Reading the Comics, June 20, 2019: Old Friends Edition


We continue to be in the summer vacation doldrums for mathematically-themed comic strips. But there’ve been a couple coming out. I could break this week’s crop into two essays, for example. All of today’s strips are comics that turn up in my essays a lot. It’s like hanging out with a couple of old friends.

Samson’s Dark Side of the Horse for the 17th uses the motif of arithmetic expressions as “difficult” things. The expressions Samson quotes seem difficult for being syntactically weird: What does the colon under the radical sign mean in \sqrt{9:}33 ? Or they’re difficult for being indirect, using a phrase like “50%” for “half”. But with some charity we can read this as Horace talking about 3:33 am to about 6:30 am. I agree that those are difficult hours.

Horace: 'I've lived through some difficult times. Especially from sqrt{9:}33 AM to 50% past sixish o'clock. Maybe I should get my watch fixed.'
Samson’s Dark Side of the Horse for the 17th of June, 2019. Some of the many essays inspired by Dark Side of the Horse are at this link.

It also puts me in mind of a gift from a few years back. An aunt sent me an Irrational Watch, with a dial that didn’t have the usual counting numbers on it. Instead there were various irrational numbers, like the Golden Ratio or the square root of 50 or the like. Also the Euler-Mascheroni Constant, a number that may or may not be irrational. Nobody knows. It’s likely that it is irrational, but it’s not proven. It’s a good bit of fun, although it does make it a bit harder to use the watch for problems like “how long is it until 4:15?” This isn’t quite what’s going on here — the square root of nine is a noticeably rational number — but it seems in that same spirit.

Mark Anderson’s Andertoons for the 18th sees Wavehead react to the terminology of the “improper fraction”. “Proper” and “improper” as words carry a suggestion of … well, decency. Like there’s something faintly immoral about having an improper fraction. “Proper” and “improper”, as words, attach to many mathematical concepts. Several years ago I wrote that “proper” amounted to “it isn’t boring”. This is a fair way to characterize, like, proper subsets or proper factors or the like. It’s less obvious that \frac{13}{12} is a boring fraction.

The teacher has on the blackboard 1/3 + 3/4 rewritten as 4/12 + 9/12 = 13/12. Wavehead: 'OK, we made it so they had something in common, added them together, and the result is *improper*? I mean, I kinda feel like we just made things worse!'
Mark Anderson’s Andertoons for the 18th of June, 2019. Essays with some mention of a topic from Andertoons are at this link.

I may need to rewrite that old essay. An “improper” form satisfies all the required conditions for the term. But it misses some of the connotation of the term. It’s true that, say, the new process takes “a fraction of the time” of the old, if the old process took one hour and the new process takes fourteen years. But if you tried telling someone that they would assume you misunderstood something. The ordinary English usage of “fraction” carries the connotation of “a fraction between zero and one”, and that’s what makes a “proper fraction”.

In practical terms, improper fractions are fine. I don’t know of any mathematicians who seriously object to them, or avoid using them. The hedging word “seriously” is in there because of a special need. That need is: how big is, say, \frac{75}{14} ? Is it bigger than five? Is it smaller than six? An improper fraction depends on you knowing, in this case, your fourteen-times tables to tell. Switching that to a mixed fraction, 5 + \frac{5}{14} , helps figure out what the number means. That’s as far as we have to worry about the propriety of fractions.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses the form of a Fermi problem for its joke. Fermi problems have a place in mathematical modeling. The idea is to find an estimate for some quantity. We often want to do this. The trick is to build a simple model, and to calculate using a tiny bit of data. The Fermi problem that has someone reached public consciousness is called the Fermi paradox. The question that paradox addresses is, how many technologically advanced species are there in the galaxy? There’s no way to guess. But we can make models and those give us topics to investigate to better understand the problem. (The paradox is that reasonable guesses about the model suggest there should be so many aliens that they’d be a menace to air traffic. Or that the universe should be empty except for us. Both alternatives seem unrealistic.) Such estimates can be quite wrong, of course. I remember a Robert Heinlein essay in which he explained the Soviets were lying about the size of Moscow, his evidence being he didn’t see the ship traffic he expected when he toured the city. I do not remember that he analyzed what he might have reasoned wrong when he republished this in a collection of essays he didn’t seem to realize were funny.

HR interviewer: 'At this company we only want geniuses. So we ask puzzles and judge how well you solve them. Quick! Estimate how many employees we have!' Job applicant: 'Given other companies use empirically validated non-annoying hiring protocols and that engineers have lots of options, I'd estimate your company has exactly one employee.' Interviewer: 'Please don't leave me.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of June, 2019. Anyone who’s been reading these for a couple weeks knows, but, Saturday Morning Breakfast Cereal features in essays at this link. Hey, every essay is somebody’s first.

So the interview question presented is such a Fermi problem. The job applicant, presumably, has not committed to memory the number of employees at the company. But there would be clues. Does the company own the whole building it’s in, or just a floor? Just an office? How large is the building? How large is the parking lot? Are there people walking the hallways? How many desks are in the offices? The question could be answerable. The applicant has a pretty good chain of reasoning too.

Bill Amend’s FoxTrot Classics for the 20th has several mathematical jokes in it. One is the use of excessively many decimal points to indicate intelligence. Grant that someone cares about the hyperbolic cosines of 15.2. There is no need to cite its wrong value to nine digits past the decimal. Decimal points are hypnotic, though, and listing many of them has connotations of relentless, robotic intelligence. That is what Amend went for in the characters here. That and showing how terrible nerds are when they find some petty issue to rage over.

Eugene: 'Lousy camp-issued calculator!' Marcus: 'What's wrong now?' Eugene: 'This thing says the hyperbolic cosine of 15.2 is 0.965016494 when any moron knows this can't be right! What kin of boneheads run this palce? See? It did it again!' Marcus: 'You need to hit the blue button first. Right now you're just getting the regular cosine. ... No need to say 'thank you'. I'm enjoying this silence.' Jason: 'Did you want to borrow mine? Some of us don't need them.'
Bill Amend’s FoxTrot Classics for the 20th of June, 2019. It originally ran the 3rd of July, 1997. Essays based on FoxTrot, either the current-run Sundays, newspaper-rerun 2000s strips, or 90s-run Classics, are at this link.

Eugene is correct about the hyperbolic cosine being wrong, there, though. He’s not wrong to check that. It’s good form to have some idea what a plausible answer should be. It lets one spot errors, for one. No mathematician is too good to avoid making dumb little mistakes. And computing tools will make mistakes too. Fortunately they don’t often, but this strip originally ran a couple years after the discovery of the Pentium FDIV bug. This was a glitch in the way certain Pentium chips handled floating-point division. It was discovered by Dr Thomas Nicely, at Lynchberg College, who found inconsistencies in some calculations when he added Pentium systems to the computers he was using. This Pentium bug may have been on Amend’s mind.

Eugene would have spotted right away that the hyperbolic cosine was wrong, though, and didn’t need nine digits for it. The hyperbolic cosine is a function. Its domain is the real numbers. It range is entirely numbers greater than or equal to one, or less than or equal to minus one. A 0.9 something just can’t happen, not as the hyperbolic cosine for a real number.

And what is the hyperbolic cosine? It’s one of the hyperbolic trigonometric functions. The other trig functions — sine, tangent, arc-sine, and all that — have their shadows too. You’ll see the hyperbolic sine and hyperbolic tangent some. You will never see the hyperbolic arc-cosecant and anyone trying to tell you that you need it is putting you on. They turn up in introductory calculus classes because you can differentiate them, and integrate them, the way you can ordinary trig functions. They look just different enough from regular trig functions to seem interesting for half a class. By the time you’re doing this, your instructor needs that.

The ordinary trig functions come from the unit circle. You can relate the Cartesian coordinates of a point on the circle described by x^2 + y^2 = 1 to the angle made between that point and the center of the circle and the positive x-axis. Hyperbolic trig functions we can relate the Cartesian coordinates of a point on the hyperbola described by x^2 - y^2 = 1 to angles instead. The functions … don’t have a lot of use at the intro-to-calculus level. Again, other than that they let you do some quite testable differentiation and integration problems that don’t look exactly like regular trig functions do. They turn up again if you get far enough into mathematical physics. The hyperbolic cosine does well in describing catenaries, that is, the shape of flexible wires under gravity. And the family of functions turn up in statistical mechanics, often, in the mathematics of heat and of magnetism. But overall, these functions aren’t needed a lot. A good scientific calculator will offer them, certainly. But it’ll be harder to get them.

There is another oddity at work here. The cosine of 15.2 degrees is about 0.965, yes. But mathematicians will usually think of trigonometric functions — regular or hyperbolic — in terms of radians. This is just a different measure of angles. A right angle, 90 degrees, is measured as \frac{1}{2}\pi radians. The use of radians makes a good bit of other work easier. Mathematicians get to accustomed to using radians that to use degrees seems slightly alien. The cosine of 15.2 radians, then, would be about -0.874. Eugene has apparently left his calculator in degree mode, rather than radian mode. If he weren’t so worked up about the hyperbolic cosine being wrong he might have noticed. Perhaps that will be another exciting error to discover down the line.

This strip was part of a several-months-long story Bill Amend did, in which Jason has adventures at Math Camp. I don’t remember the whole story. But I do expect the strip to have several more appearances here this summer.


And that’s about half of last week’s comics. A fresh Reading the Comics post should be at this link later this week. Thank you for reading along.

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Reading the Comics, June 15, 2019: School Is Out? Edition


This has not been the slowest week for mathematically-themed comic strips. The slowest would be the week nothing on topic came up. But this was close. I admit this is fine as I have things disrupting my normal schedule this week. I don’t need to write too many essays too.

On-topic enough to discuss, though, were:

Lalo Alcaraz’s La Cucaracha for the 9th features a teacher trying to get ahead of student boredom. The idea that mathematics is easier to learn if it’s about problems that seem interesting is a durable one. It agrees with my intuition. I’m less sure that just doing arithmetic while surfing is that helpful. My feeling is that a problem being interesting is separate from a problem naming an intersting thing. But making every problem uniquely interesting is probably too much to expect from a teacher. A good pop-mathematics writer can be interesting about any problem. But the pop-mathematics writer has a lot of choice about what she’ll discuss. And doesn’t need to practice examples of a problem until she can feel confident her readers have learned a skill. I don’t know that there is a good answer to this.

Teacher: 'Class, today is the last day of school. You don't want to be here, and neither do I. So, I found a way where we can learn while getting an early start on the summer break!' Next panel, they're all on surfboards. Teacher: 'Next question: whats eight sick waves times eight six waves?' Students: 'Sixty-four sick waves!'
Lalo Alcaraz’s La Cucaracha for the 9th of June, 2019. I had thought I’d mentioned this comic at least a couple times in the past, and seem to be wrong. So this is a new tag and that’s always nice to have. Any future essays which mention something inspired by La Cucaracha should be at this link.

Also part of me feels that “eight sick waves times eight sick waves” has to be “sixty-four sick-waves-squared”. This is me worrying about the dimensional analysis of a joke. All right, but if it were “eight inches times eight inches” and you came back with “sixty-four inches” you’d agree something was off, right? But it’s easy to not notice the units. That we do, mechanically, the same thing in multiplying (oh) three times $1.20 or three times 120 miles or three boxes times 120 items per box as we do multiplying three times 120 encourages this. But if we are using numbers to measure things, and if we are doing calculations about things, then the units matter. They carry information about the kinds of things our calculations represent. It’s a bad idea to misuse or ignore those tools.

Paul Trap’s Thatababy for the 14th is roughly the anthropomorphized geometry cartoon of the week. It does name the three ways to group triangles based on how many sides have the same length. Or if you prefer, how many interior angles have the same measure. So it’s probably a good choice for your geometry tip sheet. “Scalene” as a word seems to have entered English in the 1730s. Its origin traces to Late Latin “scalenus”, from the Greek “skalenos” and meaning “uneven” or “crooked”.

Thatababy drawing triangles: an equilateral triangle, an isosceles triangle, a scalene triangle, and then a love triangle, showing two isosceles triangles holding hands; one of them looks with interest at an equilateral triangle.
Paul Trap’s Thatababy for the 14th of June, 2019. Now, this strip I thought I featured more around here. It doesn’t seem to have gotten an appearance in over a year, though. Still, other appearances by Thatababy should be in essays at this link.

“Isosceles” also goes to Late Latin and, before that, the Greek “isoskeles”, with “iso” the prefix meaning “equal” and “skeles” meaning “legs”. The curious thing to me is “Isosceles”, besides sounding more pleasant, came to English around 1550. Meanwhile, “equilateral” — a simple Late Latin for “equal sides” — appeared around 1570. I don’t know what was going on that it seemed urgent to have a word for triangles with two equal sides first, and a generation later triangles with three equal sides. And then triangles with no two equal sides went nearly two centuries without getting a custom term.

But, then, I’m aware of my bias. There might have been other words for these concepts, recognized by mathematicians of the year 1600, that haven’t come to us. Or it might be that scalene triangles were thought to be so boring there wasn’t any point giving them a special name. It would take deeper mathematics history knowledge than I have to say.


Those are all the mathematically-themed comic strips I can find something to discuss from the past week. There were some others with mentions of mathematics, though. These include:

Tony Rubino and Gary Markstein’s Daddy’s Home for the 9th, in which mathematics is the last class of the school year. Francesco Marciuliano and Jim Keefe’s Sally Forth for the 11th has a study session with “math charades” mentioned. Mark Andersons Andertoons for the 11th wants in on some of my sweet Thatababy exposition. Harley Schwadron’s 9 to 5 for the 14th is trying to become the default pie chart joke around here. It won’t beat out Randolph Itch, 2 am without a stronger punch line. And Mark Tatulli’s Heart of the City for the 15th sees Dean mention hiding sleeping in algebra class.


This closes out a week’s worth of comic strips. My next Reading the Comics post should be at this link next Sunday. And now I need to think of something to post for the Thursday and, if I can, Tuesday publication dates.

Reading the Comics, May 20, 2019: I Guess I Took A Week Off Edition


I’d meant to get back into discussing continuous functions this week, and then didn’t have the time. I hope nobody was too worried.

Bill Amend’s FoxTrot for the 19th is set up as geometry or trigonometry homework. There are a couple of angles that we use all the time, and they do correspond to some common unit fractions of a circle: a quarter, a sixth, an eighth, a twelfth. These map nicely to common cuts of circular pies, at least. Well, it’s a bit of a freak move to cut a pie into twelve pieces, but it’s not totally out there. If someone cuts a pie into 24 pieces, flee.

Offscreen voice: 'So a pizza sliced into fourths has ... ' Paige: '90 degrees per slice.' Voice: 'Correct! And a pizza sliced into sixths has ... ' Page: '60 degrees per slice.' Voice: 'Good! And a pizza sliced into eighths has ... ' Paige: '45 degrees per slice.' Voice: 'Yep! I'd say you're ready for your geometry final, Paige.' Paige: 'Woo-hoo!' Voice, revealed to be Peter: 'Now help me clean up these [ seven pizza ] boxes.' Page: 'I still don't understand why teaching me this required *actual* pizzas.'
Bill Amend’s FoxTrot for the 19th of May, 2019. Essays featuring FoxTrot, either the current (Sunday-only) strips or the 1990s-vintage reruns, should be at this link.

Tom Batiuk’s vintage Funky Winkerbean for the 19th of May is a real vintage piece, showing off the days when pocket electronic calculators were new. The sales clerk describes the calculator as having “a floating decimal”. And here I must admit: I’m poorly read on early-70s consumer electronics. So I can’t say that this wasn’t a thing. But I suspect that Batiuk either misunderstood “floating-point decimal”, which would be a selling point, or shortened the phrase in order to make the dialogue less needlessly long. Which is fine, and his right as an author. The technical detail does its work, for the setup, by existing. It does not have to be an actual sales brochure. Reducing “floating point decimal” to “floating decimal” is a useful artistic shorthand. It’s the dialogue equivalent to the implausibly few, but easy to understand, buttons on the calculator in the title panel.

Calculator salesman: 'This little pocket calculator is a real beauty. It's nice and light so you can take it anywhere. It has an eight-digit readout with automatic roundoff. Not only that, but it has a floating decimal which enables you to solve ANY type of problem with it!' Les Moore: 'Amazing! May I try it out?' (To the calculator) 'Hello, pocket calculator? Why do I have so much trouble getting girls to like me?'
Tom Batiuk’s vintage Funky Winkerbean for the 19th of May, 2019. The strip originally ran the 17th of June, 1973. Comics Kingdom is printing both the current Funky Winkerbean strips and early-70s reprints. Essays that mention Funky Winkerbean, old or new, should appear at this link.

Floating point is one of the ways to represent numbers electronically. The storage scheme is much like scientific notation. That is, rather than think of 2,038, think of 2.038 times 103. In the computer’s memory are stored the 2.038 and the 3, with the “times ten to the” part implicit in the storage scheme. The advantage of this is the range of numbers one can use now. There are different ways to implement this scheme; a common one will let one represent numbers as tiny as 10-308 or as large as 10308, which is enough for most people’s needs.

The disadvantage is that floating point numbers aren’t perfect. They have only around (commonly) sixteen digits of significance. That is, the first sixteen or so nonzero numbers in the number you represent mean anything; everything after that is garbage. Most of the time, that trailing garbage doesn’t hurt. But most is not always. Trying to add, for example, a tiny number, like 10-20, to a huge number, like 1020 won’t get the right answer. And there are numbers that can’t be represented correctly anyway, including such exotic and novel numbers as \frac{1}{3} . A lot of numerical mathematics is about finding ways to compute that avoid these problems.

Back when I was a grad student I did have one casual friend who proclaimed that no real mathematician ever worked with floating point numbers, because of the limitations they impose. I could not get him to accept that no, in fact, mathematicians are fine with these limitations. Every scheme for representing numbers on a computer has limitations, and floating point numbers work quite well. At some point, you have to suspect some people would rather fight for a mistaken idea they already have than accept something new.

Matrix-O-Magic: Draw a nine-square grid on a notepad, filling in the numbers 1-9 like this: 2, 9, 4 // 7, 5, 3 // 6, 1, 8 Hand the pad and marker to a friend and tell him to pick any row of three numbers, upward, downward, or diagonal. Tell him to black out any numbers not in his row. Instruct your friend to add up his three randomly chosen numbers. Ask your friend to flip through the rest of the notepad to make sure the pages are blank. All the pages are blank except one. That one bears the number that his numbers added up to: 15. (All the rows/columns/diagonals add to 15; because the other numbers are blacked out your friend won't notice. If asked to do the trick more than once the grid can be made to look different by rotating the order of the numbers left or right, et, 6, 7, 2 // 1, 5, 9 // 8, 3, 4.)
Mac King and Bill King’s Magic in a Minute for the 19th of May, 2019. So far as I know all these panels are new ones, although they do reuse gimmicks now and then. But the arithmetic and logic tricks featured in Magic In A Minute get discussed at this link, when they get mention from me at all.

Mac King and Bill King’s Magic in a Minute for the 19th does a bit of stage magic supported by arithmetic: forecasting the sum of three numbers. The trick is that all eight possible choices someone would make have the same sum. There’s a nice bit of group theory hidden in the “Howdydoit?” panel, about how to do the trick a second time. Rotating the square of numbers makes what looks, casually, like a different square. It’s hard for human to memorize a string of digits that don’t have any obvious meaning, and the longer the string the worse people are at it. If you’ve had a person — as directed — black out the rows or columns they didn’t pick, then it’s harder to notice the reused pattern.

The different directions that you could write the digits down in represent symmetries of the square. That is, geometric operations that would replace a square with something that looks like the original. This includes rotations, by 90 or 180 or 270 degrees clockwise. Mac King and Bill King don’t mention it, but reflections would also work: if the top row were 4, 9, 2, for example, and the middle 3, 5, 7, and the bottom 8, 1, 6. Combining rotations and reflections also works.

If you do the trick a second time, your mark might notice it’s odd that the sum came up 15 again. Do it a third time, even with a different rotation or reflection, and they’ll know something’s up. There are things you could do to disguise that further. Just double each number in the square, for example: a square of 4/18/8, 14/10/6, 12/2/16 will have each row or column or diagonal add up to 30. But this loses the beauty of doing this with the digits 1 through 9, and your mark might grow suspicious anyway. The same happens if, say, you add one to each number in the square, and forecast a sum of 18. Even mathematical magic tricks are best not repeated too often, not unless you have good stage patter.

Wavehead, to classmate, over lunch: 'Did you know that every square is a rhombus, but not every rhombus is a square? I mean, you can't make this stuff up!'
Mark Anderson’s Andertoons for the 20th of May, 2019. Always glad to discuss Andertoons, as you can see from these essays.

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead’s marveling at what seems at first like an asymmetry, about squares all being rhombuses yet rhombuses not all being squares. There are similar results with squares and rectangles. Still, it makes me notice something. Nobody would write a strip where the kid marvelled that all squares were polygons but not all polygons were squares. It seems that the rhombus connotes something different. This might just be familiarity. Polygons are … well, if not a common term, at least something anyone might feel familiar. Rhombus is a more technical term. It maybe never quite gets familiar, not in the ways polygons do. And the defining feature of a rhombus — all four sides the same length — seems like the same thing that makes a square a square.


There should be another Reading the Comics post this coming week, and it should appear at this link. I’d like to publish it Tuesday but, really, Wednesday is more probable.

Reading the Comics, May 8, 2019: Strips With Art I Like Edition


Of course I like all the comics. … Well, that’s not literally true; but I have at least some affection for nearly all of the syndicated comics. This essay I bring up some strips, partly, because I just like them. This is my content hole. If you want a blog not filled with comic strips, go start your own and don’t put these things on it.

Mark Anderson’s Andertoons for the 5th is the Mark Anderson’s Andertoons for the week. Also a bit of a comment on the ability of collective action to change things. Wavehead is … well, he’s just wrong about making the number four plus the number four equal to the number seven. Not based on the numbers we mean by the words “four” and “seven”, and based on the operation we mean by “plus” and the relationship we mean by “equals”. The meaning of those things is set by, ultimately, axioms and deductive reasoning and the laws of deductive reasoning and there’s no changing the results.

Wavehead, to another student: 'If I say 4 + 4 = 7 it's wrong. If you say 4 + 4 = 7 it's wrong. But if the entire first grade says 4 + 4 = 7, well, now she has to take us seriously.
Mark Anderson’s Andertoons for the 5th of May, 2019. Essays mentioning Andertoons are at this link and also at nearly every Reading the Comics post, it feels like.

But. The thing we’re referring to when we say “seven”? Or when we write the symbol “7”? That is convention. That is a thing we’ve agreed on as a reference for this concept. And that we can change, if we decide we want to. We’ve done this. Look at a thousand-year-old manuscript and the symbol that looks like ‘4’ may represent the number we call five. And the names of numbers are just common words. They’re subject to change the way every other common word is. Which is, admittedly, not very subject. It would be almost as much bother to change the word ‘four’ as it would be to change the word ‘mom’. But that’s not impossible. Just difficult.

Viivi: 'Oh, sorry! Oh, pain!' Wagner: 'Stop worrying. 85% of fears never come through.' Viivi: 'That means 15% do! It's worse than I thought.'
Juba’s Viivi and Wagner for the 5th of May, 2019. I don’t often have chances to talk about Viivi and Wagner but when I do, it’s here.

Juba’s Viivi and Wagner for the 5th is a bit of a percentage joke. The characters also come to conclude that a thing either happens or it does not; there’s no indefinite states. This principle, the “excluded middle”, is often relied upon for deductive logic, and fairly so. It gets less clear that this can be depended on for predictions of the future, or fears for the future. And real-world things come in degrees that a mathematical concept might not. Like, your fear of the home catching fire comes true if the building burns down. But it’s also come true if a quickly-extinguished frying pan fire leaves the wall scorched, embarrassing but harmless. Anyway, relaxing someone else’s anxiety takes more than a quick declaration of statistics. Show sympathy.

Dogs in school. The dog teacher is pointing to '1 + 1' on the blackboard. A dog student whispers to the other, 'Sometimes I feel so stupid.'
Harry Bliss and Steve Martin’s Bliss for the 6th of May, 2019. Yes, by the way, it’s the Steve Martin you know and love from Looney Tunes: Back In Action and from the 1996 Sergeant Bilko movie. Anyway I haven’t had chance to write about this strip before but this and future appearances of Bliss should be here.

Harry Bliss and Steve Martin’s Bliss for the 6th is a cute little classroom strip, with arithmetic appearing as the sort of topic that students feel overwhelmed and baffled by. It could be anything, but mathematics uses the illustration space efficiently. The strip may properly be too marginal to include, but I like Bliss’s art style and want more people to see it.

Spud: 'It's official, Wallace. My socks are *too* tight. And I know it'll take at least three minutes to run home and change. Yet I can see the bus is only two stops away.' Wallace: 'I can stall for a good thirty seconds.' Spud: 'My life is a sadistic math problem.'
Will Henry’s Wallace the Brave for the 7th of May, 2019. This is one of the comic strips I’m most excited about, the last several years. Wallace the Brave appears in essays at this link.

Will Henry’s Wallace the Brave for the 7th puts up what Spud calls a sadistic math problem. And, well, it is a story problem happening in their real life. You could probably turn this into an actual exam problem without great difficulty.

Ruthie, holding up a triangle: 'What's this shape?' James: 'A square!' Ruthie: 'I already *told* you what it is, James! You're just acting dumb to hurt my feelings! Stop it! N-n-now (sob) what does this look like to you? (Sniff)?' James: 'A cryangle!'
Rick Detorie’s One Big Happy for the 8th of May, 2019. There are two strings of One Big Happy available for daily reading. Appearances by the current or the several-years-old GoComics prints of One Big Happy should be at this link.

Rick Detorie’s One Big Happy for the 8th is a bit of wordplay built around geometry, as Ruthie plays teacher. She’s a bit dramatic, but she always has been.


I’ll read some more comics for later in this week. That essay, and all similar comic strip talk, should appear at this link. Thank you.

Reading the Comics, April 24, 2019: Mic Drop Edition Edition


I can’t tell you how hard it is not to just end this review of last week’s mathematically-themed comic strips after the first panel here. It really feels like the rest is anticlimax. But here goes.

John Deering’s Strange Brew for the 20th is one of those strips that’s not on your mathematics professor’s office door only because she hasn’t seen it yet. The intended joke is obvious, mixing the tropes of the Old West with modern research-laboratory talk. “Theoretical reckoning” is a nice bit of word juxtaposition. “Institoot” is a bit classist in its rendering, but I suppose it’s meant as eye-dialect.

Cowboys at the 'Institoot of Theoretical Reckoning'. One at the whiteboard says, 'Well, boys, looks like this here's the end of the line!' The line is a long string of what looks like legitimate LaTeX
John Deering’s Strange Brew for the 20th of April, 2019. Other appearances by Strange Brew, including ones less diligent about making the blackboard stuff sensible, are at this link.

What gets it a place on office doors is the whiteboard, though. They’re writing out mathematics which looks legitimate enough to me. It doesn’t look like mathematics though. What’s being written is something any mathematician would recognize. It’s typesetting instructions. Mathematics requires all sorts of strange symbols and exotic formatting. In the old days, we’d handle this by giving the typesetters hazard pay. Or, if you were a poor grad student and couldn’t afford that, deal with workarounds. Maybe just leave space in your paper and draw symbols in later. If your university library has old enough papers you can see them. Maybe do your best to approximate mathematical symbols using ASCII art. So you get expressions that look something like this:

  / 2 pi  
 |   2
 |  x cos(theta) dx - 2 F(theta) == R(theta)
 |
/ 0

This gets old real fast. Mercifully, Donald Knuth, decades ago, worked out a great solution. It uses formatting instructions that can all be rendered in standard, ASCII-available text. And then by dark incantations and summoning of Linotype demons, re-renders that as formatted text. It handles all your basic book formatting needs — much the way HTML, used for web pages, will — and does mathematics much more easily. For example, I would enter a line like:

\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)

And this would be rendered in print as:

\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)

There are many, many expansions available to this, to handle specialized needs, hardcore mathematics among them.

Anyway, the point that makes me realize John Deering was aiming at everybody with an advanced degree in mathematics ever with this joke, using a string of typesetting instead of the usual equations here?

The typesetting language is named TeX.

Wavehead, at lunch: 'You know if I were the other shapes I'd be like, 'listen, circle, you can have a perimeter or a circumference, but you can't have both'.'
Mark Anderson’s Andertoons for the 21st of April, 2019. When do I ever not discuss this comic? All the essays at this link are about Andertoons, at least in part.

Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for the week. It’s about one of those questions that nags at you as a kid, and again periodically as an adult. The perimeter is the boundary around a shape. The circumference is the boundary around a circle. Why do we have two words for this? And why do we sound all right talking about either the circumference or the perimeter of a circle, while we sound weird talking about the circumference of a rhombus? We sound weird talking about the perimeter of a rhombus too, but that’s the rhombus’s fault.

The easy part is why there’s two words. Perimeter is a word of Greek origin; circumference, of Latin. Perimeter entered the English language in the early 15th century; circumference in the 14th. Why we have both I don’t know; my suspicion is either two groups of people translating different geometry textbooks, or some eager young Scholastic with a nickname like ‘Doctor Magnifico Triangulorum’ thought Latin sounded better. Perimeter stuck with circules early; every etymology I see about why we use the symbol π describes it as shorthand for the perimeter of the circle. Why `circumference’ ended up the word for circles or, maybe, ellipses and ovals and such is probably the arbitrariness of language. I suspect that opening “circ” sound cues people to think of it for circles and circle-like shapes, in a way that perimeter doesn’t. But that is my speculation and should not be mistaken for information.

Information panel about numerals, including a man who typed every number from one to a million, using one finger; it took over 16 years, seven months. Puzzle: add together every number that, written as a word, consists of three letters; what's the total?
Steve McGarry’s KidTown for the 21st of April, 2019. It’s rare that this panel is on-topic enough for me to bring up, but at least a few KidTown panels are discussed here.

Steve McGarry’s KidTown for the 21st is a kids’s information panel with a bit of talk about representing numbers. And, in discussing things like how long it takes to count to a million or a billion, or how long it would take to type them out, it gets into how big these numbers can be. Les Stewart typed out the English names of numbers, in words, by the way. He’d also broken the Australian record for treading water, and for continuous swimming.

Bub: 'I don't like crosswords because I'm not good at word stuff. I'm much better at math. That's why I like sudoku.' Betty: 'What math? There's no adding or subtracting or multiplying in sudoku.' Bub: 'That's my favorite kind of math.' Betty: 'If you were better at word stuff, you'd know you're confusing math with logic.'
Gary Delainey and Gerry Rasmussen’s Betty for the 24th of April, 2019. I don’t seem to have discussed this comic before. This and future appearances by Betty should be at this link.

Gary Delainey and Gerry Rasmussen’s Betty for the 24th is a sudoku comic. Betty makes the common, and understandable, conflation of arithmetic with mathematics. But she’s right in identifying sudoku as a logical rather than an arithmetic problem. You can — and sometimes will see — sudoku type puzzles rendered with icons like stars and circles rather than numerals. That you can make that substitution should clear up whether there’s arithmetic involved. Commenters at GoComics meanwhile show a conflation of mathematics with logic. Certainly every mathematician uses logic, and some of them study logic. But is logic mathematics? I’m not sure it is, and our friends in the philosophy department are certain it isn’t. But then, if something that a recognizable set of mathematicians study as part of their mathematics work isn’t mathematics, then we have a bit of a logic problem, it seems.


Come Sunday I should have a fresh Reading the Comics essay available at this link.

Reading the Comics, April 10, 2019: Grand Avenue and Luann Want My Attention Edition


So this past week has been a curious blend for the mathematically-themed comics. There were many comics mentioning some mathematical topic. But that’s because Grand Advenue and Luann Againn — reprints of early 90s Luann comics — have been doing a lot of schoolwork. There’s a certain repetitiveness to saying, “and here we get a silly answer to a story problem” four times over. But we’ll see what I do with the work.

Mark Anderson’s Andertoons for the 7th is Mark Anderson’s Andertoons for the week. Very comforting to see. It’s a geometry-vocabulary joke, with Wavehead noticing the similar ends of some terms. I’m disappointed that I can’t offer much etymological insight. “Vertex”, for example, derives from the Latin for “highest point”, and traces back to the Proto-Indo-European root “wer-”, meaning “to turn, to bend”. “Apex” derives from the Latin for “summit” or “extreme”. And that traces back to the Proto-Indo-European “ap”, meaning “to take, to reach”. Which is all fine, but doesn’t offer much about how both words ended up ending in “ex”. This is where my failure to master Latin by reading a teach-yourself book on the bus during my morning commute for three months back in 2002 comes back to haunt me. There’s probably something that might have helped me in there.

On the blackboard is a square-based pyramid with 'apex' labelled; also a circular cone with 'vertex' labelled. Wavehead: 'And if you put them together they're a duplex.'
Mark Anderson’s Andertoons for the 7th of March, 2019. I write about this strip a lot. Essays mentioning Andertoons are at this link.

Mac King and Bill King’s Magic in a Minute for the 7th is an activity puzzle this time. It’s also a legitimate problem of graph theory. Not a complicated one, but still, one. Graph theory is about sets of points, called vertices, and connections between points, called edges. It gives interesting results for anything that’s networked. That shows up in computers, in roadways, in blood vessels, in the spreads of disease, in maps, in shapes.

Here's a tough little puzzle to get your brain firing on all four cylinders. See if you can connect the matching numbered boxes with three lines. The catch is that the liens cannot cross over each other. From left to right are disjoint boxes labelled 1, 2, 1, and 2. Above and below the center of the row are two boxes labelled 3.
Mac King and Bill King’s Magic in a Minute for the 7th of March, 2019. I should have the essays mentioning Magic In A Minute at this link.

One common problem, found early in studying graph theory, is about whether a graph is planar. That is, can you draw the whole graph, all its vertices and edges, without any lines cross each other? This graph, with six vertices and three edges, is planar. There are graphs that are not. If the challenge were to connect each number to a 1, a 2, and a 3, then it would be nonplanar. That’s a famous non-planar graph, given the obvious name K3, 3. A fun part of learning graph theory — at least fun for me — is looking through pictures of graphs. The goal is finding K3, 3 or another one called K5, inside a big messy graph.

Exam Question: Jack bought seven marbles and lost six. How many additional marbles must Jack buy to equal seven? Kid's answer: 'Jack wouldn't know. He's lost his marbles.'
Mike Thompson’s Grand Avenue for the 8th of March, 2019. I’m not always cranky about this comic strips. Examples of when I’m not are at this link, as are the times I’m annoyed with Grand Avenue.

Mike Thompson’s Grand Avenue for the 8th has had a week of story problems featuring both of the kid characters. Here’s the start of them. Making an addition or subtraction problem about counting things is probably a good way of making the problem less abstract. I don’t have children, so I don’t know whether they play marbles or care about them. The most recent time I saw any of my niblings I told them about the subtleties of industrial design in the old-fashioned Western Electric Model 2500 touch-tone telephone. They love me. Also I’m not sure that this question actually tests subtraction more than it tests reading comprehension. But there are teachers who like to throw in the occasional surprisingly easy one. Keeps students on their toes.

Gunther: 'You put a question mark next to the part about using the slope-intercept form of a linear equation. What don't you understand?' Luann: 'Lemme see. Oh ... yeah. I don't understand why on earth I need to know this.'
Greg Evans’s Luann Againn for the 10th of March, 2019. This strip originally ran the 10th of March, 1991. Essays which include some mention of Luann, either current or 1990s reprints, are at this link.

Greg Evans’s Luann Againn for the 10th is part of a sequence showing Gunther helping Luann with her mathematics homework. The story started the day before, but this was the first time a specific mathematical topic was named. The point-slope form is a conventional way of writing an equation which corresponds to a particular line. There are many ways to write equations for lines. This is one that’s convenient to use if you know coordinates for one point on the line and the slope of the line. Any coordinates which make the equation true are then the coordinates for some point on the line.

How To Survive a Shark Attack (illustrated with a chicken surviving a shark.0 Keep your eye on the shark and move slowly toward safety. Don't make any sudden movements such as splashing or jazz hands. If the shark comes at you, punch it in the gills, snout, or eyes. You won't hurt the shark, but it will be surprised by your audacity. If all else fails, try to confuse it with logical paradoxes. Chicken: 'This statement is false.' Shark, wide-eyed and confused: '?'
Doug Savage’s Savage Chickens for the 10th of March, 2019. And when I think of something to write about Savage Chickens the results are at this link.

Doug Savage’s Savage Chickens for the 10th tosses in a line about logical paradoxes. In this case, using a classic problem, the self-referential statement. Working out whether a statement is true or false — its “truth value” — is one of those things we expect logic to be able to do. Some self-referential statements, logical claims about themselves, are troublesome. “This statement is false” was a good one for baffling kids and would-be world-dominating computers in science fiction television up to about 1978. Some self-referential statements seem harmless, though. Nobody expects even the most timid world-dominating computer to be bothered by “this statement is true”. It takes more than just a statement being about itself to create a paradox.


And a last note. The blog hardly needs my push to help it out, but, sometimes people will miss a good thing. Ben Orlin’s Math With Bad Drawings just ran an essay about some of the many mathematics-themed comics that Hilary Price and Rina Piccolo’s Rhymes With Orange has run. The comic is one of my favorites too. Orlin looks through some of the comic’s twenty-plus year history and discusses the different types of mathematical jokes Price (with, in recent years, Piccolo) makes.

Myself, I keep all my Reading the Comics essays at this link, and those mentioning some aspect of Rhymes With Orange at this link.

Reading the Comics, March 19, 2019: Average Edition


This time around, averages seem important.

Mark Anderson’s Andertoons for the 18th is the Mark Anderson’s Andertoons for the week. This features the kids learning some of the commonest terms in descriptive statistics. And, as Wavehead says, the similarity of names doesn’t help sorting them out. Each is a kind of average. “Mean” usually is the arithmetic mean, or the thing everyone including statisticians calls “average”. “Median” is the middle-most value, the one that half the data is less than and half the data is greater than. “Mode” is the most common value. In “normally distributed” data, these three quantities are all the same. In data gathered from real-world measurements, these are typically pretty close to one another. It’s very easy for real-world quantities to be normally distributed. The exceptions are usually when there are some weird disparities, like a cluster of abnormally high-valued (or low-valued) results. Or if there are very few data points.

On the blackboard the teacher's written Median, Mode, and Mean, with a bunch of numbers from 3 through 15. Wavehead: 'I know they're all subtly different, but I have to say, the alliteration doesn't help.'
Mark Anderson’s Andertoons for the 18th of March, 2019. Essays which discuss topics raised by Andertoons can be found at this link. Also at this link, nearly enough.

The word “mean” derives from the Old French “meien”, that is, “middle, means”. And that itself traces to the Late Latin “medianus”, and the Latin “medius”. That traces back to the Proto-Indo-European “medhyo”, meaning “middle”. That’s probably what you might expect, especially considering that the mean of a set of data is, if the data is not doing anything weird, likely close to the middle of the set. The term appeared in English in the middle 15th century.

The word “median”, meanwhile, follows a completely different path. That one traces to the Middle French “médian”, which traces to the Late Latin “medianus” and Latin “medius” and Proto-Indo-European “medhyo”. This appeared as a mathematical term in the late 19th century; Etymology Online claims 1883, but doesn’t give a manuscript citation.

The word “mode”, meanwhile, follows a completely different path. This one traces to the Old French “mode”, itself from the Latin “modus”, meaning the measure or melody or style. We get from music to common values by way of the “style” meaning. Think of something being done “á la mode”, that is, “in the [ fashionable or popular ] style”. I haven’t dug up a citation about when this word entered the mathematical parlance.

So “mean” and “median” don’t have much chance to do anything but alliterate. “Mode” is coincidence here. I agree, it might be nice if we spread out the words a little more.

Edison, pointing to a checkerboard: 'So Grandpa if you put one cookie on the first square, two on the second, four on the next, then eight, and you keep doubling them until you fill all 64 squares do you know what you'll end up with?' Grandpa: 'A stomachache for a month?'
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th of March, 2019. I’ve been talking about this strip a lot lately, it seems to me. Essays where I do discuss Edison Lee are at this link.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th has Edison introduce a sequence to his grandfather. Doubling the number of things for each square of a checkerboard is an ancient thought experiment. The notion, with grains of wheat rather than cookies, seems to be first recorded in 1256 in a book by the scholar Ibn Khallikan. One story has it that the inventor of chess requested from the ruler that many grains of wheat as reward for inventing the game.

If we followed Edison Lee’s doubling through all 64 squares we’d have, in total, need for 263-1 or 18,446,744,073,709,551,615 cookies. You can see why the inventor of chess didn’t get that reward, however popular the game was. It stands as a good display of how exponential growth eventually gets to be just that intimidatingly big.

Edison, like many a young nerd, is trying to stagger his grandfather with the enormity of this. I don’t know that it would work. Grandpa ponders eating all that many cookies, since he’s a comical glutton. I’d estimate eating all that many cookies, at the rate of one a second, eight hours a day, to take something like eighteen billion centuries. If I’m wrong? It doesn’t matter. It’s a while. But is that any more staggering than imagining a task that takes a mere ten thousand centuries to finish?

Toby, looking at his homework, and a calculator, and a textbook: '... Wow. Crazy ... Huh. How about that? ... Am I stupid if math *always* has a surprise ending?'
Greg Cravens’s The Buckets for the 19th of March, 2019. It also seems like I discuss The Buckets more these days, as seen at this link.

Greg Cravens’s The Buckets for the 19th sees Toby surprised by his mathematics homework. He’s surprised by how it turned out. I know the feeling. Everyone who does mathematics enough finds that. Surprise is one of the delights of mathematics. I had a great surprise last month, with a triangle theorem. Thomas Hobbes, the philosopher/theologian, entered his frustrating sideline of mathematics when he found the Pythagorean Theorem surprising.

Mathematics is, to an extent, about finding interesting true statements. What makes something interesting? That depends on the person surprised, certainly. A good guideline is probably “something not obvious before you’ve heard it, thatlooks inevitable after you have”. That is, a surprise. Learning mathematics probably has to be steadily surprising, and that’s good, because this kind of surprise is fun.

If it’s always a surprise there might be trouble. If you’re doing similar kinds of problems you should start to see them as pretty similar, and have a fair idea what the answers should be. So, from what Toby has said so far … I wouldn’t call him stupid. At most, just inexperienced.

Caption: Eric had good feelings about his date. It turned out, contrary to what he first thought ... [ Venn Diagram picture of two circles with a good amount of overlap ] ... They actually had quite a lot in common.
Eric the Circle for the 19th of March, 2019, this one by Janka. This and other essays with Eric the Circle, by any artist, should be at this link.

Eric the Circle for the 19th, by Janka, is the Venn Diagram joke for the week. Properly any Venn Diagram with two properties has an overlap like this. We’re supposed to place items in both circles, and in the intersection, to reflect how much overlap there is. Using the sizes of each circle to reflect the sizes of both sets, and the size of the overlap to represent the size of the intersection, is probably inevitable. The shorthand calls on our geometric intuition to convey information, anyway.

She: 'What time is it?' He: 'The clock in the hall says 7:56, the oven clock says 8:02, the DVD clock says 8:07, and the bed table clock says 8:13.' She: 'Which one's right?' He: 'I dunno.' She: 'Let's ee. Four clocks ... carry the one ... divided by four ... ' He: 'Whenever we want to know what time it is we have to do algebra.' She: 'We better hurry, it's 8:05 and five-eighths!'
Tony Murphy’s It’s All About You for the 19th of March, 2019. And the occasional time I discuss something from It’s All About You are here.

Tony Murphy’s It’s All About You for the 19th has a bunch of things going on. The punch line calls “algebra” what’s really a statistics problem, calculating the arithmetic mean of four results. The work done is basic arithmetic. But making work seem like a more onerous task is a good bit of comic exaggeration, and algebra connotes something harder than arithmetic. But Murphy exaggerates with restraint: the characters don’t rate this as calculus.

Then there’s what they’re doing at all. Given four clocks, what’s the correct time? The couple tries averaging them. Why should anyone expect that to work?

There’s reason to suppose this might work. We can suppose all the clocks are close to the correct time. If they weren’t, they would get re-set, or not looked at anymore. A clock is probably more likely to be a little wrong than a lot wrong. You’d let a clock that was two minutes off go about its business, in a way you wouldn’t let a clock that was three hours and 42 minutes off. A clock is probably as likely to show a time two minutes too early as it is two minutes too late. This all suggests that the clock errors are normally distributed, or something like that. So the error of the arithmetic mean of a bunch of clock measurements we can expect to be zero. Or close to zero, anyway.

There’s reasons this might not work. For example, a clock might systematically run late. My mantle clock, for example, usually drifts about a minute slow over the course of the week it takes to wind. Or the clock might be deliberately set wrong: it’s not unusual to set an alarm clock to five or ten or fifteen minutes ahead of the true time, to encourage people to think it’s later than it really is and they should hurry up. Similarly with watches, if their times aren’t set by Internet-connected device. I don’t know whether it’s possible to set a smart watch to be deliberately five minutes fast, or something like that. I’d imagine it should be possible, but also that the people programming watches don’t see why someone might want to set their clock to the wrong time. From January to March 2018, famously, an electrical grid conflict caused certain European clocks to lose around six minutes. The reasons for this are complicated and technical, and anyway The Doctor sorted it out. But that sort of systematic problem, causing all the clocks to be wrong in the same way, will foil this take-the-average scheme.

Murphy’s not thinking of that, not least because this comic’s a rerun from 2009. He was making a joke, going for the funnier-sounding “it’s 8:03 and five-eights” instead of the time implied by the average, 8:04 and a half. That’s all right. It’s a comic strip. Being amusing is what counts.


There were just enough mathematically-themed comic strips this past week for one more post. When that is ready, it should be at this link. I’ll likely post it Tuesday.

Reading the Comics, March 12, 2019: Back To Sequential Time Edition


Since I took the Pi Day comics ahead of their normal sequence on Sunday, it’s time I got back to the rest of the week. There weren’t any mathematically-themed comics worth mentioning from last Friday or Saturday, so I’m spending the latter part of this week covering stuff published before Pi Day. It’s got me slightly out of joint. It’ll all be better soon.

Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for this week. That’s nice to have. It’s built on the concept of story problems. That there should be “stories” behind a problem makes sense. Most actual mathematics, even among mathematicians, is done because we want to know a thing. Acting on a want is a story. Wanting to know a thing justifies the work of doing this calculation. And real mathematics work involves looking at some thing, full of the messiness of the real world, and extracting from it mathematics. This would be the question to solve, the operations to do, the numbers (or shapes or connections or whatever) to use. We surely learn how to do that by doing simple examples. The kid — not Wavehead, for a change — points out a common problem here. There’s often not much of a story to a story problem. That is, where we don’t just want something, but someone else wants something too.

On the classroom chalkboard: 'Today's number story. 2 dogs are at the park. 3 more dogs arrive. How many dogs are there?' Non-Wavehead student: 'Really? That's it? I don't mean anything by it, but where the conflict? Where's the drama? OK, try this ... a cat shows up. Or a mailman! Ooh, a cat mailman! Now we're talking!'
Mark Anderson’s Andertoons for the 11th of March, 2019. Essays which discuss topics raised by Andertoons can be found at this link. Also at this link, nearly enough.

Parker and Hart’s The Wizard of Id for the 11th is a riff on the “when do you use algebra in real life” snark. Well, no one disputes that there are fields which depend on advanced mathematics. The snark comes in from supposing that a thing is worth learning only if it’s regularly “useful”.

King: 'Wiz! We need your help! There's a giant meteor heading towards earth!' Wizard: 'oh boy! Oh boy!' (He runs off and pulls out a chalkboard full of mathematics symbols.) King: 'What are you doing?' Wizard: 'Finally! This is where algebra saves lives!'
Parker and Hart’s The Wizard of Id for the 11th of March, 2019. This is part of the current run of The Wizard of Id, such as you might find in a newspaper. Both the current and 1960s-vintage reruns get discussion at this link.

Rick Detorie’s One Big Happy for the 12th has Joe stalling class to speak to “the guy who invented zero”. I really like this strip since it’s one of those cute little wordplay jokes that also raises a legitimate point. Zero is this fantastic idea and it’s hard to imagine mathematics as we know it without the concept. Of course, we could say the same thing about trying to do mathematics without the concept of, say, “twelve”.

Teacher: 'Are there any questions before the test? Yes, Joe?' Joe: 'I would like to say a few words to the guy who invented zero. Thanks for NOTHING, dude! ... I think we can all enjoy a little math humor at a time like this.'
Rick Detorie’s One Big Happy for the 12th of March, 2019. This is part of the current run of One Big Happy, such as you might find in a newspaper. At GoComics.com strips from several years ago are reprinted. Both runs of One Big Happy get their discussion at this link.

We don’t know who’s “the guy” who invented zero. It’s probably not all a single person, though, or even a single group of people. There are several threads of thought which merged together to zero. One is the notion of emptiness, the absense of a measurable thing. That probably occurred to whoever was the first person to notice a thing wasn’t where it was expected. Another part is the notion of zero as a number, something you could add to or subtract from a conventional number. That is, there’s this concept of “having nothing”, yes. But can you add “nothing” to a pile of things? And represent that using the addition we do with numbers? Sure, but that’s because we’re so comfortable with the idea of zero that we don’t ponder whether “2 + 1” and “2 + 0” are expressing similar ideas. You’ll occasionally see people asking web forums whether zero is really a number, often without getting much sympathy for their confusion. I admit I have to think hard to not let long reflex stop me wondering what I mean by a number and why zero should be one.

And then there’s zero, the symbol. As in having a representation, almost always a circle, to mean “there is a zero here”. We don’t know who wrote the first of that. The oldest instance of it that we know of dates to the year 683, and was written in what’s now Cambodia. It’s in a stone carving that seems to be some kind of bill of sale. I’m not aware whether there’s any indication from that who the zero was written for, or who wrote it, though. And there’s no reason to think that’s the first time zero was represented with a symbol. It’s the earliest we know about.

Lemont: 'My son asked what my favorite number is. I thought, who has a favorite number? Then remembered *I* did. When I was a kid I loved the number eight, because it reminded me of infinity.' C Dog: 'Nuh-uh, Big L. You tol' me you loved it 'cause it was the only number that reminded you of food.' Lemont: 'You sure? I remember being a much more profound kid.' C Dog: 'People always re-writing e'reything.'
Darrin Bell’s Candorville for the 12th of March, 2019. Essays featuring some topic raised by of Candorville should appear here.

Darrin Bell’s Candorville for the 12th has some talk about numbers, and favorite numbers. Lemont claims to have had 8 as his favorite number because its shape, rotated, is that of the infinity symbol. C-Dog disputes Lemont’s recollection of his motives. Which is fair enough; it’s hard to remember what motivated you that long ago. What people mostly do is think of a reason that they, today, would have done that, in the past.

The ∞ symbol as we know it is credited to John Wallis, one of that bunch of 17th-century English mathematicians. He did a good bit of substantial work, in fields like conic sections and physics and whatnot. But he was also one of those people good at coming up with notation. He developed what’s now the standard notation for raising a number to a power, that x^n stuff, and showed how to define raising a number to a rational-number power. Bunch of other things. He also seems to be the person who gave the name “continued fraction” to that concept.

Wallis never explained why he picked ∞ as a shape, of all the symbols one could draw, for this concept. There’s speculation he might have been varying the Roman numeral for 1,000, which we’ve simplified to M but which had been rendered as (|) or () and I can see that. (Well, really more of a C and a mirror-reflected C rather than parentheses, but I don’t have the typesetting skills to render that.) Conflating “a thousand” with “many” or “infinitely many” has a good heritage. We do the same thing when we talk about something having millions of parts or costing trillions of dollars or such. But, Wallis never explained (so far as we’re aware), so all this has to be considered speculation and maybe mnemonic helps to remembering the symbol.

Colin: 'These math problems are impossible!' Dad: 'Calm down, I'll help you think it through. [ Reading ] Dot has 20 gumballs. Jack has twice as many, and Joe has a third as many as Jack. If Dot gives Jack half her gumballs, and Jack chews half the new amount, how many will the teacher have to take away for all the kids' gumballs to be equal?' ... Now I see why people buy the answers to these things on the Internet!' Colin: 'I'm never going to harvard, am I?'
Terry Laban and Patty LaBan’s Edge City rerun for the 12th of March, 2019. (The strip has ended.) This comic originally ran in 2004. Edge City inspires discussions in the essays at this link.

Terry LaBan and Patty LaBan’s Edge City for the 12th is another story problem joke. Curiously the joke seems to be simply that the father gets confused following the convolutions of the story. The specific story problem circles around the “participation awards are the WORST” attitude that newspaper comics are surprisingly prone to. I think the LaBans just wanted the story problem to be long and seem tedious enough that our eyes glazed over. Anyway you could not pay me to read whatever the comments on this comic are. Sorry not sorry.


I figure to have one more Reading the Comics post this week. When that’s posted it should be available at this link. Thanks for being here.

Reading the Comics, February 2, 2019: Not The February 1, 2019 Edition


The last burst of mathematically-themed comic strips last week nearly all came the 1st of the month. But the count fell just short. I can only imagine what machinations at Comic Strip Master Command went wrong, that we couldn’t get a full four comics for the same day. Well, life is messy and things will happen.

Stephen Bentley’s Herb and Jamaal for the 1st is a rerun. I discussed it last time I noticed it too. I’d previously taken Herb to be gloating about not using the calculus he’d studied. I may be reading too much into what seems like a smirk in the final panel, though. Could be he’s thinking of the strangeness that something which, at the time, is challenging and difficult and all-consuming turns out to not be such a big deal. Which could be much of high school.

Herb, sitting at his counter, thinking: 'Man, can you believe it? Another year has passed and I still haven't used the calculus I studied in high school.' (He smirks.)
Stephen Bentley’s Herb and Jamaal for the 1st of February, 2019. It previously ran the 25th of November, 2013. Essays that mention Herb and Jamaal should be at this link. Although, it happens, the last time this ran was before I started tagging comic strips by name. Whoops?

But my first instinct is still to read this as thinking of the “uselessness” of calculus. It betrays the terrible attitude that education is about job training. It should be about letting people be literate in the world’s great thoughts. Mathematics seems to get this attitude a lot, but I’m aware I may feel a confirmation bias. If I had become a French major perhaps I’d pay attention to all the comic strips where someone giggles about how they never use the foreign languages they learned in high school either.

Mathpinion City, Numerically Flexible Zones. Minion holding sign reading 1+1=3: 'Haters, man.' Another minion: 'Can't stand 'em.' 'It's all the hate hate hatet hate.' 'It always is with their kind. The hating kind.' 'I mean, the *logic* is sound.' 'You can't argue with the *logic*.' 'But still, they're total a-holes.' (Looking at minions holding up signs reading 1+1=2.) 'Just because they're right they think they can be a-holes about it.' 'Well they *can't*.' 'Technically, they can. They're right about that too.'
Jon Rosenberg’s Scenes from a Multiverse for the 1st of February, 2019. Essays with some discussion sparked by Scenes from a Multiverse are at this link.

Jon Rosenberg’s Scenes from a Multiverse for the 1st is set in a “Mathpinion City”, showing people arguing about mathematical truths. It seems to me a political commentary, about the absurdity of rejecting true things over perceived insults. The 1+1=3 partisans aren’t even insisting they’re right, just that the other side is obnoxious. Arithmetic here serves as good source for things that can’t be matters of opinion, at least provided we’ve agreed on what’s meant by ideas like ‘1’ and ‘3’.

Mathematics is a human creation, though. What we decide to study, and what concepts we think worth interesting, are matters of opinion. It’s difficult to imagine people who think 1+1=2 a statement so unimportant they don’t care whether it’s true or false. At least not ones who reason anything like we do. But that is our difficulty, not a constraint on what life could think.

Student looks at a quiz. It's full of expressions, presumably to simplify, such as [a^2 b^2/16m] x [(m^2(25^2)x24m) / (16^{-2} \delta m]. He prays. There's tapping at the window. God appears, in a tree, pointing to an answer key. The teacher runs over, 'Hey!', scaring God out of the tree.
Neil Kohney’s The Other End for the 1st of February, 2019. I thought this might be a new tag, but no. I’ve discussed The Other End at some essays linked here.

Neil Kohney’s The Other End for the 1st has a mathematics cameo. It’s the subject of a quiz so difficult that the kid begs for God’s help sorting it out. The problems all seem to be simplifying expressions. It’s a skill worth having. There are infinitely many ways to write the same quantity. Some of them are more convenient than others. Brief expressions, for example, are often easier to understand. But a longer expression might let us tease out relationships that are good to know. Many analysis proofs end up becoming simpler when you multiply by one — that is, multiplying by and dividing by the same quantity, but using the numerator to reduce one part of the expression and the denominator to reduce some other. Or by adding zero, in which you add and subtract a quantity and use either side to simplify other parts of the expression. So, y’know, just do the work. It’s better that way.

The teacher's put on the board 4 x 6 = 24 and 24 / 6 = 4. Wavehead: 'I understand how it works. But if we're just going to end up back at four, what was the point?'
Mark Anderson’s Andertoons for the 2nd of February, 2019. I’ve discussed Andertoons in so many essays like this you might think I was Mark Anderson’s publicity agent, but that I had an extremely narrow focus of what I thought marketable.

Mark Anderson’s Andertoons for the 2nd is the Mark Anderson’s Andertoons for the week. Wavehead’s learning about invertible operations: that a particular division can undo a multiplication. Or, presumably, that a particular multiplication can undo a division. Fair to wonder why you’d want to do that, though. Most of the operations we use in arithmetic have inverses, or come near it. (There’s one thing you can multiply by which you can’t divide out.) The term used in group theory for this is to say the real numbers are a “field”. This is a ring in which not just does addition have an inverse, but so does multiplication. And the operations commute; dividing by four and multiplying by four is as good as multiplying by for and dividing by four. You can build interesting mathematical structures that don’t have some of these properties. Elementary-school division, where you might describe (say) 26 divided by 4 as “6 with a remainder of 2” is one of them.


And that covers the comic strips. Come Sunday should be the next of this series, and it should be at this link.

Reading the Comics, January 26, 2019: The Week Ended Early Edition


Last week started out at a good clip: two comics with enough of a mathematical theme I could imagine writing a paragraph about them each day. Then things puttered out. The rest of the week had almost nothing. At least nothing that seemed significant enough. I’ll list those, since that’s become my habit, at the end of the essay.

Jonathan Lemon and Joey Alison Sayers’s Alley Oop for the 20th is my first chance to show off the new artist and writer team. They’ve decided to make Sunday strips a side continuity about a young Alley Oop and his friends. I’m interested. The strip is built on the bit of pop anthropology that tells us “primitive” tribes will have very few counting words. That you can express concepts like one, two, and three, but then have to give up and count “many”.

Little Alley Oop: 'You think scientists will ever invent a number bigger than three?' Garg: 'I guess it's possible. There are three scientists working around the clock trying to come up with a new number.' Oop: 'Three scientists? Wow! That's a lot.' Garg: 'Maybe someday *we'll* be scientists. Then there'll be *three* scientists.' Oop: 'Nah, I think I want to be a fire-fighter. There are only three of those in the whole world.'
Jonathan Lemon and Joey Alison Sayers’s Alley Oop for the 20th of January, 2019. I don’t seem to have had much cause to mention Alley Oop before; the only previous reference I can find is in this cameo in a different comic. Well, this and any other essays that write about Alley Oop at any length should be at this link. I don’t figure to differentiate between the weekday Alley Oop strip and the Sunday Little Oop line in using this tag.

Perhaps it’s so. Some societies have been found to have, what seem to us, rather few numerals. This doesn’t reflect on anyone’s abilities or intelligence or the like. And it doesn’t mean people who lack a word for, say, “forty-nine” would be unable to compute. It might take longer, but probably just from inexperience. If someone practiced much calculation on “forty-nine” they’d probably have a name for it. And folks raised in the western mathematics use, even enjoy, some vagueness about big numbers too. We might say there are “dozens” of a thing even if there are not precisely 24, 36, or 48 of the thing; “52” is close enough and we probably didn’t even count it up. “Hundred” similarly has gotten the connotation of being a precise number, but it’s used to mean “really quite a lot of a thing”. The words “thousands”, “millions”, and mock-numbers like “zillions” have a similar role. They suggest different ranges of what might be “many”.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a SABRmetrics joke! At least, it’s an optimization joke, built on the idea that you can find an optimum strategy for anything, whether winning baseball games or The War. The principle is hard to argue with. Nobody would doubt that different approaches to a battle affect how likely winning is. We can imagine gathering data on how different tactics affect the outcome. (We can easily imagine combat simulators running these experiments, particularly.)

War party stats nerd: 'We are taking a statistical approach combat. First, don't go for kills. Go for *stabs*. Successful stabs are *much* more valuable to victory than lethal strikes. Look at Birk over here. He averages 22.1 stabs per battle. He alone accounts for an additional 3.8 wins per campaign season. Siegwurst brings in 1.6.' Skeptic: 'But remember when Siegwurst slew two Cossacks with his Dance of the Whirling Blades?' Stats Nerd: 'NO MORE READING THE SAGAS, OK? No spin moves! They're impressive but the expected addition wins per spin is negative. NEGATIVE.' Skeptic :'This is gonna have serious negative effects on morale.' Stats nerd: 'Which correlates with EXACTLY NOTHING. Now GET STABBY!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of January, 2019. Pretty much every Reading the Comics brings up this strip. But the essays that specifically mention Saturday Morning Breakfast Cereal should be at this link.

The catch — well, one catch — is that this tempts one to reward a process. Once it’s taken for granted the process works, then whether it’s actually doing what you want gets forgotten. And once everyone knows what’s being measured it becomes possible to game the system. Famously, in the mid-1960s the United States tried to judge its progress in the Vietname War by counting the number of enemy soldiers killed. There was then little reason to care about who was killed, or why. And reason to not care whether actual enemy soldiers were being killed. There’s good to be said about testing whether the things you try to do work. There’s great danger in thinking that the thing you can measure guarantees success.

Mark Anderson’s Andertoons for the 21st is a bit of fun with definitions. Mathematicians rely on definitions. It’s hard to imagine a proof about something undefined. But definitions are hard to compose. We usually construct a definition because we want a common term to describe a collection of things, and to exclude another collection of things. And we need people like Wavehead who can find edge cases, things that seem to satisfy a definition while breaking its spirit. This can let us find unstated assumptions that we should pay attention to. Or force us to accept that the definition is so generally useful that we’ll tolerate it having some counter-intuitive implications.

On the blackboard: 'NOT A POLYGON: Fewer than three sides; not connected at ends; lines that cross.' Teacher, to student: 'True, a chicken nugget is also not a polygon, but we're going to focus more on lines and vertices.'
Mark Anderson’s Andertoons for the 21st of January, 2019. Pretty much every Reading the Comics brings up this strip, at least since last fall’s weird gap has ended. But the essays that specifically mention Andertoons should be at this link.

My favorite counter-intuitive implication is in analysis. The field has a definition for what it means that a function is continuous. It’s meant to capture the idea that you could draw a curve representing the function without having to lift the pen that does it. The best definition mathematicians have settled on allows you to count a function that’s continuous at a single point in all of space. Continuity seems like something that should need an interval to happen. But we haven’t found a better way to define “continuous” that excludes this pathological case. So we embrace the weirdness in exchange for general usefulness.

Princess Kat, awestruck at soap bubbles: 'How did you do that, Jackson? Are you a wizard?' Jackson: 'They're just bubbles. You dunk this wand into a cup of soapy water and then blow through it.' Kat: 'Gimme gimme! I wanna blow bubbles too!' Jackson: 'Just take it easy or they'll pop! It's pretty simple.' (Kat blows a soap bubble cube, then tetrahedron and cones and some optical illusion shapes.) Kat: 'I think I'm doing this wrong.' Jackson: 'Suddenly I feel like a round peg in a square hole.'
Charles Brubaker’s Ask A Cat for the 21st of January, 2019. Although I’ve mentioned this comic before, it wasn’t when I was putting up tags naming each comic. I’ll fix that now. This and future essays mentioning Ask A Cat (or its Fuzzy Princess sideline, as long as that hasn’t got its own GoComics presence) should appear at this link. And this strip previously ran the 19th of June, 2016.

Charles Brubaker’s Ask A Cat for the 21st is a guest appearance from Brubaker’s other strip, The Fuzzy Princess. It’s a rerun and I did discuss it earlier. Soap bubbles make for great mathematics. They’re easy to play with, for one thing. That’s good for capturing imagination. And the mathematics behind them is deep, and led to important results analytically and computationally. It happens when this strip first ran I’d encountered a triplet of essays about the mathematics of soap bubbles and wireframe surfaces. My introduction to those essays is here.

Benita Epstein’s Six Chix for the 25th I wasn’t sure I’d include. But Roy Kassinger asked about it, so that tipped the scales. The dog tries to blame his bad behavior on “the algorithm”, bringing up one of the better monsters of the last couple years. An algorithm is just the procedure by which you do something. Mathematically, that’s usually to solve a problem. That might be finding some interesting part of the domain or range of a function. That might be putting a collection of things in order. that might be any of a host of things. And then we go make a decision based on the results of the algorithm.

Dog, explaining a messy room to its horrified humans: 'The algorithm made me do it!'
Benita Epstein’s Six Chix for the 25th of January, 2019. The essays wherein I mention Six Chix, from any of the shared strip’s authors, should appear at this link.

What earns The Algorithm its deserved bad name is mindlessness. The idea that once you have an algorithm that a problem is solved. Worse, that once an algorithm is in place it would be irrational to challenge it. I have seen the process termed “mathwashing”, by analogy with whitewashing, and it’s a good one. The notion that because something is done by computer it must be done correctly is absurd. We knew it was absurd before there were computers as we knew them, as see anyone for the past century who has spoken of a “Kafkaesque” interaction with a large organization. It’s impossible to foresee all the outcomes of any reasonably complicated process, much less to verify that all the outcomes are handled correctly. This is before we consider that there will always be mistakes made in the handling of data. Or in the carrying out of the process. And that’s before we consider bad actors. I’m sure there must be research into algorithms designed to handle gaming of the system. I don’t know that there are any good results yet, though. We certainly need them.


There were a couple comics that didn’t seem to be substantial enough for me to write at length about. You might like them anyway. Connie Sun’s Connie to the Wonnie for the 21st shows off a Venn Diagram. Hector D Cantú and Carlos Castellanos’s Baldo for the 23rd is a bit of wordplay about what mathematicians do. Jonathan Lemon’s Rabbits Against Magic for the 23rd similarly is a bit of wordplay built around percentages. (Lemon is the new artist for Alley Oop.) And Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips features Albert Einstein, and a joke based on one of the symmetries which make relativity such a useful explanation of the world’s workings.


I don’t plan to have another Reading the Comics post until next Sunday. But when I do, it’ll be here.

Reading the Comics, January 12, 2019: A Edition


As I said Sunday, last week was a slow one for mathematically-themed comic strips. Here’s the second half of them. They’re not tightly on point. But that’s all right. They all have titles starting with ‘A’. I mean if you ignore the article ‘the’, the way we usually do when alphabetizing titles.

Tony Cochran’s Agnes for the 11th is basically a name-drop of mathematics. The joke would be unchanged if the teacher asked Agnes to circle all the adjectives in a sentence, or something like that. But there are historically links between religious thinking and mathematics. The Pythagoreans, for example, always a great and incredible starting point for any mathematical topic or just some preposterous jokes that might have nothing to do with their reality, were at least as much a religious and philosophical cult. For a long while in the Western tradition, the people with the time and training to do advanced mathematics work were often working for the church. Even as people were more able to specialize, a mystic streak remained. It’s easy to understand why. Mathematics promises to speak about things that are universally true. It encourages thinking about the infinite. It encourages thinking about the infinitely tiny. It courts paradoxes as difficult as any religious Mystery. It’s easy to snark at someone who takes numerology seriously. But I’m not sure the impulse that sees magic in arithmetic is different to the one that sees something supernatural in a “transfinite” item.

Teacher: 'Agnes, what is 18 divided by 7?' Agnes: 'It matters not. I am being called to a religious life of heavy piety, general holiness, and things of that nature. I want none of math's Satanity to taint my walk.' Later, Agnes, to the principal: 'I'm toying with starting a compound in New Mexico, if you want to blow this cheap pop stand and find the light.'
Tony Cochran’s Agnes for the 11th of January, 2019. I’m always glad to have the chance to write about Agnes, and you can see things I’ve written about it at this link.

Scott Hilburn’s The Argyle Sweater for the 11th is another mistimed Pi Day joke. π is, famously, an irrational number. But so is every number, except for a handful of strange ones that we’ve happened to find interesting. That π should go on and on follows from what an irrational number means. It’s a bit surprising the 4 didn’t know all this before they married.

Cartoon Labelled 'Wife of Pi'. At a therapist's office. The therapist is a division sign. A pi with a face rolls its eyes. A 4 with a face complains, 'He's irrational and he goes on and on.'
Scott Hilburn’s The Argyle Sweater for the 11th of January, 2019. It’s not quite the most common strip mentioned here. But you can find other topics raised by The Argyle Sweater at this link.

I appreciate the secondary joke that the marriage counselor is a “Hugh Jripov”, and the counselor’s being a ripoff is signaled by being a ÷ sign. It suggests that maybe successful reconciliation isn’t an option. I’m curious why the letters ‘POV’ are doubled, in the diploma there. In a strip with tighter drafting I’d think it was suggesting the way a glass frame will distort an image. But Hilburn draws much more loosely than that. I don’t know if it means anything.

Wavehead working on the problem 17 + 7 at the blackboard. He's circled the 17, put a ? under the equals sign, added an ! before the +, and a * after it. Teacher: 'This is math; you don't need to annotate.'
Mark Anderson’s Andertoons for the 12th of January, 2019. There’s more essays with Andertoons at this link.

Mark Anderson’s Andertoons for the 12th is the Mark Anderson’s Andertoons for the essay. I’m so relieved to have a regular stream of these again. The teacher thinks Wavehead doesn’t need to annotate his work. And maybe so. But writing down thoughts about a problem is often good practice. If you don’t know what to do, or you aren’t sure how to do what you want? Absolutely write down notes. List the things you’d want to do. Or things you’d want to know. Ways you could check your answer. Ways that you might work similar problems. Easier problems that resemble the one you want to do. You find answers by thinking about what you know, and the implications of what you know. Writing these thoughts out encourages you to find interesting true things.

And this was too marginal a mention of mathematics even for me, even on a slow week. But Georgia Dunn’s Breaking Cat News for the 12th has a cat having a nightmare about mathematics class. And it’s a fun comic strip that I’d like people to notice more.


And that’s as many comics as I have to talk about from last week. Sunday, I should have another Reading the Comics post and it’ll be at this link.

Reading the Comics, January 9, 2018: I Go On About Johnny Appleseed Edition


This was a slow week for mathematically-themed comic strips. Such things happen. I put together a half-dozen that see on-topic enough to talk about, but I stretched to do it. You’ll see.

Mark Anderson’s Andertoons for the 6th mentions addition as one of the things you learn in an average day of elementary school. I can’t help noticing also the mention of Johnny Appleseed, who’s got a weird place in my heart as he and I share a birthday. He got to it first. Although Johnny Appleseed — John Champan — is legendary for scattering apple seeds, that’s not what he mostly did. He would more often grow apple-tree nurseries, from which settlers could buy plants and demonstrate they were “improving” their plots. He was also committed to spreading the word of Emanuel Swedenborg’s New Church, one of those religious movements that you somehow don’t hear about. But there was this like 200-year-long stretch where a particular kind of idiosyncratic thinker was Swedenborgian, or at least influenced by that. I don’t know offhand of any important Swedenborgian mathematicians, I admit, but I’m glad to hear if someone has news.

Wavehead, walking home, talking to another kid: 'Today we learned about Columbus planting apple seeds using two-digit addition. I also daydreamed a lot.'
Mark Anderson’s Andertoons for the 6th of January, 2019. Andertoons often appears in these essays. You can see the proof of that Andertoons claim at this link.

Justin Thompson’s MythTickle rerun for the 9th mentions “algebra” as something so dreadful that even being middle-aged is preferable. Everyone has their own tastes, yes, although it would be the same joke if it were “gym class” or something. (I suppose that’s not one word. “Dodgeball” would do, but I never remember playing it. It exists just as a legendarily feared activity, to me.) Granting, though, that I had a terrible time with the introduction to algebra class I had in middle school.

Karma: 'What's wrong, Dziva?' Dziva: 'Watching Boody act so young and carefree makes me long for my own youth. I could run faster then, eat more and care less. I'm getting sad about it. Karma, isn't there some magical word that could make me quit wanting to be young again? Some profound reminder that being a kid wasn't so --- ' Karma: 'Algebra.' (Both rest, happy.)
Justin Thompson’s MythTickle rerun for the 9th of January, 2019. MythTickle has only barely appeared before in these essays. You can see the proof of that MythTickle claim at this link.

Tom Wilson’s Ziggy for the 9th is a very early Pi Day joke, so, there’s that. There’s not much reason a take-a-number dispenser couldn’t give out π, or other non-integer numbers. What the numbers are doesn’t matter. It’s just that the dispensed numbers need to be in order. It should be helpful if there’s a clear idea how uniformly spaced the numbers are, so there’s some idea how long a wait to expect between the currently-serving number and whatever number you’ve got. But that only helps if you have a fair idea of how long an order should on average take.

Ziggy, at a pie counter, takes a number. It's pi.
Tom Wilson’s Ziggy for the 9th of January, 2019. Ziggy has turned up once or twice in these essays. You can see the proof of that Ziggy claim at this link.

I’ll close out last week’s comics soon. The next Reading the Comics post, like all the earlier ones, should be at this link.

Reading the Comics, January 5, 2019: Start of the Year Edition


With me wrapping up the mathematically-themed comic strips that ran the first of the year, you can see how far behind I’m falling keeping everything current. In my defense, Monday was busier than I hoped it would be, so everything ran late. Next week is looking quite slow for comics, so maybe I can catch up then. I will never catch up on anything the rest of my life, ever.

Scott Hilburn’s The Argyle Sweater for the 2nd is a bit of wordplay about regular and irregular polygons. Many mathematical constructs, in geometry and elsewhere, come in “regular” and “irregular” forms. The regular form usually has symmetries that make it stand out. For polygons, this is each side having the same length, and each interior angle being congruent. Irregular is everything else. The symmetries which constrain the regular version of anything often mean we can prove things we otherwise can’t. But most of anything is the irregular. We might know fewer interesting things about them, or have a harder time proving them.

Teacher: 'Well, class, who'd like to show Mr Hoffmeyer how to correctly make an irregular polygon regular?' On the blackboard is an irregular pentagon and, drawn by Mr Hoffmeyer, a box of Ex-Lax.
Scott Hilburn’s The Argyle Sweater for the 2nd of January, 2019. The many appearances of Argyle Sweater in these pages are at this link.

I’m not sure what the teacher would be asking for in how to “make an irregular polygon regular”. I mean if we pretend that it’s not setting up the laxative joke. I can think of two alternatives that would make sense. One is to draw a polygon with the same number of sides and the same perimeter as the original. The other is to draw a polygon with the same number of sides and the same area as the original. I’m not sure of the point of either. I suppose polygons of the same area have some connection to quadrature, that is, integration. But that seems like it’s higher-level stuff than this class should be doing. I hate to question the reality of a comic strip but that’s what I’m forced to do.

Mutt, to Jeff in the hospital bed: 'Don't be afraid! Surgery on the tonsils is very simple!' Doctor: 'Don't you worry about the results!' Jeff: 'How do you know I'll be all right?' Doctor: 'Well, I lost my last eleven patients! So if the law of probabilities doesn't lie, you'll be all right! May I do something for you before I begin?' Jeff: 'Oh, yes, Doc! Help me put on my trousers and my jacket!'
Bud Fisher’s Mutt and Jeff rerun for the 4th of January, 2019. The several appearances of Mutt and Jeff in these pages are at this link.

Bud Fisher’s Mutt and Jeff rerun for the 4th is a gambler’s fallacy joke. Superficially the gambler’s fallacy seems to make perfect sense: the chance of twelve bad things in a row has to be less than the chance of eleven bad things in a row. So after eleven bad things, the twelfth has to come up good, right? But there’s two ways this can go wrong.

Suppose each attempted thing is independent. In this case, what if each patient is equally likely to live or die, regardless of what’s come before? And in that case, the eleven deaths don’t make it more likely that the next will live.

Suppose each attempted thing is not independent, though. This is easy to imagine. Each surgery, for example, is a chance for the surgeon to learn what to do, or not do. He could be getting better, that is, more likely to succeed, each operation. Or the failures could reflect the surgeon’s skills declining, perhaps from overwork or age or a loss of confidence. Impossible to say without more data. Eleven deaths on what context suggests are low-risk operations suggest a poor chances of surviving any given surgery, though. I’m on Jeff’s side here.

On the blackboard: 'Ratios: Apples 9, Oranges 6'. Wavehead, to teacher: 'Technically the ratio is 3:2, but as a practical matter we shouldn't even really be considering this.'
Mark Anderson’s Andertoons for the 5th of January, 2019. The amazingly many appearances of Andertoons in these pages are at this link.

Mark Anderson’s Andertoons for the 5th is a welcome return of Wavehead. It’s about ratios. My impression is that ratios don’t get much attention in themselves anymore, except to dunk on stupid Twitter comments. It’s too easy to jump right into fractions, and division. Ratios underlie this, at least historically. It’s even in the name, ‘rational numbers’.

Wavehead’s got a point in literally comparing apples and oranges. It’s at least weird to compare directly different kinds of things. This is one of those conceptual gaps between ancient mathematics and modern mathematics. We’re comfortable stripping the units off of numbers, and working with them as abstract entities. But that does mean we can calculate things that don’t make sense. This produces the occasional bit of fun on social media where we see something like Google trying to estimate a movie’s box office per square inch of land in Australia. Just because numbers can be combined doesn’t mean they should be.

Kid: 'Dad, I need help with a math problem. If striking NFL players who get $35,000 a game are replaced by scab players who get $1,000 a game ... what will be the point spread in a game between the Lions and the Packers?'
Larry Wright’s Motley rerun for the 5th of January, 2019. The occasional appearances of Motley in these pages are at this link.

Larry Wright’s Motley rerun for the 5th has the form of a story problem. And one timely to the strip’s original appearance in 1987, during the National Football League players strike. The setup, talking about the difference in weekly pay between the real players and the scabs, seems like it’s about the payroll difference. The punchline jumps to another bit of mathematics, the point spread. Which is an estimate of the expected difference in scoring between teams. I don’t know for a fact, but would imagine the scab teams had nearly meaningless point spreads. The teams were thrown together extremely quickly, without much training time. The tools to forecast what a team might do wouldn’t have the data to rely on.


The at-least-weekly appearances of Reading the Comics in these pages are at this link.

Reading the Comics, December 19, 2018: Andertoons Is Back Edition


I had not wanted to mention, for fear of setting off a panic. But Mark Anderson’s Andertoons, which I think of as being in every Reading the Comics post, hasn’t been around lately. If I’m not missing something, it hasn’t made an appearance in three months now. I don’t know why, and I’ve been trying not to look too worried by it. Mostly I’ve been forgetting to mention the strange absence. This even though I would think any given Tuesday or Friday that I should talk about the strip not having anything for me to write about. Fretting about it would make a great running theme. But I have never spotted a running theme before it’s finished. In any event the good news is that the long drought has ended, and Andertoons reappears this week. Yes, I’m hoping that it won’t be going to long between appearances this time.

Mrs Olsen: 'How do you know I haven't got my flu shot?' Caulfield: 'Just playing the odds.' Mrs Olsen: 'Maybe I was playing some odds myself. Maybe I got to the pharmacy and remembered that this year's vaccine is 30-40% effective.' Caulfield: 'I'd take those odds.' Mrs Olsen: 'They're not my kind of odds.' Caulfield: 'And what are the odds you bought a lottery ticket on your way out?' (Pause.) Mrs Olsen: 'You are getting under my skin.' Caulfield: 'That's good news. Now there's a 30-40% chance you'll develop a resistance.'
Jef Mallett’s Frazz for the 16th of December, 2018. Other essays discussing topics raised by Frazz are at this link.

Jef Mallett’s Frazz for the 16th talks about probabilities. This in the context of assessing risks. People are really bad at estimating probabilities. We’re notoriously worse at assessing risks, especially when it’s a matter of balancing a present cost like “fifteen minutes waiting while the pharmacy figures out whether insurance will pay for the flu shot” versus a nebulous benefit like “lessened chance of getting influenza, or at least having a less severe influenza”. And it’s asymmetric, too. We view improbable but potentially enormous losses differently from the way we view improbable but potentially enormous gains. And it’s hard to make the rationally-correct choice reliably, not when there are so many choices of this kind every day.

Guard, to new prisoner: 'Never mind Professor Phillip. He's always preoccupied with some theory of escape probability.' The cell walls are covered with mathematical scrawls.
Tak Bui’s PC and Pixel for the 16th of December, 2018. This and other essays, when they’re written, inspired by PC and Pixel should be at this link. It’s a new tag, which surprises me.

Tak Bui’s PC and Pixel for the 16th features a wall full of mathematical symbols, used to represent deep thought about a topic. The symbols are gibberish, yes. I’m not sure that an actual “escape probability” could be done in a legible way, though. Or even what precisely Professor Phillip might be calculating. I imagine it would be an estimate of the various ways he might try to escape, and what things might affect that. This might be for the purpose of figuring out what he might do to maximize his chances of a successful escape. Although I wouldn’t put it past the professor to just be quite curious what the odds are. There’s a thrill in having a problem solved, even if you don’t use the answer for anything.

Amazing Yet Tautological strip: 'Each year America consumes enough EGG SALAD ... ' (Picture of a woman holding up a lumpy pile that context indicates is egg salad.) ' ... to give EACH AMERICAN an annualized national-average serving of the tasty concoction!'
Ruben Bolling’s Super-Fun-Pak Comix for the 18th of December, 2018. Essays based on Super-Fun-Pak Comix are at this link. (Amazing Yet Tautological is one of the features that turns up in Super-Fun-Pak Comix, which is why it doesn’t rate a tag on its own)

Ruben Bolling’s Super-Fun-Pak Comix for the 18th has a trivia-panel-spoof dubbed Amazing Yet Tautological. One could make an argument that most mathematics trivia fits into this category. At least anything about something that’s been proven. Anyway, whether this is a tautological strip depends on what the strip means by “average” in the phrase “average serving”. There’s about four jillion things dubbed “average” and each of them has a context in which they make sense. The thing intended here, and the thing meant if nobody says anything otherwise, is the “arithmetic mean”. That’s what you get from adding up everything in a sample (here, the amount of egg salad each person in America eats per year) and dividing it by the size of the sample (the number of people in America that year). Another “average” which would make sense, but would break this strip, would be the median. That would be the amount of egg salad that half of all Americans eat more than, and half eat less than. But whether every American could have that big a serving really depends on what that median is. The “mode”, the most common serving, would also be a reasonable “average” to expect someone to talk about.

Teacher showing solid geometry to the class. Wavehead: 'I saw a movie where the robot monster came right at me. If you want me to get excited about 3D shapes, you're going to have to do better than that.'
Mark Anderson’s Andertoons for the 19th of December, 2018. The many essays which discuss Andertoons are at this link.

Mark Anderson’s Andertoons for the 19th is that strip’s much-awaited return to my column here. It features solid geometry, which is both an important part of geometry and also a part that doesn’t get nearly as much attention as plane geometry. It’s reductive to suppose the problem is that it’s harder to draw solids than planar figures. I suspect that’s a fair part of the problem, though. Mathematicians don’t get much art training, not anymore. And while geometry is supposed to be able to rely on pure reasoning, a good picture still helps. And a bad picture will lead us into trouble.


Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

Reading the Comics, September 17, 2018: Hard To Credit Edition


Two of the four comic strips I mean to feature here have credits that feel unsatisfying to me. One of them is someone’s pseudonym and, yeah, that’s their business. One is Dennis the Menace, for which I find an in-strip signature that doesn’t match the credentials on Comics Kingdom’s web site, never mind Wikipedia. I’ll go with what’s signed in the comic as probably authoritative. But I don’t like it.

R Ferdinand and S Ketcham’s Dennis the Menace for the 16th is about calculation. One eternally surprising little thing about calculators and computers is that they don’t do anything you can’t do by hand. Or, for that matter, in your head. They do it faster, typically, and more reliably. They can seem magical. But the only difference between what they do and what we do is the quantity with which they do this work. You can take this as humbling or as inspirational, as fits your worldview.

Dad: 'The battery in my calculator is dead!' Dennis: 'Here! Use my chalkboard!' Dad: 'Uhhh ... thanks, but ... ' Denis: 'There's no batteries, so you can always COUNT on it! Well! Go ahead, Dad!' Dad: 'Okay! Okay! .. Let's see here ... just a give me a minute ... and carry the one ... ' (Dennis looks at the reader.)
R Ferdinand and S Ketcham’s Dennis the Menace for the 16th of September, 2018. I know it’s a standard bit of Dennis the Menace snarking to rate Dennis’s actual menacing nature. But forcing his father to show that he’s lost his proficiency at arithmetic in the guise of helping him? … It’s kind of a long-game bit of menace, I suppose.

Ham’s Life on Earth for the 16th is a joke about the magical powers we attribute to mathematics. It’s also built on one of our underlying assumptions of the world, that it must be logically consistent. If one has an irrefutable logical argument that something isn’t so, then that thing must not be so. It’s hard to imagine how an illogical world would work. But it is hard not to wonder if there’s some arrogance involved in supposing the world has to square with the rules of logic that we find sensible. And to wonder whether we perceive world consistent with that logic because our expectations frame what we’re able to perceive.

Man at chalkboard writing out the 'Mathematical Proof That I Don't Exist'. As he finishes it, he disappears.
Ham’s Life on Earth for the 16th of September, 2018. Raise your hand if you’ve been there. Hah! You’re fibbing.

In any case, as we frame logic, an argument’s validity shouldn’t depend on the person making the argument. Or even whether the argument has been made. So it’s hard to see how simply voicing the argument that one doesn’t exist could have that effect. Except that mathematics has got magical connotations, and vice-versa. That’ll be good for building jokes for a while yet.

Wavehead responding to the blackboard question 36 / 6: 'It's not that I can't divide that, but I've always fancied myself more of a uniter.'
Mark Anderson’s Andertoons for the 17th of September, 2018. So is this a substitute or does Wavehead just have a new mathematics teacher?

Mark Anderson’s Andertoons for the 17th is the Mark Anderson’s Andertoons for the week. It’s wordplay, built on the connotation that division is a bad thing. It seems less dire if we think of division as learning how to equally share something that’s been held in common, though. Or if we think of it as learning what to multiply a thing by to get a particular value. Most mathematical operations can be taken to mean many things. Surely division has some constructive and happy interpretations.

Poncho, the dog, looking over his owner's laptop: 'They say if you let an infinite number of cats walk on an infinite number of keyboards, they'll eventually type all the great works of Shakespeare.' The cat walks across the laptop, connecting to their owner's bank site and entering the correct password. Poncho: 'I'll take it.'
Paul Gilligan’s Pooch Cafe for the 17th of September, 2018. For my money the cat hasn’t walked across the keyboard right if it hasn’t got you Freakazoid powers.

Paul Gilligan’s Pooch Cafe for the 17th is a variation of the monkeys-on-keyboards joke. If what you need is a string of nonsense characters then … well, a cat on the keys is at least famous for producing some gibberish. It’s likely not going to be truly random, though. If a cat’s paw has stepped on, say, the ‘O’, there’s a good chance the cat is also stepping on ‘P’ or ‘9’. It also suggests that if the cat starts from the right, they’re more likely to have a character like ‘O’ early in the string of characters and less likely at the end. A completely random string would be as likely to have an ‘O’ at the start as at the end of the string.

And even if a cat on the keyboard did produce good-quality randomness, well. How likely a randomly-generated string of characters is to match a thing depends on the length of the thing. If the meaning of the symbols doesn’t matter, then ‘Penny Lane’ is as good as ‘*2ft,2igFIt’. This is not to say you can just use, say, ‘asdfghjkl’ as your password, at least not for anything that would hurt you if it were cracked. If everyone picked all passwords with no regard for what the symbols meant, these would be. But passwords that seem easy to think get used more often than they should be. It’s not that they’re easier to guess, but that guessing them is more likely to be correct.


Later this week I’ll host this month’s Playful Mathematics Blog Carnival! If you know of any mathematics that teaches or delights or both please share it with me, and we’ll let the world know. Also this week I should finally start my 2018 Mathematics A To Z, explaining words from mathematics one at a time.

And there’ll be another Reading the Comics Post before next Sunday. It and all my other Reading the Comics posts should be at this tag. Other appearances of Dennis the Menace should be at this link. This and other essays mentioning Life On Earth are at this link. The many appearances of Andertoons are at this link And other essays with Pooch Cafe should be at this link. Thanks for reading along.

Reading the Comics, September 11, 2018: 60% Reruns Edition


Three of the five comic strips I review today are reruns. I think that I’ve only mentioned two of them before, though. But let me preface all this with a plea I’ve posted before: I’m hosting the Playful Mathematics Blog Carnival the last week in September. Have you run across something mathematical that was educational, or informative, or playful, or just made you glad to know about? Please share it with me, and we can share it with the world. It can be for any level of mathematical background knowledge. Thank you.

Tom Batiuk’s Funky Winkerbean vintage rerun for the 10th is part of an early storyline of Funky attempting to tutor football jock Bull Bushka. Mathematics — geometry, particularly — gets called on as a subject Bull struggles to understand. Geometry’s also well-suited for the joke because it has visual appeal, in a way that English or History wouldn’t. And, you know, I’ll take “pretty” as a first impression to geometry. There are a lot of diagrams whose beauty is obvious even if their reasons or points or importance are obscure.

Funky: 'Okay, Bull, I'm going to help you with your math! Now this is a hexagon.' Bull: 'Pretty!' Funky, to camera: 'This is going to take a little longer than I thought!'
Tom Batiuk’s Funky Winkerbean vintage rerun for the 10th of September, 2018. It originally ran the 28th of September, 1972.

Dan Collins’s Looks Good on Paper for the 10th is about everyone’s favorite non-orientable surface. The first time this strip appeared I noted that the road as presented isn’t a Möbius strip. The opossums and the car are on different surfaces. Unless there’s a very sudden ‘twist’ in the road in the part obscured from the viewer, anyway. If I’d drawn this in class I would try to save face by saying that’s where the ‘twist’ is, but none of my students would be convinced. But we’d like to have it that the car would, if it kept driving, go over all the pavement.

Mobius Trip. A car, loaded for vacation, with someone in it asking 'Are we there yet? Are we there yet? Are we there yet?' The road is along a Mobius strip, with roadside bits like deer or road signs or opossums crossing the road on the margins.
Dan Collins’s Looks Good on Paper for the 10th of September, 2018. It originally ran the 10th of September, 2016.

Bud Fisher’s Mutt and Jeff for the 10th is a joke about story problems. The setup suggests that there’s enough information in what Jeff has to say about the cop’s age to work out what it must be. Mutt isn’t crazy to suppose there is some solution possible. The point of this kind of challenge is realizing there are constraints on possible ages which are not explicit in the original statements. But in this case there’s just nothing. We would call the cop’s age “underdetermined”. The information we have allows for many different answers. We’d like to have just enough information to rule out all but one of them.

Jeff: 'If Mario the policeman at our corner has been in the force for nine years ... he has been married for 17 years and has two kids, seven and twelve years old ... HOW OLD is Mario?' Mutt: 'I give up! How old is he?' Jeff: 'He's 40 years old!' Mutt: 'How did you calculate that?' Jeff: 'Oh, I didn't! I just asked him and he told me!' (Jeff, fleeing.) Jeff: 'I might as well stop running! Sooner or later he'll catch me anyway!'
Bud Fisher’s Mutt and Jeff for the 10th of September, 2018. No guessing when this originally ran, and the lettering has been re-done. This is probably why Jeff’s word balloons in the first and second panel don’t quite low logically together.

John Rose’s Barney Google and Snuffy Smith for the 11th is here by popular request. Jughead hopes that a complicated process of dubious relevance will make his report card look not so bad. Loweezey makes a New Math joke about it. This serves as a shocking reminder that, as most comic strip characters are fixed in age, my cohort is now older than Snuffy and Loweezey Smith. At least is plausibly older than them.

Jughead, explaining his report card: 'If ya add the two B's together, subtract th' C an' cancel out th' F, then I got mostly A's!' Ma, skeptical: 'Did they replace that new math wif even newer math?'
John Rose’s Barney Google and Snuffy Smith for the 11th of September, 2018. Not a rerun! As far as I know. But see how Henry fooled me about its rerun cycle lately. Not answered: a report card in early-to-mid September?

Anyway it’s also a nice example of the lasting cultural reference of the New Math. It might not have lasted long as an attempt to teach mathematics in ways more like mathematicians do. But it’s still, nearly fifty years on, got an unshakable and overblown reputation for turning mathematics into doubletalk and impossibly complicated rules. I imagine it’s the name; “New Math” is a nice, short, punchy name. But the name also looks like what you’d give something that was being ruined, under the guise of improvement. It looks like that terrible moment of something familiar being ruined even if you don’t know that the New Math was an educational reform movement. Common Core’s done well in attracting a reputation for doing problems the complicated way. But I don’t think its name is going to have the cultural legacy of the New Math.

Wavehead, facing a set of blackboard subtraction problems: 'This is why no one likes math; it's a branding issue. Everything's a problem.'
Mark Anderson’s Andertoons for the 11th of September, 2018. All right, but another problem they have: no chalk.

Mark Anderson’s Andertoons for the 11th is another kid-resisting-the-problem joke. Wavehead’s obfuscation does hit on something that I have wondered, though. When we describe things, we aren’t just saying what we think of them. We’re describing what we think our audience should think of them. This struck me back around 1990 when I observed to a friend that then-current jokes about how hard VCRs were to use failed for me. Everyone in my family, after all, had no trouble at all setting the VCR to record something. My friend pointed out that I talked about setting the VCR. Other people talk about programming the VCR. Setting is what you do to clocks and to pots on a stove and little things like that; an obviously easy chore. Programming is what you do to a computer, an arcane process filled with poor documentation and mysterious problems. We framed our thinking about the task as a simple, accessible thing, and we all found it simple and accessible. Mathematics does tend to look at “problems”, and we do, especially in teaching, look at “finding solutions”. Finding solutions sounds nice and positive. But then we just go back to new problems. And the most interesting problems don’t have solutions, at least not ones that we know about. What’s enjoyable about facing these new problems?


One thing that’s not a problem: finding other Reading the Comics posts. They should all appear at this link. Appearances by the current-run and the vintage Funky Winkerbean are at this link. Essays with a mention of Looks Good On Paper are at this link. Meanwhile, essays with Mutt and Jeff in the are at this link. Other appearances by Barney Google and Snuffy Smith — current and vintage, if vintage ever does something on-topic — are at this link. And the many appearances by Andertoons are at this link, or just use any Reading the Comics post, really. Thank you.

Reading the Comics, August 24, 2018: Delayed But Eventually There Edition


Now I’ve finally had the time to deal with the rest of last week’s comics. I’ve rarely been so glad that Comic Strip Master Command has taken it easy on me for this week.

Tom Toles’s Randolph Itch, 2am for the 20th is about a common daydream, that of soap bubbles of weird shapes. There’s fun mathematics to do with soap bubbles. Most of these fall into the “calculus of variations”, which is good at finding minimums and maximums. The minimum here is a surface with zero mean curvature that satisfies particular boundaries. In soap bubble problems the boundaries have a convenient physical interpretation. They’re the wire frames you dunk into soap film, and pull out again, to see what happens. There’s less that’s proven about soap bubbles than you might think. For example: we know that two bubbles of the same size will join on a flat common surface. Do three bubbles? They seem to, when you try blowing bubbles and fitting them together. But this falls short of mathematical rigor.

Randolph blows several soap bubbles. One is a perfect cube. Caption; 'But naturally nobody was around.' Footer joke: Randolph has a video camera but asks, 'What does 'battery, battery, battery' mean?'
Tom Toles’s Randolph Itch, 2am for the 20th of August, 2018. Little does Randolph realize that the others are all five-dimensional hyperspheres too.

Parker and Hart’s Wizard of Id Classics for the 21st is a joke about the ignorance of students. Of course they don’t know basic arithmetic. Curious thing about the strip is that you can read it as an indictment of the school system, failing to help students learn basic stuff. Or you can read it as an indictment of students, refusing the hard work of learning while demanding a place in politics. Given the 1968 publication date I have a suspicion which was more likely intended. But it’s hard to tell; 1968 was a long time ago. And sometimes it’s just so easy to crack an insult there’s no guessing what it’s supposed to mean.

King: 'I'm addressing a bunch of students tomorrow. What can I tell them that they don't already know?' Speechwriter: 'How about, two and two is four?'
Parker and Hart’s Wizard of Id Classics for the 21st of August, 2018. I’m curious when the speechwriter character disappeared from the comic strip. I remember it doing inexplicable election-year jokes at least to 1980 or maybe 1984. (I mean, he is a King, and we know he’s not King of Poland, so, you know?)

Gene Mora’s Graffiti for the 22nd mentions what’s probably the most famous equation after that thing with two times two in it. It does cry out something which seems true, that E = mc^2 was there before Albert Einstein noticed it. It does get at one of those questions that, I say without knowledge, is probably less core to philosophers of mathematics than the non-expert would think. But are mathematical truths discovered or invented? There seems to be a good argument that mathematical truths are discovered. If something follows by deductive logic from the axioms of the field, and the assumptions that go into a question, then … what’s there to invent? Anyone following the same deductive rules, and using the same axioms and assumptions, would agree on the thing discovered. Invention seems like something that reflects an inventor.

On a cement wall: 'e = mc^2 was there before Einstein discovered it'
Gene Mora’s Graffiti for the 22nd of August, 2018. Still have no idea whether this comic is still in production.

But it’s hard to shake the feeling that there is invention going on. Anyone developing new mathematics decides what things seem like useful axioms. She decides that some bundle of properties is interesting enough to have a name. She decides that some consequences of these properties are so interesting as to be named theorems. Maybe even the Fundamental Theorem of the field. And there was the decision that this is a field with a question interesting enough to study. I’m not convinced that isn’t invention.

On the blackboard: 49. 40% is ___. 50% is ___. 60% is ___. Non-Wavehead kid: 'I say we wait a few days and see if it doesn't hit 70%'.
Mark Anderson’s Andertoons for the 23rd of August, 2018. Ew, I wouldn’t want to do that problem on the board either.

Mark Anderson’s Andertoons for the 23rd sees Wavehead — waaait a minute. That’s not Wavehead! This throws everything off. Well, it’s using mathematics as the subject that Not-Wavehead is trying to avoid. And it’s not using arithmetic as the subject easiest to draw on the board. It needs some kind of ascending progression to make waiting for some threshold make sense. Numbers rising that way makes sense.

Leader of a (sideways) V-flock of birds: 'Roman ya newbies! A Roman five!' The other flock of birds is in a (sideways) 5 shape.

Scott Hilburn’s The Argyle Sweater for the 24th of August, 2018. Of course, we’re assuming that they’re flying off to the right. If they’re flying towards the lower left corner, then the 5-birds are doing a bit better.

Scott Hilburn’s The Argyle Sweater for the 24th is the Roman numerals joke for this week. Oh, and apparently it’s a rerun; I hadn’t noticed before that the strip was rerunning. This isn’t a complaint. Cartoonists need vacations too.

That birds will fly in V-formation has long captured people’s imaginations. We’re pretty confident we know why they do it. The wake of one bird’s flight can make it easier for another bird to stay aloft. This is especially good for migrating birds. The fluid-dynamic calculations of this are hard to do, but any fluid-dynamic calculations are hard to do. Verifying the work was also hard, but could be done. I found and promptly lost an article about how heartbeat monitors were attached to a particular flock of birds whose migration path was well-known, so the sensors could be checked and data from them gathered several times over. (Birds take turns as the lead bird, the one that gets no lift from anyone else’s efforts.)

So far as I’m aware there’s still some mystery as to how they do it. That is, how they know to form this V-formation. A particularly promising line of study in the 80s and 90s was to look at these as self-organizing structures. This would have each bird just trying to pay attention to what made sense for itself, where to fly relative to its nearest-neighbor birds. And these simple rules created, when applied to the whole flock, that V pattern. I do not know whether this reflects current thinking about bird formations. I do know that the search for simple rules that produce rich, complicated patterns goes on. Centuries of mathematics, physics, and to an extent chemistry have primed us to expect that everything is the well-developed result of simple components.

God, on a cloud, holding a lightning bolt: 'Whenever someone talks about the odds of winning the lottery, I like to hit 'em with one of these bad boys.'
Dave Whamond’s Reality Check for the 24th of August, 2018. Well, I guess the squirrel didn’t have anything to add to this one either.

Dave Whamond’s Reality Check for the 24th is apparently an answer to The Wandering Melon‘s comic earlier this month. So now we know what kind of lead time Dave Whamond is working on.


My next, and past, Reading the Comics posts are available at this link. Other essays with Randolph Itch, 2 a.m., are at this link. Essays that mention The Wizard of Id, classic or modern, are at this link. Essays mentioning Graffiti are at this link. Other appearances by Andertoons are at this link, or just read about half of all Reading the Comics posts. The Argyle Sweater is mentioned in these essays. And other essays with Reality Check are at this link. And what the heck; here’s other essays with The Wandering Melon in them.

Reading the Comics, August 18, 2018: Ragged Ends Edition


I apologize for the ragged nature of this entry, but I’ve had a ragged sort of week and it’s all I can do to keep up. Alert calendar-watchers might have figured out I would have rather had this posted on Thursday or Friday, but I couldn’t make that work. I’m trying. Thanks for your patience.

Mark Anderson’s Andertoons for the 17th feeds rumors that I just reflexively include Mark Anderson’s Andertoons in these posts whenever I see one. But it features the name of something dear to me, so that’s worthwhile. And I love etymology, although not enough to actually learn anything substantive about it. I just enjoy trivia about where some words come from, and sometimes how they change over time. (The average English word meant the exact opposite thing about two hundred years ago, and it meant something hilariously unrelated two centuries before that.)

Wavehead, talking to another student: 'I forgot to do my geometry, but I got an extra day by distracting Mrs Wilson with whether 'geo' is a root or a prefix.'
Mark Anderson’s Andertoons for the 17th of August, 2018. Also so I guess now we know Wavehead’s teacher’s name. Please make a note so that when I forget this you can tell me and I’ll act like it’s a discovery.

So I’m not sure how real word-studyers would regard the “geo” in “geometry”. The word is more or less Ancient Greek, given a bit of age and worn down into common English forms. It’s fair enough to describe it as originally meaning “land survey” or “land measure”. This might seem eccentric. But much of the early use of geometry was to figure out where things were, and how far they were from each other. It seems likely the earliest uses, for example, of the Pythagorean Theorem dealt with how to draw right angles on the surface of the Earth. And how to draw boundaries. The Greek fascination with compass-and-straightedge construction — work done without a ruler, so that you know distance only as a thing relative to other things in your figure — obscures how much of the field is about measurement.

Herman: 'I just joined a new club!' Lil: 'What's it called?' Herman: 'The society of parallel lines!' Lil: 'When do you meet?' Herman: 'Never!'
Brett Koth’s Diamond Lil for the 17th of August, 2018. Will admit that I’m at the stage in life that social clubs I don’t have to actually do anything for sound attractive.

Brett Koth’s Diamond Lil for the 17th is another geometry joke, and a much clearer one. And if there’s one thing we can say about parallel lines it’s that they don’t meet. There are some corners of geometry in which it’s convenient to say they “meet at infinity”, that is, they intersect at some point an infinite distance away. I don’t recommend bringing this up in casual conversation. I’m not sure I wanted to bring it up here.

Peter, ice skating, draws a '2' with his skates, and then falls through the ice. Other, non-Peter guy: 'Now there's a figure 8 the hard way. A 2 and a deep 6.'
Johnny Hart’s Back to BC for the 18th of August, 2018. It originally ran the 21st of February, 1961.

Johnny Hart’s Back to BC for the 18th is … hm. Well, I’ll call it a numerals joke. It’s part of the continuum of jokes made about ice skating in figure-eights.


Other essays about comic strips are at this link. When I’ve talked about Andertoons I’ve tried to make sure it turns up at this link. Essays in which I’ve discussed Diamond Lil should be at this link when there are other ones. Turns out this is a new tag. The times I’ve discussed B.C., old or new, should be at this link.

Reading the Comics, August 11, 2018: Strips For The Week Edition


The other half of last week’s mathematically-themed comics were on familiar old themes. I’ll see what I can do with them anyway.

Scott Hilburn’s The Argyle Sweater for the 9th is the anthropomorphic numerals joke for the week. I’m curious why the Middletons would need multiple division symbols, but I suppose that’s their business. It does play on the idea that “division” and “splitting up” are the same thing. And that fits the normal use of these words. We’re used to thinking, say, of dividing a desired thing between several parties. While that’s probably all right in introducing the idea, I do understand why someone would get very confused when they first divide by one-half or one-third or any number between zero and one. And then negative numbers make things even more confusing.

5, looking out the window and speaking to 3: 'Oh dear. Looks like the Middletons are getting a divorce.' 3: 'How can you tell?' (Next door a 4 has driven up with two division obeluses in the car.)
Scott Hilburn’s The Argyle Sweater for the 9th of August, 2018. The division symbol ÷ is the “obelus”, by the way. And no, the dots above and below the line are not meant to represent where you would fit a numerator and denominator into a fraction. That’s a useful trick to remember what the symbol does, but it’s not how the symbol was “designed”.

Thaves’s Frank and Ernest for the 9th is the anthropomorphic geometric figures joke for the week. I think I can wrangle a way by which Circle’s question has deeper mathematical context. Mathematicians use the idea of “space” a lot. The use is inspired by how, you know, the geometry of a room works. Euclidean space, in the trade. A Euclidean space is a collection of points that obey a couple simple rules. You can take two points and add them, and get something in the space. You can take any scalar and multiply it by any point and get a point in the space. A scalar is something that acts like a real number. For example, real numbers. Maybe complex numbers, if you’re feeling wild.

Circle, and triangle, speaking to a cube: 'Three-dimensional, eh --- what makes you so spatial?
Thaves’s Frank and Ernest for the 9th of August, 2018. Idly curious if they’ve done this same joke in Eric the Circle.

A Euclidean space can be two-dimensional. This is the geometry of stuff you draw on paper. It can be three-dimensional. This is the geometry of stuff in the real world, or stuff you draw on paper with shading. It can be four-dimensional. This is the geometry of stuff you draw on paper with big blobby lines around it. Each of these is an equally good space, though, as legitimate and as real as any other. Context usually puts an implicit “three dimensional” before most uses of the word “space”. But it’s not required to be there. There’s many kinds of spaces out there.

And “space” describes stuff that doesn’t look anything like rooms or table tops or sheets of paper. These are spaces built of things like functions, or of sets of things, or of ways to manipulate things. Spaces built of the ways you can subdivide the integers. The details vary. But there’s something in common in all these ideas that communicates.

Wavehead at the blackboard, speaking to his teacher. On the board is '14 - x = 5'. Wavehead: 'I'm just saying --- sooner or later X is going to have to solve these things for itself.'
Mark Anderson’s Andertoons for the 11th of August, 2018. Why do they always see it as x needing solving, and not, say, 14 needing solving?

Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. I think we’ve all seen this joke go across our social media feed and it’s reassuring to know Mark Anderson has social media too. We do talk about solving for x, using the language of describing how we help someone get past a problem. I wonder if people might like this kind of algebra more if we talked more about finding out what values ‘x’ could have that make the equation true. Well, it won’t stop people feeling they don’t like the mathematics they learned in school. But it might help people feel like they know why they’re doing it.


You can see this and more essays about comic strips by following this link. Other essays describing The Argyle Sweater are at this link. Essays inspired by Frank and Ernest are at this link. And some of the very many essays about Andertoons are at this link. Enjoy responsibly.

Reading the Comics, August 2, 2018: Non-Euclidean Geometry Edition


There’s really only the one strip that I talk about today that gets into non-Euclidean geometries. I was hoping to have the time to get into negative temperatures. That came up in the comics too, and it’s a subject close to my heart. But I didn’t have time to write that and so must go with what I did have. I’ve surely used “Non-Euclidean Geometry Edition” as a name before too, but that name and the date of August 2, 2018? Just as surely not.

Mark Anderson’s Andertoons for the 29th is the Mark Anderson’s Andertoons for the week, at last. Wavehead gets to be disappointed by what a numerator and denominator are. Common problem; there are many mathematics things with great, evocative names that all turn out to be mathematics things.

Both “numerator” and “denominator”, as words, trace to the mid-16th century. They come from Medieval Latin, as you might have guessed. “Denominator” parses out roughly as “to completely name”. As in, break something up into some number of equal-sized pieces. You’d need the denominator number of those pieces to have the whole again. “Numerator” parses out roughly as “count”, as in the count of how many denominator-sized pieces you have. So for all that numerator and denominator look like one another, with with the meat of the words being the letters “n-m–ator”, their centers don’t have anything to do with one another. (I would believe a claim that the way the words always crop up together encouraged them to harmonize their appearances.)

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 29th of July, 2018. Poor Wavehead is never going to get over his disappointment when he learns about the Fredholm Alternative. I still insist it’s an underrated mid-70s paranoia-thriller.

Johnny Hart’s Back to BC for the 29th is a surprisingly sly joke about non-Euclidean geometries. You wouldn’t expect that given the reputation of the comic the last decade of Hart’s life. And I did misread it at first, thinking that after circumnavigating the globe Peter had come back to have what had been the right line touch the left. That the trouble was his stick wearing down I didn’t notice until I re-read.

But Peter’s problem would be there if his stick didn’t wear down. “Parallel” lines on a globe don’t exist. One can try to draw a straight line on the surface of a sphere. These are “great circles”, with famous map examples of those being the equator and the lines of longitude. They don’t keep a constant distance from one another, and they do meet. Peter’s experiment, as conducted, would be a piece of proof that they have to live on a curved surface.

Peter, holding up a Y-shaped stick: 'In proving to you rather dense individuals that parallel lines never meet I am about to embark on a heretofore unprecedented expedition which will encompass the globe. See you.' (He walks off, dragging both ends of the stick in the ground, creating parallel lines. He walks several panels; 'Fifty Thousand Miles Later' as the stick wears down and the parallel lines get less far apart ... he gets back where he started, with the stick worn down to a single ribbon, and the surviving line left.)
Johnny Hart’s Back to BC for the 29th of July, 2018. It originally ran it looks like, the 13th of August, 1961. Or I’m reading the second row, second panel wrong.

And this gets at one of those questions that bothers mathematicians, cosmologists, and philosophers. How do we know the geometry of the universe? If we could peek at it from outside we’d have some help, but that is a big if. So we have to rely on what we can learn from inside the universe. And we can do some experiments that tell us about the geometry we’re in. Peter’s line example would be one; he can use that to show the world’s curved in at least one direction. A couple more lines and he’d be confident the world was a sphere. If we could make precise enough measurements we could do better, with geometric experiments smaller than the circumference of the Earth. (Or universe.) Famously, the sum of the interior angles of a triangle tell us something about the space the triangle’s inscribed in. There are dangers in going from information about one point, or a small area, to information about the whole. But we can tell some things.

Dr Strange-y type doing mind stuff: 'Using my MENTALISTIC powers of the occult, I shall attempt to DOUBLE your brain power, Captain Victorious! Peruse the potency of ... PROFESSOR PECULIAR!' Captain Victorious: 'Mind ... reeling! So much ... information!! ... Two plus two equals ... four! Hey! It worked!' Professor Peculiar: 'I guess double was too relative a term.'
Phil Dunlap’s Ink Pen rerun for the 29th of July, 2018. This one originally ran the 21st of August, 2011.

Phil Dunlap’s Ink Pen for the 29th is another use of arithmetic as shorthand for intelligence. Might be fun to ponder how Captain Victorious would know that he was right about two plus two equalling four, if he didn’t know that already. But we all are in the same state, for mathematical truths. We know we’ve got it right because we believe we have a sound logical argument for the thing being true.

A 'Newton Enterprises' boat pulls up to a desert island. A skeleton's under the tree; beside it, a whiteboard that starts with 'Coconut' and proceeds through a few lines of text to, finally, 'F = GMm/R^2'. Caption: 'How Newton actually stumbled across his formula for the theory of gravity.'
Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 30th of July, 2018. All right, so the rubber boat is an obvious anachronism. But Newton’s pal Edmond Halley made some money building diving bells for people to excavate shipwrecks and if that doesn’t mess with your idea of what the 17th century was you’re a stronger one than I am.

Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 30th is a riff on the story of Isaac Newton and the apple. The story of Newton starting his serious thinking of gravity by pondering why apples should fall while the Moon did not is famous. And it seems to trace to Newton. We have a good account of it from William Stukeley, who in the mid-18th century wrote Memoirs of Sir Isaac Newton’s Life. Stukeley knew Newton, and claimed to get the story right from him. He also told it to his niece’s husband, John Conduitt. Whether this is what got Newton fired with the need to create such calculus and physics, or whether it was a story he composed to give his life narrative charm, is beyond my ability to say. It’s an important piece of mathematics history anyway.


If you’d like more Reading the Comics essays you can find them at this link. Some of the many essays to mention Andertoons are at this link. Other essays mentioning B.C. (vintage and current) are at this link. The comic strip Ink Pen gets its mentions at this link, although I’m surprised to learn it’s a new tag today. And the Chuckle Brothers I discuss at this link. Thank you.