## Reading the Comics, March 17, 2018: Pi Day 2018 Edition

So today I am trying out including images for all the mathematically-themed comic strips here. This is because of my discovery that some links even on GoComics.com vanish without warning. I’m curious how long I can keep doing this. Not for legal reasons. Including comics for the purpose of an educational essay about topics raised by the strips is almost the most fair use imaginable. Just because it’s a hassle copying the images and putting them up on WordPress.com and that’s even before I think about how much image space I have there. We’ll see. I might try to figure out a better scheme.

Also in this batch of comics are the various Pi Day strips. There was a healthy number of mathematically-themed comics on the 14th of March. Many of those were just coincidence, though, with no Pi content. I’ll group the Pi Day strips together.

Tom Batiuk’s Funky Winkerbean for the 2nd of April, 1972 is, I think, the first appearance of Funky Winkerbean around here. Comics Kingdom just started running the strip, as well as Bud Blake’s Tiger and Bill Hoest’s Lockhorns, from the beginning as part of its Vintage Comics roster. And this strip really belonged in Sunday’s essay, but I noticed the vintage comics only after that installment went to press. Anyway, this strip — possibly the first Sunday Funky Winkerbean — plays off a then-contemporary fear of people being reduced to numbers in the face of a computerized society. If you can imagine people ever worrying about something like that. The early 1970s were a time in American society when people first paid attention to the existence of, like, credit reporting agencies. Just what they did and how they did it drew a lot of critical examination. Josh Lauer’s recently published Creditworthy: a History of Consumer Surveillance and Financial Identity in America gets into this.

Bob Scott’s Bear With Me for the 14th sees Molly struggling with failure on a mathematics test. Could be any subject and the story would go as well, but I suppose mathematics gets a connotation of the subject everybody has to study for, even the geniuses. (The strip used to be called Molly and the Bear. In either name this seems to be the first time I’ve tagged it, although I only started tagging strips by name recently.)

Bud Fisher’s Mutt and Jeff rerun for the 14th is a rerun from sometime in 1952. I’m tickled by the problem of figuring out how many times Fisher and his uncredited assistants drew Mutt and Jeff. Mutt saying that the boss “drew us 14,436 times” is the number of days in 45 years, so that makes sense if he’s counting the number of strips drawn. The number of times that Mutt and Jeff were drawn is … probably impossible to calculate. There’s so many panels each strip, especially going back to earlier and earlier times. And how many panels don’t have Mutt or don’t have Jeff or don’t have either in them? Jeff didn’t appear in the strip until March of 1908, for example, four months after the comic began. (With a different title, so the comic wasn’t just dangling loose all that while.)

Doug Savage’s Savage Chickens for the 14th is a collection of charts. Not all pie charts. And yes, it ran the 14th but avoids the pun it could make. I really like the tart charts, myself.

And now for the Pi Day strips proper.

Scott Hilburn’s The Argyle Sweater for the 14th starts the Pi Day off, of course, with a pun and some extension of what makes 3/14 get its attention. And until Hilburn brought it up I’d never thought about the zodiac sign for someone born the 14th of March, so that’s something.

Mark Parisi’s Off The Mark for the 14th riffs on one of the interesting features of π, that it’s an irrational number. Well, that its decimal representation goes on forever. Rational numbers do that too, yes, but they all end in the infinite repetition of finitely many digits. And for a lot of them, that digit is ‘0’. Irrational numbers keep going on with more complicated patterns. π sure seems like it’s a normal number. So we could expect that any finite string of digits appears somewhere in its decimal expansion. This would include a string of digits that encodes any story you like, The Neverending Story included. This does not mean we might ever find where that string is.

Michael Cavna’s Warped for the 14th combines the two major joke threads for Pi Day. Specifically naming Archimedes is a good choice. One of the many things Archimedes is famous for is finding an approximation for π. He’d worked out that π has to be larger than 310/71 but smaller than 3 1/7. Archimedes used an ingenious approach: we might not know the precise area of a circle given only its radius. But we can know the area of a triangle if we know the lengths of its legs. And we can draw a series of triangles that are enclosed by a circle. The area of the circle has to be larger than the sum of the areas of those triangles. We can draw a series of triangles that enclose a circle. The area of the circle has to be less than the sum of the areas of those triangles. If we use a few triangles these bounds are going to be very loose. If we use a lot of triangles these bounds can be tight. In principle, we could make the bounds as close together as we could possibly need. We can see this, now, as a forerunner to calculus. They didn’t see it as such at the time, though. And it’s a demonstration of what amazing results can be found, even without calculus, but with clever specific reasoning. Here’s a run-through of the process.

John Zakour and Scott Roberts’s Working Daze for the 15th is a response to Dr Stephen Hawking’s death. The coincidence that he did die on the 14th of March made for an irresistibly interesting bit of trivia. Zakour and Roberts could get there first, thanks to working on a web comic and being quick on the draw. (I’m curious whether they replaced a strip that was ready to go for the 15th, or whether they normally work one day ahead of publication. It’s an exciting but dangerous way to go.)

## What I’ve Been Reading, Mid-March 2018

So here’s some of the stuff I’ve noticed while being on the Internet and sometimes noticing interesting mathematical stuff.

Here from the end of January is a bit of oddball news. A story problem for 11-year-olds in one district of China set up a problem that couldn’t be solved. Not exactly, anyway. The question — “if a ship had 26 sheep and 10 goats onboard, how old is the ship’s captain?” — squares nicely with that Gil comic strip I discussed the other day. After seeing 26 (something) and 10 (something else) it’s easy to think of what answers might be wanted: 36 (total animals) or 16 (how many more sheep there are than goats) or maybe 104 (how many hooves there are, if they all have the standard four hooves). That the question doesn’t ask anything that the given numbers matter for barely registers unless you read the question again. I like the principle of reminding people not to calculate until you know what you want to do and why that. And it’s possible to give partial answers: the BBC News report linked above includes a mention from one commenter that allowed a reasonable lower bound to be set on the ship’s captain’s age.

In something for my mathematics majors, here’s A Regiment of Monstrous Functions as assembled by Rob J Low. This is about functions with a domain and a range that are both real numbers. There’s many kinds of these functions. They match nicely to the kinds of curves you can draw on a sheet of paper. So take a sheet of paper and draw a curve. You’ve probably drawn a continuous curve, one that can be drawn without lifting your pencil off the paper. Good chance you drew a differentiable one, one without corners. But most functions aren’t continuous. And aren’t differentiable. Of those few exceptions that are, many of them are continuous or differentiable only in weird cases. Low reviews some of the many kinds of functions out there. Functions discontinuous at a point. Functions continuous only on one point, and why that’s not a crazy thing to say. Functions continuous on irrational numbers but discontinuous on rational numbers. This is where mathematics majors taking real analysis feel overwhelmed. And then there’s stranger stuff out there.

Here’s a neat one. It’s about finding recognizable, particular, interesting pictures in long enough prime numbers. The secret to it is described in the linked paper. The key is that the eye is very forgiving of slightly imperfect images. This fact should reassure people learning to draw, but will not. And there’s a lot of prime numbers out there. If an exactly-correct image doesn’t happen to be a prime number that’s all right. There’s a number close enough to it that will be. That latter point is something that anyone interested in number theory “knows”, in that we know some stuff about the biggest possible gaps between prime numbers. But that fact isn’t the same as seeing it.

And finally there’s something for mathematics majors. Differential equations are big and important. They appear whenever you want to describe something that changes based on its current state. And this is so much stuff. Finding solutions to differential equations is a whole major field of mathematics. The linked PDF is a slideshow of notes about one way to crack these problems: find symmetries. The only trouble is it’s a PDF of a Powerpoint presentation, one of those where each of the items gets added on in sequence. So each slide appears like eight times, each time with one extra line on it. It’s still good, interesting stuff.

## Reading the Comics, March 13, 2018: One Of My Assumptions Is Shaken Edition

I learn, from reading not-yet-dead Usenet group rec.arts.comics.strips, that Rick Stromoski is apparently ending the comic Soup To Nutz. This is sad enough. But worse, GoComics.com has removed all but the current day’s strip from its archives. I had trusted that GoComics.com links were reliable in a way that Comics Kingdom and Creators.com weren’t. Now I learn that maybe I need to include images of the comics I review and discuss here lest my essays become unintelligible in the future? That’s not a good sign. I can do it, mind you. I just haven’t got started. You’ll know when I swing into action.

Norm Feuti, of Retail, still draws Sunday strips for Gil. They’re to start appearing on GoComics.com soon, and I can talk about them from my regular sources after that. But for now I follow the strip on Twitter. And last Sunday he posted this one.

It’s sort of a protesting-the-problem question. It’s also a reaction a lot of people have to “explain how you found the answer” questions. In a sense, yeah, the division shows how the answer was found. But what’s wanted — and what’s actually worth learning — is to explain why you did this calculation. Why, in this case, 216 divided by 8? Why not 216 times 8? Why not 8 divided by 216? Why not 216 minus 8? “How you found your answer” is probably a hard question to make interesting on arithmetic, unfortunately. If you’re doing a long sheet of problems practicing division, it’s not hard to guess that dividing is the answer. And that it’s the big number divided by the small. It can be good training to do blocks of problems that use the same approach, for the same reason it can be good training to focus on any exercise a while. But this does cheat someone of the chance to think about why one does this rather than that.

Patrick Roberts’s Todd the Dinosaur for the 11th has mathematics as the thing Todd’s trying to get out of doing. (I suppose someone could try to argue the Y2K bug was an offshoot of mathematics, on the grounds that computer science has so much to do with mathematics. I wouldn’t want to try defending that, though.) I grant that most fraction-to-decimal conversion problems hit that sweet spot of being dull, tedious, and seemingly pointless. There’s some fun decimal expansions of fractions. The sevenths and the elevenths and 1/243 have charm to them. There’s some kid who’ll become a mathematician because at the right age she was told about $\frac{1}{8991}$. 3/16th? Eh.

Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. I don’t remember seeing a spinny wheel like this used to introduce probability. It’s a good prop, though. I would believe in a class having it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 11th is built on the Travelling Salesman Problem. It’s one of the famous unsolved and hard problems of mathematics. Weinersmith’s joke is a nice gag about one way to “solve” the problem, that of making it irrelevant. But even if we didn’t need to get to a collection of places efficiently mathematicians would still like to know good ways to do it. It turns out that finding the shortest (quickest, cheapest, easiest, whatever) route connecting a bunch of places is great problem. You can phrase enormously many problems about doing something as well as possible as a Travelling Salesman Problem. It’s easy conceptually to find the answer: try out all the possibilities and pick the best one. But if there’s more than a handful of cities, there are so many possible routes there’s no checking them all, not before you die of old age. We can do very well finding approximate answers, including by my specialization of Monte Carlo methods. In those you take a guess at an answer. Then make, randomly, a change. You’ll either have made things better or worse. If you’ve made it better, keep the change. If you’ve made it worse, usually you reject the change but sometimes you keep it. And repeat. In surprisingly little time you’ll get a really good answer. Maybe not the best possible, but a great answer for how straightforward setting it up was.

Dan Thompson’s Brevity for the 12th is a Rubik’s Cube joke. There’s not a lot of mathematics to that. But I do admire how Thompson was careful enough to draw a Rubik’s Cube that actually looks like the real article; it’s not just an isometric cube with thick lines partitioning it. Look at the corners of each colored sub-cube. I may be the only reader to notice this but I’m glad Thompson did the work.

Mason Mastroianni’s The Wizard of Id for the 12th gets Sir Rodney in trouble with the King for doing arithmetic. I haven’t read the comments on GoComics.com. I’d like to enter “three” as my guess for how many comments one would have to read before finding the “weapons of math instruction” joke in there.

Jef Mallett’s Frazz for the 13th has mathematics homework given as the thing lost by the time change. It’s just a cameo mention.

Steve Moore’s In The Bleachers for the 13th features a story problem as a test of mental acuity. When the boxer can’t work out what the heck the trains-leaving-Penn-Station problem even means he’s ruled unfit to keep boxing. The question is baffling, though. As put, the second train won’t ever overtake the first. The question: did Moore just slip up? If the first train were going 30 miles per hour and the second 40 there would be a perfectly good, solvable question in this. Or was Moore slipping in an extra joke, making the referee’s question one that sounds like it was given wrong? Don’t know, so I’ll suppose the second.

## Is A Basketball Tournament Interesting? My Thoughts

It’s a good weekend to bring this back. I have some essays about information theory and sports contests and maybe you missed them earlier. Here goes.

And then for a follow-up I started looking into actual scoring results from major sports. This let me estimate the information-theory content of the scores of soccer, (US) football, and baseball scores, to match my estimate of basketball scores’ information content.

Don’t try to use this to pass your computer science quals. But I hope it gives you something interesting to talk about while sulking over your brackets, and maybe to read about after that.

## Reading the Comics, March 10, 2018: I Will Get To Pi Day Edition

There were fewer Pi Day comic strips than I had expected for this year. It’s gotten much more public mention than I had expected a pop-mathematics bit of whimsy might. But I’m still working off last week’s strips; I’ll get to this week’s next week. This makes sense to me, which is as good as making sense at all.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 7th is a percentages joke, as applied to hair. Lard doesn’t seem clear whether this would be 10% off the hair by individual strand length or by total volume. Either way, Lard’s right to wonder about the accuracy.

Mark Pett’s Mr Lowe rerun for the 7th is a standardized test joke. Part of the premise of Pett’s strip is that Mister Lowe is a brand-new teacher, which is why he makes mistakes like this problem. (This is touchy to me, as in grad school I hoped to make some spare money selling questions to a standardized testing company. I wasn’t good enough at it, and ultimately didn’t have the time to train up to their needs.) A multiple-choice question needs to clear and concise and to have one clearly best answer. As the given question’s worded, though, I could accept ‘2’ or ’12’ as a correct answer. With a bit of experience Lowe would probably clarify that Tommy and Suzie are getting the same number of apples and that together they should have 20 total.

Then on the 9th Mr Lowe has a joke about cultural bias in standardized tests. It uses an arithmetic problem as the type case. Mathematicians like to think of themselves as working in a universal, culturally independent subject. I suppose it is, but only in ways that aren’t interesting: if you suppose these rules of logic and these axioms and these definitions then these results follow, and it doesn’t matter who does the supposing. But start filtering that by stuff people care about, such as the time it takes for two travelling parties to meet, and you’ve got cultural influence. (Back when this strip was new the idea that a mathematics exam could be culturally biased was a fresh new topic of mockery among people who don’t pay much attention to the problems of teaching but who know what those who do are doing wrong.)

Ralph Hagen’s The Barn for the 8th — a new tag for my comics, by the way — lists a bunch of calculation tools and techniques as “obsolete” items. I’m assuming Rory means that longhand multiplication is obsolete. I’m not sure that it is, but I have an unusual perspective on this.

Thaves’s Frank and Ernest for the 8th is an anthropomorphic-numerals joke. I was annoyed when I first read this because I thought, wait, 97 isn’t a prime number. It is, of course. I have no explanation for my blunder.

Jon Rosenberg’s Scenes from a Multiverse has restarted its run on GoComics. The strip for the 8th is a riff on Venn Diagrams. And, it seems to me, about those logic-bomb problems about sets consisting of sets that don’t contain themselves and the like. You get weird and apparently self-destructive results pondering that stuff. The last time GoComics ran the Scenes from a Multiverse series I did not appreciate right away that there were many continuing stories. There might be follow-ups to this Former Venn Prime Universe story.

Brian Fies’s The Last Mechanical Monster for the 9th has the Mad Scientist, struggling his way into the climax of the story, testing his mind by calculating a Fibonacci Sequence. Whatever keeps you engaged and going. You can build a Fibonacci Sequence from any two starting terms. Each term after the first two is the sum of the previous two. If someone just says “the Fibonacci Sequence” they mean the sequence that starts with 0, 1, or perhaps with 1, 1. (There’s no interesting difference.) Fibonacci Sequences were introduced to the west by Leonardo of Pisa, who did so much to introduce Hindu-Arabic Numerals to a Europe that didn’t know it wanted this stuff. They touch on some fascinating stuff: the probability of not getting two tails in a row of a set number of coin tosses. Chebyshev polynomials. Diophantine equations. They also touch on the Golden Ratio, which isn’t at all important but that people like.

Nicholas Gurewitch’s Perry Bible Fellowship for the 9th just has a blackboard of arithmetic to stand in for schoolwork.

## And My Pi Day Stuff

The 14th of March offers many things. A chance for calendar nerds to get all excited about the meaning of “ides”. A chance to bring out Pi-related content on mathematics blogs. I’ll take advantage of the latter. There’s not a lot in public dispute about the ides of March. The ides of February, now, that I’m not sure I can talk about coherently. But, for Pi:

• The End 2016 Mathematics A To Z: Normal Numbers which is relevant because π is probably a normal number. We don’t know, but it would be really weird if it weren’t. Normal numbers are weird, but most numbers are normal.
• Calculating Pi Terribly was my first, big, and basically sour essay about π. It describes the Buffon needle drop experiment, which is a real experiment you could do with actual physical objects if you wanted to eventually, someday, calculate the digits of π. You should use basically any other approach before this if you actually need to know them.
• Calculating Pi Less Terribly is a follow-up, about finding the digits of π using way less work. It gets into alternating series, which are mathematically interesting enough and very useful.

Enjoy, I hope!

## Reading the Comics, March 5, 2018: If It’s Even Mathematics Edition

Many of the strips from the first half of last week are ones that just barely touch on mathematical content. I’m not sure how relevant they all are. I hope you like encountering them anyway.

Bill Griffith’s Zippy the Pinhead for the 4th of March offers “an infinite number of mathematicians walk into a bar” as a joke’s setup. Mathematics popularizers have a small set of jokes about infinite numbers of mathematicians, often arriving at hotels. They’re used to talk about how we now understand infinitely large sets. There’s often counter-intuitive or just plain weird results that follow. And presenting it as a joke works surprisingly well in introducing the ideas. There’s a kind of joke that is essentially a tall tale, spinning out an initial premise to as far and as absurd a consequence as you can get. In structure, that’s not much different to a proof, a discussion of the consequences of an idea. It’s a shame that it’s hard to make jokes or anecdotes about more fields of mathematics. Somehow infinitely large groups of people are funnier than, say, upper-bounded nondecreasing sequences.

Mike Baldwin’s Cornered for the 4th has a bit of fraction-based wordplay. I’m not sure how mathematical this is, but I grinned.

Bill Amend’s FoxTrot for the 4th has Jason try to make a “universal” loot box that consists of zeroes and ones. As he says, accumulate enough and put them in the right order and you have any digital prize imaginable. Implementation is, as joked, the problem. Assembling ones and zeroes at random isn’t likely to turn up anything you might care about in a reasonable time. (It’s the monkeys-at-typewriters problem.) If you know how to assemble ones and zeroes to get what you want, well, what do you need Jason’s boxes for? As with most clever ideas by computer-oriented boys it shouldn’t really be listened to.

Mark Pett’s Lucky Cow rerun for the 4th has Neil make an order-of-magnitude error estimating what animal power can do. We’ve all made them. They’re particularly easy to make when switching the unit measure. Trying to go from meters to kilometers and multiplying the distance by a thousand, say. Which is annoying since often it’s easiest to estimate the order of magnitude of something first. I can’t find easily an estimate of how many calories a hamster eats over the course of the day. That seems like it would give an idea of how much energy a hamster could possibly be expected to provide, and so work out whether the estimate of four million hamsters to power a car is itself plausible. If someone has information, I’d take it.

Jonathan Lemon’s Rabbits Against Magic for the 4th is a Rubik’s Cube joke. Also a random processes joke. If a blender could turn the faces of a cube, and could turn them randomly, and could run the right period of time … well, yeah, it could unscramble a cube. But see the previous talk about Jason Fox and the delivery of ones and zeroes.

Mark Tatulli’s Lio for the 5th is a solid geometry joke. I’ve put more thought into whether and where to put hyphens in the last three words of that sentence than is worth it.

Steve Sicula’s Home and Away rerun for the 6th has the father and son happily doing some mathematics. It’s in the service of better gambling on sports. But at least they know why they would like to do these calculations.

## Did The Greatest Generation Hosts Get As Drunk As I Expected?

I finally finished listening to Benjamin Ahr Harrison and Adam Pranica’s Greatest Generation podcast reviews of the first season of Star Trek: Deep Space Nine. (We’ve had fewer long car trips for this.) So I can return to my projection of how their drinking game would turn out.

Their plan was to make more exciting the discussion of some of Deep Space Nine‘s episodes by recording their reviews while drinking a lot. The plan was, for the fifteen episodes they had in the season, there would be a one-in-fifteen chance of doing any particular episode drunk. So how many drunk episodes would you expect to get, on this basis?

It’s a well-formed expectation value problem. There could be as few as zero or as many as fifteen, but some cases are more likely than others. Each episode could be recorded drunk or not-drunk. There’s an equal chance of each episode being recorded drunk. Whether one episode is drunk or not doesn’t depend on whether the one before was, and doesn’t affect whether the next one is. (I’ll come back to this.)

The most likely case was for there to be one drunk episode. The probability of exactly one drunk episode was a little over 38%. No drunk episodes was also a likely outcome. There was a better than 35% chance it would never have turned up. The chance of exactly two drunk episodes was about 19%. There drunk episodes had a slightly less than 6% chance of happening. Four drunk episodes a slightly more than 1% chance of happening. And after that you get into the deeply unlikely cases.

As the Deep Space Nine season turned out, this one-in-fifteen chance came up twice. It turned out they sort of did three drunk episodes, though. One of the drunk episodes turned out to be the first of two they planned to record that day. I’m not sure why they didn’t just swap what episode they recorded first, but I trust they had logistical reasons. As often happens with probability questions, the independence of events — whether a success for one affects the outcome of another — changes calculations.

There’s not going to be a second-season update to this. They’ve chosen to make a more elaborate recording game of things. They’ve set up a modified Snakes and Ladders type board with a handful of spots marked for stunts. Some sound like fun, such as recording without taking any notes about the episode. Some are, yes, drinking episodes. But this is all a very different and more complicated thing to project. If I were going to tackle that it’d probably be by running a bunch of simulations and taking averages from that.

Also I trust they’ve been warned about the episode where Quark has a sex change so he can meet a top Ferengi soda magnate after accidentally giving his mother a heart attack because gads but that was a thing that happened somehow.

## Reading the Comics, March 2, 2018: Socks Edition

There were enough comics last week to justify splitting them across two posts. But several of them were on a single theme. So they’re bundled together and you see what the theme is already if you pay attention to the edition titles.

Jeff Mallet’s Frazz on the 26th of February had a joke about a story problem going awry. Properly this should’ve been included in the Sunday update, but the theme was riffed on the next several days, and so I thought moving this made for a better split. In this case the kids resist the problem on the grounds that the cost ($1.50 for a pair of socks) is implausibly low. And now I’m reminded that a couple months ago I wondered if a comic strip (possibly Frazz again) gave a plausible price for apples. And I go to a great farmer’s market nearly every week and look at the apple prices and never think to write them down so I can check. But the topic, and the attempt to use the price of socks as a joke, continued on the 27th. Here the resistance was on the grounds there might be a sale on. Fair enough, although the students should feel free to ask about sales. And the teacher ought to be able to offer that. Also, it seems to me that “twice$5” is a different problem to “twice $1.50”, at least at this level. An easier one, I’d say, too. If the pair of socks were$4.50 it would preserve what I imagine is the point being tested. I think that’s how to multiply a compound fraction or a number with a decimal. But Frazz’s characters know the objectives better than I do.

The topic gets clarified on the 28th, which doesn’t end the students’ resistance on the grounds of plausibility. This seems to portray the kids as more conscious of clothing prices than I think I was as a kid, but it’s Mallet’s comic strip. He knows what his kids care about. The sequence closes out the 1st of March with a coda that’s the sort of joke every academic department tells about the others.

Julie Larson’s Dinette Set rerun for the 27th is an extended bit of people not understanding two-for-one sales. I’m tickled by it, but I won’t think ill of you if you decide you don’t want to read all those word balloons. There’s some further jokes in the signs and the t-shirts people are wearing, but they’re not part of the main joke. (Larson would often include stray extra jokes like that. It always confuses people who didn’t get the strip’s humor style.)

Dan Thompson’s Brevity for the 1st of March is close enough to the anthropomorphic numerals joke of the week.

Jeffery Lambros’s Domestic Abuse for the 1st is the spare numerical symbols joke for the week, too.

## How February 2018 Treated My Mathematics Blog

It was a less riotously popular month here in February than it was in January. I’d like to blame the shortness of February, but that isn’t it. I know. I’ve got statistics.

The big one that I worry excessively over: total page views. 1,062 of them in February, down from January’s 1,274 but up from December 2017’s 899. And hey, anything above a thousand feels gratifying enough. The count of unique visitors dropped to 611. It had been at 670 in January, but then it was at 599 in December. I’m working on stuff that might affect this. We’ll see. I’d wondered if the readership drop might entirely represent February being such a short month. But WordPress’s insights page lets me know the average number of pages viewed per day. 41 in January (part of a three-way tie for third-highest, alongside September 2017 and November 2015). 38 in February. Still, not bad for a month that went by without a major overarching theme to pull people back in.

It was still a pretty likable month: 102 things clicked on over the course of the month. Down from January’s 112, but still, well ahead of December’s 71. It’s still in the range of liked-essays that I haven’t seen since the last A To Z project. There were 30 comments, once more down from January’s total (39) but up from December’s (24). It seems obvious that all these three data points should track together, although I’ve never tested that and maybe I could have some fun rambling about curve-fitting with it.

Oh, for the one data point wholly within my control: I posted 13 things in February. 14 in January. 11 in December, which was an awful month. (We haven’t found our next rabbit yet. I’ve been gently calling this one rescue every couple days to mention how the person fostering a Flemish Giant we find appealing hasn’t called us back to set a time when we might meet. I have a suspicion the person fostering has decided to quietly adopt the rabbit. And that’s fine, but not being told that gets in the emotional way of looking elsewhere.)

So what all was popular? … Pretty much what I would have guessed without knowing anything about the month:

I’m kind of seriously thinking to take some time off this month and just improve the graphics of the Record Grooves and the Trapezoids articles. And I’m always tickled when what amounts to a self-reblog, like the buy-a-theorem post, comes out more popular than the original post it references. I’m also thinking about setting some day aside to just reblog something from my archives.

What countries sent me readers? This bunch, says WordPress.

United States 703
United Kingdom 44
India 42
Philippines 42
Australia 14
Sweden 14
Singapore 12
France 9
Germany 9
Mexico 8
Pakistan 8
Brazil 6
Puerto Rico 6
Slovenia 6
Netherlands 5
Turkey 5
Algeria 4
Hungary 4
Italy 4
Spain 4
Bulgaria 3
Finland 3
Greece 3
Indonesia 3
Nepal 3
New Zealand 3
Portugal 3
South Africa 3
Switzerland 3
Belgium 2
Hong Kong SAR China 2
Japan 2
Mongolia 2
Romania 2
South Korea 2
Taiwan 2
Uruguay 2
Bahamas 1
Costa Rica 1
Cyprus 1
Denmark 1
Egypt 1
European Union 1
Ireland 1 (*)
Israel 1
Kenya 1
Lebanon 1
Mozambique 1
Poland 1
Russia 1 (**)
United Arab Emirates 1

That’s 54 countries altogether, if we don’t ask serious questions about the European Union and, for that matter, Hong Kong or Puerto Rico. There’d been 50 countries give or take in January, and 53 in December. There were 16 single-reader countries in February, up from the 14 in January and 15 in December. Ireland was a single-reader country in January too. Russia’s been a single-reader country two months running. And otherwise there’s been a turnover in single-readership countries.

The Insights panel says March started with 58,654 page views here, from an admitted 27,772 unique viewers and aw, isn’t that sweet number? The insights panel is also threatening to ruin me as a person by giving me some new interesting year-to-date statistics. According to these, as of the 5th of March (I didn’t have the chance to check on the 1st, and I don’t know how to find a year-to-specified date) I’ve published 25,359 total words, at an average 845 words per post. 30 posts to date for the year. 207 total likes, 77 total comments. And an average of 2.6 comments and 6.9 likes per post. I just know I’m going to obsess on these, what with how they’re numbers that have decimal points. But this is way more interesting than tracking the most popular day and hour.

## Reading the Comics, February 26, 2018: Possible Reruns Edition

Comic Strip Master Command spent most of February making sure I could barely keep up. It didn’t slow down the final week of the month either. Some of the comics were those that I know are in eternal reruns. I don’t think I’m repeating things I’ve already discussed here, but it is so hard to be sure.

Bill Amend’s FoxTrot for the 24th of February has a mathematics problem with a joke answer. The approach to finding the area’s exactly right. It’s easy to find areas of simple shapes like rectangles and triangles and circles and half-circles. Cutting a complicated shape into known shapes, finding those areas, and adding them together works quite well, most of the time. And that’s intuitive enough. There are other approaches. If you can describe the outline of a shape well, you can use an integral along that outline to get the enclosed area. And that amazes me even now. One of the wonders of calculus is that you can swap information about a boundary for information about the interior, and vice-versa. It’s a bit much for even Jason Fox, though.

Jef Mallett’s Frazz for the 25th is a dispute between Mrs Olsen and Caulfield about whether it’s possible to give more than 100 percent. I come down, now as always, on the side that argues it depends what you figure 100 percent is of. If you mean “100% of the effort it’s humanly possible to expend” then yes, there’s no making more than 100% of an effort. But there is an amount of effort reasonable to expect for, say, an in-class quiz. It’s far below the effort one could possibly humanly give. And one could certainly give 105% of that effort, if desired. This happens in the real world, of course. Famously, in the right circles, the Space Shuttle Main Engines normally reached 104% of full throttle during liftoff. That’s because the original specifications for what full throttle would be turned out to be lower than was ultimately needed. And it was easier to plan around running the engines at greater-than-100%-throttle than it was to change all the earlier design documents.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 25th straddles the line between Pi Day jokes and architecture jokes. I think this is a rerun, but am not sure.

Matt Janz’s Out of the Gene Pool rerun for the 25th tosses off a mention of “New Math”. It’s referenced as a subject that’s both very powerful but also impossible for Pop, as an adult, to understand. It’s an interesting denotation. Usually “New Math”, if it’s mentioned at all, is held up as a pointlessly complicated way of doing simple problems. This is, yes, the niche that “Common Core” has taken. But Janz’s strip might be old enough to predate people blaming everything on Common Core. And it might be character, that the father is old enough to have heard of New Math but not anything in the nearly half-century since. It’s an unusual mention in that “New” Math is credited as being good for things. (I’m aware this strip’s a rerun. I had thought I’d mentioned it in an earlier Reading the Comics post, but can’t find it. I am surprised.)

Mark Anderson’s Andertoons for the 26th is a reassuring island of normal calm in these trying times. It’s a student-at-the-blackboard problem.

Morrie Turner’s Wee Pals rerun for the 26th just mentions arithmetic as the sort of homework someone would need help with. This is another one of those reruns I’d have thought has come up here before, but hasn’t.

## Were Story Problems Ever Any Good?

I have been reading Mapping In Michigan and the Great Lakes Region, edited by David I Macleod, because — look, I understand that I have a problem. I just live with it. The book is about exactly what you might imagine from the title. And it features lots of those charming old maps where, you know, there wasn’t so very much hard data available and everyone did the best with what they had. So you get these maps with spot-on perfect Lake Eries and the eastern shore of Lake Huron looking like you pulled it off of Open Street Maps. And then Michigan looks like a kid’s drawing of a Thanksgiving turkey. Also sometimes they drop a mountain range in the middle of the state because I guess it seemed a little empty without.

The first chapter, by Mary Sponberg Pedley, is a biography and work-history of Louis Charles Karpinski, 1878-1956. Karpinski did a lot to bring scholastic attention to maps of the Great Lakes area. He was a professor of mathematics for the University of Michigan. And he commented a good bit about the problems of teaching mathematics. Pedley quoted this bit that I thought was too good not to share. It’s from Arithmetic For The Farm. It’s about the failure of textbooks to provide examples that actually reflected anything anyone might want to know. I quote here Pedley’s endnote:

Karpinski disparaged the typical “story problems” found in contemporary textbooks, such as the following: “How many sacks, holding 2 bushels, 3 pecks and 2 quarts each can be filled from a bin containing 366 bushels, 3 pecks, 4 quarts of what?” Karpinski comments: “How carefully would you have to fill a sack to make it hold 3 pecks 2 quarts of anything? And who filled the bin so marvelously that the capacity is known with an accuracy of one-25th of 1% of the total?” He recommended an easier, more practical means of doing such problems, noting that a bushel is about 1 & 1/4 or 5/4 cubic feet. Therefore the number of bushels in the bin is the length times width times 4/5; the easiest way to get 4/5 of anything is to take away one-fifth of it.

This does read to me like Pedley jumped a track somewhere. It seems to go from the demolition of the plausibility of one problem’s setup to demolishing the plausibility of how to answer a problem. Still, the core complaint is with us yet. It’s hard to frame problems that might actually come up in ways that clearly test specific mathematical skills.

And on another note. This is the 1,000th mathematical piece that I’ve published since I started in September of 2011. If I’m not misunderstanding this authorship statistic on WordPress, which is never a safe bet. I’m surprised that it has taken as long as this to get to a thousand posts. Also I’m surprised that I should be surprised. I know roughly how many days there are in a year. And I know I need special circumstances to post something more often than every other day. Still, I’m glad to reach this milestone, and gratified that there’s anyone interested in what I have to say. In my next thousand posts I hope to say something.

## Wronski’s Formula For Pi: My Boring Mistake

Previously:

So, I must confess failure. Not about deciphering Józef Maria Hoëne-Wronski’s attempted definition of π. He’d tried this crazy method throwing a lot of infinities and roots of infinities and imaginary numbers together. I believe I translated it into the language of modern mathematics fairly. And my failure is not that I found the formula actually described the number -½π.

Oh, I had an error in there, yes. And I’d found where it was. It was all the way back in the essay which first converted Wronski’s formula into something respectable. It was a small error, first appearing in the last formula of that essay and never corrected from there. This reinforces my suspicion that when normal people see formulas they mostly look at them to confirm there is a formula there. With luck they carry on and read the sentences around them.

My failure is I wanted to write a bit about boring mistakes. The kinds which you make all the time while doing mathematics work, but which you don’t worry about. Dropped signs. Constants which aren’t divided out, or which get multiplied in incorrectly. Stuff like this which you only detect because you know, deep down, that you should have gotten to an attractive simple formula and you haven’t. Mistakes which are tiresome to make, but never make you wonder if you’re in the wrong job.

The trouble is I can’t think of how to make an essay of that. We don’t tend to rate little mistakes like the wrong sign or the wrong multiple or a boring unnecessary added constant as important. This is because they’re not. The interesting stuff in a mathematical formula is usually the stuff representing variations. Change is interesting. The direction of the change? Eh, nice to know. A swapped plus or minus sign alters your understanding of the direction of the change, but that’s all. Multiplying or dividing by a constant wrongly changes your understanding of the size of the change. But that doesn’t alter what the change looks like. Just the scale of the change. Adding or subtracting the wrong constant alters what you think the change is varying from, but not what the shape of the change is. Once more, not a big deal.

But you also know that instinctively, or at least you get it from seeing how it’s worth one or two points on an exam to write -sin where you mean +sin. Or how if you ask the instructor in class about that 2 where a ½ should be, she’ll say, “Oh, yeah, you’re right” and do a hurried bit of erasing before going on.

Thus my failure: I don’t know what to say about boring mistakes that has any insight.

For the record here’s where I got things wrong. I was creating a function, named ‘f’ and using as a variable ‘x’, to represent Wronski’s formula. I’d gotten to this point:

$f(x) = -4 \imath x 2^{\frac{1}{2}\cdot \frac{1}{x}} \left\{ e^{\imath \frac{\pi}{4}\cdot\frac{1}{x}} - e^{- \imath \frac{\pi}{4}\cdot\frac{1}{x}} \right\}$

And then I observed how the stuff in curly braces there is “one of those magic tricks that mathematicians know because they see it all the time”. And I wanted to call in this formula, correctly:

$\sin\left(\phi\right) = \frac{e^{\imath \phi} - e^{-\imath \phi}}{2\imath }$

So here’s where I went wrong. I took the $4\imath$ way off in the front of that first formula and combined it with the stuff in braces to make 2 times a sine of some stuff. I apologize for this. I must have been writing stuff out faster than I was thinking about it. If I had thought, I would have gone through this intermediate step:

$f(x) = -4 \imath x 2^{\frac{1}{2}\cdot \frac{1}{x}} \left\{ e^{\imath \frac{\pi}{4}\cdot\frac{1}{x}} - e^{- \imath \frac{\pi}{4}\cdot\frac{1}{x}} \right\} \cdot \frac{2\imath}{2\imath}$

Because with that form in mind, it’s easy to take the stuff in curled braces and the $2\imath$ in the denominator. From that we get, correctly, $\sin\left(\frac{\pi}{4}\cdot\frac{1}{x}\right)$. And then the $-4\imath$ on the far left of that expression and the $2\imath$ on the right multiply together to produce the number 8.

So the function ought to have been, all along:

$f(x) = 8 x 2^{\frac{1}{2}\cdot \frac{1}{x}} \sin\left(\frac{\pi}{4}\cdot \frac{1}{x}\right)$

Not very different, is it? Ah, but it makes a huge difference. Carry through with all the L’Hôpital’s Rule stuff described in previous essays. All the complicated formula work is the same. There’s a different number hanging off the front, waiting to multiply in. That’s all. And what you find, redoing all the work but using this corrected function, is that Wronski’s original mess —

$\pi = \frac{4\infty}{\sqrt{-1}}\left\{ \left(1 + \sqrt{-1}\right)^{\frac{1}{\infty}} - \left(1 - \sqrt{-1}\right)^{\frac{1}{\infty}} \right\}$

— should indeed equal:

$2\pi$

All right, there’s an extra factor of 2 here. And I don’t think that is my mistake. Or if it is, other people come to the same mistake without my prompting.

Possibly the book I drew this from misquoted Wronski. It’s at least as good to have a formula for 2π as it is to have one for π. Or Wronski had a mistake in his original formula, and had a constant multiplied out front which he didn’t want. It happens to us all.

Fin.

## Reading the Comics, February 24, 2018: My One Boring Linear Algebra Anecdote Edition

Wait for it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st mentions mathematics — geometry, primarily — as something a substitute teacher has tried teaching with the use of a cucumber and condom. These aren’t terrible examples to use to make concrete the difference between volumes and surface areas. There are limitations, though. It’s possible to construct a shape that has a finite volume but an infinitely large surface area, albeit not using cucumbers.

There’s also a mention of the spring constant, and physics. This isn’t explicitly mathematical. But the description of movement on a spring are about the first interesting differential equation of mathematical physics. The solution is that of simple harmonic motion. I don’t think anyone taking the subject for the first time would guess at the answer. But it’s easy enough to verify it’s right. And this motion — sine waves — just turns up everywhere in mathematical physics.

Bud Blake’s Tiger rerun for the 23rd just mentions mathematics as a topic Hugo finds challenging, and what’s challenging about it. So a personal story: when I took Intro to Linear Algebra my freshman year one day I spaced on the fact we had an exam. So, I put the textbook on the shelf under my desk, and then forgot to take it when I left. The book disappeared, of course, and the professor never heard of it being turned in to lost-and-found or anything. Fortunately the homework was handwritten questions passed out on photocopies (ask your parents), so I could still do the assignments, but for all those, you know, definitions and examples I had to rely on my own notes. I don’t know why I couldn’t ask a classmate. Shyness, probably. Came through all right, though.

Cathy Law’s Claw for the 23rd technically qualifies as an anthropomorphic-numerals joke, in this panel about the smothering of education by the infection of guns into American culture.

Jim Meddick’s Monty for the 23rd has wealthy child Wedgwick unsatisfied with a mere ball of snow. He instead has a snow Truncated Icosahedron (the hyphens in Jarvis’s word balloon may baffle the innocent reader). This is a real shape, one that’s been known for a very long time. It’s one of the Archimedean Solids, a set of 13 solids that have convex shapes (no holes or indents or anything) and have all vertices the same, the identical number of edges coming in to each point in the same relative directions. The truncated icosahedron you maybe also know as the soccer ball shape, at least for those old-style soccer balls made of patches that were hexagons and pentagons. An actual truncated icosahedron needs twelve pentagons, so the figure drawn in the third panel isn’t quite right. At least one pentagonal face would be visible. But that’s also tricky to draw. The aerodynamics of a truncated icosahedron are surely different from those of a sphere. But in snowball-fight conditions, probably not different enough to even notice.

Mark Litzler’s Joe Vanilla for the 24th uses a blackboard full of formulas to represent an overcomplicated answer. The formulas look, offhand, like gibberish to me. But I’ll admit uncertainty since the odd capitalization of “iG(p)” at the start makes me think of some deeper group theory or knot theory symbols. And to see an “m + p” and an “m – p” makes me think of quantum mechanics of atomic orbitals. (But then an “m – p2” is weird.) So if this were anything I’d say it was some quantum chemistry formula. But my gut says if Litzler did take the blackboard symbols from anything, it was without going back to references. (Which he has no need to do, I should point out; the joke wouldn’t be any stronger — or weaker — if the blackboard meant anything.)

## Reading the Comics, February 20, 2018: Bob the Squirrel Edition

So one comic strip was technically on point all this week, without ever quite giving me a specific thing to talk about. And I came to conclude there was another comic strip I could drop from my consideration. Which all were they? Read on.

Frank Page’s Bob the Squirrel for the 18th of February isn’t really about the Rubik’s Cube. It’s just something to occupy Bob’s mind until a deeper mystery emerges. Rubik’s Cubes, meanwhile, are everyone’s favorite group theory pastime, although I’m not sure how many people have learned group theory starting from that point. Where flies come from in the middle of winter I don’t know. We’ve been dealing with box elder bugs ourselves. (We’ve been scooping them up and tossing them outside where they can hopefully find the trees they should be using instead.)

Bob the Squirrel went on, during the week, to start a sequence about Lauren needing a geometry tutor. The story hasn’t done much that geometry-specific — Saturday’s was the most approximately on point — but it’s a comic strip I like. Squirrel fans might agree. (The strip for the 22nd has most tickled me.)

Allison Barrows’s PreTeena rerun for the 19th has a student teacher starting off her experience with a story problem. Your classic time-estimation problem.

Jack Pullan’s Boomerangs rerun for the 20th is one that mentions entropy and that I’ve already talked about at least twice before. These were times in January 2017 and also in November 2013. Given that the strip’s no longer in production and that I’m clearly on at least my third go-round I suppose I’ll retire it from my daily read. I’m curious why, if it was about 14 months between the last appearance and this appearance of this strip, why I didn’t have it at all in 2015 or 2016. Maybe I missed it, or it came a week there was enough to write about that I didn’t need to include a marginal strip.

Christopher Grady’s Lunarbaboon for the 20th is intended to be a heartwarming little story of encouragement and warm feelings. (Most Lunarbaboon strips are intended to be a heartwarming little story of encouragement and warm feelings.) That it’s mathematics the kid struggles with is incidental to the story setup. But it does make it easy to picture a kid struggling and a couple kind words offering some motivation, or at least better feelings.

Richard Thompson’s Richard’s Poor Almanac for the 20th is a casual mention of sudoku and a publication error that would supposedly have made it impossible. If the numbers were transposed consistently — everything that ought to have been a ‘2’ printed as a ‘5’, and everything that ought to have been ‘5’ printed as ‘2’ — the problem would be exactly as solvable. This is why you can sometimes see sudoku-type puzzles that use symbols or letters or other characters. But if, say, the third and the second rows were transposed then there’s a chance the incorrect puzzle would be solvable. Transposing a bunch of squares, like, the top three rows with the bottom three rows, wouldn’t make the puzzle unsolvable. This serves as a reminder that if you make enough mistakes you can still turn out all right, a comforting message for our times. Also I know I’ve featured Richard’s Poor Almanac several times over, but I’m a Richard Thompson fan so I’m not dropping that from my feed.

Will Henry’s Wallace the Brave — to be newspaper-syndicated from the 26th of March, by the way, and I’m glad for that as Wallace and I share the same favorite pinball game — just mentions mathematics as a subject Wallace isn’t thinking enough about. I’m also fond of the Loch Ness Monster, so, all the better.

I’m not surprised that this seems to be the first time I’ve had Lunarbabboon tagged. I am surprised that Bob the Squirrel seems not to have been tagged here before. Maybe I didn’t give the tag suggested-completion enough time to figure out what to do with ‘bob the’. We’ve been having odd little net glitches that mostly pass quickly, but that kill any sort of client-side Javascript-based page rendering. You know, like every web page does anymore because somehow “the web server puts together a bunch of stuff and transmits that to the reader” is too inefficient a system.

## How Often Should Records Break? A Puzzle For You

Lansing got some record-breaking rain this week. Tuesday we got over two inches of rain, doubling the hundred-plus-year-old previous record. I mention because it got me to wondering how often we should expect records to break. I mean if the thing being measured probably isn’t changing. So my inspiration is out, as there’s no serious question about the climate changing. Measures of sports performance are also no good.

But we can imagine there’s something with an underlying property that isn’t changing. So if you keep getting samples of some independent, normally-distributed property in, how often should you expect to go between record-setting values? New records should start pretty thick on the ground. The first value is necessarily both a new high and low. The second is either a high or a low. The third seems to have a good chance of being a new extreme. Fourth, too. But somewhere along the way extremes should get rarer. Even if the 10,000th sample recorded is a new record high or low, what are the odds the 10,001st is? The 10,010th?

Haven’t got an answer offhand, although it’s surely available. Just mulling over how to attack the problem before I do what I always do and write a Matlab program to do a bunch of simulations. Easier than thinking. But I’ll leave the problem out for someone needing the challenge.

## Reading the Comics, February 17, 2018: Continuing Deluge Month

February’s been a flooding month. Literally (we’re about two blocks away from the Voluntary Evacuation Zone after the rains earlier this week) and figuratively, in Comic Strip Master Command’s suggestions about what I might write. I have started thinking about making a little list of the comics that just say mathematics in some capacity but don’t give me much to talk about. (For example, Bob the Squirrel having a sequence, as it does this week, with a geometry tutor.) But I also know, this is unusually busy this month. The problem will recede without my having to fix anything. One of life’s secrets is learning how to tell when a problem’s that kind.

Patrick Roberts’s Todd the Dinosaur for the 12th just shows off an arithmetic problem — fractions — as the thing that can be put on the board and left for students to do.

Ham’s Life on Earth for the 12th has a science-y type giving a formula as “something you should know”. The formula’s gibberish, so don’t worry about it. I got a vibe of it intending to be some formula from statistics, but there’s no good reason for that. I’ve had some statistical distribution problems on my mind lately.

Eric Teitelbaum and Bill Teitelbaum’s Bottomliners for the 12th maybe influenced my thinking. It has a person claiming to be a former statistician, and his estimate of how changing his job’s affected his happiness. Could really be any job that encourages people to measure and quantify things. But “statistician” is a job with strong connotations of being able to quantify happiness. To have that quantity feature a decimal point, too, makes him sound more mathematical and thus, more surely correct. I’d be surprised if “two and a half times” weren’t a more justifiable estimate, given the margin for error on happiness-measurement I have to imagine would be there. (This seems to be the first time I’ve featured Bottomliners at least since I started tagging the comic strips named. Neat.)

Ruben Bolling’s Super-Fun-Pak Comix for the 12th reprinted a panel called The Uncertainty Principal that baffled commenters there. It’s a pun on “Uncertainty Principle”, the surprising quantum mechanics result that there are some kinds of measurements that can’t be taken together with perfect precision. To know precisely where something is destroys one’s ability to measure its momentum. To know the angular momentum along one axis destroys one’s ability to measure it along another. This is a physics result (note that the panel’s signed “Heisenberg”, for the name famously attached to the Uncertainty Principle). But the effect has a mathematical side. The operations that describe finding these incompatible pairs of things are noncommutative; it depends what order you do them in.

We’re familiar enough with noncommutative operations in the real world: to cut a piece of paper and then fold it usually gives something different to folding a piece of paper and then cutting it. To pour batter in a bowl and then put it in the oven has a different outcome than putting batter in the oven and then trying to pour it into the bowl. Nice ordinary familiar mathematics that people learn, like addition and multiplication, do commute. These come with partners that don’t commute, subtraction and division. But I get the sense we don’t think of subtraction and division like that. It’s plain enough that ‘a’ divided by ‘b’ and ‘b’ divided by ‘a’ are such different things that we don’t consider what’s neat about that.

In the ordinary world the Uncertainty Principle’s almost impossible to detect; I’m not sure there’s any macroscopic phenomena that show it off. I mean, that atoms don’t collapse into electrically neutral points within nanoseconds, sure, but that isn’t as compelling as, like, something with a sodium lamp and a diffraction grating and an interference pattern on the wall. The limits of describing certain pairs of properties is about how precisely both quantities can be known, together. For everyday purposes there’s enough uncertainty about, say, the principal’s weight (and thus momentum) that uncertainty in his position won’t be noticeable. There’s reasons it took so long for anyone to suspect this thing existed.

Samson’s Dark Side of the Horse for the 13th uses a spot of arithmetic as the sort of problem coffee helps Horace solve. The answer’s 1.

Mike Baldwin’s Cornered for the 14th is a blackboard-full-of-symbols panel. Well, a whiteboard. It’s another in the line of mathematical proofs of love.

Dana Simpson’s Ozy and Millie rerun for the 14th has the title characters playing “logical fallacy tag”. Ozy is, as Millie says, making an induction argument. In a proper induction argument, you characterize something with some measure of size. Often this is literally a number. You then show that if it’s true that the thing is true for smaller problems than you’re interested in, then it has to also be true for the problem you are interested in. Add to that a proof that it’s true for some small enough problem and you’re done. In this case, Ozy’s specific fallacy is an appeal to probability: all but one of the people playing tag are not it, and therefore, any particular person playing the game isn’t it. That it’s fallacious really stands out when there’s only two people playing.

Alex Hallatt’s Arctic Circle for the 16th riffs on the mathematics abilities of birds. Pigeons, in this case. The strip starts from their abilities understanding space and time (which are amazing) and proposes pigeons have some insight into the Grand Unified Theory. Animals have got astounding mathematical abilities, should point out. Don’t underestimate them. (This also seems to be the first time I’ve tagged Arctic Circle which doesn’t seem like it could be right. But I didn’t remember naming the penguins before so maybe I haven’t? Huh. Mind, I only started tagging the comic strip titles a couple months ago.)

Tony Cochrane’s Agnes for the 17th has the title character try bluffing her way out of mathematics homework. Could there be a fundamental flaw in mathematics as we know it? Possibly. It’s hard to prove that any field complicated enough to be interesting is also self-consistent. And there’s a lot of mathematics out there. And mathematics subjects often develop with an explosion of new ideas and then a later generation that cleans them up and fills in logical gaps. Symplectic geometry is, if I’m following the news right, going into one of those cleaning-up phases now. Is it likely to be uncovered by a girl in elementary school? I’m skeptical, and also skeptical that she’d have a replacement system that would be any better. I admire Agnes’s ambition, though.

Mike Baldwin’s Cornered for the 17th plays on the reputation for quantum mechanics as a bunch of mathematically weird, counter-intuitive results. In fairness to the TV program, I’ve had series run longer than I originally planned too.

## Some Mathematics Things I Read On Twitter

I had thought I’d culled some more pieces from my Twitter and other mathematics-writing-reading the last couple weeks and I’m not sure where it all went. I think I might be baffled by the repostings of things on Quanta Magazine (which has a lot of good mathematics articles, but not, like, a 3,000-word piece every day, and they showcase their archive just as anyone ought).

So, here, first.

It reviews Kim Plofker’s 2008 text Mathematics In India, a subject that I both know is important — I love to teach with historic context included — and something that I very much bluff my way through. I mean, I do research things I expect I’ll mention, but I don’t learn enough of the big picture and a determined questioner could prove how fragile my knowledge was. So Plofker’s book should go on my reading list at least.

These are lecture notes about analysis. In the 19th century mathematicians tried to tighten up exactly what we meant by things like “functions” and “limits” and “integrals” and “numbers” and all that. It was a lot of good solid argument, and a lot of surprising, intuition-defying results. This isn’t something that a lay reader’s likely to appreciate, and I’m sorry for that, but if you do know the difference between Riemann and Lebesgue integrals the notes are likely to help.

And this, Daniel Grieser and Svenja Maronna’s Hearing The Shape Of A Triangle, follows up on a classic mathematics paper, Mark Kac’s Can One Hear The Shape Of A Drum? This is part of a class of problems in which you try to reconstruct what kinds of things can produce a signal. It turns out to be impossible to perfectly say what shape and material of a drum produced a certain sound of a drum. But. A triangle — the instrument, that is, but also the shape — has a simpler structure. Could we go from the way a triangle sounds to knowing what it looks like?

And I mentioned this before but if you want to go reading every Calvin and Hobbes strip to pick out the ones that mention mathematics, you can be doing someone a favor too.

## Reading the Comics, February 11, 2018: February 11, 2018 Edition

And it’s not always fair to say that the gods mock any plans made by humans, but Comic Strip Master Command has been doing its best to break me of reading and commenting on any comic strip with a mathematical theme. I grant that I could make things a little easier if I demanded more from a comic strip before including it here. But even if I think a theme is slight that doesn’t mean the reader does, and it’s easy to let the eye drop to the next paragraph if the reader does think it’s too slight. The anthology nature of these posts is part of what works for them. And then sometimes Comic Strip Master Command sends me a day like last Sunday when everybody was putting in some bit of mathematics. There’ll be another essay on the past week’s strips, never fear. But today’s is just for the single day.

Susan Camilleri Konar’s Six Chix for the 11th illustrates the Lemniscate Family. The lemniscate is a shape well known as the curve made by a bit of water inside a narrow tube by people who’ve confused it with a meniscus. An actual lemniscate is, as the chain of pointing fingers suggests, a figure-eight shape. You get — well, I got — introduced to them in prealgebra. They’re shapes really easy to describe in polar coordinates but a pain to describe in Cartesian coordinates. There are several different kinds of lemniscates, each satisfying slightly different conditions while looking roughly like a figure eight. If you’re open to the two lobes of the shape not being the same size there’s even a kind of famous-ish lemniscate called the analemma. This is the figure traced out by the sun if you look at its position from a set point on the surface of the Earth at the same clock time each day over the course of the year. That the sun moves north and south from the horizon is easy to spot. That it is sometimes east or west of some reference spot is a surprise. It shows the difference between the movement of the mean sun, the sun as we’d see it if the Earth had a perfectly circular orbit, and the messy actual thing. Dr Helmer Aslasken has a fine piece about this, and how it affects when the sun rises earliest and latest in the year.

There’s also a thing called the “polynomial lemniscate”. This is a level curve of a polynomial. That is, what are all the possible values of the independent variable which cause the polynomial to evaluate to some particular number? This is going to be a polynomial in a complex-valued variable, in order to get one or more closed and (often) wriggly loops. A polynomial of a real-valued variable would typically give you a boring shape. There’s a bunch of these polynomial lemniscates that approximate the boundary of the Mandelbrot Set, that fractal that you know from your mathematics friend’s wall in 1992.

Mark Anderson’s Andertoons took care of being Mark Anderson’s Andertoons early in the week. It’s a bit of optimistic blackboard work.

Lincoln Pierce’s Big Nate features the formula for calculating the wind chill factor. Francis reads out what is legitimately the formula for estimating the wind chill temperature. I’m not going to get into whether the wind chill formula makes sense as a concept because I’m not crazy. The thinking behind it is that a windless temperature feels about the same as a different temperature with a particular wind. How one evaluates those equivalences offers a lot of room for debate. The formula as the National Weather Service, and Francis, offer looks frightening, but isn’t really hard. It’s not a polynomial, in terms of temperature and wind speed, but it’s close to that in form. The strip is rerun from the 15th of February, 2009, as Lincoln Pierce has had some not-publicly-revealed problem taking him away from the comic for about a month and a half now.

Jim Scancarelli’s Gasoline Alley included a couple of mathematics formulas, including the famous E = mc2 and the slightly less famous πr2, as part of Walt Wallet’s fantasy of advising scientists and inventors. (Scientists have already heard both.) There’s a curious stray bit in the corner, writing out 6.626 x 102 x 3 that I wonder about. 6.626 is the first couple digits of Planck’s Constant, as measured in Joule-seconds. (This is h, not h-bar, I say for the person about to complain.) It’d be reasonable for Scancarelli to have drawn that out of a physics book or reference page. But the exponent is all wrong, even if you suppose he mis-wrote 1023. It should be 6.626 x 10-34. So I don’t know whether Scancarelli got things very garbled, or if he just picked a nice sciencey-looking number and happened to hit on a significant one. (There’s enough significant science numbers that he’d have a fair chance of finding something.) The strip is a reprint from the 4th of February, 2007, as Jim Scancarelli has been absent for no publicly announced reason for four months now.

Greg Evans and Karen Evans’s Luann is not perfectly clear. But I think it’s presenting Gunther doing mathematics work to support his mother’s contention that he’s smart. There’s no working out what work he’s doing. But then we might ask how smart his mother is to have made that much food for just the two of them. Also that I think he’s eating a potato by hand? … Well, there are a lot of kinds of food that are hard to draw.

Greg Evans’s Luann Againn reprints the strip from the 11th of February (again), 1990. It mentions as one of those fascinating things of arithmetic an easy test to see if a number’s a multiple of nine. There are several tricks like this, although the only ones anybody can remember are finding multiples of 3 and finding multiples of 9. Well, they know the rules for something being a multiple of 2, 5, or 10, but those hardly look like rules, and there’s no addition needed. Similarly with multiples of 4.

Modular arithmetic underlies all these rules. Once you know the trick you can use it to work out your own add-up-the-numbers rules to find what numbers are multiples of small numbers. Here’s an example. Think of a three-digit number. Suppose its first digit is ‘a’, its second digit ‘b’, and its third digit ‘c’. So we’d write the number as ‘abc’, or, 100a + 10b + 1c. What’s this number equal to, modulo 9? Well, 100a modulo 9 has to be equal to whatever a modulo 9 is: (100 a) modulo 9 is (100) modulo 9 — that is, 1 — times (a) modulo 9. 10b modulo 9 is (10) modulo 9 — again, 1 — times (b) modulo 9. 1c modulo 9 is … well, (c) modulo 9. Add that all together and you have a + b + c modulo 9. If a + b + c is some multiple of 9, so must be 100a + 10b + 1c.

The rules about whether something’s divisible by 2 or 5 or 10 are easy to work with since 10 is a multiple of 2, and of 5, and for that matter of 10, so that 100a + 10b + 1c modulo 10 is just c modulo 10. You might want to let this settle. Then, if you like, practice by working out what an add-the-digits rule for multiples of 11 would be. (This is made a lot easier if you remember that 10 is equal to 11 – 1.) And if you want to show off some serious arithmetic skills, try working out an add-the-digits rule for finding whether something’s a multiple of 7. Then you’ll know why nobody has ever used that for any real work.

J C Duffy’s Lug Nuts plays on the equivalence people draw between intelligence and arithmetic ability. Also on the idea that brain size should have something particularly strong link to intelligence. Really anyone having trouble figuring out 15% of \$10 is psyching themselves out. They’re too much overwhelmed by the idea of percents being complicated to realize that it’s, well, ten times 15 cents.

## Reading the Comics, February 10, 2018: I Meant To Post This Thursday Edition

Ah, yes, so, in the midst of feeling all proud that I’d gotten my Reading the Comics workflow improved, I went out to do my afternoon chores without posting the essay. I’m embarrassed. But it really only affects me looking at the WordPress Insights page. It publishes this neat little calendar-style grid that highlights the days when someone’s posted and this breaks up the columns. This can only unnerve me. I deserve it.

Tom Thaves’s Frank and Ernest for the 8th of February is about the struggle to understand zero. As often happens, the joke has a lot of truth to it. Zero bundles together several ideas, overlapping but not precisely equal. And part of that is the idea of “nothing”. Which is a subtly elusive concept: to talk about the properties of a thing that does not exist is hard. As adults it’s easy to not notice this anymore. Part’s likely because mastering a concept makes one forget what it took to understand. Part is likely because if you don’t have to ponder whether the “zero” that’s “one less than one” is the same as the “zero” that denotes “what separates the count of thousands from the count of tens in the numeral 2,038” you might not, and just assume you could explain the difference or similarity to someone who has no idea.

John Zakour and Scott Roberts’s Maria’s Day for the 8th has maria and another girl bonding over their hatred of mathematics. Well, at least they’re getting something out of it. The date in the strip leads me to realize this is probably a rerun. I’m not sure just when it’s from.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th proposes a prank based on mathematical use of the word “arbitrarily”. This is a word that appears a lot in analysis, and the strip makes me realize I’m not sure I can give a precise definition. An “arbitrarily large number”, for example, would be any number that’s large enough. But this also makes me realize I’m not sure precisely what joke Weinersmith is going for. I suppose that if someone were to select an arbitrarily large number they might pick 53, or a hundred, or million billion trillion. I suppose Weinersmith’s point is that in ordinary speech an arbitrarily made choice is one selection from all the possible alternatives. In mathematical speech an arbitrarily made choice reflects every possible choice. To speak of an arbitrarily large number is to say that whatever selection is made, we can go on to show this interesting stuff is true. We’d typically like to prove the most generically true thing possible. But picking a single example can be easier to prove. It can certainly be easier to visualize. 53 is probably easier to imagine than “every number 52 or larger”, for example.

Ted Shearer’s Quincy for the 16th of December, 1978 was rerun the 9th of February. It just shows Quincy at work on his mathematics homework, and considering dedicating it to his grandmother. Mathematics books have dedications, just as any other book does. I’m not aware of dedications of proofs or other shorter mathematics works, but there’s likely some. There’s often a note of thanks, usually given to people who’ve made the paper’s writers think harder about the subjects. But I don’t think there’s any reason a paper wouldn’t thank someone who provided “mere” emotional support. I just don’t have examples offhand.

Jef Mallet’s Frazz for the 9th looks like one of those creative-teaching exercises I sometimes see in Mathematics Education Twitter: the teacher gives answers and the students come up with story problems to match. That’s not a bad project. I’m not sure how to grade it, but I haven’t done anything that creative when I’ve taught. I’m sorry I haven’t got more to say about it since the idea seems fun.

Gordon Bess’s Redeye for the 30th of September, 1971 was rerun the 10th. It’s a bit of extremely long division and I don’t blame Pokey for giving up on that problem. Starting from 5,967,342 divided by 973 I’d say, well, that’s about six million divided by a thousand, so the answer should be near six thousand. I don’t think the last digits of 2 and 3 suggest anything about what the final digit should be, if this divides evenly. So the only guidance I have is that my answer ought to be around six thousand and then we have to go into actually working. It turns out that 973 doesn’t go into 5,967,342 a whole number of times, so I sympathize more with Pokey. The answer is a little more than 6,132.9311.