Iva Sallay teaches you how to host the Playful Math Education Blog Carnival


Iva Sallay, creator of the Find The Factors recreational mathematics puzzle and a kind friend to my blog, posted Yes, YOU Can Host a Playful Math Education Blog Carnival. It explains in quite good form how to join in Denise Gaskins’s roaming blog event. It tries to gather educational or recreational or fun or just delightful mathematics links.

Hosting the blog carnival is a great experience I recommend for mathematics bloggers at least once. I seem to be up to hosting it about once a year, most recently in September 2020. Most important in putting one together is looking at your mathematics reading with different eyes. Sallay, though, goes into specifics about what to look for, and how to find that.

If you’d like to host a carnival you can sign up now for the June slot, blog #147, or for most of the rest of the year.

History of Philosophy podcast has episode on Nicholas of Cusa


I continue to share things I’ve heard, rather than created. Peter Adamson’s podcast The History Of Philosophy Without Any Gaps this week had an episode about Nicholas of Cusa. There’s another episode on him scheduled for two weeks from now.

Nicholas is one of those many polymaths of the not-quite-modern era. Someone who worked in philosophy, theology, astronomy, mathematics, with a side in calendar reform. He’s noteworthy in mathematics and theology and philosophy for trying to understand the infinite and the infinitesimal. Adamson’s podcast — about a half-hour — focuses on the philosophical and theological sides of things. But the mathematics can’t help creeping in, with questions like, how can you tell the difference between a straight line and the edge of a circle with infinitely large diameter? Or between a circle and a regular polygon with infinitely many sides?

The St Andrews Maths History site has an article on Nicholas that focuses more on the kinds of work he did.

How May 2021 Treated My Mathematics Blog


I’ll take this chance now to look over my readership from the past month. It’s either that or actually edit this massive article I’ve had sitting for two months. I keep figuring I’ll edit it this next weekend, and then the week ends before I do. This weekend, though, I’m sure to edit it into coherence. Just you watch.

According to WordPress I had 3,068 page views in May of 2021. That’s an impressive number: my 12-month running mean, leading up to May, was 2,366.0 views per month. The 12-month running median is a similar 2,394 views per month. That startles me, especially as I don’t have any pieces that obviously drew special interest. Sometimes there’s a flood of people to a particular page, or from a particular site. That didn’t happen this month, at least as far as I can tell. There was a steady flow of readers to all kinds of things.

There were 2,085 unique visitors, according to WordPress. That’s down from April, but still well above the running mean of 1,671.9 visitors. And above the median of 1,697 unique visitors.

When we rate things per post the dominance of the past month gets even more amazing. That’s an average 340.9 views per posting this month, compared to a mean of 202.5 or a median of 175.5. (Granted, yes, the majority of those were to things from earlier months; there’s almost ten years of backlog and people notice those too.) And it’s 231.7 unique visitors per posting, versus a mean of 144.7 and a median of 127.4.

Bar chart of two and a half years's worth of monthly readership figures. The last several months have seen a steady roughly 3,000 page views and 2,000 unique visitors a month, an increase over the preceding years.
The most important thing in tracking all this is I hope to someday catch WordPress giving me the same readership statistics two months in a row.

There were 48 likes given in May. That’s below the running mean of 56.3 and median of 55.5. Per-posting, though, these numbers look better. That’s 5.3 likes per posting over the course of May. The mean per posting was 4.5 and the median 4.1 over the previous twelve months. There were 20 comments, barely above the running mean of 19.4 and running median of 18. But that’s 2.2 comments per posting, versus a mean per posting of 1.7 and a median per posting of 1.4. I make my biggest impact with readers by shutting up more.

I got around to publishing nine things in May. A startling number of them were references to other people’s work or, in one case, me talking about using an earlier bit I wrote. Here’s the posts in descending order of popularity. I’m surprised how much this differs from simple chronological order. It suggests there are things people are eager to see, and one of them is Reading the Comics posts. Which I don’t do on a schedule anymore.

As that last and least popular post says, I plan to do an A-to-Z this year. A shorter one than usual, though, one of only fifteen week’s duration, and covering only ten different letters. It’s been a hard year and I need to conserve my energies. I’ll begin appealing for subjects soon.

In May 2021 I posted 4,719 words here, figures WordPress, bringing me to a total of 22,620 words this year. This averages out at 524.3 words per posting in May, and 552 words per post for the year.

As of the start of June I’ve had 1,623 posts to here, which gathered a total 135,779 views from a logged 79,646 unique visitors.

I’d be glad to have you as a regular reader. To be one that never appears in my statistics you can use the RSS feed for my essays. If you don’t have an RSS reader you can sign up for a free account at Dreamwidth or Livejournal. You can add any RSS feed by https://www.dreamwidth.org/feeds/ or https://www.livejournal.com/syn and have it appear on your Friends page.

If you have a WordPress account, you can add my posts to your Reader. Use the “Follow NebusResearch” button to do that. Or you can use “Follow NebusResearch by E-mail” to get posts sent to your mailbox. That’s the way to get essays before I notice their most humiliating typos.

I’m @nebusj on Twitter, but don’t read or interact with it. It posts announcements of essays is all. I do read @nebusj@mathstodon.xyz, on the mathematics-themed Mastodon instance.

Thank you for reading, however it is you’re doing, and I hope you’ll do more of that. If you’re not reading, I suppose I don’t have anything more to say.

Announcing my 2021 Mathematics A-to-Z


I enjoy the tradition of writing an A-to-Z, a string of essays about topics from across the alphabet and mostly chosen by readers and commenters. I’ve done at least one each year since 2015 and it’s a thrilling, exhausting performance. I didn’t want to miss this year, too.

But note the “exhausting” there. It’s been a heck of a year and while I’ve been more fortunate than many, I also know my limits. I don’t believe I have the energy to do the whole alphabet. I tell myself these essays don’t have to be big productions, and then they turn into 2,500 words a week for 26 weeks. It’s nice work but it’s also a (slender) pop mathematics book a year, on top of everything else I write in the corners around my actual work.

So how to do less, and without losing the Mathematics A-to-Z theme? And Iva Sallay, creator of Find the Factors and always a kind and generous reader, had the solution. This year I’ll plan on a subset of the alphabet, corresponding to a simple phrase. That phrase? I’m embarrassed to say how long it took me to think of, but it must be the right one.

I plan to do, in this order, the letters of “MATHEMATICS A-TO-Z”.

That is still a 15-week course of essays, but I did want something that would still be a worthwhile project. I intend to keep the essays shorter this year, aiming at a 1,000-word cap, so look forward to me breaking 4,000 words explaining “saddle points”. This also implies that I’ll be doubling and even tripling letters, for the first time in one of these sequences. There’s to be three A’s, three T’s, and two M’s. Also one each of C, E, H, I, O, S, and Z. I figure I have one Z essay left before I exhaust the letter. I may deal with that problem in 2022.

I plan to set my call for topics soon. I’d like to get the sequence started publishing in July, so I have to do that soon. But to give some idea the range of things I’ve discussed before, here’s the roster of past, full-alphabet, A-to-Z topics:

I, too, am fascinated by the small changes in how I titled these posts and even chose whether to capitalize subject names in the roster. By “am fascinated by the small changes” I mean “am annoyed beyond reason by the inconsistencies”. I hope you too have an appropriate reaction to them.

Reading the Comics, May 25, 2021: Hilbert’s Hotel Edition


I have only a couple strips this time, and from this week. I’m not sure when I’ll return to full-time comics reading, but I do want to share strips that inspire something.

Carol Lay’s Lay Lines for the 24th of May riffs on Hilbert’s Hotel. This is a metaphor often used in pop mathematics treatments of infinity. So often, in fact, a friend snarked that he wished for any YouTube mathematics channel that didn’t do the same three math theorems. Hilbert’s Hotel was among them. I think I’ve never written a piece specifically about Hilbert’s Hotel. In part because every pop mathematics blog has one, so there are better expositions available. I have a similar restraint against a detailed exploration of the different sizes of infinity, or of the Monty Hall Problem.

Narration, with illustrations to match: 'Hilbert's Hotel: The infinite hotel was always filled to capacity. Yet if a new guest arrived, she was always given a room. After all, there were an infinite number of rooms. This paradox assumed that management could always add one or more to infinity. The brain-bruising hotel attracted a lot of mathematicians and philosophers. They liked to argue into the wee hours abou the nature of infinity. Unfortunately, they were a bunch of slobs. Management had to hire a new maid to keep up with the mess. Daunted by the number of rooms to clean... the maid set fire to the joint. The philosophers escaped ... but the hotel burned forever.'
Carol Lay’s Lay Lines for the 24th of May, 2021. This and a couple other essays inspired by something in Lay Lines are at this link. This comic is, per the copyright notice, from 2002. I don’t know anything of its publication history past that.

Hilbert’s Hotel is named for David Hilbert, of Hilbert problems fame. It’s a thought experiment to explore weird consequences of our modern understanding of infinite sets. It presents various cases about matching elements of a set to the whole numbers, by making it about guests in hotel rooms. And then translates things we accept in set theory, like combining two infinitely large sets, into material terms. In material terms, the operations seem ridiculous. So the set of thought experiments get labelled “paradoxes”. This is not in the logician sense of being things both true and false, but in the ordinary sense that we are asked to reconcile our logic with our intuition.

So the Hotel serves a curious role. It doesn’t make a complex idea understandable, the way many demonstrations do. It instead draws attention to the weirdness in something a mathematics student might otherwise nod through. It does serve some role, or it wouldn’t be so popular now.

It hasn’t always been popular, though. Hilbert introduced the idea in 1924, though per a paper by Helge Kragh, only to address one question. A modern pop mathematician would have a half-dozen problems. George Gamow’s 1947 book One Two Three … Infinity brought it up again, but it didn’t stay in the public eye. It wasn’t until the 1980s that it got a secure place in pop mathematics culture, and that by way of philosophers and theologians. If you aren’t up to reading the whole of Kragh’s paper, I did summarize it a bit more completely in this 2018 Reading the Comics essay.

Anyway, Carol Lay does an great job making a story of it.

Two people stand in front of a chalkboard which contains a gibberish equation: 'sqrt(PB+J(ax pi)^2) * Z/y { = D/8 + H} - 17^4 x G + z x 2 / 129 \div +/o + exp(null set mickey-mouse-ears), et cetera. One person says: 'Oh, it definitely proves something, all right ... when it comes to actual equations, at least one cartoonist doesn't know squat.'
Leigh Rubin’s Rubes for the 25th of May, 2021. This and other essays mentioning Rubes are at this link. I’m not sure whether that symbol at the end of the second line is meant to be Mickey Mouse ears, or a Venn diagram, or a symbol that I’m not recognizing.

Leigh Rubin’s Rubes for the 25th of May I’ll toss in here too. It’s a riff on the art convention of a blackboard equation being meaningless. Normally, of course, the content of the equation doesn’t matter. So it gets simplified and abstracted, for the same reason one draws a brick wall as four separate patches of two or three bricks together. It sometimes happens that a cartoonist makes the equation meaningful. That’s because they’re a recovering physics major like Bill Amend of FoxTrot. Or it’s because the content of the blackboard supports the joke. Which, in this case, it does.

The essays I write about comic strips I tag so they appear at this link. You may enjoy some more pieces there.

In Which I Get To Use My Own Work


We have goldfish, normally kept in an outdoor pond. It’s not a deep enough pond that it would be safe to leave them out for a very harsh winter. So we keep as many as we can catch in a couple 150-gallon tanks in the basement.

Recently, and irritatingly close to when we’d set them outside, the nitrate level in the tanks grew too high. Fish excrete ammonia. Microorganisms then turn the ammonia into nitrates and then nitrates. In the wild, the nitrates then get used by … I dunno, plants? Which don’t thrive enough hin our basement to clean them out. To get the nitrate out of the water all there is to do is replace the water.

We have six buckets, each holding five gallons, of water that we can use for replacement. So there’s up to 30 gallons of water that we could change out in a day. Can’t change more because tap water contains chloramines, which kill bacteria (good news for humans) but hurt fish (bad news for goldfish). We can treat the tap water to neutralize the chloramines, but want to give that time to finish. I have never found a good reference for how long this takes. I’ve adopted “about a day” because we don’t have a water tap in the basement and I don’t want to haul more than 30 gallons of water downstairs any given day.

So I got thinking, what’s the fastest way to get the nitrate level down for both tanks? Change 15 gallons in each of them once a day, or change 30 gallons in one tank one day and the other tank the next?

Several dozen goldfish, most of them babies, within a 150-gallon rubber stock tank, their wintering home.
Not a current picture, but the fish look about like this still.

And, happy to say, I realized this was the tea-making problem I’d done a couple months ago. The tea-making problem had a different goal, that of keeping as much milk in the tea as possible. But the thing being studied was how partial replacements of a solution with one component affects the amount of the other component. The major difference is that the fish produce (ultimately) more nitrates in time. There’s no tea that spontaneously produces milk. But if nitrate-generation is low enough, the same conclusions follow. So, a couple days of 30-gallon changes, in alternating tanks, and we had the nitrates back to a decent level.

We’d have put the fish outside this past week if I hadn’t broken, again, the tool used for cleaning the outside pond.

Homologies and Cohomologies explained quickly


I’d hoped to have a pretty substantial post today. I fell short of having time to edit the beast into shape. I apologize but hope to have that soon.

I also hope to soon have an announcement about a Mathematics A-to-Z for this year. But until then, here’s this.

Several years ago in an A-to-Z I tried to explain cohomologies. I wasn’t satisfied with it, as, in part, I couldn’t think of a good example. You know, something you could imagine demonstrating with specific physical objects. I can reel off definitions, once I look up the definitions, but there’s only so many people who can understand something from that.

Quanta Magazine recently ran an article about homologies. It’s a great piece, if we get past the introduction of topology with that doughnut-and-coffee-cup joke. (Not that it’s wrong, just that it’s tired.) It’s got pictures, too, which is great.

This I came to notice because Refurio Anachro on Mathstodon wrote a bit about it. This in a thread of toots talking about homologies and cohomologies. The thread at this link is more for mathematicians than the lay audience, unlike the Quanta Magazine article. If you’re comfortable reading about simplexes and linear operators and multifunctions you’re good. Otherwise … well, I imagine you trust that cohomologies can take care of themselves. But I feel better-informed for reading the thread. And it includes a link to a downloadable textbook in algebraic topology, useful for people who want to give that a try on their own.

In Our Time podcast has an episode on Longitude


The BBC’s In Our Time program, and podcast, did a 50-minute chat about the longitude problem. That’s the question of how to find one’s position, east or west of some reference point. It’s an iconic story of pop science and, I’ll admit, I’d think anyone likely to read my blog already knows the rough outline of the story. But you never know what people don’t know. And even if you do know, it’s often enjoyable to hear the story told a different way.

The mathematics content of the longitude problem is real, although it’s not discussed more than in passing during the chat. The core insight Western mapmakers used is that the difference between local (sun) time and a reference point’s time tells you how far east or west you are of that reference point. So then the question becomes how you know what your reference point’s time is.

This story, as it’s often told in pop science treatments, tends to focus on the brilliant clockmaker John Harrison, and the podcast does a fair bit of this. Harrison spent his life building a series of ever-more-precise clocks. These could keep London time on ships sailing around the world. (Or at least to the Caribbean, where the most profitable, slavery-driven, British interests were.) But he also spent decades fighting with the authorities he expected to reward him for his work. It makes for an almost classic narrative of lone genius versus the establishment.

But, and I’m glad the podcast discussion comes around to this, the reality more ambiguous than this. (Actual history is always more ambiguous than whatever you think.) Part of the goal of the goal of the British (and other powers) was finding a practical way for any ship to find longitude. Granted Harrison could build an advanced, ingenious clock more accurate than anyone else could. Could he build the hundreds, or thousands, of those clocks that British shipping needed? Could anyone?

And the competing methods for finding longitude were based on astronomy and calculation. The moment when, say, the Moon passes in front of Jupiter is the same for everyone on Earth. (At least for the accuracy needed here.) It can, in principle, be forecast years, even decades ahead of time. So why not print up books listing astronomical events for the next five years and the formulas to turn observations into longitudes? Books are easy to print. You already train your navigators in astronomy so that they can find latitude. (This by how far above the horizon the pole star, or the sun, or another identifiable feature is.) And, incidentally, you gain a way of computing longitude that you don’t lose if your clock breaks. I appreciated having some of that perspective shown.

(The problem of longitude on land gets briefly addressed. The same principles that work at sea work on land. And land offers some secondary checks. For an unmentioned example there’s triangulation. It’s a great process, and a compelling use of trigonometry. I may do a piece about that myself sometime.)

Also a thing I somehow did not realize: British English pronounces “longitude” with a hard G sound. Huh.

Reading the Comics update: Wavehead does not have a name


So this is not a mathematics-themed comic update, not really. It’s just a bit of startling news about frequent Reading the Comics subject Andertoons. A comic strip back in December revealed that Wavehead had a specific name. According to the strip from the 3rd of December, the student most often challenging the word problem or the definition on the blackboard is named Tommy.

And then last week we got this bombshell:

Wavehead on the telephone at the school office: 'Mom, it's Charlie. Boy, you'd think with everything in the news that Mrs Philips would have more to worry about than me talking in class, but here we are.'
Mark Anderson’s Andertoons for the 4th of May, 2021. This strip previously ran the 23rd of July, 2018, and don’t think I’m not surprised to discover Andertoons has been in reruns.

So, also, it turns out I should have already known this since the strip ran in 2018 also. All I can say is I have a hard enough time reading nearly every comic strip in the world. I can’t be expected to understand them too.

So as not to leave things too despairing let me share a mathematics-mentioning Andertoons from yesterday and also from July 2018.

On the board, the fraction 3/4 with the numerator and denominator labelled. Wavehead: 'You know, for something that sounds like two killer robots, this is really disappointing.'
Mark Anderson’s Andertoons for the 10th of May, 2021. This strip previously ran the 29th of July, 2018, and I discussed it then.

I don’t know if it’s run before that.

At this link are my essays discussing Andertoons. And my Reading the Comics essays are at this link.

How April 2021 Treated My Mathematics Blog, and a question about my A-to-Z’s


I grant that I’m later even than usual in doing my readership recap. That news about how to get rid of the awful awful awful Block Editor was too important to not give last Wednesday’s publication slot. But let me get back to the self-preening and self-examination that people always seem to like and that I never take any lessons from.

In April 2021 there were 3,016 page views recorded here, according to WordPress. These came from 2,298 unique visitors. These are some impressive-looking numbers, especially given that in April I only published nine pieces. And one of those was the readership report for March.

The 3,016 page views is appreciably above the running mean of 2,267.9 views per month for the twelve months leading up to April. It’s also above the running median of 2,266.5 for the twelve months before. And, per posting, the apparent growth is the more impressive. This averages at 335.1 views per posting. The twelve-month running mean was 185.5 views per posting, and twelve-month running median 161.0.

Similarly, unique visitors are well above the averages. 2,298 unique visitors in April is well above the running mean of 1,589.9, and the running median of 1,609.5. The total comes out to 255.3 unique visitors per posting. The running mean, per posting, for the twelve months prior to April was 130.7 unique visitors per posting. The median was a mere 114.1 views per posting.

There were even nice results in the things that show engagement. There were 70 things liked in April, compared to the mean of 54.1 and median of 49. That’s 7.8 likes per posting, well above the mean of 4.1 and median of 4.0. There were for a wonder even more comments than average, 22 given in April compared to a mean of 18.3 and median of 18. Per-posting, that’s 2.4 comments per posting, comfortably above the 1.5 comments per posting mean and 1.2 comments per posting median. It all suggests that I’m finally finding readers who appreciate my genius, or at least style.

Bar chart showing two and a half years' worth of of monthly readership figures. There's a huge spike in October 2019. Beyond that, the past several months of 2021 have shown a fair rise to around 3,000 page views and 2,000 visitors per month.
I would have sworn I’d managed ten posts in April. No way to tell, really, except by counting.

I have doubts, of course, because I don’t have the self-confidence to be a successful writer. But I also notice, for example, that quite a few of these views, and visitors, came in a rush from about the 12th through 16th of April. That’s significant because my humor blog logged an incredible number of visits that week. Someone on the Fandom Drama reddit, explaining James Allen’s departure from Mark Trail, linked to a comic strip I’d saved for my own plot recaps. I’m not sure that this resulted in anyone on the Fandom Drama reddit reading a word I wrote. I also don’t know how this would have brought even a few people to my mathematics blog. The most I can find is several hundred people coming to the mathematics blog from Facebook. As far as I know Facebook had nothing to do with the Fandom Drama reddit. But the coincidence is hard to ignore.


As said, I posted nine things in April. Here they are in decreasing order of popularity. This isn’t quite chronological order, even though pieces from earlier in the month have more time to gather views. It likely means something that one of the more popular pieces is a Reading the Comics post for a comic strip which has run in no newspapers since the 1960s.

My writing plans? I do keep reading the comics. I’m trying to read more for comic strips that offer interesting mathematics points or puzzles to discuss. There’ve been few of those, it seems. But I’m burned out on pointing out how a student got a story problem. And it does seem there’ve been fewer of those, too. But since I don’t want to gather the data needed to do statistics I’ll go with my impression. If I am wrong, what harm will it do?

For each of the past several years I’ve done an A-to-Z, writing an essay for each letter in the alphabet. I am almost resolved to do one for this year. My reservation is that I have felt close to burnout for a long while. This is part of why I am posting two or even one things per week, and have since the 2020 A-to-Z finished. I think that if I do a 2021 A-to-Z it will have to be under some constraints. First is space. A 2,500-word essay lets me put in a lot of nice discoveries and thoughts about topics. It also takes forever to write. Planning to write an 800-word essay trains me to look at smaller scopes, and be easier to find energy and time to write.

Then, too, I may forego making a complete tour of the alphabet. Some letters are so near tapped out that they stop being fun. Some letters end up getting more subject nominations than I can fulfil. It feels a bit off to start an A-to-Z that won’t ever hit Z, but we do live in difficult times. If I end up doing only thirteen essays? That is probably better than none at all.

If you have thoughts about how I could do a different A-to-Z, or better, please let me know. I’m open to outside thoughts about what’s good in these series and what’s bad in them.


In April 2021 I posted 5,057 words here, by WordPress’s estimate. Over nine posts that averages 561,9 words per post. Things brings me to a total of 17,901 words for the year and an average 559 words per post for 2021.

As of the start of May I’ve posted 1,614 things here. They had gathered 131,712 views from 77,564 logged unique visitors.

If you’d like to be a regular reader here, you have options. One is, if you have an RSS reader, to follow essays from the RSS feed. If you don’t have an RSS reader but want one, good news! Sign up for a free account at Dreamwidth or Livejournal. You can use their Reading/Friends page as an RSS reader. Add any RSS feed using https://www.dreamwidth.org/feeds/ or https://www.livejournal.com/syn.

If you have a WordPress account, you can use the “Follow NebusResearch” button, and posts will appear in your Reader here. If you’d rather get posts in e-mail, typos and all, you can click the “Follow NebusResearch by E-mail” button.

On Twitter my @nebusj account still exists, and posts announcements of things. But Safari doesn’t want to reliably let me read Twitter and I don’t care enough to get that sorted out, so you can’t use it to communicate with me. If you’re on Mastodon, you can find me as @nebusj@mathstodon.xyz, the mathematics-themed server there. Safari does mostly like and let me read that. (It has an annoying tendency to jump back to the top of the timeline. But since Mathstodon is a quiet neighborhood this jumping around is not a major nuisance.)

Thank you for reading. I hope you’re enjoying it. And if you do have thoughts for a 2021 A-to-Z, I hope you’ll share them.

Here’s how to get rid of WordPress’s Block Editor and get the good editor back


So I have to skip my planned post for right now, in favor of good news for WordPress bloggers. I apologize for the insular nature of this, but, it’s news worth sharing.

This is how to dump the Block Editor and get the classic, or ‘good’, editor back. WordPress’s ‘Classic Editor Guide’ explains that you go to your — not your blog’s — account settings. That would be https://wordpress.com/me/account. Under ‘Account Settings’ look for the ‘Interface Settings’ section. There’s a toggle for ‘Dashboard appearance’. Click it to ‘Show wp-admin pages if available’, and save that setting. There! Now you have the usable editor again.

Here’s what it looks like:

Screenshot of https://wordpress.com/me/account showing the Account Settings / Interface Settings section. A red ellipse outlines the 'Show wp-admin pages if available' toggle.
There it is! ‘Show wp-admin pages if available’ and if they ever stop being available, I’m out of here.

Now for how I came to this knowledge.

About two months ago WordPress pushed this update where I had no choice but to use their modern ‘Block’ editor. Its main characteristics are that everything takes longer and behaves worse. And more unpredictably. This is part of a site-wide reorganization where everything is worse. Like, it dumped the old system where you could upload several pictures, put in captions and alt-text for them, and have the captions be saved. And somehow the Block Editor kept getting worse. It has two modes, a ‘Visual Editor’ where it shows roughly what your post would look like, and a ‘Code Editor’ where it shows the HTML code you’re typing in. And this past week it decided anything put in as Code Editor should preview as ‘This block has encountered an error and cannot be previewed’.

It’s sloppy, but everything about the Block Editor is sloppy. There is no guessing, at any point, what clicking the mouse will do, much less why it would do that. The Block Editor is a master class in teaching helplessness. I would pay ten dollars toward an article that studied the complex system of failures and bad decisions that created such a bad editor.

This is not me being a cranky old man at a web site changing. I gave it around two months, plenty of time to get used to the scheme and to understand what it does well. It does nothing well.

For example, if I have an article and wish to insert a picture between two paragraphs? And I click at the space between the two paragraphs where I want the picture? There are at least four different things that the mouse click might cause to happen, one of them being “the editor jumps to the very start of the post”. Which of those four will happen? Why? I don’t know, and you know what? I should not have to know.

In the Classic Editor, if I want to insert a picture, I click in my post where I want the picture to go. I click the ‘Insert Media’ button. I select the picture I want, and that’s it. Any replacement system should be no less hard for me, the writer, to use. Last week, I had to forego putting a picture in one of my Popeye cartoon reviews because nothing would allow me to insert a picture. This is WordPress’s failure, not mine.

With the latest change, and thinking seriously whether WordPress blogging is worth the aggravation, I went to WordPress’s help pages looking for how to get the old editor back. And, because their help pages are also a user-interface clusterfluff, ended up posting this question to a forum that exists somewhere. And, wonderfully, musicdoc1 saw my frustrated pleas and gave me the answer. I am grateful to them and I cannot exaggerate how much difference this makes. Were I forced to choose between the Block Editor and not blogging at all, not blogging would win.

I am so very grateful to musicdoc1 for this information and I am glad to be able to carry on here.

If you are one of the WordPress programmers behind the Block Editor, first, shame on you, and second, I am willing to offer advice on how to make an editor. First bit of advice: it should be less hard than using a scrap of metal to carve a message into Commander Data’s severed head for recovery 500 years in the future. There’s more that’s necessary, but get back to me when you’ve managed that at least.

Iva Sallay’s published the 146th Playful Math Education Blog Carnival


Iva Sallay, who kindly runs the FindTheFactors mathematics puzzle, is the host of the Playful Math Education Blog Carnival this month. She’s just published the current, 146th, edition.

These carnivals often feature recreational mathematics. Sallay’s collection this month has even more than usual, and (to my tastes) more delightful ones than usual. Even if you aren’t an educator or parent it’s worth reading, as there’s surely something you haven’t thought about before.

And if you have a blog, and would like to host the carnival some month? Denise Gaskins, who organizes the project, is taking volunteers. The 147th carnival needs a host yet, and there’s all of fall and winter available too. Hosting is an exciting and challenging thing to do, and I do recommend anyone with pop-mathematics inclinations trying it at least once.

How do I do a matrix in WordPress LaTeX?


I wanted to walk through the calculation that Atlas the Mental Giant did in that installment of Barnaby which I shared on Monday. I’ve been stymied, though. To have anything comprehensible, I need a matrix. (To be precise, I need the determinant of a matrix.) It needs to be typeset in a particular way.

In normal mathematics communications this is easy. We can use the LaTeX typesetting standard, and I would write something like this:

\left(U - TS\right)\left|\begin{tabular}{cc} -dT^2 & S \\ e^{\imath \pi} & \zeta(0) L \end{tabular} \right|

I haven’t checked that I have the syntax precisely right, but it’s something like that.

WordPress includes a bit of support for LaTeX expressions. Here I mean the standard free account that I have; I can write in some line like

\int_0^M \sum_{j=1}^{N} a_j x^j dx

and it will get displayed neat and clean as

\int_0^M \sum_{j=1}^{N} a_j x^j dx


Thing is, the standard installation only has a subset of LaTeX’s commands. This is fair enough. It’s ridiculous to bring the entire workshop out when all you need is one hammer. What I can’t find, though, is a description of what LaTeX tools are available to the standard default WordPress free-account user. My experiments in my own comments suggest that the tabular, and the table, structures aren’t supported. But I can’t find a reference that says what’s allowed and what isn’t. I might, after all, be making a silly error in syntax, over and over. When you make an error in WordPress LaTeX you get a sulky note that the formula does not parse. There’s no hint given to what went wrong, or where. You have to remove symbols until the error disappears, and then reverse-engineer what should have been there.

(And the new WordPress editor does not help either. There is not a single point in the new editor where I am fully sure what clicking the mouse will do, or why. Whether it’ll pop up a toolbar I don’t need, or open a new section I don’t want, or pop up a menu where items have moved around from the last time, or whether it’ll jump back to the start of my post and challenge me to remember what I was doing. I realize it is always popular to complain about a web site change, but usually the changes make at least one thing better than it used to be. I can’t find the thing this has made at all better.)

So I’m hoping to attract information. Does anyone have a list of what LaTeX commands WordPress can use? And how the set of what’s available differs between the original post and the comments on the post? And what, for a basic subscription, you can use to represent a matrix?

Incidentally, here’s how to make WordPress print a line of LaTeX larger. Put a &s=N just before the closing $ of your symbol. That N can be 1, 2, 3, or 4. The bigger the N, the bigger the print. You can also put in 0 or negative numbers, if you want the expression to be smaller. I can’t imagine wanting that, but it’s out there.

Reading the Comics, December 20, 1948: What is Barnaby’s friend’s name Edition?


Have a special one today. I’ve been reading a compilation of Crockett Johnson’s 1940s comic Barnaby. The title character, an almost too gentle child, follows his fairy godfather Mr O’Malley into various shenanigans. Many (the best ones, I’d say) involve the magical world. The steady complication is that Mr O’Malley boasts abilities beyond his demonstrated competence. (Although most of the magic characters are shown to be not all that good at their business.) It’s a gentle strip and everything works out all right, if farcically.

This particular strip comes from a late 1948 storyline. Mr O’Malley’s gone missing, coincidentally to a fairy cop come to arrest the pixie, who is a con artist at heart. So this sees the entry of Atlas, the Mental Giant, who’s got some pleasant gimmicks. One of them is his requiring mnemonics built on mathematical formulas to work out names. And this is a charming one, with a great little puzzle: how do you get A-T-L-A-S out of the formula Atlas has remembered?

While Barnaby and Jane look on a Fairy Cop asks: 'Sergeant Ausdauer is the name. With a baffling problem. Baffling for the police, that is. But I'm sure that if a Mental Giant like you, sir, will apply his direct scientific mind to --- ' Atlas: 'Eh? How do you do. My name is --- er --- my name is --- er --- Where's my slide rule?' While he works on this Jane says to Barnaby, 'He forgot his name.' Atlas mutters: '(U - TS) * det( -dT^2 S \ e^{i*pi} zeta(0) L) = ... ' Walking off panel, Atlas says, 'A-T-L-A-S --- my name is Atlas. I never forget a name. With my memory system --- good day. Sorry to have bothered you --- ' Barnaby, calling him back: 'Hey! Wait!'
Crockett Johnson and Jack Morley’s Barnaby for the 20th of December, 1948. (Morley drew the strip at this point.) I haven’t had cause to discuss other Barnaby strips but if I do, I’ll put them in an essay here. Sergeant Ausdauer reasons that “one of those upper-class amateur detectives with scientific minds who solve all the problems for Scotland Yard” could get him through this puzzle. If they were in London they could just ring any doorbell … which gives you a further sense of the comic strip’s sensibility.

I’m sorry the solution requires a bit of abusing notation, so please forgive it. But it’s a fun puzzle, especially as the joke would not be funnier if the formula didn’t work. I’m always impressed when a comic strip goes to that extra effort.

Johnson, who also wrote the Harold and the Purple Crayon books, painted over a hundred canvasses with theorem-based pictures. There’s a selection of them at the Smithsonian Institute’s web site, here.

There’s a new tiny sci.math archive out there


My friend Porsupah Rhee — you might know her work from a sometimes-viral photo of rabbits fighting, available on some fun merchandise — tipped me off to this. It’s a new attempt at archiving Usenet, and also Fidonet and other bulletin boards. These are the things we used for communicating before web forums and then Facebonk took over everything everywhere. There were sprawling and often messy things, moderated only by the willingness of people to not violate social norms. Sometimes this worked; sometimes it didn’t.

Usenet was a most important piece of my Internet history; for many years it was very nearly the thing to use the Internet for. For several years it had a great archive, in the form of Deja News, which kept its many conversations researchable. Google bought this up, and as is their way, made it worse. Part of this was trying to confuse people about the difference between Usenet and their own Google Groups, a discussion-board system that I assume they have shut down. If it’s possible to search Usenet through Google anymore, I can’t find how to do it.

So I’m eager to see this archive at I Ping Therefore I Am. I don’t know where it’s getting its records from, or how new ones are coming in. What it has got is a bunch of messages from 1986. This makes for a great, weird peek at a time when the Internet was much smaller, and free of advertising, but still recognizable.

The archives do extend already to sci.math, a group for the discussion of mathematics topics. Also for discovering how people write out mathematics expressions when they don’t have LaTeX, or at least Word’s Equation Editor, to format things. This also covers two subordinate groups, sci.math.stat (for statistics) and sci.math.symbolic (for symbolic algebra discussions).

It would be bad form to join any of these conversations, even if you could figure a way how. But there may be some revealing pieces there now. And I hope the archive will grow, especially to cover the heights of 1990s Usenet. You do not have permission to look up anything I wrote longer than, oh, six weeks ago.

No, You Can’t Say What 6/2(1+2) Equals


I am made aware that a section of Twitter argues about how to evaluate an expression. There may be more than one of these going around, but the expression I’ve seen is:

6 \div 2\left(1 + 2\right) =

Many people feel that the challenge is knowing the order of operations. This is reasonable. That is, that to evaluate arithmetic, you evaluate terms inside parentheses first. Then terms within exponentials. Then multiplication and division. Then addition and subtraction. This is often abbreviated as PEMDAS, and made into a mnemonic like “Please Excuse My Dear Aunt Sally”.

That is fine as far as it goes. Many people likely start by adding the 1 and 2 within the parentheses, and that’s fair. Then they get:

6 \div 2(3) =

Putting two quantities next to one another, as the 2 and the (3) are, means to multiply them. And then comes the disagreement: does this mean take 6\div 2 and multiply that by 3, in which case the answer is 9? Or does it mean take 6 divided by 2\cdot 3, in which case the answer is 1?

And there is the trick. Depending on which way you choose to parse these instructions you get different answers. But you don’t get to do that, not and have arithmetic. So the answer is that this expression has no answer. The phrasing is ambiguous and can’t be resolved.

I’m aware there are people who reject this answer. They picked up along the line somewhere a rule like “do multiplication and division from left to right”. And a similar rule for addition and subtraction. This is wrong, but understandable. The left-to-right “rule” is a decent heuristic, a guide to how to attack a problem too big to do at once. The rule works because multiplication-and-division associates. The quantity a-times-b, multiplied by c, has to be the same number as the quantity a multiplied by the quantity b-times-c. The rule also works for addition-and-subtraction because addition associates too. The quantity a-plus-b, plus the quantity c, has to be the same as the quantity a plus the quantity b-plus-c.

This left-to-right “rule”, though, just helps you evaluate a meaningful expression. It would be just as valid to do all the multiplications-and-divisions from right-to-left. If you get different values working left-to-right from right-to-left, you have a meaningless expression.

But you also start to see why mathematicians tend to avoid the \div symbol. We understand, for example, a \div b to mean a \cdot \frac{1}{b} . Carry that out and then there’s no ambiguity about

6 \cdot \frac{1}{2} \cdot 3 =

I understand the desire to fix an ambiguity. Believe me. I’m a know-it-all; I only like ambiguities that enable logic-based jokes. (“Would you like ice cream or cake?” “Yes.”) But the rules that could remove the ambiguity in 6\div 2(1 + 2) also remove associativity from multiplication. Once you do that, you’re not doing arithmetic anymore. Resist the urge.

(And the mnemonic is a bit dangerous. We can say division has the same priority as multiplication, but we also say “multiplication” first. I bet you can construct an ambiguous expression which would mislead someone who learned Please Excuse Dear Miss Sally Andrews.)

And now a qualifier: computer languages will often impose doing a calculation in some order. Usually left-to-right. The microchips doing the work need to have some instructions. Spotting all possible ambiguous phrasings ahead of time is a challenge. But we accept our computers doing not-quite-actual-arithmetic. They’re able to do not-quite-actual-arithmetic much faster and more reliably than we can. This makes the compromise worthwhile. We need to remember the difference between what the computer does and the calculation we intend.

And another qualifier: it is possible to do interesting mathematics with operations that aren’t associative. But if you are it’s in your research as a person with a postgraduate degree in mathematics. It’s possible it might fit in social media, but I would be surprised. It won’t draw great public attention, anyway.

Reading the Comics Follow-up: Where Else Is A Tetrahedron’s Centroid Edition


A Reading the Comics post a couple weeks back inspired me to find the centroid of a regular tetrahedron. A regular tetrahedron, also known as “a tetrahedron”, is the four-sided die shape. A pyramid with triangular base. Or a cone with a triangle base, if you prefer. If one asks a person to draw a tetrahedron, and they comply, they’ll likely draw this shape. The centroid, the center of mass of the tetrahedron, is at a point easy enough to find. It’s on the perpendicular between any of the four faces — the equilateral triangles — and the vertex not on that face. Particularly, it’s one-quarter the distance from the face towards the other vertex. We can reason that out purely geometrically, without calculating, and I did in that earlier post.

But most tetrahedrons are not regular. They have centroids too; where are they?

In a boxing ring. Facing off and wearing boxing gloves are a tetrahedron and a cube. The umpire, a sphere, says into the microphone, 'And remember: nothing below the centroid.'
Ben Zaehringer’s In The Bleachers for the 16th of March, 2021. This and other essays featuring In The Bleachers are gathered at this link.

Thing is I know the correct answer going in. It’s at the “average” of the vertices of the tetrahedron. Start with the Cartesian coordinates of the four vertices. The x-coordinate of the centroid is the arithmetic mean of the x-coordinates of the four vertices. The y-coordinate of the centroid is the mean of the y-coordinates of the vertices. The z-coordinate of the centroid is the mean of the z-coordinates of the vertices. Easy to calculate; but, is there a way to see that this is right?

What’s got me is I can think of an argument that convinces me. So in this sense, I have an easy proof of it. But I also see where this argument leaves a lot unaddressed. So it may not prove things to anyone else. Let me lay it out, though.

So start with a tetrahedron of your own design. This will be less confusing if I have labels for the four vertices. I’m going to call them A, B, C, and D. I don’t like those labels, not just for being trite, but because I so want ‘C’ to be the name for the centroid. I can’t find a way to do that, though, and not have the four tetrahedron vertices be some weird set of letters. So let me use ‘P’ as the name for the centroid.

Where is P, relative to the points A, B, C, and D?

And here’s where I give a part of an answer. Start out by putting the tetrahedron somewhere convenient. That would be the floor. Set the tetrahedron so that the face with triangle ABC is in the xy plane. That is, points A, B, and C all have the z-coordinate of 0. The point D has a z-coordinate that is not zero. Let me call that coordinate h. I don’t care what the x- and y-coordinates for any of these points are. What I care about is what the z-coordinate for the centroid P is.

The property of the centroid that was useful last time around was that it split the regular tetrahedron into four smaller, irregular, tetrahedrons, each with the same volume. Each with one-quarter the volume of the original. The centroid P does that for the tetrahedron too. So, how far does the point P have to be from the triangle ABC to make a tetrahedron with one-quarter the volume of the original?

The answer comes from the same trick used last time. The volume of a cone is one-third the area of the base times its altitude. The volume of the tetrahedron ABCD, for example, is one-third times the area of triangle ABC times how far point D is from the triangle. That number I’d labelled h. The volume of the tetrahedron ABCP, meanwhile, is one-third times the area of triangle ABC times how far point P is from the triangle. So the point P has to be one-quarter as far from triangle ABC as the point D is. It’s got a z-coordinate of one-quarter h.

Notice, by the way, that while I don’t know anything about the x- and y- coordinates of any of these points, I do know the z-coordinates. A, B, and C all have z-coordinate of 0. D has a z-coordinate of h. And P has a z-coordinate of one-quarter h. One-quarter h sure looks like the arithmetic mean of 0, 0, 0, and h.

At this point, I’m convinced. The coordinates of the centroid have to be the mean of the coordinates of the vertices. But you also see how much is not addressed. You’d probably grant that I have the z-coordinate coordinate worked out when three vertices have the same z-coordinate. Or where three vertices have the same y-coordinate or the same x-coordinate. You might allow that if I can rotate a tetrahedron, I can get three points to the same z-coordinate (or y- or x- if you like). But this still only gets one coordinate of the centroid P.

I’m sure a bit of algebra would wrap this up. But I would like to avoid that, if I can. I suspect the way to argue this geometrically depends on knowing the line from vertex D to tetrahedron centroid P, if extended, passes through the centroid of triangle ABC. And something similar applies for vertexes A, B, and C. I also suspect there’s a link between the vector which points the direction from D to P and the sum of the three vectors that point the directions from D to A, B, and C. I haven’t quite got there, though.

I will let you know if I get closer.

In Our Time podcast has episode on Pierre-Simon Laplace


I have another mathematics-themed podcast to share. It’s again from the BBC’s In Our Time, a 50-minute program in which three experts discuss a topic. Here they came back around to mathematics and physics. And along the way chemistry and mensuration. The topic here was Pierre-Simon Laplace, who’s one of those people whose name you learn well as a mathematics or physics major. He doesn’t quite reach the levels of Euler — who does? — but he’s up there.

Laplace might be best known for his work in celestial mechanics. He (independently of Immanuel Kant) developed the nebular hypothesis, that the solar system formed from the contraction of a great cloud of dust. We today accept a modified version of this. And for studying the question of whether the solar system is stable. That is, whether the perturbations every planet has on one another average out to nothing, or to something catastrophic. And studying probability, which has more to do with these questions than one might imagine. And then there’s general mechanics, and differential equations, and if that weren’t enough, his role in establishing the Metric system. This and more gets discussion.

How March 2020 Treated My Mathematics Blog


March was the first time in three-quarters of a year that I did any Reading the Comics posts. One was traditional, a round-up of comics on a particular theme. The other was new for me, a close look at a question inspired by one comic. Both turned out to be popular. Now see if I learn anything from that.

I’d left the Reading the Comics posts on hiatus when I started last year’s A-to-Z. Given the stress of the pandemic I did not feel up to that great a workload. For this year I am considering whether I feel up to an A-to-Z again. An A-to-Z is enjoyable work, yes, and I like the work. But I am still thinking over whether this is work I want to commit to just now.

That’s for the future. What of the recent past? WordPress’s statistics page suggests that the comics were very well-received. It tells me there were 2,867 page views in March. That’s the greatest number since November, the last full month of the 2020 A-to-Z. This is well above the twelve-month running average of 2,199.8 views per month. And as far above the twelve-month running median of 2,108 views per month. Per posting — there were ten postings in March — the figures are even greater. There were 286.7 views per posting in March. The running mean is 172.9 views per posting, and the running median 144.8.

Bar chart showing monthly readership for the preceding two and a half years. After a loose deecline in January and February both readership and visitor counts were up sharply in March, though still far below the October 2019 peak.
While I am curious what kind of insights Cloudflare Analytics could hope to offer me, I suspect what it really amounts to is “Cloudflare will allow me to give them some money in exchange for confirmation that the most popular stuff is the things that lots of people like”.

There were 1,993 unique visitors in March. This is well above the running averages. The twelve-month running mean was 1,529.4 unique visitors, and the running median 1,491.5. This is 199.3 unique visitors per March posting, not a difficult calculation to make. The twelve-month running mean was 121.1 viewers per posting, though, and the mean a mere 99.8 viewers per posting. So that’s popular.

Not popular? Talking to me. We all feel like that sometimes but I have data. After a chatty February things fell below average for March. There were 30 likes given in March, below the running mean of 56.7 and median of 55.5. There were 3.0 likes per posting. The running mean for the twelve months leading in to this was 4.2 likes per posting. The running median was 4.0.

And actual comments? There were 10 of them in March, below the mean of 14.3 and median of 10. This averaged 1.0 comments per posting, which is at least something. The running per-post mean is 1.6 comments, though, and median is 1.4. It could be the centroids of regular tetrahedrons are not the hot, debatable topic I had assumed.

Pi Day was, as I’d expected, a good day for reading Pi Day comics. And miscellaneous other articles about Pi Day. I need to write some more up for next year, to enjoy those search engine queries. There are some things in differential equations that would be a nice different take.

As mentioned, I posted ten things in March. Here they are in decreasing order of popularity. I would expect this to be roughly a chronological list of when things were posted. It doesn’t seem to be, but I haven’t checked whether the difference is statistically significant.

In March I posted 5,173 words here, for an average 517.3 words per post. That’s shorter than my average January and February posts were. My average words-per-posting for the year has dropped to 558. And despite my posts being on average shorter, this was still my most verbose month of 2021. I’ve had 12,844 words posted this year, through the start of April, and more than two-fifths of them were March.

As of the start of April I’ve posted 1,605 things to the blog here. They’ve gathered 129,696 page views from an acknowledged 75,266 visitors.

If you’d like to be a regular reader, there’s a couple approaches. One is to read regularly. The best way for you to do that is using the RSS feed in whatever reader you prefer. I won’t see you show up in my statistics, and that’s fine. If you don’t have an RSS reader, you can open a free account at Dreamwidth or Livejournal and add any RSS feed you like. This from https://www.dreamwidth.org/feeds/ or https://www.livejournal.com/syn depending on what you sign up for. If that’s too much, you can use the “Follow NebusResearch By E-mail” button, which will send you essays after they’ve appeared and before I’ve fixed typos.

If you have a WordPress account you can use the “Follow NebusResearch” button to add me to your Reader. If you have Twitter, congratulations; I don’t exactly. My account at @nebusj is still there, but it only has an automated post announcement. I don’t know when that will break. If you’re on Mastodon, you can find me as @nebusj@mathstodon.xyz.

One last thing. WordPress imposed their awful, awful, awful ‘Block’ editor on my blog. I used to be able to us the classic, or ‘good’, editor, where I could post stuff without it needing twelve extra mouse clicks. If anyone knows hacks to get the good editor back please leave a comment.

Thank you all for reading.

A quick reading recommendation


I’ve been reading The Disordered Cosmos: A Journey Into Dark Matter, Spacetime, and Dreams Deferred, by Chanda Prescod-Weinstein. It’s the best science book I’ve read in a long while.

Part of it is a pop-science discussion of particle physics and cosmology, as they’re now understood. It may seem strange that the tiniest things and the biggest thing are such natural companion subjects. That is what seems to make sense, though. I’ve fallen out of touch with a lot of particle physics since my undergraduate days and it’s wonderful to have it discussed well. This sort of pop physics is for me a pleasant comfort read.

The other part of the book is more memoir, and discussion of the culture of science. This is all discomfort reading. It’s an important discomfort.

I discuss sometimes how mathematics is, pretensions aside, a culturally-determined thing. Usually this is in the context of how, for example, that we have questions about “perfect numbers” is plausibly an idiosyncrasy. I don’t talk much about the culture of working mathematicians. In large part this is because I’m not a working mathematician, and don’t have close contact with working mathematicians. And then even if I did — well, I’m a tall, skinny white guy. I could step into most any college’s mathematics or physics department, sit down in a seminar, and be accepted as belonging there. People will assume that if I say anything, it’s worth listening to.

Chanda Prescod-Weinstein, a Black Jewish agender woman, does not get similar consideration. This despite her much greater merit. And, like, I was aware that women have it harder than men. And Black people have it harder than white people. And that being open about any but heterosexual cisgender inclinations is making one’s own life harder. What I hadn’t paid attention to was how much harder, and how relentlessly harder. Most every chapter, including the comfortable easy ones talking about families of quarks and all, is several needed slaps to my complacent face.

Her focus is on science, particularly physics. It’s not as though mathematics is innocent of driving women out or ignoring them when it can’t. Or of treating Black people with similar hostility. Much of what’s wrong is passively accepting patterns of not thinking about whether mathematics is open to everyone who wants in. Prescod-Weinstein offers many thoughts and many difficult thoughts. They are worth listening to.

Reading the Comics, April 1, 2021: Why Is Gunther Taking Algebraic Topology Edition


I’m not yet looking to discuss every comic strip with any mathematics mention. But something gnawed at me in this installment of Greg Evans and Karen Evans’s Luann. It’s about the classes Gunther says he’s taking.

The main characters in Luann are in that vaguely-defined early-adult era. They’re almost all attending a local university. They’re at least sophomores, since they haven’t been doing stories about the trauma and liberation of going off to school. How far they’ve gotten has been completely undefined. So here’s what gets me.

Gunther, looking at sewing patterns: 'You want me to sew pirate outfits?' Bets: 'I'm thinking satin brocade doublet and velvet pantaloons.' Les, not in the conversation: 'Nerd.' Gunther: 'I'm thinking algebraic topology and vector calculus homework.' (He shows his textbooks.) Les: 'And nerdier. (Les pets a cat.)
Greg Evans and Karen Evans’s Luann for the 1st of April, 2021. This and other essays discussing topics raised by Luann are at this link. The overall story here is that Bets wants to have this pirate-themed dinner and trusts Gunther, who’s rather good at clothes-making, to do the decorating.

Gunther taking vector calculus? That makes sense. Vector calculus is a standard course if you’re taking any mathematics-dependent major. It might be listed as Multivariable Calculus or Advanced Calculus or Calculus III. It’s where you learn partial derivatives, integrals along a path, integrals over a surface or volume. I don’t know Gunther’s major, but if it’s any kind of science, yeah, he’s taking vector calculus.

Algebraic topology, though. That I don’t get. Topology at all is usually an upper-level course. It’s for mathematics majors, maybe physics majors.  Not every mathematics major takes topology.   Algebraic topology is a deeper specialization of the subject. I’ve only seen courses listed as algebraic topology as graduate courses. It’s possible for an undergraduate to take a graduate-level course, yes. And it may be that Gunther is taking a regular topology course, and the instructor prefers to focus on algebraic topology.

But even a regular topology course relies on abstract algebra. Which, again, is something you’ll get as an undergraduate. If you’re a mathematics major you’ll get at least two years of algebra. And, if my experience is typical, still feel not too sure about the subject. Thing is that Intro to Abstract Algebra is something you’d plausibly take at the same time as Vector Calculus.  Then you’d get Abstract Algebra and then, if you wished, Topology.

So you see the trouble. I don’t remember anything in algebra-to-topology that would demand knowing vector calculus. So it wouldn’t mean Gunther took courses without taking the prerequisites. But it’s odd to take an advanced mathematics course at the same time as a basic mathematics course. Unless Gunther’s taking an advanced vector calculus course, which might be. Although since he wants to emphasize that he’s taking difficult courses, it’s odd to not say “advanced”. Especially if he is tossing in “algebraic” before topology.

And, yes, I’m aware of the Doylist explanation for this. The Evanses wanted courses that sound impressive and hard. And that’s all the scene demands. The joke would not be more successful if they picked two classes from my actual Junior year schedule. None of the characters have a course of study that could be taken literally. They’ve been university students full-time since 2013 and aren’t in their senior year yet. It would be fun, is all, to find a way this makes sense.


This and my other essays discussing something from the comic strips are at this link.

When is Easter likely to happen?


It’s a natural question to wonder this time of year. The date when Easter falls is calculated by some tricky numerical rules. These come from the desire to make Easter an early-spring (in the Northern hemisphere) holiday, while tying it to the date of Passover, as worked out by people who did not know the exact rules by which the Jewish calendar worked. The result is that some dates are more likely than others to be Easter.

A few years ago I wrote a piece finding how often Easter would be on different dates, in the possible range from the 22nd of March through the 25th of April. And discussed some challenges in the problem. Calendars are full of surprising subtle problems. Easter creates a host of new challenges.

The 145th Playful Math Education Blog Carnival is posted


John Golden, MathHombre, was host this month for the Playful Math Education Blog Carnival. And this month’s collection of puzzles, essays, and creative mathematics projects. Among them are some quilts and pattern-block tiles, which manifest all that talk about the structure of mathematical objects and their symmetries in easy-to-see form. There’s likely to be something of interest there.

Among the wonderful things I discovered there is Math Zine Fest 2021. It’s as the name suggests, a bunch of zines — short printable magazines on a niche topic — put together for the end of February. I had missed this organizing, but hope to get to see later installments. I don’t know what zine I might make, but I must have something I could do.

Denise Gaskins, who organizes the carnival, has hosting slots available for later this year. Hosting is an exciting challenge I encourage people to try at least the once.

Reading the Comics, March 16, 2021: Where Is A Tetrahedron’s Centroid Edition


Comic Strip Master Command has not, to appearances, been distressed by my Reading the Comics hiatus. There are still mathematically-themed comic strips. Many of them are about story problems and kids not doing them. Some get into a mathematical concept. One that ran last week caught my imagination so I’ll give it some time here. This and other Reading the Comics essays I have at this link, and I figure to resume posting them, at least sometimes.

Ben Zaehringer’s In The Bleachers for the 16th of March, 2021 is an anthropomorphized-geometry joke. Here the centroid stands in for “the waist”, the height below which boxers may not punch.

In a boxing ring. Facing off and wearing boxing gloves are a tetrahedron and a cube. The umpire, a sphere, says into the microphone, 'And remember: nothing below the centroid.'
Ben Zaehringer’s In The Bleachers for the 16th of March, 2021. This and other essays featuring In The Bleachers are gathered at this link. There haven’t been many, so far. One of the few appearances was another boxing joke, though.

The centroid is good geometry, something which turns up in plane and solid shapes. It’s a center of the shape: the arithmetic mean of all the points in the shape. (There are other things that can, with reason, be called a center too. Mathworld mentions the existence of 2,001 things that can be called the “center” of a triangle. It must be only a lack of interest that’s kept people from identifying even more centers for solid shapes.) It’s the center of mass, if the shape is a homogenous block. Balance the shape from below this centroid and it stays balanced.

For a complicated shape, finding the centroid is a challenge worthy of calculus. For these shapes, though? The sphere, the cube, the regular tetrahedron? We can work those out by reason. And, along the way, work out whether this rule gives an advantage to either boxer.

The sphere first. That’s the easiest. The centroid has to be the center of the sphere. Like, the point that the surface of the sphere is a fixed radius from. This is so obvious it takes a moment to think why it’s obvious. “Why” is a treacherous question for mathematics facts; why should 4 divide 8? But sometimes we can find answers that give us insight into other questions.

Here, the “why” I like is symmetry. Look at a sphere. Suppose it lacks markings. There’s none of the referee’s face or bow tie here. Imagine then rotating the sphere some amount. Can you see any difference? You shouldn’t be able to. So, in doing that rotation, the centroid can’t have moved. If it had moved, you’d be able to tell the difference. The rotated sphere would be off-balance. The only place inside the sphere that doesn’t move when the sphere is rotated is the center.

This symmetry consideration helps answer where the cube’s centroid is. That also has to be the center of the cube. That is, halfway between the top and bottom, halfway between the front and back, halfway between the left and right. Symmetry again. Take the cube and stand it upside-down; does it look any different? No, so, the centroid can’t be any closer to the top than it can the bottom. Similarly, rotate it 180 degrees without taking it off the mat. The rotation leaves the cube looking the same. So this rules out the centroid being closer to the front than to the back. It also rules out the centroid being closer to the left end than to the right. It has to be dead center in the cube.

Now to the regular tetrahedron. Obviously the centroid is … all right, now we have issues. Dead center is … where? We can tell when the regular tetrahedron’s turned upside-down. Also when it’s turned 90 or 180 degrees.

Symmetry will guide us. We can say some things about it. Each face of the regular tetrahedron is an equilateral triangle. The centroid has to be along the altitude. That is, the vertical line connecting the point on top of the pyramid with the equilateral triangle base, down on the mat. Imagine looking down on the shape from above, and rotating the shape 120 or 240 degrees if you’re still not convinced.

And! We can tip the regular tetrahedron over, and put another of its faces down on the mat. The shape looks the same once we’ve done that. So the centroid has to be along the altitude between the new highest point and the equilateral triangle that’s now the base, down on the mat. We can do that for each of the four sides. That tells us the centroid has to be at the intersection of these four altitudes. More, that the centroid has to be exactly the same distance to each of the four vertices of the regular tetrahedron. Or, if you feel a little fancier, that it’s exactly the same distance to the centers of each of the four faces.

It would be nice to know where along this altitude this intersection is, though. We can work it out by algebra. It’s no challenge to figure out the Cartesian coordinates for a good regular tetrahedron. Then finding the point that’s got the right distance is easy. (Set the base triangle in the xy plane. Center it, so the coordinates of the highest point are (0, 0, h) for some number h. Set one of the other vertices so it’s in the xz plane, that is, at coordinates (0, b, 0) for some b. Then find the c so that (0, 0, c) is exactly as far from (0, 0, h) as it is from (0, b, 0).) But algebra is such a mass of calculation. Can we do it by reason instead?

That I ask the question answers it. That I preceded the question with talk about symmetry answers how to reason it. The trick is that we can divide the regular tetrahedron into four smaller tetrahedrons. These smaller tetrahedrons aren’t regular; they’re not the Platonic solid. But they are still tetrahedrons. The little tetrahedron has as its base one of the equilateral triangles that’s the bigger shape’s face. The little tetrahedron has as its fourth vertex the centroid of the bigger shape. Draw in the edges, and the faces, like you’d imagine. Three edges, each connecting one of the base triangle’s vertices to the centroid. The faces have two of these new edges plus one of the base triangle’s edges.

The four little tetrahedrons have to all be congruent. Symmetry again; tip the big tetrahedron onto a different face and you can’t see a difference. So we’ll know, for example, all four little tetrahedrons have the same volume. The same altitude, too. The centroid is the same distance to each of the regular tetrahedron’s faces. And the four little tetrahedrons, together, have the same volume as the original regular tetrahedron.

What is the volume of a tetrahedron?

If we remember dimensional analysis we may expect the volume should be a constant times the area of the base of the shape times the altitude of the shape. We might also dimly remember there is some formula for the volume of any conical shape. A conical shape here is something that’s got a simple, closed shape in a plane as its base. And some point P, above the base, that connects by straight lines to every point on the base shape. This sounds like we’re talking about circular cones, but it can be any shape at the base, including polygons.

So we double-check that formula. The volume of a conical shape is one-third times the area of the base shape times the altitude. That’s the perpendicular distance between P and the plane that the base shape is in. And, hey, one-third times the area of the face times the altitude is exactly what we’d expect.

So. The original regular tetrahedron has a base — has all its faces — with area A. It has an altitude h. That h must relate in some way to the area; I don’t care how. The volume of the regular tetrahedron has to be \frac{1}{3} A h .

The volume of the little tetrahedrons is — well, they have the same base as the original regular tetrahedron. So a little tetrahedron’s base is A. The altitude of the little tetrahedron is the height of the original tetrahedron’s centroid above the base. Call that h_c . How can the volume of the little tetrahedron, \frac{1}{3} A h_c , be one-quarter the volume of the original tetrahedron, \frac{1}{3} A h ? Only if h_c is one-quarter h .

This pins down where the centroid of the regular tetrahedron has to be. It’s on the altitude underneath the top point of the tetrahedron. It’s one-quarter of the way up from the equilateral-triangle face.

(And I’m glad, checking this out, that I got to the right answer after all.)

So, if the cube and the tetrahedron have the same height, then the cube has an advantage. The cube’s centroid is higher up, so the tetrahedron has a narrower range to punch. Problem solved.


I do figure to talk about comic strips, and mathematics problems they bring up, more. I’m not sure how writing about one single strip turned into 1300 words. But that’s what happens every time I try to do something simpler. You know how it goes.

How to crumple paper


I intend to post something inspired by the comics. I’m not ready just yet. Until then, though, I’d like to share a neat article published in Nature. It’s about paper.

In particular, it’s about how paper crumples. When paper is crumpled, and flattened out again, it looks different. When it’s crumpled and flattened out again, it looks even more different. But you reach a point where crumping and flattening the paper stops making it look all that different. A model for the fragmentation kinetics of crumpled thin sheets, by Jovana Andrejevic, Lisa M Lee, Shmuel M Rubinstein, and Chris H Rycroft, tries to explain the process.

The skeptical reader might say this is obvious. They’re invited to write a simulation that takes a set of fold lines and predicts which sides of the paper are angled out and which are angled in. The skeptical reader may also ask who cares about paper. It’s paper because many mathematics problems start from the kinds of things one sets one’s hands on. Anyone who’s seen a crack growing across their sidewalk, though — or across their countertop, or their grandfather’s desk — realizes there are things we don’t understand about how things break. And why they break that way. And, more generally, there’s a lot we don’t understand about how complicated “natural” shapes form. The big interest in this is how long molecules crumple up. The shapes of these govern how they behave, and it’d be nice to understand that more.

The New York Times has an article explaining the paper, with more of the story of what the research is and why it’s important. This if you don’t feel comfortable reading formulas about compaction ratios or skipping over formulas to get to text again.

Reading the Comics, March 14, 2021: Pi Day Edition


I was embarrassed, on looking at old Pi Day Reading the Comics posts, to see how often I observed there were fewer Pi Day comics than I expected. There was not a shortage this year. This even though if Pi Day has any value it’s as an educational event, and there should be no in-person educational events while the pandemic is still on. Of course one can still do educational stuff remotely, mathematics especially. But after a year of watching teaching on screens and sometimes doing projects at home, it’s hard for me to imagine a bit more of that being all that fun.

But Pi Day being a Sunday did give cartoonists more space to explain what they’re talking about. This is valuable. It’s easy for the dreadfully online, like me, to forget that most people haven’t heard of Pi Day. Most people don’t have any idea why that should be a thing or what it should be about. This seems to have freed up many people to write about it. But — to write what? Let’s take a quick tour of my daily comics reading.

Agnes: 'Today we do that 'Daylight Savings Time' thing again.' Trout: 'Do we add an hour or subtract an hour?' Agnes: 'Um ... good question. Let's see. 'Spring around, fall over.' Trout: 'That makes *no* sense.' Agnes: 'You're right. Hey! It's also pie day!' Trout: 'Now *that* makes sense!'
Tony Cochran’s Agnes for the 14th of March, 2021. My essays exploring something mentioned in Agnes appear at this link.

Tony Cochran’s Agnes starts with some talk about Daylight Saving Time. Agnes and Trout don’t quite understand how it works, and get from there to Pi Day. Or as Agnes says, Pie Day, missing the mathematics altogether in favor of the food.

A numeral 3 stands in front of a mirror and says, 'Hope this works.' He swallows several pills. Beside the 3 appear a decimal point, and then a 1, 4, 1, 5, 9, 2, and so on. The 3 says 'Whoa!' while looking at the decimal train as the reader finally sees the prescription was for 'Pi-Agra'.
Scott Hilburn’s The Argyle Sweater for the 14th of March, 2021. Essays featuring some discussion of The Argyle Sweater are at this link.

Scott Hilburn’s The Argyle Sweater is an anthropomorphic-numerals joke. It’s a bit risqué compared to the sort of thing you expect to see around here. The reflection of the numerals is correct, but it bothered me too.

Lupin, white cat, reporting: 'It's National Pie Day! [ Handed a bulletin ] Excuse me?' A chart shows a circle, diameter, circumference, and radius. Puck, black cat, interrupts, wearing a T-shirt with pi on it: 'It's Pi Day! When folks celebrate the mathematical constant pi! Not to be confused with the pastry dessert pie! Though people celebrate it by baking and eating pies.' (See The Woman and her child rolling out pie crust in the kitchen.) 'Which is very confusing! Just like math ... ' Agnes, mouse: 'No way, Puck! Check it out!' Agnes shows the pi = C / D formula on a card, and with some other mice demonstrates: 'Measure the circumference of the pie with a ribbon. Now, measure the diameter across and cut the ribbon each time. You should be left with three equal ribbons and a little extra! 3.14, that's pi!' Puck looks at the pie, with a slice cut out: 'Hey, where did this radius-slice go?' Agnes: 'OUR WORK HERE IS DONE!' Other mouse: 'MATH RULES!'
Georgia Dunn’s Breaking Cat News for the 14th of March, 2021. I don’t seem to have ever discussed this strip before. This essay, and any future ones mentioning Breaking Cat News, should be at this link.

Georgia Dunn’s Breaking Cat News is a delightful cute comic strip. It doesn’t mention mathematics much. Here the cat reporters do a fine job explaining what Pi Day is and why everybody spent Sunday showing pictures of pies. This could almost be the standard reference for all the Pi Day strips.

Jason: 'Today's 3-14.' Dad: 'So it is.' Jason: 'I don't have a watch. Can you let me know when it's exactly 1:50 ... ' Dad: 'I'll try.' Jason: 'And 26 secones ... and 535 milliseconds ... and 897 microseconds ... and 932 nanoseconds ... and 384 picoseconds ... does your watch do femtoseconds?' Dad: 'There's such a thing as taking Pi day too seriously, son.
Bill Amend’s FoxTrot for the 14th of March, 2021. Essays that have some mention of FoxTrot are gathered at this link.

Bill Amend’s FoxTrot is one of the handful that don’t mention pie at all. It focuses on representing the decimal digits of π. At least within the confines of something someone might write in the American dating system. The logic of it is a bit rough but if we’ve accepted 3-14 to represent 3.14, we can accept 1:59 as standing in for the 0.00159 of the original number. But represent 0.0015926 (etc) of a day however you like. If we accept that time is continuous, then there’s some moment on the 14th of March which matches that perfectly.

Frazz: 'The ratio of any circle's circumference to its diameter is commonly rounded off to 3.14 and iconically represented by pi. Which linguistically sounds like 'pie'. Hence 3-14 is Pi Day.' Caulfield, walking a dog: 'Everybody knows that, Frazz. What's your point?' Frazz: 'Is it a language gag or a math gag?' Caulfield: 'Oh, I see. Wait for it.' Frazz, eyes bugged out and covering his nose: 'Good Lord. Was that you?' Caulfield: 'I fed cheese pizza to the dog. Now it's a biology gag.'
Jef Mallett’s Frazz for the 14th of March, 2021. Essays inspired by something in Frazz should be gathered at this link.

Jef Mallett’s Frazz talks about the eliding between π and pie for the 14th of March. The strip wonders a bit what kind of joke it is exactly. It’s a nerd pun, or at least nerd wordplay. If I had to cast a vote I’d call it a language gag. If they celebrated Pi Day in Germany, there would not be any comic strips calling it Tortentag.

Heart, to her friends: 'I do love a good pie. But why are we doing this again?' Kat: 'It's Pi Day!' Heart: 'Which is?' Kat start stalking in long swirly dialogue all around an extra-wide panel explaining pi and even reciting digits. Heart: 'Mhmm. Right. Gotcha.'
Steenz’s Heart of the City for the 14th of March, 2021. Essays which mention Heart of the City — which until summer last year was written by Mark Tatulli, and is now by Steenz — are at this link.

Steenz’s Heart of the City is another of the pi-pie comics. I do feel for Heart’s bewilderment at hearing π explained at length. Also Kat’s desire to explain mathematics overwhelming her audience. It’s a feeling I struggle with too. The thing is it’s a lot of fun to explain things. It’s so much fun you can lose track whether you’re still communicating. If you set off one of these knowledge-floods from a friend? Try to hold on and look interested and remember any single piece anywhere of it. You are doing so much good for your friend. And if you realize you’re knowledge-flooding someone? Yeah, try not to overload them, but think about the things that are exciting about this. Your enthusiasm will communicate when your words do not.

Two mathematicians at a wide chalkboard have written out digits of pi. One says, 'There's always room for more Pi.' The other, at the right end says, 'Or not.' In the corner the Reality Check Squirrel holds up a slice of pie and offers, 'Happy Pi Day!' to the right-side scientist.
Dave Whamond’s Reality Check for the 14th of March, 2021. Essays featuring something discussed in Reality Check are at this link.

Dave Whamond’s Reality Check is a pi-pie joke that doesn’t rely on actual pie. Well, there’s a small slice in the corner. It relies on the infinite length of the decimal representation of π. (Or its representation in any integer base.)

A man digs his way through a deep snow heap to a piece of pie in a glass-covered jar. Caption: 'Sunday! Mr Lux digs out of snow on his way to pie day!' s
Michael Jantze’s Studio Jantze for the 15th of March, 2021. This is a new comic so there’s no old essays about it. But this and any future essays about Studio Jantze should appear at this link.

Michael Jantze’s Studio Jantze ran on Monday instead, although the caption suggests it was intended for Pi Day. So I’m including it here. And it’s the last of the strips sliding the day over to pie.

But there were a couple of comic strips with some mathematics mention that were not about Pi Day. It may have been coincidence.

Sandra Bell-Lundy's Between Friends for the 14th of March, 2021. This and a bunch of other appearances of Venn Diagrams should be gathered under the Between Friends tag here.
Sandra Bell-Lundy’s Between Friends for the 14th of March, 2021. This and a bunch of other appearances of Venn Diagrams should be gathered under the Between Friends tag here.

Sandra Bell-Lundy’s Between Friends is of the “word problem in real life” kind. It’s a fair enough word problem, though, asking about how long something would take. From the premises, it takes a hair seven weeks to grow one-quarter inch, and it gets trimmed one quarter-inch every six weeks. It’s making progress, but it might be easier to pull out the entire grey hair. This won’t help things though.

Bucky, cat: 'I've been thinkin' about the whole infinite monkey thing lately.' Satchell, dog: 'You lost me.' Bucky: 'It's the theory that if you get a load of monkeys on typewriters, one will accidentally type Shakespeare at some point.' Satchell: 'Mm-hm, mm-hm.' Bucky: 'Well, the whole theory is flawed. 'Infinite' is too many monkeys. Over 8 monkeys and you're running into discipline and hygiene issues. And who's gonna read infinite monkey scripts? Some chimp could hvae written the next Da Vinci code, but newsflash: he's eating that script before you ever see it. Here's what you do: you buy a $2 bag of nuts, you go trap yourself some squirrels. You put them on word processors --- WITH SPELLCHECK --- and you shoot for a 'Two and a Half Men' script ... you pocket the infinite monkey allocation money, sell the script, and retire to Hawaii.' Satchell: 'So now it's finite squirrels at word processors? ... I'm still lost.' Bucky: 'Never mind. You got two dollars?'
Darby Conley’s Get Fuzzy for the 14th of March, 2021. Essays mentioning Get Fuzzy are gathered at this link.

Darby Conley’s Get Fuzzy is a rerun, as all Get Fuzzy strips are. It first (I think) ran the 13th of September, 2009. And it’s another Infinite Monkeys comic strip, built on how a random process should be able to create specific outcomes. As often happens when joking about monkeys writing Shakespeare, some piece of pop culture is treated as being easier. But for these problems the meaning of the content doesn’t count. Only the length counts. A monkey typing “let it be written in eight and eight” is as improbable as a monkey typing “yrg vg or jevggra va rvtug naq rvtug”. It’s on us that we find one of those more impressive than the other.

And this wraps up my Pi Day comic strips. I don’t promise that I’m back to reading the comics for their mathematics content regularly. But I have done a lot of it, and figure to do it again. All my Reading the Comics posts appear at this link. Thank you for reading and I hope you had some good pie.

I don’t know how Andertoons didn’t get an appearance here.

My Things for Pi Day


I regret not having the time or energy to write something original about π for today. I hope you’ll accept this offering of past Reading the Comics posts covering the day, and some of my other π-related writings:

For the Pi Day Of The Century (3/14/15) I wrote Calculating Pi Terribly. It’s about a legitimate way to calculate the digits of π, using so very much work that nobody will ever do it. But the wonderful thing about it is it’s experimental. And it doesn’t involve something with an obvious circle. A couple months later I followed up with Calculating Pi Less Terribly, using one of the most famous numerical methods for calculating the digits of π. It’s still not that good, but it’s far better than the experimental approach.

In 2019, as part of that year’s A-to-Z, I wrote more extensively about Buffon’s Needle problem, the core of this experimental method for finding digits of π.

And then there’s comic strips. I seem to complain every year that there’s fewer Pi Day comic strips than I expected, which invites the question of just what I expect. Here’s, as best I can tell, the actual record:

I have not yet read today’s comics, so don’t know what they’ll offer. We shall see! Also, I apologize but some of the comics may have been removed from GoComics or Comics Kingdom, and so the links may be dead. I’m not happy about that. But if I wanted the essays discussing these strips to stay permanently sensible I’d have posted the comics on my own web site.

And one last thing, bringing up an essay I’ve shared before. The End 2016 Mathematics A To Z: Normal Numbers is … maybe … about π. Nobody knows whether π is a normal number. It most likely is, but we haven’t been able to prove it.

And the last thing. When I thought I would have time this March, I hoped to write something about how π can be defined starting from differential equations. Things changed my plans out from under me. But my 2020 A-to-Z essay on the Exponential gets at some of why π should turn up in the correct differential equation. That essay sets you up more to understand a famous equation, that e^{\pi \imath} + 1 = 0 . But it’s not too far to getting π out of solving y''(t) = -y(t) in the right circumstances. I may get to writing that one yet.

Some Mathematics Books By Women


David Eppstein, on Mathstodon (the mathematics-themed Mastodon instance), offered a nice essay for people looking for more mathematics reading. It’s a collection of mathematics books written by women.

The post is a collection of titles and brief descriptions. Some of them are general-interest books, such as one about the Inca system of knotted strings for recording numbers, or about how non-Euclidean geometries work. Others are textbooks or histories or biographies. And some are research monographs or other highly specialized work.

It’s a follow-up to a similar list published in 2020, though that earlier one focused on more recently published books.

I’ve found several that I intend to borrow from the library; you might have similar good fortune.

The 144th Playful Math Education Blog Carnival has published


I apologize for not noticing sooner. The 144th edition of the Playful Math Education Blog Carnival published a couple weeks ago. I’ve been distracted. Denise Gaskins, organizer of the blog carnival, hosted it this time.

The Playful Math Education Blog Carnival is a collection of posts about mathematics that are educational or recreational or delightful or just fun. This isn’t an exclusive or. There’s a good chance at least some posts will interest you. Some may be useful if you ever need to teach or communicate mathematics to an audience.

The blog has a new host every month. Math Hombre is scheduled to host this month’s. And if you’d like to host yourself, the rest of the year is available. Hosting one is a challenging exercise, but worthwhile. I’ve hosted a couple of times, most recently in September. It encourages me, at least, to look at my mathematics reading with different goals in mind. Even if you’re not up to the challenge of hosting, you can submit a blog — your own, or another’s — for inclusion in the monthly carnival. It helps people who might see something educational or delightful. And it helps whoever the host is.

The Fundamentals of Mathematics


A toot on Mathstodon made me aware of this. It’s a listing, and brief description, of 243 theorems, as compiled by Oliver Knill. As the title implies they’re all intended to be fundamental theorems of some area of mathematics.

Many areas of mathematics have something called their Fundamental Theorem. The one that comes first to my mind is always the Fundamental Theorem of Calculus. That one connects derivatives and indefinite integrals in a way that saves a lot of work. But also commonly in my mind are the Fundamental Theorem of Algebra, which assures one of how many roots a polynomial should have, and the Fundamental Theorem of Arithmetic, about factoring counting numbers into primes.

The list does not stop there. And it gets into areas where “Fundamental Theorem Of ___ ” is not the common phrasing. They are, where I know something about the area, certainly core, fundamental theorems as promised, though. Or important mathematical principles, such as the pigeon-hole principle. It’s worth skimming around; even if you don’t know anything about the area, Knill provides some context, so you can understand why this might be of interest.

And then after the many theorems Knill provides some thoughts about why these theorems. What makes a theorem “fundamental”. This is something which shows off how culturally dependent and human the construction of mathematics is. And then, from page 147, a set of short lecture notes about the history of mathematics. Even if your eyes glaze over at torsion groups, it’s worth going into those notes at the end.

How February 2021 Treated My Mathematics Blog


I hadn’t quite intended it, but February was another low-power month here. No big A-to-Z project and no resumption of Reading the Comics. The high points were sharing things that I’d seen elsewhere, and a mathematics problem that occurred to me while making tea. Very low-scale stuff. Still, I like to check on how that’s received.

I did put together seven posts for February — the same as January — and here’s a list of them in descending order of popularity:

I assume the essay setting out the tea question was more popular than the answer because it had a week more to pick up readers. That or people reading the answer checked back on what the question was. It couldn’t be that people are that uninterested in my actually explaining a mathematics thing.

That’s it for relative popularity. How about for total readership?

Bar chart of monthly readership statistics for two and a half years worth of blogging. After a rise in late 2020 the readership's been dwindling the last several months, though not precipitously.
Oh, WordPress has given up on that little window promising a strategy to make money while blogging. I’m gratified they admitted it so swiftly.

I had expected readership to continue declining, since I’m publishing fewer things and having my name out there seems to matter. But the decline’s less drastic than I expected. There were 2,167 page views here in February. But in the twelve months from February 2020 through January 2021? I had a mean of 2,137.4 page views, and a median of 2,044.5. That is, I’m still on the high side of my popularity.

There were 1,576 logged unique visitors in February. In the twelve months leading up to that the mean was 1,480.7 unique visitors, and the median 1,395.5.

The figures look more impressive if you rate them by number of postings. In that case in February I gathered 309.6 views per posting, way above the mean of 157.9 and median of 135.6. There were also 225.1 unique visitors per posting, again way above the running mean of 109.9 and median of 90.7.

I’ll dig unpopularity out of any set of numbers, though. There were only 47 likes granted here in February, down from the running mean of 55.8 and median of 55.5. That is still 6.7 likes per posting, above the mean of 3.9 and median of 4.0, but it’s still sparse likings. There were a hearty 39 comments given — my highest number since October 2018 — and that’s well above the mean of 17.0 and median of 18. Per posting, that’s 5.6 comments per posting, the highest I have since I started calculating this figure back in July of 2018. The mean and median comments per posting, for the twelve months leading up to this, were both 1.2.

WordPress’s insights panel tells me I published seven things in February, which matches my experience. I still can’t explain the discrepancy back in January. It says also that I published 3,440 words over February, my quietest month since I started tracking those numbers. It put my average post at 590 words for February, and 573.3 words for the whole year to date.

I start March, if WordPress is reliable, having gathered 126,829 views from 73,273 logged unique visitors. This after 1,595 posts in total.

If you’d like to be a regular reader, I’d be glad to have you. You can read without showing up in my statistics by adding my RSS feed to whatever your RSS reader is. If you don’t have an RSS reader, you can make one by signing up for a free account at Dreamwidth or Livejournal. Then you can add feeds from here, or any web site with an RSS feed. Use https://www.dreamwidth.org/feeds/ for Dreamwidth or at https://www.livejournal.com/syn for Livejournal. More blogs than you think have RSS feeds, even if they’re not advertised. Try adding /feed or /?feed=atom to the end of a blog’s URL. It may work!

If you have a WordPress account you can add me to your Reader by clicking the “Follow Nebusresearch” button on this page. I’ve also re-enabled the “Follow NebusResearch By E-mail” option, for people who want to see posts before I’ve fixed the typos. The typos will never be fixed. Every time an author looks at an old blog post there are three more typos, even if they’ve corrected the typos before.

My Twitter account is still feral; it announces posts but I don’t read it. If you want to social-media-engage me the way to go is @nebusj@mathstodon.xyz on the Mastodon microblogging network.

Thank you all for reading, whatever way you do that.

My best tea-refilling strategy


The problem I’d set out last week: I have a teapot good for about three cups of tea. I want to put milk in the once, before the first cup. How much should I drink before topping up the cup, to have the most milk at the end?

I have expectations. Some of this I know from experience, doing other problems where things get replaced at random. Here, tea or milk particles get swallowed at random, and replaced with tea particles. Yes, ‘particle’ is a strange word to apply to “a small bit of tea”. But it’s not like I can call them tea molecules. “Particle” will do and stop seeming weird someday.

Random replacement problems tend to be exponential decays. That I know from experience doing problems like this. So if I get an answer that doesn’t look like an exponential decay I’ll doubt it. I might be right, but I’ll need more convincing.

I also get some insight from extreme cases. We can call them reductios. Here “reductio” as in the word we usually follow with “ad absurdum”. Make the case ridiculous and see if that offers insight. The first reductio is to suppose I drink the entire first cup down to the last particle, then pour new tea in. By the second cup, there’s no milk left. The second reductio is to suppose I drink not a bit of the first cup of milk-with-tea. Then I have the most milk preserved. It’s not a satisfying break. But it leads me to suppose the most milk makes it through to the end if I have a lot of small sips and replacements of tea. And to look skeptically if my work suggests otherwise.

So that’s what I expect. What actually happens? Here, I do a bit of reasoning. Suppose that I have a mug. It can hold up to 1 unit of tea-and-milk. And the teapot, which holds up to 2 more units of tea-and-milk. What units? For the mathematics, I don’t care.

I’m going to suppose that I start with some amount — call it a — of milk. a is some number between 0 and 1. I fill the cup up to full, that is, 1 unit of tea-and-milk. And I drink some amount of the mixture. Call the amount I drink x . It, too, is between 0 and 1. After this, I refill the mug up to full, so, putting in x units of tea. And I repeat this until I empty the teapot. So I can do this \frac{2}{x} times.

I know you noticed that I’m short on tea here. The teapot should hold 3 units of tea. I’m only pouring out 3 - a . I could be more precise by refilling the mug \frac{2 + a}{x} times. I’m also going to suppose that I refill the mug with x amount of tea a whole number of times. This sounds necessarily true. But consider: what if I drank and re-filled three-quarters of a cup of tea each time? How much tea is poured that third time?

I make these simplifications for good reasons. They reduce the complexity of the calculations I do without, I trust, making the result misleading. I can justify it too. I don’t drink tea from a graduated cylinder. It’s a false precision to pretend I do. I drink (say) about half my cup and refill it. How much tea I get in the teapot is variable too. Also, I don’t want to do that much work for this problem.

In fact, I’m going to do most of the work of this problem with a single drawing of a square. Here it is.

A unit square representing the milk-and-tea problem. There's a horizontal strip, of height 'a', representing the milk. There's a vertical strip, of width 'x', representing how much is drunk between refills and replaced with tea.
You may protest that my tea is not layered so the milk is all on the bottom, and that I do not drink a vertical column from it. To this I say: how do you know how I drink my tea?

So! I start out with a units of tea in the mixture. After drinking x units of milk-and-tea, what’s left is a\cdot(1 - x) units of milk in the mixture.

How about the second refill? The process is the same as the first refill. But where, before, there had been a units of milk in the tea, now there are only a\cdot(1 - x) units in. So that horizontal strip is a little narrower is all. The same reasoning applies and so, after the second refill, there’s a\cdot(1 - x)\cdot(1 - x) milk in the mixture.

If you nodded to that, you’d agree that after the third refill there’s a\cdot(1 - x)\cdot(1 - x)\cdot(1 - x) . And are pretty sure what happens at the fourth and fifth and so on. If you didn’t nod to that, it’s all right. If you’re willing to take me on faith we can continue. If you’re not, that’s good too. Try doing a couple drawings yourself and you may convince yourself. If not, I don’t know. Maybe try, like, getting six white and 24 brown beads, stir them up, take out four at random. Replace all four with brown beads and count, and do that several times over. If you’re short on beads, cut up some paper into squares and write ‘B’ and ‘W’ on each square.

But anyone comfortable with algebra can see how to reduce this. The amount of milk remaining after j refills is going to be

a\cdot(1 - x)^j

How many refills does it take to run out of tea? That we knew from above: it’s \frac{2}{j} refills. So my last full mug of tea will have left in it

a\cdot(1 - x)^{\frac{2}{x}}

units of milk.

Anyone who does differential equations recognizes this. It’s the discrete approximation of the exponential decay curve. Discrete, here, because we take out some finite but nonzero amount of milk-and-tea, x , and replace it with the same amount of pure tea.

Now, again, I’ve seen this before so I know its conclusions. The most milk will make it to the end of x is as small as possible. The best possible case would be if I drink and replace an infinitesimal bit of milk-and-tea each time. Then the last mug would end with a\cdot e^{-2} of milk. That’s e as in the base of the natural logarithm. Every mathematics problem has an e somewhere in it and I’m not exaggerating much. All told this would be about 13 and a half percent of the original milk.

Drinking more realistic amounts, like, half the mug before refilling, makes the milk situation more dire. Replacing half the mug at a time means the last full mug has only one-sixteenth what I started with. Drinking a quarter of the mug and replacing it lets about one-tenth the original milk survive.

But all told the lesson is clear. If I want milk in the last mug, I should put some in each refill. Putting all the milk in at the start and letting it dissolve doesn’t work.

Why not make an iceberg?


A post on Mathstodon made me aware there’s a bit of talk about iceberg shapes. Particularly that one of the iconic photographs of an iceberg above-and-below water, is a imaginative work. A real iceberg wouldn’t be stable in that orientation. Which, I’ll admit, isn’t something I had thought about. I also hadn’t thought about the photography challenge of getting a clear picture of something in sunlight and in water at once. There was a lot I hadn’t thought about. In my defense, I spend a lot of time noticing comic strips had a character complain about the New Math.

But this all leads me to a fun little play tool: Iceberger, designed to let you sketch in a potential iceberg and see what it does. Often, that’s roll over to a more stable orientation. It’s fun to play with, and to watch shapes tilt over, gradually or rapidly. And playing with it may help one develop a sense for what kinds of shapes should be stable in water, and what kinds should not.

When should I refill my tea?


I’ve been taking milk in my tea lately. I have a teapot good for about three cups of tea. So that’s got me thinking about how to keep the most milk in the last of my tea. You may ask why I don’t just get some more milk when I refill the cup. I answer that if I were willing to work that hard I wouldn’t be a mathematician.

It’s easy to spot the lowest amount of milk I could have. If I drank the whole of the first cup, there’d be only whatever milk was stuck by surface tension to the cup for the second. And so even less than that for the third. But if I drank half a cup, poured more tea in, drank half again, poured more in … without doing the calculation, that’s surely more milk for the last full cup.

So what’s the strategy for the most milk I could get in the final cup? And how much is in there?

I haven’t done the calculations yet. Wanted to put the problem out and see if my intuition about this matches anyone else’s, and how close that might be to right. Or at least calculated. I suspect it’s one of a particular kind of problem, though.

Monte Carlo pioneer Arianna Wright Rosenbluth dead at 93


The New York Times carried an obituary for Dr Arianna Wright Rosenbluth. She died in December from Covid-19 and the United States’s mass-murderous handling of Covid-19. And she’s a person I am not sure I knew anything substantial about. I had known her name, but not anything more. This is a chance to correct that a bit.

Rosenbluth was a PhD in physics (and an Olympics-qualified fencer). Her postdoctoral work was with the Atomic Energy Commission, bringing her to a position at Los Alamos National Laboratory in the early 1950s. And a moment in computer science that touches very many people’s work, my own included. This is in what we call Metropolis-Hastings Markov Chain Monte Carlo.

Monte Carlo methods are numerical techniques that rely on randomness. The name references the casinos. Markov Chain refers to techniques that create a sequence of things. Each thing exists in some set of possibilities. If we’re talking about Markov Chain Monte Carlo this is usually an enormous set of possibilities, too many to deal with by hand, except for little tutorial problems. The trick is that what the next item in the sequence is depends on what the current item is, and nothing more. This may sound implausible — when does anything in the real world not depend on its history? — but the technique works well regardless. Metropolis-Hastings is a way of finding states that meet some condition well. Usually this is a maximum, or minimum, of some interesting property. The Metropolis-Hastings rule has the chance of going to an improved state, one with more of whatever the property we like, be 1, a certainty. The chance of going to a worsened state, with less of the property, be not zero. The worse the new state is, the less likely it is, but it’s never zero. The result is a sequence of states which, most of the time, improve whatever it is you’re looking for. It sometimes tries out some worse fits, in the hopes that this leads us to a better fit, for the same reason sometimes you have to go downhill to reach a larger hill. The technique works quite well at finding approximately-optimum states when it’s hard to find the best state, but it’s easy to judge which of two states is better. Also when you can have a computer do a lot of calculations, because it needs a lot of calculations.

So here we come to Rosenbluth. She and her then-husband, according to an interview he gave in 2003, were the primary workers behind the 1953 paper that set out the technique. And, particularly, she wrote the MANIAC computer program which ran the algorithm. It’s important work and an uncounted number of mathematicians, physicists, chemists, biologists, economists, and other planners have followed. She would go on to study statistical mechanics problems, in particular simulations of molecules. It’s still a rich field of study.