Terry in 1901 proved there were none.

The solution discussed by Ripley’s is of a slightly different problem. So I’m not saying to not believe it, just, that you need to believe it with reservations. The modified problem casts this as a quantum-entanglement, in which the rank and regiment of an officer in one position is connected to that of their neighbors. I admit I’m not sure I understand this well enough to explain; I’m not confident I can give a clear answer why a solution of the entangled problem can’t be used for the classical problem.

The problem, at this point, isn’t about organizing officers anymore. It never was, since that started as an idle pastime. Legend has it that it started as a challenge about organizing cards; if you look at the paper you’ll see it presenting states as card suits and values. But the problem emerged from idle curiosity into practicality. These turn out to be applicable to quantum error detection codes. I’m not certain I can explain how myself. You might be able to convince yourself of this by thinking how you know that someone who tells you the sum of six odd numbers is itself an odd number made a mistake somewhere, and you can then look for what went wrong.

And that’s as many comics from last week as I feel like discussing. All my Reading the Comics posts should be gathered at this link. Thanks for reading this and I hope to do this again soon.

## Reading the Comics, April 2, 2022: Pi Day Extra Edition

I’m not sure that I will make a habit of this. It’s been a while since I did a regular Reading the Comics post, looking for mathematics topics in syndicated newspaper comic strips. I thought I might dip my toes in those waters again. Since my Pi Day essay there’ve been only a few with anything much to say. One of them was a rerun I’ve discussed before, too, a Bloom County Sunday strip that did an elaborate calculation to conceal the number 1. I’ve written about that strip twice before, in May 2016 and then in October 2016, so that’s too well-explained to need revisiting.

As it happens two of the three strips remaining were repeats, though ones I don’t think I’ve addressed before here.

Bill Amend’s FoxTrot Classics for the 18th of March looks like a Pi Day strip. It’s not, though: it originally ran the 16th of March, 2001. We didn’t have Pi Day back then.

What Peter Fox is doing is drawing a unit circle — a circle of radius 1 — and dividing it into a couple common angles. Trigonometry students are expected to know the sines and cosines and tangents of a handful of angles. If they don’t know them, they can work these out from first principles. Draw a line from the center of the unit circle at an angle measured counterclockwise from the positive x-axis. Find where that line you’ve just drawn intersects the unit circle. The x-coordinate of that point has the same value as the cosine of that angle. The y-coordinate of that point has the same value as the sine of that angle. And for a handful of angles — the ones Peter marks off in the second panel — you can work them out by reason alone.

These angles we know as, like, 45 degrees or 120 degrees or 135 degrees. Peter writes them as $\frac{\pi}{4}$ or $\frac{2}{3}\pi$ or $\frac{9}{8}\pi$, because these are radian measure rather than degree measure. It’s a different scale, one that’s more convenient for calculus. And for some ordinary uses too: an angle of (say) $\frac{3}{4}\pi$ radians sweeps out an arc of length $\frac{3}{4}\pi$ on the unit circle. You can see where that’s easier to keep straight than how long an arc of 135 degrees might be.

Drawing this circle is a good way to work out or remember sines and cosines for the angles you’re expected to know, which is why you’d get them on a trig test.

Scott Hilburn’s The Argyle Sweater for the 27th of March summons every humorist’s favorite piece of topology, the Möbius strip. Unfortunately the line work makes it look to me like Hilburn’s drawn a simple loop of a steak. Follow the white strip along the upper edge. Could be the restaurant does the best it can with a challenging presentation.

August Ferdinand Möbius by the way was an astronomer, working most of his career at the Observatory at Leipzig. (His work as a professor was not particularly successful; he was too poor a lecturer to keep students.) His father was a dancing teacher, and his mother was a descendant of Martin Luther, although I imagine she did other things too.

Rina Piccolo’s Tina’s Groove for the 2nd of April makes its first appearance in a Reading the Comics post in almost a decade. The strip ended in 2017 and only recently has Comics Kingdom started showing reprints. The strip is about the numerical coincidence between 3.14 of a thing and the digits of π. It originally ran at the end of March, 2007, which like the vintage FoxTrot reminds us how recent a thing Pi Day is to observe.

3.14 hours is three hours, 8.4 minutes, which implies that she clocked in at about 9:56.

And that’s this installment. All my Reading the Comics posts should be at this link. I don’t know when I’ll publish a next one, but it should be there, too. Thanks for reading.

## How March 2022 Treated My Mathematics Blog

I expected readers to be happy I was finishing the Little 2021 Mathematics A-to-Z. My doubt was how happy they would be. Turns out they were a middling amount of happy. So this is my regular review of the readership statistics for the past month, as provided by WordPress.

I published eight things in March, which is average for me the past twelve months. It was a long, long time ago that I went whole months posting something every day. But my twelve-month running mean has been 8.5 posts per month, and the median 8, so that’s just in line. There were 2,272 page views recorded in March, which is below the running mean of 2,336.4 and above the running median of 2,122. So, average, like I said. There were 1,545 unique visitors, below the running mean of 1,640.0 and above the running median of 1,479.

Prorated by posting, the showing is a little worse. There were 284.0 views and 193.1 unique visitors per posting in March. The running mean is 301.9 views and 211.6 visitors per posting. The median, 302.8 views and 211.3 visitors. I have no explanation for this phenomenon.

I have a hypothesis. There were 32 likes given in the month, below the mean of 39.3 and median of 35. But several of the posts were pointers to other essays and those are naturally less well-liked. That came to 4.0 likes per posting, below the mean of 4.9 likes per posting and median of 4.5 likes per posting. Comments were anemic again, with only four given in the month. The mean is an impossible-seeming 11.8 and median 10. Per posting, there were 0.5 comments here in March, compared to a mean of 1.4 and median of 1.2. So it goes.

What was popular in March? Pi Day comic strips, of course, and my making something out of the NCAA March Madness basketball tournament. Here’s the March postings in descending order of popularity.

Stuff from before this past month was popular too, including several of the individual Pi Day pages. And my post about the most and least likely dates for Easter, which is sure to be a seasonal favorite.

WordPress figures that I posted 6,655 words in March, for an average post length of 1,128. If that number seems familiar it does to me too. I had 1,128 words per posting, on average, in January too, an event that caused me to go check that I hadn’t recorded something wrong. But that was also a month with many more posts (many repeats). This brought my average words per post for the year down to 831.9, close to half what my average was at the end of February.

WordPress figures that I started April 2022 with a total of 1,705 posts here. They’d drawn 3,317 comments, with a total 157,138 views from 94,502 recorded unique visitors.

If you’d like to be a regular reader around here, please read. There’s a button at the upper right of the page, “Follow Nebusresearch”. That adds this blog to your WordPress reader. There’s a field below that to get posts e-mailed as they’re published. I do nothing with the e-mail except send those posts. WordPress probably has some incomprehensible page where they say what the do with your e-mails. And if you have an RSS reader, you can put the essays feed into that.

## What I Learned Writing the Little 2021 Mathematics A-to-Z

I try, at the end of each of these A-to-Z sessions, to think about what I’ve learned from the experience. The challenge is reliably interesting, thanks to the kind readers who suggest topics. While I reserve the right to choose my own subject for any letter, I usually go for what of the suggestions sounds most interesting. That nudges me out of my comfortable, familiar thoughts and into topics I know less well. I would never have written about cohomologies if I waited to think I had something to say about them.

I didn’t have any deep experiences like that this time, although I did get a better handle on tangent spaces and why we like them. Most of what I did learn was about process, and about how to approach writing here.

For example, I started appealing for topics more letters ahead than I had previous projects. The goal was to let myself build a reserve, so that I would have a week or more to let an essay sit while I re-thought what I’d said. Early on, this worked well and I liked the results. It also made it easier to tie essays together; multiplication and addition could complement one another. This is something I could expand on.

And varying from the strict alphabetical order seems to have worked too. The advantage of doing every letter in order is that I’m pushed into some unpromising letters, like ‘Q’ or ‘Y’. It’s fantastic when I get a good essay out of that. But that’s harder work. This time around I did three topics starting with A, and three with T, and there’s so many more I could write.

The biggest and hardest thing I learned was related to how my plans went awry. How I lost the several-weeks lead time I started with, and how I had to put the project on hold for nearly three months.

2021 was a hard year, after another hard year, after a succession of hard years. Mostly, these were hard years because the world had been hard. Wearying, which is why I started out doing a mere 15 essays instead of the full 26. But not things that too directly hit my personal comfort. During the Little 2021 A-to-Z, though, the hard got intimate. Personal disasters hit starting in mid-August, and kept progressing — or dragging out — through to the new year. Just in time for the world-hardness of the first Omicron wave of the pandemic.

I have always thought of myself as a Sabbath-is-made-for-Man person. That is, schedules are ways to help you get done what you want or need; they’re not of value in themselves. Yet I do value them. I like their hold, and I thrive within them. Part of my surviving the pandemic, when all normal activities stopped, was the schedule of things I write here and on my humor blog. They offered a reason to do something particular. If I were not living up to this commitment, then what was I doing?

The answer is I would be not stressing myself past what I can do. I like these A-to-Z essays, and all the writing I do, or I wouldn’t do it. It’s nourishing and often exciting. But it is labor, and it is stress. Exercising a bit longer or a bit harder than one feels able to helps one build endurance and strength. But there are times one’s muscles are exhausted, or one’s joints are worked too much, and you must rest. Not just stick to the routine exercise, but take a break so that you can recover. I had not taken a serious break since starting this blog, and hadn’t realized I would need to. Over the course of this A-to-Z I learned I sometimes need to, and I should.

I need also to think of what I will do next. I’m not sure when I will feel confident that I can do a full A-to-Z, or even a truncated version. My hunch is I need to do more mathematical projects here that are fun and playful. This implies thinking of fun and playful projects, and thinking is the hard part again. But I understand, in a way I had not before, that I can let go.

The whole of the Little 2021 Mathematics A-to-Z sequence should be at this link. And then at this link should be all of the A-to-Z essays from all past years. Thank you.

## What I Wrote About In My Little 2021 Mathematics A to Z

It’s good to have an index of the topics I wrote about for each of my A-to-Z sequences. It’s good for me, at least. It makes my future work much easier. And it might help people find past essays. I hope to have my essay about what I learned from a project that was supposed to be nearly one-third shorter, and ended up sprawling past its designated year, next week.

All of the Little 2021 Mathematics A-to-Z essays should be at this link. And gathered at this link should be all of the A-to-Z essays from all past years. Thank you for your reading.

## Reading the Comics, March 14, 2022: Pi Day Edition

As promised I have the Pi Day comic strips from my reading here. I read nearly all the comics run on Comics Kingdom and on GoComics, no matter how hard their web sites try to avoid showing comics. (They have some server optimization thing that makes the comics sometimes just not load.) (By server optimization I mean “tracking for advertising purposes”.)

Pi Day in the comics this year saw the event almost wholly given over to the phonetic coincidence that π sounds, in English, like pie. So this is not the deepest bench of mathematical topics to discuss. My love, who is not as fond of wordplay as I am, notes that the ancient Greeks likely pronounced the name of π about the same way we pronounce the letter “p”. This may be etymologically sound, but that’s not how we do it in English, and even if we switched over, that would not make things better.

Scott Hilburn’s The Argyle Sweater is one of the few strips not to be about food. It is set in the world of anthropomorphized numerals, the other common theme to the day.

John Hambrook’s The Brilliant Mind of Edison Lee leads off with the food jokes, in this case cookies rather than pie. The change adds a bit of Abbott-and-Costello energy to the action.

Mick Mastroianni and Mason Mastroianni’s Dogs of C Kennel gets our first pie proper, this time tossed in the face. One of the commenters observes that the middle of a pecan pie can really hold heat, “Ouch”. Will’s holding it in his bare paw, though, so it can’t be that bad.

Jules Rivera’s Mark Trail makes the most casual Pi Day reference. If the narrator hadn’t interrupted in the final panel no one would have reason to think this referenced anything.

Mark Parisi’s Off The Mark is the other anthropomorphic numerals joke for the day. It’s built on the familiar fact that the digits of π go on forever. This is true for any integer base. In base π, of course, the representation of π is just “10”. But who uses that? And in base π, the number six would be something with infinitely many digits. There’s no fitting that in a one-panel comic, though.

Doug Savage’s Savage Chickens is the one strip that wasn’t about food or anthropomorphized numerals. There is no practical reason to memorize digits of π, other than that you’re calculating something by hand and don’t want to waste time looking them up. In that case there’s not much call go to past 3.14. If you need more than about 3.14159, get a calculator to do it. But memorizing digits can be fun, and I will not underestimate the value of fun in getting someone interested in mathematics.

For my part, I memorized π out to 3.1415926535787932, so that’s sixteen digits past the decimal. Always felt I could do more and I don’t know why I didn’t. The next couple digits are 8462, which has a nice descending-fifths cadence to it. The 626 following is a neat coda. My describing it this way may give you some idea to how I visualize the digits of π. They might help you, if you figure for some reason you need to do this. You do not, but if you enjoy it, enjoy it.

Bianca Xunise’s Six Chix for the 15th ran a day late; Xunise only gets the comic on Tuesdays and the occasional Sunday. It returns to the food theme.

And this brings me to the end of this year’s Pi Day comic strips. All of my Reading the Comics posts, past and someday future, should be at this link. And my various Pi Day essays should be here. Thank you for reading.

## Let Me Remind You How Interesting a Basketball Tournament Is

Several years ago I stumbled into a nice sequence. All my nice sequences have been things I stumbled upon. This one looked at the most basic elements of information theory by what they tell us about the NCAA College Basketball tournament. This is (in the main) a 64-team single-elimination playoff. It’s been a few years since I ran through the sequence. But it’s been a couple years since the tournament could be run with a reasonably clear conscience too. So here’s my essays:

And this spins off to questions about other sports events.

And I still figure to get to this year’s Pi Day comic strips. Soon. It’s been a while since I felt I had so much to write up.

## Here Are Past Years’ Pi Day Comic Strips

I haven’t yet read today’s comics; it takes a while to get through them. But I hope to summarize what Comic Strip Master Command has sent out for the syndicated comics for today. In the meanwhile, here’s Pi Day strips of past years.

And I have to offer a warning. GoComics.Com has discontinued a lot of comics in the past couple years. They’ve been brutal about removing the archives of strips they’ve discontinued. Comics Kingdom is similarly ruthless in removing strips not in production. And a recent and, to the user, bad code update broke a lot of what had been non-expiring links. But my discussions of the themes in the comic are still there. And, as I got more into the Reading the Comics project I got more likely to include the original comic. So that’s some compensation.

Here’s the past several years in comics from on or around the 14th of March:

• 2015, featuring The Argyle Sweater, Baldo, The Chuckle Brothers, Dog Eat Doug, FoxTrot Classics, Herb and Jamaal, Long Story Short, The New Adventures of Queen Victoria, Off The Mark, and Working Daze.
• 2016, featuring The Argyle Sweater, B.C., Brewster Rockit, The Brilliant Mind of Edison Lee, Curtis, Dog Eat Doug, F Minus, Free Range, and Holiday Doodles.
• 2017, featuring 2 Cows and a Chicken, Archie, The Argyle Sweater, Arlo and Janis, Lard’s World Peace Tips, Loose Parts, Off The Mark, Saturday Morning Breakfast Cereal, TruthFacts, and Working Daze.
• 2018, featuring The Argyle Sweater, Bear With Me, Funky Winterbean Classic, Mutt and Jeff, Off The Mark, Savage Chickens, Warped, and Working Daze.
• 2019, featuring The Brilliant Mind of Edison Lee, Liz Climo’s Cartoons, The Grizzwells, Off The Mark, and Working Daze.
• 2020, featuring Baldo, Calvin and Hobbes, Off The Mark, Real Life Adventures, Reality Check, and Warped.
• 2021, featuring Agnes, The Argyle Sweater, Between Friends, Breaking Cat News, FoxTrot, Frazz, Get Fuzzy, Heart of the City, Reality Check, and Studio Jantze.

As mentioned, I have yet to read today’s comics. I’m looking forward to it, at least to learn what Funky Winkerbean character I’m going to be most annoyed with this week. It will be Les Moore. I was also going to look forward to seeing if there would ever be a Pi Day strips roundup without The Argyle Sweater or Reality Check. It turns out there was one in 2019. Weird how you can get the impression something is always there even when it’s not.

## My Little 2021 Mathematics A-to-Z: Zorn’s Lemma

The joke to which I alluded last week was a quick pun. The setup is, “What is yellow and equivalent to the Axiom of Choice?” It’s the topic for this week, and the conclusion of the Little 2021 Mathematics A-to-Z. I again thank Mr Wu, of Singapore Maths Tuition, for a delightful topic.

# Zorn’s Lemma

Max Zorn did not name it Zorn’s Lemma. You expected that. He thought of it just as a Maximal Principle when introducing it in a 1934 presentation and 1935 paper. The word “lemma” connotes that some theorem is a small thing. It usually means it’s used to prove some larger and more interesting theorem. Zorn’s Lemma is one of those small things. With the right background, a rigorous proof is a couple not-too-dense paragraphs. Without the right background? It’s one of those proofs you read the statement of and nod, agreeing, that sounds reasonable.

The lemma is about partially ordered sets. A set’s partially ordered if it has a relationship between pairs of items in it. You will sometimes see a partially ordered set called a “poset”, a term of mathematical art which make me smile too. If we don’t know anything about the ordering relationship we’ll use the ≤ symbol, just like this was ordinary numbers. To be partially ordered, whenever x ≤ y and y ≤ x, we know that x and y must be equal. And the converse: if x = y then x ≤ y and y ≤ x. What makes this partial is that we’re not guaranteed that every x and y relate in some way. It’s a totally ordered set if we’re guaranteed that at least one of x ≤ y and y ≤ x is always true. And then there is such a thing as a well-ordered set. This is a totally ordered set for which every subset (unless it’s empty) has a minimal element.

If we have a couple elements, each of which we can put in some order, then we can create a chain. If x ≤ y and y ≤ z, then we can write x ≤ y ≤ z and we have at least three things all relating to one another. This seems like stuff too basic to notice, if we think too literally about the relationship being “is less than or equal to”. If the relationship is, say, “divides wholly into”, then we get some interesting different chains. Like, 2 divides into 4, which divides into 8, which divides into 24. And 3 divides into 6 which divides into 24. But 2 doesn’t divide into 3, nor 3 into 2. 4 doesn’t divide into 6, nor 6 into either 8 or 4.

So what Zorn’s Lemma says is, if all the chains in a partially ordered set each have an upper bound, then, the partially ordered set has a maximal element. “Maximal element” here means an element that doesn’t have a bigger comparable element. (That is, m is maximal if there’s no other element b for which m ≤ b. It’s possible that m and b can’t be compared, though, the way 6 doesn’t divide 8 and 8 doesn’t divide 6.) This is a little different from a “maximum” . It’s possible for there to be several maximal elements. But if you parse this as “if you can always find a maximum in a string of elements, there’s some maximum element”? And remember there could be many maximums? Then you’re getting the point.

You may also ask how this could be interesting. Zorn’s Lemma is an existence proof. Most existence proofs assure us a thing we thought existed does, but don’t tell us how to find it. This is all right. We tend to rely on an existence proof when we want to talk about some mathematical item but don’t care about fussy things like what it is. It is much the way we might talk about “an odd perfect number N”. We can describe interesting things that follow from having such a number even before we know what value N has.

A classic example, the one you find in any discussion of using Zorn’s Lemma, is about the basis for a vector space. This is like deciding how to give directions to a point in space. But vector spaces include some quite abstract things. One vector space is “the set of all functions you can integrate”. Another is “matrices whose elements are all four-dimensional rotations”. There might be literally infinitely many “directions” to go. How do we know we can find a set of directions that work as well as, for guiding us around a city, the north-south-east-west compass rose does? So there’s the answer. There are other things done all the time, too. A nontrivial ring-with-identity, for example, has to have a maximal ideal. (An ideal is a subset of the ring that’s still a ring.) This is handy to know if you’re working with rings a lot.

The joke in my prologue was built on the claim Zorn’s Lemma is equivalent to the Axiom of Choice. The Axiom of Choice is a piece of set theory that surprised everyone by being independent of the Zermelo-Fraenkel axioms. The Axiom says that, if you have a collection of disjoint nonempty sets, then there must exist at least one set with exactly one element from each of those sets. That is, you can pick one thing out of each of a set of bins. It’s easy to see how this has in common with Zorn’s Lemma being too obvious to imagine proving. That’s the sort of thing that makes a good axiom. Thing about a lemma, though, is we do prove it. That’s how we know it’s a lemma. How can a lemma be equivalent to an axiom?

I’l argue by analogy. In Euclidean geometry one of the axioms is this annoying statement about on which side of a line two other lines that intersect it will meet. If you have this axiom, you can prove some nice results, like, the interior angles of a triangle add up to two right angles. If you decide you’d rather make your axiom that bit about the interior angles adding up? You can go from that to prove the thing about two lines crossing a third line.

So it is here. If you suppose the Axiom of Choice is true, you can get Zorn’s Lemma: you can pick an element in your set, find a chain for which that’s the minimum, and find your maximal element from that. If you make Zorn’s Lemma your axiom? You can use x ≤ y to mean “x is a less desirable element to pick out of this set than is y”. And then you can choose a maximal element out of your set. (It’s a bit more work than that, but it’s that kind of work.)

There’s another theorem, or principle, that’s (with reservations) equivalent to both Zorn’s Lemma and the Axiom of Choice. It’s another piece that seems so obvious it should defy proof. This is the well-ordering theorem, which says that every set can be well-ordered. That is, so that every non-empty subset has some minimum element. Finally, a mathematical excuse for why we have alphabetical order, even if there’s no clear reason that “j” should come after “i”.

(I said “with reservations” above. This is because whether these are equivalent depends on what, precisely, kind of deductive logic you’re using. If you are not using ordinary propositional logic, and are using a “second-order logic” instead, they differ.)

Ermst Zermelo introduced the Axiom of Choice to set theory so that he could prove this in a way that felt reasonable. I bet you can imagine how you’d go from “every non-empty set has a minimum element” right back to “you can always pick one element of every set”, though. And, maybe if you squint, can see how to get from “there’s always a minimum” to “there has to be a maximum”. I’m speaking casually here because proving it precisely is more work than we need to do.

I mentioned how Zorn did not name his lemma after himself. Mathematicians typically don’t name things for themselves. Nor did he even think of it as a lemma. His name seems to have adhered to the principle in the late 30s. Credit the nonexistent mathematician Bourbaki writing about “le théorème de Zorn”. By 1940 John Tukey, celebrated for the Fast Fourier Transform, wrote of “Zorn’s Lemma”. Tukey’s impression was that this is how people in Princeton spoke of it at the time. He seems to have been the first to put the words “Zorn’s Lemma” in print, though. Zorn isn’t the first to have stated this. Kazimierez Kuratowski, in 1922, described what is clearly Zorn’s Lemma in a different form. Zorn remembered being aware of Kuratowski’s publication but did not remember noticing the property. The Hausdorff Maximal Principle, of Felix Hausdorff, has much the same content. Zorn said he did not know about Hausdorff’s 1927 paper until decades later.

Zorn’s lemma, the Axiom of Choice, the well-ordering theorem, and Hausdorff’s Maximal Principle all date to the early 20th century. So do a handful of other ideas that turn out to be equivalent. This was an era when set theory saw an explosive development of new and powerful ideas. The point of describing this chain is to emphasize that great concepts often don’t have a unique presentation. Part of the development of mathematics is picking through several quite similar expressions of a concept. Which one do we enshrine as an axiom, or at least the canonical presentation of the idea?

We have to choose.

And with this I at last declare the hard work Little 2021 Mathematics A-to-Z at an end. I plan to follow up, as traditional, with a little essay about what I learned while doing this project. All of the Little 2021 Mathematics A-to-Z essays should be at this link. And then all of the A-to-Z essays from all eight projects should be at this link. Thank you so for your support in these difficult times.

## How February 2022 Treated My Mathematics Blog

This past month I finished my hiatus, the one where I reran old A-to-Z pieces instead of finishing off what I thought would be a simple, small project for 2021. And, after a mishap, got back to finishing things. As a result I published fewer pieces in February than I had since October. I had an inflated posting record in December and January, from reposting old material. I expected that end to shrink my readership again. And, yes, that’s what happened.

In February, according to WordPress, I attracted 1,875 page views. That’s below the twelve-month running mean of 2,360.8 page views leading up to February 2022. It’s also below the running median of 2,151.5 page views. In fact, it’s the lowest number of page views in a month going back to July 2020, around here.

Ah, but what about unique visitors? There were 1,313 of those, figures WordPress. That’s below the twelve-month running mean of 1,661.9 and the running median of 1,534.5. It happens that’s also the lowest monthly figure going back to July 2020. (Although that by a whisker: July 2021 had a couple more views, and unique visitors, than did February 2022. I don’t know what’s wrong with Julys around here.)

The number of likes dropped to 28, way below the mea of 40.9 and median of 39.5. And that was the lowest count since November of 2021. And there were only two comments, way below the mean of 14.9 and median of 10, I haven’t been below that figure since December of 2019. At least these are non-July dates to deal with.

This would all be too sad to bear except that if you look at these figures per posting? Then they snap right back into line. Like, this was in February an average of 312.5 page views every time I posted something. The twelve months leading up to that saw a mean of 301.6 page views per posting and a median of 302.8 page views per posting. February saw 218.8 unique visitors per posting. The running mean was 212.2 and running median 211.3. Even the likes become not so bad: 4.7 per posting. The mean was 5.1 and the median 4.9. In this figuring, the only dire number was comments, a scant 0.3 per posting, compared to mean of 1.9 and median of 1.4. So in that light, you know, things aren’t so bad.

What are the popular things of February? It’s worth running the whole list down. In decreasing order of popularity we have:

Other stuff, from before February, was even more popular, though. It’s getting to be the time of year people look to learn what the most and least likely dates of Easter are, for example. (Easter 2022 is set for the 17th of April. This is on the less-likely side of the band from the 28th of March through 21st of April when Easter is most likely. However, it is one of the most likely dates for Easter in the lifetime of anyone reading this blog, that is, for the span from 1925 to 2100.)

WordPress credits me with publishing 9,163 words in February, for an average post length of 1,527.2 words. This brings my average post length for the year up to 1,237. This is impressive considering I’ve been trying to write my A-to-Zs short for 2021.

WordPress figures that I started March 2022 having posted 1,697 things here. They’ve altogether drawn 3,313 comments from a total 154,866 page views and 92,956 logged unique visitors.

If you’d like to be a regular reader around here, please keep reading. There’s a button at the upper right of the page, “Follow Nebusresearch”, to add this blog to your WordPress reader. There’s a field below that to get posts sent to you in e-mail as they’re published. I do nothing with the e-mail except send those posts; I can’t say what WordPress Master Command does with them. And if you have an RSS reader, you can put the essays feed into that.

## My Little 2021 Mathematics A-to-Z: Ordinary Differential Equations

Mr Wu, my Singapore Maths Tuition friend, has offered many fine ideas for A-to-Z topics. This week’s is another of them, and I’m grateful for it.

# Ordinary Differential Equations

As a rule, if you can do something with a number, you can do the same thing with a function. Not always, of course, but the exceptions are fewer than you might imagine. I’ll start with one of those things you can do to both.

A powerful thing we learn in (high school) algebra is that we can use a number without knowing what it is. We give it a name like ‘x’ or ‘y’ and describe what we find interesting about it. If we want to know what it is, we (usually) find some equation or set of equations and find what value of x could make that true. If we study enough (college) mathematics we learn its equivalent in functions. We give something a name like f or g or Ψ and describe what we know about it. And then try to find functions which make that true.

There are a couple common types of equation for these not-yet-known functions. The kind you expect to learn as a mathematics major involves differential equations. These are ones where your equation (or equations) involve derivatives of the not-yet-known f. A derivative describes the rate at which something changes. If we imagine the original f is a position, the derivative is velocity. Derivatives can have derivatives also; this second derivative would be the acceleration. And then second derivatives can have derivatives also, and so on, into infinity. When an equation involves a function and its derivatives we have a differential equation.

(The second common type is the integral equation, using a function and its integrals. And a third involves both derivatives and integrals. That’s known as an integro-differential equation, and isn’t life complicated enough? )

Differential equations themselves naturally divide into two kinds, ordinary and partial. They serve different roles. Usually an ordinary differential equation we can describe the change for from knowing only the current situation. (This may include velocities and accelerations and stuff. We could ask what the velocity at an instant means. But never mind that here.) Usually a partial differential equation bases the change where you are on the neighborhood of where your location. If you see holes you can pick in that, you’re right. The precise difference is about the independent variables. If the function f has more than one independent variable, it’s possible to take a partial derivative. This describes how f changes if one variable changes while the others stay fixed. If the function f has only the one independent variable, you can only take ordinary derivatives. So you get an ordinary differential equation.

But let’s speak casually here. If what you’re studying can be fully represented with a dashboard readout? Like, an ordered list of positions and velocities and stuff? You probably have an ordinary differential equation. If you need a picture with a three-dimensional surface or a color map to understand it? You probably have a partial differential equation.

One more metaphor. If you can imagine the thing you’re modeling as a marble rolling around on a hilly table? Odds are that’s an ordinary differential equation. And that representation covers a lot of interesting problems. Marbles on hills, obviously. But also rigid pendulums: we can treat the angle a pendulum makes and the rate at which those change as dimensions of space. The pendulum’s swinging then matches exactly a marble rolling around the right hilly table. Planets in space, too. We need more dimensions — three space dimensions and three velocity dimensions — for each planet. So, like, the Sun-Earth-and-Moon would be rolling around a hilly table with 18 dimensions. That’s all right. We don’t have to draw it. The mathematics works about the same. Just longer.

[ To be precise we need three momentum dimensions for each orbiting body. If they’re not changing mass appreciably, and not moving too near the speed of light, velocity is just momentum times a constant number, so we can use whichever is easier to visualize. ]

We mostly work with ordinary differential equations of either the first or the second order. First order means we have first derivatives in the equation, but never have to deal with more than the original function and its first derivative. Second order means we have second derivatives in the equation, but never have to deal with more than the original function or its first or second derivatives. You’ll never guess what a “third order” differential equation is unless you have experience in reading words. There are some reasons we stick to these low orders like first and second, though. One is that we know of good techniques for solving most first- and second-order ordinary differential equations. For higher-order differential equations we often use techniques that find a related normal old polynomial. Its solution helps with the thing we want. Or we break a high-order differential equation into a set of low-order ones. So yes, again, we search for answers where the light is good. But the good light covers many things we like to look at.

There’s simple harmonic motion, for example. It covers pendulums and springs and perturbations around stable equilibriums and all. This turns out to cover so many problems that, as a physics major, you get a little sick of simple harmonic motion. There’s the Airy function, which started out to describe the rainbow. It turns out to describe particles trapped in a triangular quantum well. The van der Pol equation, about systems where a small oscillation gets energy fed into it while a large oscillation gets energy drained. All kinds of exponential growth and decay problems. Very many functions where pairs of particles interact.

This doesn’t cover everything we would like to do. That’s all right. Ordinary differential equations lend themselves to numerical solutions. It requires considerable study and thought to do these numerical solutions well. But this doesn’t make the subject unapproachable. Few of us could animate the “Pink Elephants on Parade” scene from Dumbo. But could you draw a flip book of two stick figures tossing a ball back and forth? If you’ve had a good rest, a hearty breakfast, and have not listened to the news yet today, so you’re in a good mood?

The flip book ball is a decent example here, too. The animation will look good if the ball moves about the “right” amount between pages. A little faster when it’s first thrown, a bit slower as it reaches the top of its arc, a little faster as it falls back to the catcher. The ordinary differential equation tells us how fast our marble is rolling on this hilly table, and in what direction. So we can calculate how far the marble needs to move, and in what direction, to make the next page in the flip book.

Almost. The rate at which the marble should move will change, in the interval between one flip-book page and the next. The difference, the error, may not be much. But there is a difference between the exact and the numerical solution. Well, there is a difference between a circle and a regular polygon. We have many ways of minimizing and estimating and controlling the error. Doing that is what makes numerical mathematics the high-paid professional industry it is. Our game of catch we can verify by flipping through the book. The motion of four dozen planets and moons attracting one another is harder to be sure we calculate it right.

I said at the top that most anything one can do with numbers one can do with functions also. I would like to close the essay with some great parallel. Like, the way that trying to solve cubic equations made people realize complex numbers were good things to have. I don’t have a good example like that for ordinary differential equations, where the study expanded our ideas of what functions could be. Part of that is that complex numbers are more accessible than the stranger functions. Part of that is that complex numbers have a story behind them. The story features titanic figures like Gerolamo Cardano, Niccolò Tartaglia and Ludovico Ferrari. We see some awesome and weird personalities in 19th century mathematics. But their fights are generally harder to watch from the sidelines and cheer on. And part is that it’s easier to find pop historical treatments of the kinds of numbers. The historiography of what a “function” is is a specialist occupation.

But I can think of a possible case. A tool that’s sometimes used in solving ordinary differential equations is the “Dirac delta function”. Yes, that Paul Dirac. It’s a weird function, written as $\delta(x)$. It’s equal to zero everywhere, except where $x$ is zero. When $x$ is zero? It’s … we don’t talk about what it is. Instead we talk about what it can do. The integral of that Dirac delta function times some other function can equal that other function at a single point. It strains credibility to call this a function the way we speak of, like, $sin(x)$ or $\sqrt{x^2 + 4}$ being functions. Many will classify it as a distribution instead. But it is so useful, for a particular kind of problem, that it’s impossible to throw away.

So perhaps the parallels between numbers and functions extend that far. Ordinary differential equations can make us notice kinds of functions we would not have seen otherwise.

And with this — I can see the much-postponed end of the Little 2021 Mathematics A-to-Z! You can read all my entries for 2021 at this link, and if you’d like can find all my A-to-Z essays here. How will I finish off the shortest yet most challenging sequence I’ve done yet? Will it be yellow and equivalent to the Axiom of Choice? Answers should come, in a week, if all starts going well.