I’ve had several weeks since my last Reading the Comics post. They’ve been quiet enough weeks. Let me share some of the recent offerings from Comic Strip Master Command that I enjoyed, though. I enjoy many comic strips but not all of them mention something on-point here.
Bill Amend’s FoxTrot for the 24th of July gets a bit more solidly mathematical, as it’s natural to think of this sort of complicated polyhedron as something mathematicians do. Geometers at least. There’s a comfortable bit of work to be done in these sorts of shapes. They sometimes have appealing properties, for instance balancing weight loads well. Building polyhedrons out of toothpicks and gumdrops, or straws and marshmallows, or some other rigid-and-soft material, is a fine enough activity. I think every mathematics department has some dusty display shelf with a couple of these.
There are many shapes that Paige Fox’s construction might be. To my eye Paige Fox seems to be building a truncated icosahedron, that is to say, the soccer-ball shape. It’s an Archimedean solid, one of the family of thirteen shapes made of nonintersecting regular convex polygons. These are the shapes you discover if you go past Platonic solids. The family is named for that Archimedes, although the work in which he discussed them is now lost.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th of July is a different take on the challenge of motivating students to care about mathematics. The “right approach” argument has its appeal, although it’s obviously thinking of “professor” as the only job a mathematician can hold. And even granting that none of the people who understand your work have the power to fire you, that means people who don’t understand your work do have it. Also, people shut down very hard at your promise that they could understand anything about what you do, even when you know how to express it in common-language terms. This gets old fast. I am also skeptical that women are impressed by men at bars who claim to be employed for their intellect.
Anyway, academic jobs are great and more jobs should work by their rules, which, yes, do include extremely loose set hours and built-in seasons where the amount and kind of work you do varies.
Patrick Roberts’s Todd the Dinosaur for the 5th of August is a mathematics-anxiety dream, represented with the sudden challenge to do mental arithmetic. 47 times 342 is an annoying problem, yes, but one can at least approximate it fairly well quickly. 47 is almost 50, which is half a hundred. So 47 times 342 has to be nearly half of a hundred times 342, that is, half of 34,200. This is an easy number to cut in half, though: 17,100. To get this exactly? 47 is three less than fifty, so, subtract three times 342. 342 is about a third of a thousand, we can make a better estimate by subtracting a thousand: 16,100.
If you’re really good you notice that 342 is nine more than 333, so, three times 342 is three times nine more than three times 333. That is, it’s 27 more than 999. So the 16,100 estimate is 26 more than the correct number, 16,074. And I believe if you check, you will find the card in your hand is the ace of clubs. Am I not right, professor?
As promised I have the Pi Day comic strips from my reading here. I read nearly all the comics run on Comics Kingdom and on GoComics, no matter how hard their web sites try to avoid showing comics. (They have some server optimization thing that makes the comics sometimes just not load.) (By server optimization I mean “tracking for advertising purposes”.)
Pi Day in the comics this year saw the event almost wholly given over to the phonetic coincidence that π sounds, in English, like pie. So this is not the deepest bench of mathematical topics to discuss. My love, who is not as fond of wordplay as I am, notes that the ancient Greeks likely pronounced the name of π about the same way we pronounce the letter “p”. This may be etymologically sound, but that’s not how we do it in English, and even if we switched over, that would not make things better.
Scott Hilburn’s The Argyle Sweater is one of the few strips not to be about food. It is set in the world of anthropomorphized numerals, the other common theme to the day.
Jules Rivera’s Mark Trail makes the most casual Pi Day reference. If the narrator hadn’t interrupted in the final panel no one would have reason to think this referenced anything.
Mark Parisi’s Off The Mark is the other anthropomorphic numerals joke for the day. It’s built on the familiar fact that the digits of π go on forever. This is true for any integer base. In base π, of course, the representation of π is just “10”. But who uses that? And in base π, the number six would be something with infinitely many digits. There’s no fitting that in a one-panel comic, though.
Doug Savage’s Savage Chickens is the one strip that wasn’t about food or anthropomorphized numerals. There is no practical reason to memorize digits of π, other than that you’re calculating something by hand and don’t want to waste time looking them up. In that case there’s not much call go to past 3.14. If you need more than about 3.14159, get a calculator to do it. But memorizing digits can be fun, and I will not underestimate the value of fun in getting someone interested in mathematics.
For my part, I memorized π out to 3.1415926535787932, so that’s sixteen digits past the decimal. Always felt I could do more and I don’t know why I didn’t. The next couple digits are 8462, which has a nice descending-fifths cadence to it. The 626 following is a neat coda. My describing it this way may give you some idea to how I visualize the digits of π. They might help you, if you figure for some reason you need to do this. You do not, but if you enjoy it, enjoy it.
This week had a pretty good crop. I think Comic Strip Master Command is warming its people up for Pi Day. Better, there’s one that’s a good open-ended topic. We’ll get there.
Bill Amend’s FoxTrot for the 3rd (not a rerun) has Jason trying to teach his pet iguana algebra. Animals have some number sense, certainly. It depends on the animal. But we do see evidence of animals that can count, and that understand some geometrical truths. The level of abstraction needed for algebra — to discuss numbers when we don’t know, or don’t care, about their value — seems likely beyond what we could expect from animals. I say this aware that the last fifty years of animal cognition research have been, mostly, “yeah, so remember how we all agreed only humans could do this thing? Well, we looked at some nutrias here and … ”
Jason’s diagnosis that Quincy needs something more challenging is fair enough though. Teaching needs a couple of elements to succeed. The student’s confidence that this is worth the attention is one of them. A lot of teaching focuses on things that are, yes, beyond what the student now knows. But that the student can work out without feeling too lost. Feeling a bit lost helps. But there is great motivation in the moment when you feel less lost. Setting up such moments is among the things skilled teachers do.
(And I say “among”. There can be great joy in teaching a topic someone already knows, if what you’re really doing is showing some new perspective on it. And teaching things someone already knows is a good way to reassure that they have got it. Nothing is ever just the one thing.)
Mac King and Bill King’s Magic in a Minute for the 3rd is a variation of a trick from mid-January and mentioned here. It is, like many mathematics problems on a clock face, or a clock-like face, a modular numbers game in disguise. The trick is to give every starting, blue, bubble a path that ends at the same spot. There are tricks to get there, hidden in the network. For example, the first step is to start at any magician’s name in the outer ring, and move clockwise a number of steps equal to the number of letters in their name. All right: where would you start to finish on ‘Roy’ or ‘Thurston’? Given the levels of work needed for this I find it more impressive than I do January’s clock trick.
Frank Page’s Bob the Squirrel for the 4th sees Lauren working on a multiple-choice mathematics question. (It’s SAT prep work.) She’s startled that Bob can spot the answer right away. But there’s reasons it’s not so shocking Bob would be so fast.
The first thing I notice in this problem is f(x). For positive values of x this is an “increasing” function. That is, if you have two positive numbers x and y, and x is less than y, then f(x) is less than f(y). You can see that from how is an increasing function. Multiply an increasing function by a positive number and it stays increasing. Add a constant to an increasing function and it stays increasing. So this right away rules out f(4) as a possible answer. If Lauren guessed wildly at this point, she’d have a one-in-three chance of getting it right. If the SAT still scores by the rules in place when I took it, that’s a chance worth taking.
That is another tip. This value grows, and pretty fast. It grows even faster the bigger x gets. The difference between f(10) and f(11) is 42. The difference between f(11) and f(12) is 46. The difference between f(12) and f(13) is 50. So just from that alone it’s hard to imagine f(15) being the right answer. Easier to imagine f(10) being right. Less hard to imagine f(6) being right. If I had to guess, f(6) would be it. If I must know which is right? I’d start by calculating f(5) and f(6). Then check their difference. If that seems close to what f(3) must be, good, call it done. If that didn’t work I’d move reluctantly on to calculating f(10). But, bleah. Seems tedious. I’m glad to be past having to work that out.
S Camilleri Konar’s Six Chix for the 6th name-drops Fibonacci. This fellow is Leonardo of Pisa, who lived from around 1175 to around 1240 or so. He’s famous for — well, a bunch of things. One is his book explaining Arabic numerals to Western Europe and why they’re really better for so much calculation work. But another is what we now call the Fibonacci Sequence. We now call him Fibonacci, although that name’s a 19th century retronym. He belonged to the Bonacci family (‘Fibonacci’ would mean ‘child of Bonacci’) and, at least sometimes, called himself Leonardo Bigollo. Bigollo here meaning a traveller or a good-for-nothing.
His sequence is famous; it starts 1, 1, 2, 3, 5, 8, and so on, with each term in the sequence being the sum of the two terms before it. He was using this as a toy problem about breeding rabbits, meant to demonstrate ways to calculate better. This toy problem turns up in surprising contexts. Sometimes in algorithms. Sometimes in growth of natural objects; plant leaves and genes moving around on chromosomes and such. Sometimes in number theory. It’s even got links to the Golden Ratio, if we count that as interesting mathematics. And it inspires an activity problem. Per John Golden, a friend on Twitter:
The joke is all right as it is. The thing someone might associate with the name Fibonacci is the sequence, and it’s true that one never ends. But never ending isn’t a particularly distinctive feature of the Fibonacci sequence. Can the joke be rewritten so that the mathematics referenced is important?
There’s several properties of the sequence that might be useful. One is the thing that defined the sequence. Each term in it is the sum of the two preceding terms. The Golden Ratio offers another. Take any term in the sequence. The next term in the sequence is, approximately, the golden ratio of 1.618(etc) times the current term. The approximation gets better and better the more terms you go on.
That’s … really probably all you can expect to work with. There are fascinating other properties but you have to be really into number theory to know them. A positive number x is a Fibonacci number if and only if either or , or both, are perfect squares, for example. 1, 8, and 144 are the only Fibonacci numbers that are perfect powers of a whole number. Any Fibonacci number besides 1, 2, and 3 is the largest number of a Pythagorean triplet. Building a joke on any of these facts aims it at a particularly narrow audience.
If you feel the essential part of the joke is “this thing is never-ending” rather than “this involves Fibonacci” you have other options. How you might rewrite the joke depends on what you think the joke is.
And to speak of rewriting the joke is not to say Konar was wrong to make the joke she did, of course. We all understood what was being referenced and why it made for a punch line. Rewriting the joke to more tightly use its mathematical content does not necessarily make it funnier. This is especially so if a rewrite makes the joke too inaccessible. A comic strip is an optimization problem of how to compose a funny idea and to express it to a broad audience quickly. And then you have to solve it again.
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.
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.)
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 Vietnam 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.
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.
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.
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.
And it’s not always fair to say that the gods mock any plans made by humans, but Comic Strip Master Command has been doing its best to break me of reading and commenting on any comic strip with a mathematical theme. I grant that I could make things a little easier if I demanded more from a comic strip before including it here. But even if I think a theme is slight that doesn’t mean the reader does, and it’s easy to let the eye drop to the next paragraph if the reader does think it’s too slight. The anthology nature of these posts is part of what works for them. And then sometimes Comic Strip Master Command sends me a day like last Sunday when everybody was putting in some bit of mathematics. There’ll be another essay on the past week’s strips, never fear. But today’s is just for the single day.
Susan Camilleri Konar’s Six Chix for the 11th illustrates the Lemniscate Family. The lemniscate is a shape well known as the curve made by a bit of water inside a narrow tube by people who’ve confused it with a meniscus. An actual lemniscate is, as the chain of pointing fingers suggests, a figure-eight shape. You get — well, I got — introduced to them in prealgebra. They’re shapes really easy to describe in polar coordinates but a pain to describe in Cartesian coordinates. There are several different kinds of lemniscates, each satisfying slightly different conditions while looking roughly like a figure eight. If you’re open to the two lobes of the shape not being the same size there’s even a kind of famous-ish lemniscate called the analemma. This is the figure traced out by the sun if you look at its position from a set point on the surface of the Earth at the same clock time each day over the course of the year. That the sun moves north and south from the horizon is easy to spot. That it is sometimes east or west of some reference spot is a surprise. It shows the difference between the movement of the mean sun, the sun as we’d see it if the Earth had a perfectly circular orbit, and the messy actual thing. Dr Helmer Aslasken has a fine piece about this, and how it affects when the sun rises earliest and latest in the year.
There’s also a thing called the “polynomial lemniscate”. This is a level curve of a polynomial. That is, what are all the possible values of the independent variable which cause the polynomial to evaluate to some particular number? This is going to be a polynomial in a complex-valued variable, in order to get one or more closed and (often) wriggly loops. A polynomial of a real-valued variable would typically give you a boring shape. There’s a bunch of these polynomial lemniscates that approximate the boundary of the Mandelbrot Set, that fractal that you know from your mathematics friend’s wall in 1992.
Mark Anderson’s Andertoons took care of being Mark Anderson’s Andertoons early in the week. It’s a bit of optimistic blackboard work.
Lincoln Pierce’s Big Nate features the formula for calculating the wind chill factor. Francis reads out what is legitimately the formula for estimating the wind chill temperature. I’m not going to get into whether the wind chill formula makes sense as a concept because I’m not crazy. The thinking behind it is that a windless temperature feels about the same as a different temperature with a particular wind. How one evaluates those equivalences offers a lot of room for debate. The formula as the National Weather Service, and Francis, offer looks frightening, but isn’t really hard. It’s not a polynomial, in terms of temperature and wind speed, but it’s close to that in form. The strip is rerun from the 15th of February, 2009, as Lincoln Pierce has had some not-publicly-revealed problem taking him away from the comic for about a month and a half now.
Jim Scancarelli’s Gasoline Alley included a couple of mathematics formulas, including the famous E = mc2 and the slightly less famous πr2, as part of Walt Wallet’s fantasy of advising scientists and inventors. (Scientists have already heard both.) There’s a curious stray bit in the corner, writing out 6.626 x 102 x 3 that I wonder about. 6.626 is the first couple digits of Planck’s Constant, as measured in Joule-seconds. (This is h, not h-bar, I say for the person about to complain.) It’d be reasonable for Scancarelli to have drawn that out of a physics book or reference page. But the exponent is all wrong, even if you suppose he mis-wrote 1023. It should be 6.626 x 10-34. So I don’t know whether Scancarelli got things very garbled, or if he just picked a nice sciencey-looking number and happened to hit on a significant one. (There’s enough significant science numbers that he’d have a fair chance of finding something.) The strip is a reprint from the 4th of February, 2007, as Jim Scancarelli has been absent for no publicly announced reason for four months now.
Greg Evans and Karen Evans’s Luann is not perfectly clear. But I think it’s presenting Gunther doing mathematics work to support his mother’s contention that he’s smart. There’s no working out what work he’s doing. But then we might ask how smart his mother is to have made that much food for just the two of them. Also that I think he’s eating a potato by hand? … Well, there are a lot of kinds of food that are hard to draw.
Greg Evans’s Luann Againn reprints the strip from the 11th of February (again), 1990. It mentions as one of those fascinating things of arithmetic an easy test to see if a number’s a multiple of nine. There are several tricks like this, although the only ones anybody can remember are finding multiples of 3 and finding multiples of 9. Well, they know the rules for something being a multiple of 2, 5, or 10, but those hardly look like rules, and there’s no addition needed. Similarly with multiples of 4.
Modular arithmetic underlies all these rules. Once you know the trick you can use it to work out your own add-up-the-numbers rules to find what numbers are multiples of small numbers. Here’s an example. Think of a three-digit number. Suppose its first digit is ‘a’, its second digit ‘b’, and its third digit ‘c’. So we’d write the number as ‘abc’, or, 100a + 10b + 1c. What’s this number equal to, modulo 9? Well, 100a modulo 9 has to be equal to whatever a modulo 9 is: (100 a) modulo 9 is (100) modulo 9 — that is, 1 — times (a) modulo 9. 10b modulo 9 is (10) modulo 9 — again, 1 — times (b) modulo 9. 1c modulo 9 is … well, (c) modulo 9. Add that all together and you have a + b + c modulo 9. If a + b + c is some multiple of 9, so must be 100a + 10b + 1c.
The rules about whether something’s divisible by 2 or 5 or 10 are easy to work with since 10 is a multiple of 2, and of 5, and for that matter of 10, so that 100a + 10b + 1c modulo 10 is just c modulo 10. You might want to let this settle. Then, if you like, practice by working out what an add-the-digits rule for multiples of 11 would be. (This is made a lot easier if you remember that 10 is equal to 11 – 1.) And if you want to show off some serious arithmetic skills, try working out an add-the-digits rule for finding whether something’s a multiple of 7. Then you’ll know why nobody has ever used that for any real work.
J C Duffy’s Lug Nuts plays on the equivalence people draw between intelligence and arithmetic ability. Also on the idea that brain size should have something particularly strong link to intelligence. Really anyone having trouble figuring out 15% of $10 is psyching themselves out. They’re too much overwhelmed by the idea of percents being complicated to realize that it’s, well, ten times 15 cents.
It was an ordinary enough week when I realized I wasn’t sure about the name of the schoolmarm in Barney Google and Snuffy Smith. So I looked it up on Comics Kingdom’s official cast page for John Rose’s comic strip. And then I realized something about the Smiths’ next-door neighbor Elviney and Jughaid’s teacher Miss Prunelly:
Are … are they the same character, just wearing different glasses? I’ve been reading this comic strip for like forty years and I’ve never noticed this before. I’ve also never heard any of you all joking about this, by the way, so I stand by my argument that if they’re prominent enough then, yes, glasses could be an adequate disguise for Superman. Anyway, I’m startled. (Are they sisters? Cousins? But wouldn’t that make mention on the cast page? There are missing pieces here.)
Mac King and Bill King’s Magic In A Minute feature for the 10th sneaks in here yet again with a magic trick based in arithmetic. Here, they use what’s got to be some Magic Square-based technology for a card trick. This probably could be put to use with other arrangements of numbers, but cards have the advantage of being stuff a magician is likely to have around and that are expected to do something weird.
Thom Bluemel’s Birdbrains for the 13th is an Albert Einstein Needing Help panel. It’s got your blackboard full of symbols, not one of which is the famous E = mc2 equation. But given the setup it couldn’t feature that equation, not and be a correct joke.
John Rose’s Barney Google for the 14th does a little more work than necessary for its subtraction-explained-with-candy joke. I non-sarcastically appreciate Rose’s dodging the obvious joke in favor of a guy-is-stupid joke.
Niklas Eriksson’s Carpe Diem for the 14th is a kind of lying-with-statistics joke. That’s as much as it needs to be. Still, thought always should go into exactly how one presents data, especially visually. There are connotations to things. Just inverting an axis is dangerous stuff, though. The convention of matching an increase in number to moving up on the graph is so ingrained that it should be avoided only for enormous cause.
This joke also seems conceptually close, to me, to the jokes about the strangeness of how a “negative” medical test is so often the good news.
Olivia Walch’s Imogen Quest for the 15th is not about solitaire. But “solving” a game by simulating many gameplays and drawing strategic advice from that is a classic numerical mathematics trick. Whether a game is fun once it’s been solved so is up to you. And often in actual play, for a game with many options at each step, it’s impossible without a computer to know the best possible move. You could use simulations like this to develop general guidelines, and a couple rules that often pan out.
I’ve gotten enough comics, I think, to justify a fresh roundup of mathematics appearances in the comic strips. Unfortunately the first mathematics-linked appearance since my most recent entry is also the most badly dated. Pab Sugenis’s The New Adventures of Queen Victoria took (the appropriate) day to celebrate the birthday of Tom Lehrer, but fails to mention his actual greatest contribution to American culture, the “Silent E” song for The Electric Company. He’s also author of the humorous song “Lobachevsky”, which is pretty much the only place to go if you need a mathematics-based song and can’t use They Might Be Giants for some reason. (I regard Lehrer’s “New Math” song as not having a strong enough melody to count.)