Last week Comic Strip Master Command sent out just enough on-theme comics for two essays, the way I do them these days. The first half has some multiplication in two of the strips. So that’s enough to count as a theme for me.
Aaron Neathery’s Endtown for the 26th depicts a dreary, boring school day by using arithmetic. A lot of times tables. There is some credible in-universe reason to be drilling on multiplication like this. The setting is one where the characters can’t expect to have computers available. That granted, I’m not sure there’s a point to going up to memorizing four times 27. Going up to twelve-times seems like enough for common uses. For multiplying two- and longer-digit numbers together we usually break the problem up into a string of single-digit multiplications.
There are a handful of bigger multiplications that can make your life easier to know, like how four times 25 is 100. Or three times 33 is pretty near 100. But otherwise? … Of course, the story needs the class to do something dull and seemingly pointless. Going deep into multiplication tables communicates that to the reader quickly.
Thaves’s Frank and Ernest for the 26th is a spot of wordplay. Also a shout-out to my friends who record mathematics videos for YouTube. It is built on the conflation between the ideas of something multiplying and the amount of something growing. It’s easy to see where the idea comes from; just keep hitting ‘x 2’ on a calculator and the numbers grow excitingly fast. You get even more exciting results with ‘x 3’ or ‘x π’. But multiplying by 1 is still multiplication. As is multiplying by a number smaller than 1. Including negative numbers. That doesn’t hurt the joke any. That multiplying two things together doesn’t necessarily give you something larger is a consideration when you’re thinking rigorously about what multiplication can do. It doesn’t have to be part of normal speech.
Nate Frakes’s Break of Day for the 27th is the anthropomorphic numerals joke for the week. I don’t know that there’s anything in the other numerals being odds rather than evens, or a mixture of odds and evens. It might just be that they needed to be anything but 1.
And now I’ve got caught up with last week’s comics. I can get to readying for this coming Sunday looking at … so far … nine comic strips that made the preliminary cut. Whimper.
This time the name does mean something.
Thaves’s Frank and Ernest for the 31st complains about not being treated as a “prime number”. There’s a lot of linguistic connotation gone into this strip. The first is the sense that to be a number is to be stripped of one’s humanity, to become one of a featureless horde. Each number is unique, of course; Iva Sallay’s Find the Factors page each day starts with some of the features of each whole number in turn. But one might look at, oh, 84,644 and not something very different from 84,464.
And yet there’s the idea that there are prime numbers, celebrities within the anonymous counting numbers. The name even says it; a prime something is especially choice. And we speak of prime numbers as somehow being the backbone of numbers. This reflects that we find unique factorizations to be a useful thing to do. But being a prime number doesn’t make a number necessarily better. There are reasons most (European) currencies, before decimalization, divided their currency unit into 20 parts of 12 parts each. And nobody divided them into 19 parts of 13 parts each. As often happens, whether something is good depends on what you’re hoping it’s good for.
Nate Fakes’s Break of Day for the 1st of June is more or less the anthropomorphized numerals installment for the week. It’s also a bit of wordplay, so, good on them. There’s not so many movies about mathematics. Darren Aronofsky’s Pi, Ron Howard’s A Beautiful Mind, and Theodore Melfi’s Hidden Figures are the ones that come to mind, at least in American cinema. And there was the TV detective series Numbers. It seems odd that there wasn’t, like, some little studio prestige thing where Paul Muni played Évariste Galois back in the day. But a lot of the mathematical process isn’t cinematic. People scribbling notes, typing on a computer, or arguing about something you don’t understand are all hard to make worth watching. And the parts that anyone could understand — obsession, self-doubt, arguments over priority, debates about implications — are universal to any discovery or invention. Note that the movies listed are mostly about people who happen to be doing mathematics. You could change the specialties to, say, chemical engineering without altering the major plot beats. Well, Pi would need more alteration. But you could make it about any process that seems to offer reliable forecasting in a new field.
Greg Evans’s Luann Againn for the 1st takes place in mathematics class. The subject doesn’t matter for the joke. It could be anything that doesn’t take much word-balloon space but that someone couldn’t bluff their way through.
Ted Shearer’s Quincy for the 7th of April, 1979 has Quincy thinking what he’ll do with his head for figures. He sees accounting as plausible. Good for him. Society always needs accountants. And they probably do more of society’s mathematics than the mathematicians do.
Bill Abbott’s Spectickles for the 1st features the blackboard-full-of-mathematics to represent the complicated. It shows off the motif that an advanced mathematical formula will be a long and complicated one. This has good grounds behind it. If you want to model something interesting that hasn’t been done before, chances are it’s because you need to consider many factors. And trying to represent them will be clumsily done. It takes reflection and consideration and, often, new mathematical tools to make a formula pithy. Famously, James Clerk Maxwell introduced his equations about electricity and magnetism as a set of twenty equations. By 1873 Maxwell, making some use of quaternions, was able to reduce this to eight equations. Oliver Heaviside, in the late 19th century, used the still-new symbols of vector mechanics. This let him make an attractive quartet. We still see that as the best way to describe electromagnetic fields. As with writing, much of mathematics is rewriting.
And this should clear out last week’s mathematically-themed comic strips. I didn’t realize just how busy last week had been until I looked at what I thought was a backlog of just two days’ worth of strips and it turned out to be about two thousand comics. I exaggerate, but as ever, not by much. This current week seems to be a more relaxed pace. So I’ll have to think of something to write for the Tuesday and Thursday slots. Hm. (I’ll be all right. I’ve got one thing I need to stop bluffing about and write, and there’s usually a fair roundup of interesting tweets or articles I’ve seen that I can write. Those are often the most popular articles around here.)
Mark Tatulli’s Heart of the City rerun for the 1st finally has some specific mathematics mentioned in Heart’s efforts to avoid a mathematics tutor. The bit about the sum of adjacent angles forming a right line being 180 degrees is an important one. A great number of proofs rely on it. I can’t deny the bare fact seems dull, though. I know offhand, for example, that this bit about adjacent angles comes in handy in proving that the interior angles of a triangle add up to 180 degrees. At least for Euclidean geometry. And there are non-Euclidean geometries that are interesting and important and for which that’s not true. Which inspires the question: on a non-Euclidean surface, like say the surface of the Earth, is it that adjacent angles don’t add up to 180 degrees? Or does something else in the proof of a triangle’s interior angles adding up to 180 degrees go wrong?
Bill Whitehead’s Free Range for the 2nd features the classic page full of equations to demonstrate some hard mathematical work. And it is the sort of subject that is done mathematically. The equations don’t look to me anything like what you’d use for asteroid orbit projections. I’d expect forecasting just where an asteroid might hit the Earth to be done partly by analytic formulas that could be done on a blackboard. And then made precise by a numerical estimate. The advantage of the numerical estimate is that stuff like how air resistance affects the path of something in flight is hard to deal with analytically. Numerically, it’s tedious, but we can let the computer deal with the tedium. So there’d be just a boring old computer screen to show on-panel.
Bud Fisher’s Mutt and Jeff reprint for the 2nd is a little baffling. And not really mathematical. It’s just got a bizarre arithmetic error in it. Mutt’s fiancee Encee wants earrings that cost ten dollars (each?) and Mutt takes this to be fifty dollars in earring costs and I have no idea what happened there. Thomas K Dye, the web cartoonist who’s done artwork for various article series, has pointed out that the lettering on these strips have been redone with a computer font. (Look at the letters ‘S’; once you see it, you’ll also notice it in the slightly lumpy ‘O’ and the curly-arrow ‘G’ shapes.) So maybe in the transcription the earring cost got garbled? And then not a single person reading the finished product read it over and thought about what they were doing? I don’t know.
Zach Weinersmith’s Saturday Morning Breakfast Cereal reprint for the 2nd is based, as his efforts to get my attention often are, on a real mathematical physics postulate. As the woman postulates: given a deterministic universe, with known positions and momentums of every particle, and known forces for how all these interact, it seems like it should be possible to predict the future perfectly. It would also be possible to “retrodict” the past. All the laws of physics that we know are symmetric in time; there’s no reason you can’t predict the motion of something one second into the past just as well as you an one second into the future. This fascinating observation took a lot of battery in the 19th century. Many physical phenomena are better described by statistical laws, particularly in thermodynamics, the flow of heat. In these it’s often possible to predict the future well but retrodict the past not at all.
But that looks as though it’s a matter of computing power. We resort to a statistical understanding of, say, the rings of Saturn because it’s too hard to track the billions of positions and momentums we’d need to otherwise. A sufficiently powerful mathematician, for example God, would be able to do that. Fair enough. Then came the 1890s. Henri Poincaré discovered something terrifying about deterministic systems. It’s possible to have chaos. A mathematical representation of a system is a bit different from the original system. There’s some unavoidable error. That’s bound to make some, larger, error in any prediction of its future. For simple enough systems, this is okay. We can make a projection with an error as small as we need, at the cost of knowing the current state of affairs with enough detail. Poincaré found that some systems can be chaotic, though, ones in which any error between the current system and its representation will grow to make the projection useless. (At least for some starting conditions.) And so many interesting systems are chaotic. Incredibly simplified models of the weather are chaotic; surely the actual thing is. This implies that God’s projection of the universe would be an amusing but almost instantly meaningless toy. At least unless it were a duplicate of the universe. In which case we have to start asking our philosopher friends about the nature of identity and what a universe is, exactly.
Ruben Bolling’s Super-Fun-Pak Comix for the 2nd is an installment of Guy Walks Into A Bar featuring what looks like an arithmetic problem to start. It takes a turn into base-ten jokes. There are times I suspect Ruben Bolling to be a bit of a nerd.
Percy Crosby’s Skippy for the 3rd originally ran the 8th of December, 1930. It alludes to one of those classic probability questions: what’s the chance that in your lungs is one of the molecules exhaled by Julius Caesar in his dying gasp? Or whatever other event you want: the first breath you ever took, or something exhaled by Jesus during the Sermon on the Mount, or exhaled by Sue the T-Rex as she died. Whatever. The chance is always surprisingly high, which reflects the fact there’s a lot of molecules out there. This also reflects a confidence that we can say one molecule of air is “the same” as some molecule if air in a much earlier time. We have to make that supposition to have a problem we can treat mathematically. My understanding is chemists laugh at us if we try to suggest this seriously. Fair enough. But whether the air pumped out of a bicycle tire is ever the same as what’s pumped back in? That’s the same kind of problem. At least some of the molecules of air will be the same ones. Pretend “the same ones” makes sense. Please.
And today I bring the last couple mathematically-themed comic strips sent my way last week. GoComics has had my comics page working intermittently this week. And I was able to get a response from them, by e-mailing their international sales office, the only non-form contact I could find. Anyway, this flood of comics does take up the publishing spot I’d figured for figuring how I messed up Wronski’s formula. But that’s all right, as I wanted to spend more time thinking about that. Here’s hoping spending more time thinking works out for me.
Mark Tatulli’s Heart of the City from the 24th got into a storyline about Heart needing a mathematics tutor. It’s a rerun sequence, although if you remember a particular comic storyline from 2009 you’re doing pretty well. Nothing significantly mathematical has turned up in the story so far, past the mention of fractions as things that exist and torment students. But the stories are usually pretty good for this sort of strip.
Mikael Wulff and Anders Morganthaler’s WuMo for the 24th includes a story problems freak out. I’m not sure what’s particularly implausible about buying nine apples. I’d agree a person is probably more likely to buy an even number of things, since we seem to like numbers like “ten” and “eight” so well, but it’s hardly ridiculous.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a reminder that most of my days are spent seeing how Zach Weinersmith wants my attention. It also includes what I suppose is a legitimate attempt to offer a definition for what all mathematics is. It’s hard to come up with something that does cover all the stuff mathematicians do. Bear in mind, this includes counting, calculating how far the Sun is based on the appearance of a lunar eclipse, removing static from a recording, and telling how many queens it’s possible to place eight queens on a chess board that’s wrapped around a torus without any being able to capture another, among other problems. My instinct is to dismiss the proposed “anything you can think deeply about that has no reference to the real world”. That seems over-broad, and to cover a lot of areas that are really philosophy’s beat. And I think there’s something unseemly in mathematicians gloating about their work having no “practical” use. I grant I come from an applied school, and I came to there through an interest in physics. But to build up “inapplicability to the real word” as if it were some ideal, as opposed to just how something has turned out to be right now, strikes me as silly. Applicability is so dependent on context, on culture, and accidents of fate that there’s no way it can be important to characterizing mathematics. And it would imply that once we found a use for something it would stop being mathematically interesting. I don’t see evidence of that in mathematical history.
The rest of last week had more mathematically-themed comic strips than Sunday alone did. As sometimes happens, I noticed an objectively unimportant detail in one of the comics and got to thinking about it. Whether I could solve the equation as posted, or whether at least part of it made sense as a mathematics problem. Well, you’ll see.
Patrick McDonnell’s Mutts for the 25th of September I include because it’s cute and I like when I can feature some comic in these roundups. Maybe there’s some discussion that could be had about what “equals” means in ordinary English versus what it means in mathematics. But I admit that’s a stretch.
Olivia Walch’s Imogen Quest for the 25th uses, and describes, the mathematics of a famous probability problem. This is the surprising result of how few people you need to have a 50 percent chance that some pair of people have a birthday in common. It then goes over to some other probability problems. The examples are silly. But the reasoning is sound. And the approach is useful. To find the chance of something happens it’s often easiest to work out the chance it doesn’t. Which is as good as knowing the chance it does, since a thing can either happen or not happen. At least in probability problems, which define “thing” and “happen” so there’s not ambiguity about whether it happened or not.
Piers Baker’s Ollie and Quentin rerun for the 26th I’m pretty sure I’ve written about before, although back before I included pictures of the Comics Kingdom strips. (The strip moved from Comics Kingdom over to GoComics, which I haven’t caught removing old comics from their pages.) Anyway, it plays on a core piece of probability. It sets out the world as things, “events”, that can have one of multiple outcomes, and which must have one of those outcomes. Coin tossing is taken to mean, by default, an event that has exactly two possible outcomes, each equally likely. And that is near enough true for real-world coin tossing. But there is a little gap between “near enough” and “true”.
Rick Stromoski’s Soup To Nutz for the 27th is your standard sort of Dumb Royboy joke, in this case about him not knowing what percentages are. You could do the same joke about fractions, including with the same breakdown of what part of the mathematics geek population ruins it for the remainder.
Nate Fakes’s Break of Day for the 28th is not quite the anthropomorphic-numerals joke for the week. Anthropomorphic mathematics problems, anyway. The intriguing thing to me is that the difficult, calculus, problem looks almost legitimate to me. On the right-hand-side of the first two lines, for example, the calculation goes from
This is a little sloppy. The first line ought to end in a ‘dt’, and the second ought to have a constant of integration. If you don’t know what these calculus things are let me explain: they’re calculus things. You need to include them to express the work correctly. But if you’re just doing a quick check of something, the mathematical equivalent of a very rough preliminary sketch, it’s common enough to leave that out.
It doesn’t quite parse or mean anything precisely as it is. But it looks like the sort of thing that some context would make meaningful. That there’s repeated appearances of , or , particularly makes me wonder if Frakes used a problem he (or a friend) was doing for some reason.
Can’t say this was too fast or too slow a week for mathematically-themed comic strips. A bunch of the strips were panel comics, so that’ll do for my theme.
Norm Feuti’s Retail for the 21st mentions every (not that) algebra teacher’s favorite vague introduction to group theory, the Rubik’s Cube. Well, the ways you can rotate the various sides of the cube do form a group, which is something that acts like arithmetic without necessarily being numbers. And it gets into value judgements. There exist algorithms to solve Rubik’s cubes. Is it a show of intelligence that someone can learn an algorithm and solve any cube? — But then, how is solving a Rubik’s cube, with or without the help of an algorithm, a show of intelligence? At least of any intelligence more than the bit of spatial recognition that’s good for rotating cubes around?
I don’t see that learning an algorithm for a problem is a lack of intelligence. No more than using a photo reference shows a lack of drawing skill. It’s still something you need to learn, and to apply, and to adapt to the cube as you have it to deal with. Anyway, I never learned any techniques for solving it either. Would just play for the joy of it. Here’s a page with one approach to solving the cube, if you’d like to give it a try yourself. Good luck.
Nate Fakes’s Break of Day for the 24th features the traditional whiteboard full of mathematics scrawls as a sign of intelligence. The scrawl on the whiteboard looks almost meaningful. The integral, particularly, looks like it might have been copied from a legitimate problem in polar or cylindrical coordinates. I say “almost” because while I think that some of the r symbols there are r’ I’m not positive those aren’t just stray marks. If they are r’ symbols, it’s the sort of integral that comes up when you look at surfaces of spheres. It would be the electric field of a conductive metal ball given some charge, or the gravitational field of a shell. These are tedious integrals to solve, but fortunately after you do them in a couple of introductory physics-for-majors classes you can just look up the answers instead.