I have been reading Pierre-Simon LaPlace, 1749 – 1827, A Life In Exact Science, by Charles Coulson Gillispie with Robert Fox and Ivor Grattan-Guinness. It’s less of a biography than I expected and more a discussion of LaPlace’s considerable body of work. Part of LaPlace’s work was in giving probability a logically coherent, rigorous meaning. Laplace discusses the gambler’s fallacy and the tendency to assign causes to random events. That, for example, if we came across letters from a printer’s font reading out ‘INFINITESIMAL’ we would think that deliberate. We wouldn’t think that for a string of letters in no recognized language. And that brings up this neat quote from Gillispie:
The example may in all probability be adapted from the chapter in the Port-Royal La Logique (1662) on judgement of future events, where Arnauld points out that it would be stupid to bet twenty sous against ten thousand livres that a child playing with printer’s type would arrange the letters to compose the first twenty lines of Virgil’s Aenid.
I was embarrassed, on looking at old Pi Day Reading the Comics posts, to see how often I observed there were fewer Pi Day comics than I expected. There was not a shortage this year. This even though if Pi Day has any value it’s as an educational event, and there should be no in-person educational events while the pandemic is still on. Of course one can still do educational stuff remotely, mathematics especially. But after a year of watching teaching on screens and sometimes doing projects at home, it’s hard for me to imagine a bit more of that being all that fun.
But Pi Day being a Sunday did give cartoonists more space to explain what they’re talking about. This is valuable. It’s easy for the dreadfully online, like me, to forget that most people haven’t heard of Pi Day. Most people don’t have any idea why that should be a thing or what it should be about. This seems to have freed up many people to write about it. But — to write what? Let’s take a quick tour of my daily comics reading.
Tony Cochran’s Agnes starts with some talk about Daylight Saving Time. Agnes and Trout don’t quite understand how it works, and get from there to Pi Day. Or as Agnes says, Pie Day, missing the mathematics altogether in favor of the food.
Scott Hilburn’s The Argyle Sweater is an anthropomorphic-numerals joke. It’s a bit risqué compared to the sort of thing you expect to see around here. The reflection of the numerals is correct, but it bothered me too.
Georgia Dunn’s Breaking Cat News is a delightful cute comic strip. It doesn’t mention mathematics much. Here the cat reporters do a fine job explaining what Pi Day is and why everybody spent Sunday showing pictures of pies. This could almost be the standard reference for all the Pi Day strips.
Bill Amend’s FoxTrot is one of the handful that don’t mention pie at all. It focuses on representing the decimal digits of π. At least within the confines of something someone might write in the American dating system. The logic of it is a bit rough but if we’ve accepted 3-14 to represent 3.14, we can accept 1:59 as standing in for the 0.00159 of the original number. But represent 0.0015926 (etc) of a day however you like. If we accept that time is continuous, then there’s some moment on the 14th of March which matches that perfectly.
Jef Mallett’s Frazz talks about the eliding between π and pie for the 14th of March. The strip wonders a bit what kind of joke it is exactly. It’s a nerd pun, or at least nerd wordplay. If I had to cast a vote I’d call it a language gag. If they celebrated Pi Day in Germany, there would not be any comic strips calling it Tortentag.
Steenz’s Heart of the City is another of the pi-pie comics. I do feel for Heart’s bewilderment at hearing π explained at length. Also Kat’s desire to explain mathematics overwhelming her audience. It’s a feeling I struggle with too. The thing is it’s a lot of fun to explain things. It’s so much fun you can lose track whether you’re still communicating. If you set off one of these knowledge-floods from a friend? Try to hold on and look interested and remember any single piece anywhere of it. You are doing so much good for your friend. And if you realize you’re knowledge-flooding someone? Yeah, try not to overload them, but think about the things that are exciting about this. Your enthusiasm will communicate when your words do not.
Michael Jantze’s Studio Jantze ran on Monday instead, although the caption suggests it was intended for Pi Day. So I’m including it here. And it’s the last of the strips sliding the day over to pie.
But there were a couple of comic strips with some mathematics mention that were not about Pi Day. It may have been coincidence.
Sandra Bell-Lundy’s Between Friends is of the “word problem in real life” kind. It’s a fair enough word problem, though, asking about how long something would take. From the premises, it takes a hair seven weeks to grow one-quarter inch, and it gets trimmed one quarter-inch every six weeks. It’s making progress, but it might be easier to pull out the entire grey hair. This won’t help things though.
Darby Conley’s Get Fuzzy is a rerun, as all Get Fuzzy strips are. It first (I think) ran the 13th of September, 2009. And it’s another Infinite Monkeys comic strip, built on how a random process should be able to create specific outcomes. As often happens when joking about monkeys writing Shakespeare, some piece of pop culture is treated as being easier. But for these problems the meaning of the content doesn’t count. Only the length counts. A monkey typing “let it be written in eight and eight” is as improbable as a monkey typing “yrg vg or jevggra va rvtug naq rvtug”. It’s on us that we find one of those more impressive than the other.
And this wraps up my Pi Day comic strips. I don’t promise that I’m back to reading the comics for their mathematics content regularly. But I have done a lot of it, and figure to do it again. All my Reading the Comics posts appear at this link. Thank you for reading and I hope you had some good pie.
I don’t know how Andertoons didn’t get an appearance here.
Dana Simpson’s Ozy and Millie rerun for the 9th is part of a sequence of Ozy being home-schooled. The joke puts the transient nature of knowledge up against the apparent permanent of arithmetic. The joke does get at one of those fundamental questions in the philosophy of mathematics: is mathematics created or discovered? The expression of mathematics is unmistakably created. There is nothing universal in declaring “six times eight is forty-eight” and if you wish to say there is, then ask someone who speaks only Tamil and not a word of English whether they agree with exactly that proposition.
But, grant that while we may have different representations of the concept, it is the case that “eight” exists, right? We get right back into trouble if we follow up by asking, all right, will “eight” fit in my hand? Is “eight” larger than the weather? Is “eight” more or less red than nominalism? I chose nouns that made those questions obviously ridiculous. But if we want to talk about a mathematical construct existing, someone’s going to ask what traits that existence implies. It’s convenient for mathematicians, and good publicity, for us to think that we work on things that exist independently of the accidental facts of the universe. But then we’re stuck when we’re asked how we, stuck in the universe, can have anything to do with a thing that’s not part of it.
Not mentioned in this particular Ozy and Millie strip is that the characters are Buddhist. The (American) pop culture interpretation of Buddhism includes an emphasis on understanding the transient nature of … everything … which would seem to include mathematical knowledge. Still, there is a long history of great mathematical work done by Buddhist scholars; the oldest known manuscript of Indian mathematics is written in a Buddhist Hybrid Sanskrit. The author of that manuscript is unknown, but it’s not as if that were the lone piece of mathematical writing.
My limited understanding is that Indian mathematics used an interesting twist on the problem of the excluded middle. This is a question important to proofs. Can we take every logical proposition as being either true or false? If we can, then we are able to prove statements by contradiction: suppose the reverse of what we want to prove and show that implies nonsense. This is common in western mathematics. But there is a school of thought that we should not do this, and only allow as true statements we have directly proven to be true. My understanding is that at least one school of Indian mathematics allowed proof by contradiction if it proved that a thing did not exist. It would not be used to show that a thing existed. So, for example, it would allow the ordinary proof that the square root of two can’t be a rational number; it would not allow an indirect proof that, say, a kind of mapping must have a fixed point. (It would allow a proof that showed you how to find that point, though.) It’s an interesting division, and a reminder that even what counts as a logical derivation is a matter of custom.
I concede I am late in wrapping up last week’s mathematically-themed comics. But please understand there were important reasons for my not having posted this earlier, like, I didn’t get it written in time. I hope you understand and agree with me about this.
Bill Griffith’s Zippy the Pinhead for the 9th brings up mathematics in a discussion about perfection. The debate of perfection versus “messiness” begs some important questions. What I’m marginally competent to discuss is the idea of mathematics as this perfect thing. Mathematics seems to have many traits that are easy to think of as perfect. That everything in it should follow from clearly stated axioms, precise definitions, and deductive logic, for example. This makes mathematics seem orderly and universal and fair in a way that the real world never is. If we allow that this is a kind of perfection then … does mathematics reach it?
Even the idea of a “precise definition” is perilous. If it weren’t there wouldn’t be so many pop mathematics articles about why 1 isn’t a prime number. It’s difficult to prove that any particular set of axioms that give us interesting results are also logically consistent. If they’re not consistent, then we can prove absolutely anything, including that the axioms are false. That seems imperfect. And few mathematicians even prepare fully complete, step-by-step proofs of anything. It takes ridiculously long to get anything done if you try. The proofs we present tend to show, instead, the reasoning in enough detail that we’re confident we could fill in the omitted parts if we really needed them for some reason. And that’s fine, nearly all the time, but it does leave the potential for mistakes present.
Zippy offers up a perfect parallelogram. Making it geometry is of good symbolic importance. Everyone knows geometric figures, and definitions of some basic ideas like a line or a circle or, maybe, a parallelogram. Nobody’s ever seen one, though. There’s never been a straight line, much less two parallel lines, and even less the pair of parallel lines we’d need for a parallellogram. There can be renderings good enough to fool the eye. But none of the lines are completely straight, not if we examine closely enough. None of the pairs of lines are truly parallel, not if we extend them far enough. The figure isn’t even two-dimensional, not if it’s rendered in three-dimensional things like atoms or waves of light or such. We know things about parallelograms, which don’t exist. They tell us some things about their shadows in the real world, at least.
Mark Litzler’s Joe Vanilla for the 9th is a play on the old joke about “a billion dollars here, a billion dollars there, soon you’re talking about real money”. As we hear more about larger numbers they seem familiar and accessible to us, to the point that they stop seeming so big. A trillion is still a massive number, at least for most purposes. If you aren’t doing combinatorics, anyway; just yesterday I was doing a little toy problem and realized it implied 470,184,984,576 configurations. Which still falls short of a trillion, but had I made one arbitrary choice differently I could’ve blasted well past a trillion.
Ruben Bolling’s Super-Fun-Pak Comix for the 9th is another monkeys-at-typewriters joke, that great thought experiment about probability and infinity. I should add it to my essay about the Infinite Monkey Theorem. Part of the joke is that the monkey is thinking about the content of the writing. This doesn’t destroy the prospect that a monkey given enough time would write any of the works of William Shakespeare. It makes the simple estimates of how unlikely that is, and how long it would take to do, invalid. But the event might yet happen. Suppose this monkey decided there was no credible way to delay Hamlet’s revenge to Act V, and tried to write accordingly. Mightn’t the monkey make a mistake? It’s easy to type a letter you don’t mean to. Or a word you don’t mean to. Why not a sentence you don’t mean to? Why not a whole act you don’t mean to? Impossible? No, just improbable. And the monkeys have enough time to let the improbable happen.
Eric the Circle for the 10th, this one by Kingsnake, declares itself set in “the 20th dimension, where shape has no meaning”. This plays on a pop-cultural idea of dimensions as a kind of fairyland, subject to strange and alternate rules. A mathematician wouldn’t think of dimensions that way. 20-dimensional spaces — and even higher-dimensional spaces — follow rules just as two- and three-dimensional spaces do. They’re harder to draw, certainly, and mathematicians are not selected for — or trained in — drawing, at least not in United States schools. So attempts at rendering a high-dimensional space tend to be sort of weird blobby lumps, maybe with a label “N-dimensional”.
And a projection of a high-dimensional shape into lower dimensions will be weird. I used to have around here a web site with a rotatable tesseract, which would draw a flat-screen rendition of what its projection in three-dimensional space would be. But I can’t find it now and probably it ran as a Java applet that you just can’t get to work anymore. Anyway, non-interactive videos of this sort of thing are common enough; here’s one that goes through some of the dimensions of a tesseract, one at a time. It’ll give some idea how something that “should” just be a set of cubes will not look so much like that.
Steve Kelly and Jeff Parker’s Dustin for the 11th is a variation on the “why do I have to learn this” protest. This one is about long division and the question of why one needs to know it when there’s cheap, easily-available tools that do the job better. It’s a fair question and Hayden’s answer is a hard one to refute. I think arithmetic’s worth knowing how to do, but I’ll also admit, if I need to divide something by 23 I’m probably letting the computer do it.
I went a little wild explaining the first of last week’s mathematically-themed comic strips. So let me split the week between the strips that I know to have been reruns and the ones I’m not so sure were.
Bill Amend’s FoxTrot for the 23rd — not a rerun; the strip is still new on Sundays — is a probability question. And a joke about story problems with relevance. Anyway, the question uses the binomial distribution. I know that because the question is about doing a bunch of things, homework questions, each of which can turn out one of two ways, right or wrong. It’s supposed to be equally likely to get the question right or wrong. It’s a little tedious but not hard to work out the chance of getting exactly six problems right, or exactly seven, or exactly eight, or so on. To work out the chance of getting six or more questions right — the problem given — there’s two ways to go about it.
One is the conceptually easy but tedious way. Work out the chance of getting exactly six questions right. Work out the chance of getting exactly seven questions right. Exactly eight questions. Exactly nine. All ten. Add these chances up. You’ll get to a number slightly below 0.377. That is, Mary Lou would have just under a 37.7 percent chance of passing. The answer’s right and it’s easy to understand how it’s right. The only drawback is it’s a lot of calculating to get there.
So here’s the conceptually harder but faster way. It works because the problem says Mary Lou is as likely to get a problem wrong as right. So she’s as likely to get exactly ten questions right as exactly ten wrong. And as likely to get at least nine questions right as at least nine wrong. To get at least eight questions right as at least eight wrong. You see where this is going: she’s as likely to get at least six right as to get at least six wrong.
There’s exactly three possibilities for a ten-question assignment like this. She can get four or fewer questions right (six or more wrong). She can get exactly five questions right. She can get six or more questions right. The chance of the first case and the chance of the last have to be the same.
So, take 1 — the chance that one of the three possibilities will happen — and subtract the chance she gets exactly five problems right, which is a touch over 24.6 percent. So there’s just under a 75.4 percent chance she does not get exactly five questions right. It’s equally likely to be four or fewer, or six or more. Just-under-75.4 divided by two is just under 37.7 percent, which is the chance she’ll pass as the problem’s given. It’s trickier to see why that’s right, but it’s a lot less calculating to do. That’s a common trade-off.
Ruben Bolling’s Super-Fun-Pax Comix rerun for the 23rd is an aptly titled installment of A Million Monkeys At A Million Typewriters. It reminds me that I don’t remember if I’d retired the monkeys-at-typewriters motif from Reading the Comics collections. If I haven’t I probably should, at least after making a proper essay explaining what the monkeys-at-typewriters thing is all about.
Ted Shearer’s Quincy from the 28th of February, 1978 reveals to me that pocket calculators were a thing much earlier than I realized. Well, I was too young to be allowed near stuff like that in 1978. I don’t think my parents got their first credit-card-sized, solar-powered calculator that kind of worked for another couple years after that. Kids, ask about them. They looked like good ideas, but you could use them for maybe five minutes before the things came apart. Your cell phone is so much better.
Bil Watterson’s Calvin and Hobbes rerun for the 24th can be classed as a resisting-the-word-problem joke. It’s so not about that, but who am I to slow you down from reading a Calvin and Hobbes story?
Garry Trudeau’s Doonesbury rerun for the 24th started a story about high school kids and their bad geography skills. I rate it as qualifying for inclusion here because it’s a mathematics teacher deciding to include more geography in his course. I was amused by the week’s jokes anyway. There’s no hint given what mathematics Gil teaches, but given the links between geometry, navigation, and geography there is surely something that could be relevant. It might not help with geographic points like which states are in New England and where they are, though.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is built on a plot point from Carl Sagan’s science fiction novel Contact. In it, a particular “message” is found in the digits of π. (By “message” I mean a string of digits that are interesting to us. I’m not sure that you can properly call something a message if it hasn’t got any sender and if there’s not obviously some intended receiver.) In the book this is an astounding thing because the message can’t be; any reasonable explanation for how it should be there is impossible. But short “messages” are going to turn up in π also, as per the comic strips.
I assume the peer review would correct the cartoon mathematicians’ unfortunate spelling of understanding.
Comic Strip Master Command sent out a big mass of comics this past week. Today’s installment will only cover about half of them. This half does feature a number of comics that show off jokes that’ve run here before. I’m sure it was coincidence. Comic Strip Master Command must have heard I was considering alerting cartoonists that I was talking about them. That’s fine for something like last week when I could talk about NP-complete problems or why we call something a “hypotenuse”. It can start a conversation. But “here’s a joke treating numerals as if they were beings”? All they can do is agree, that is what the joke is. If they disagree at that point they’re just trying to start a funny argument.
Scott Metzger’s The Bent Pinky for the 14th sees the return of anthropomorphic numerals humor. I’m a bit surprised Metzger goes so far as to make every numeral either a 3 or a 9. I’d have expected a couple of 2’s and 4’s. I understand not wanting to get into two-digit numbers. The premise of anthropomorphic numerals is troublesome if you need multiple-digit numbers.
Jon Rosenberg’s Goats for the 14th doesn’t directly mention a mathematical topic. But the story has the characters transported to a world with monkeys at typewriters. We know where that is. So we see that return after no time away, really.
Rick Detorie’s One Big Happy rerun for the 14th sees the return of “110 percent”. Happily the joke’s structured so that we can dodge arguing about whether it’s even possible to give 110 percent. I’m inclined to say of course it’s possible. “Giving 100 percent” in the context of playing a sport would mean giving the full reasonable effort. Or it does if we want to insist on idiomatic expressions making sense. It seems late to be insisting on that standard, but some people like it as an idea.
Terri Libenson’s Pajama Diaries for the 16th sees the return of Venn Diagram jokes. And it’s a properly-formed Venn Diagram, with the three circles coming together to indicate seven different conditions.
Gary Wise and Lance Aldrich’s Real Life Adventures for the 16th just name-drops rhomboids, using them as just a funny word. Geometry is filled with wonderful, funny-sounding words. I’m fond of “icosahedron” myself. But “rhomboid” and its related words are good ones. I think they hit that sweet spot between being uncommon in ordinary language without being so exotic that a reader’s eye trips over it. However funny a “triacontahedron” might be, no writer should expect the reader to forgive that pile of syllables. A rhomboid is a kind of parallelogram, so it’s got four sides. The sides come in two parallel pairs. Both members of a pair have the same length, but the different pairs don’t. They look like the kitchen tiles you’d get for a house you couldn’t really afford, not with tiling like that.
Two things made repeat appearances in the mathematically-themed comics this week. They’re the comic strip Frazz and the idea of having infinitely many monkeys typing. Well, silly answers to word problems also turned up, but that’s hard to say many different things about. Here’s what I make the week in comics out to be.
Sandra Bell-Lundy’s Between Friends for the 6th of November, 2016. I’m surprised Bell-Lundy used the broader space of a Sunday strip for a joke that doesn’t need that much illustration, but I understand sometimes you just have to go with the joke that you have. And it isn’t as though Sunday comics get that much space anymore either. Anyway, I suppose we have all been there, although for me that’s more often because I used to have a six-digit pin, and a six-digit library card pin, and those were just close enough to each other that I could never convince myself I was remembering the right one in context, so I would guess wrong.
Sandra Bell-Lundy’s Between Friends for the 6th introduces the infinite monkeys problem. I wonder sometimes why the monkeys-on-typewriters thing has so caught the public imagination. And then I remember it encourages us to stare directly into infinity and its intuition-destroying nature from the comfortable furniture of the mundane — typewriters, or keyboards, for goodness’ sake — with that childish comic dose of monkeys. Given that it’s a wonder we ever talk about anything else, really.
Monkeys writing Shakespeare has for over a century stood as a marker for what’s possible but incredibly improbable. I haven’t seen it compared to finding a four-digit PIN. It has got me wondering about the chance that four randomly picked letters will be a legitimate English word. I’m sure the chance is more than the one-in-a-thousand chance someone would guess a randomly drawn PIN correctly on one try. More than one in a hundred? I’m less sure. The easy-to-imagine thing to do is set a computer to try out all 456,976 possible sets of four letters and check them against a dictionary. The number of hits divided by the number of possibilities would be the chance of drawing a legitimate word. If I had a less capable computer, or were checking even longer words, I might instead draw some set number of words, never minding that I didn’t get every possibility. The fraction of successful words in my sample would be something close to the chance of drawing any legitimate word.
If I thought a little deeper about the problem, though, I’d just count how many four-letter words are already in my dictionary and divide that into 456,976. It’s always a mistake to start programming before you’ve thought the problem out. The trouble is not being able to tell when that thinking-out is done.
Richard Thompson’s Poor Richard’s Almanac for the 7th is the other comic strip to mention infinite monkeys. Well, chimpanzees in this case. But for the mathematical problem they’re not different. I’ve featured this particular strip before. But I’m a Thompson fan. And goodness but look at the face on the T S Eliot fan in the lower left corner there.
Jeff Mallet’s Frazz for the 6th gives Caulfield one of those flashes of insight that seems like it should be something but doesn’t mean much. He’s had several of these lately, as mentioned here last week. As before this is a fun discovery about Roman Numerals, but it doesn’t seem like it leads to much. Perhaps a discussion of how the subtractive principle — that you can write “four” as “IV” instead of “IIII” — evolved over time. But then there isn’t much point to learning Roman Numerals at all. It’s got some value in showing how much mathematics depends on culture. Not just that stuff can be expressed in different ways, but that those different expressions make different things easier or harder to do. But I suspect that isn’t the objective of lessons about Roman Numerals.
Frazz got my attention again the 12th. This time it just uses arithmetic, and a real bear of an arithmetic problem, as signifier for “a big pile of hard work”. This particular problem would be — well, I have to call it tedious, rather than hard. doing it is just a long string of adding together two numbers. But to do that over and over, by my count, at least 47 times for this one problem? Hardly any point to doing that much for one result.
Patrick Roberts’s Todd the Dinosaur for the 7th calls out fractions, and arithmetic generally, as the stuff that ruins a child’s dreams. (Well, a dinosaur child’s dreams.) Still, it’s nice to see someone reminding mathematicians that a lot of their field is mostly used by accountants. Actuaries we know about; mathematics departments like to point out that majors can get jobs as actuaries. I don’t know of anyone I went to school with who chose to become one or expressed a desire to be an actuary. But I admit not asking either.
Mike Thompson’s Grand Avenue started off a week of students-resisting-the-test-question jokes on the 7th. Most of them are hoary old word problem jokes. But, hey, I signed up to talk about it when a comic strip touches a mathematics topic and word problems do count.
Zach Weinersmith’s Saturday Morning Breakfast Cereal reprinted the 7th is a higher level of mathematical joke. It’s from the genre of nonsense calculation. This one starts off with what’s almost a cliche, at least for mathematics and physics majors. The equation it starts with, , is true. And famous. It should be. It links exponentiation, imaginary numbers, π, and negative numbers. Nobody would have seen it coming. And from there is the sort of typical gibberish reasoning, like writing “Pi” instead of π so that it can be thought of as “P times i”, to draw to the silly conclusion that P = 0. That much work is legitimate.
From there it sidelines into “P = NP”, which is another equation famous to mathematicians and computer scientists. It’s a shorthand expression of a problem about how long it takes to find solutions. That is, how many steps it takes. How much time it would take a computer to solve a problem. You can see why it’s important to have some study of how long it takes to do a problem. It would be poor form to tie up your computer on a problem that won’t be finished before the computer dies of old age. Or just take too long to be practical.
Most problems have some sense of size. You can look for a solution in a small problem or in a big one. You expect searching for the solution in a big problem to take longer. The question is how much longer? Some methods of solving problems take a length of time that grows only slowly as the size of the problem grows. Some take a length of time that grows crazy fast as the size of the problem grows. And there are different kinds of time growth. One kind is called Polynomial, because everything is polynomials. But there’s a polynomial in the problem’s size that describes how long it takes to solve. We call this kind of problem P. Another is called Non-Deterministic Polynomial, for problems that … can’t. We assume. We don’t know. But we know some problems that look like they should be NP (“NP Complete”, to be exact).
It’s an open question whether P and NP are the same thing. It’s possible that everything we think might be NP actually can be solved by a P-class algorithm we just haven’t thought of yet. It would be a revolution in our understanding of how to find solutions if it were. Most people who study algorithms think P is not NP. But that’s mostly (as I understand it) because it seems like if P were NP then we’d have some leads on proving that by now. You see how this falls short of being rigorous. But it is part of expertise to get a feel for what seems to make sense in light of everything else we know. We may be surprised. But it would be inhuman not to have any expectations of a problem like this.
Mark Anderson’s Andertoons for the 8th gives us the Andertoons content for the week. It’s a fair question why a right triangle might have three sides, three angles, three vertices, and just the one hypotenuse. The word’s origin, from Greek, meaning “stretching under” or “stretching between”. It’s unobjectionable that we might say this is the stretch from one leg of the right triangle to another. But that leaves unanswered why there’s just the one hypothenuse, since the other two legs also stretch from the end of one leg to another. Dr Sarah on The Math Forum suggests we need to think of circles. Draw a circle and a diameter line on it. Now pick any point on the circle other than where the diameter cuts it. Draw a line from one end of the diameter to your point. And from your point to the other end of the diameter. You have a right triangle! And the hypothenuse is the leg stretching under the other two. Yes, I’m assuming you picked a point above the diameter. You did, though, didn’t you? Humans do that sort of thing.
I don’t know if Dr Sarah’s explanation is right. It sounds plausible and sensible. But those are weak pins to hang an etymology on. But I have no reason to think she’s mistaken. And the explanation might help people accept there is the one hypothenuse and there’s something interesting about it.
The first (and as I write this only) commenter, Kristiaan, has a good if cheap joke there.
Last week had a whole bundle and a half of mathematically-themed comics so let me finish off the set. Also let me refresh my appeal for words for my End Of 2016 Mathematics A To Z. There’s all sorts of letters not yet claimed; please think of a mathematical term and request it!
David L Hoyt and Jeff Knurek’s Jumble for the 19th gives us a chance to do some word puzzle games again. If you like getting the big answer without doing the individual words then pay attention to the blackboard in the comic. Just saying.
Gabby describes her shape as an isocahedron, which must be a typo. We all make them. There’s icosahedrons which look like that figure and I’ve certainly slipped consonants around that way.
I’m surprised and delighted to find there are ways to make an origami icosahedron. Her figure doesn’t look much like the origami icosahedron of those instructions, but there are many icosahedrons. The name just means there are 20 faces to the polyhedron so there’s a lot of room for variants.
If you were wondering, yes, the Francis of the title is meant to be the Pope. It’s kind of a Pope Francis fan comic. I cannot explain this phenomenon.
Rick Detorie’s One Big Happy rerun for the 21st retells one of the standard jokes you can always make about fractions. Fortunately it uses that only as part of the setup, which shows off why I’ve long liked Detorie’s work. Good cartoonists — good writers — take a stock joke and add something to make it fit their characters.
I’ve featured Richard Thompson’s Poor Richard’s Almanac rerun from the 21st before. I’ll surely feature it again. I just like Richard Thompson art like this. This is my dubious inclusion of the essay. In “What’s New At The Zoo” he tosses off a mention of chimpanzees now typing at 120 words per minute. A comic reference to the famous thought experiment of a monkey, or a hundred monkeys, or infinitely many monkeys given typewriters and time to write all the works of literature? Maybe. Or it might just be that it’s a funny idea. It is, of course.
In Rick Kirkman and Jerry Scott’s Baby Blues for the 22nd Hammie offers multiple answers to each mathematics problem. “I like to increase my odds,” he says. For arithmetic problems, that’s not really helping. But it is often useful, especially in modeling complicated systems, to work out multiple answers. If you’re not sure how something should behave, and it’s troublesome to run experiments, then try develop several different models. If the models all describe similar behavior, then, good! It’s reason to believe you’re probably right, or at least close to right. If the models disagree about their conclusions then you need information. You need experimental results. The ways your models disagree can inspire new experiments.
Mark Leiknes’s Cow and Boy rerun for the 22nd is another with one of the standard jokes you can make about fractions. I suspect I’ve featured this before too, but I quite like Cow and Boy. It’s sad that the strip was cancelled, and couldn’t make a go of it as web comic. I’m not surprised; the strip had so many running jokes it might as well have had a deer and an orca shooting rocket-propelled grenades at new readers. But it’s grand seeing the many, many, many running jokes as they were first established. This is part of the sequence in which Billy, the Boy of the title, discovers there’s another kid named Billy in the class, quickly dubbed Smart Billy for reasons the strip makes clear.
Comic Strip Master Command followed up its slow times with a rush of comic strips I can talk about. Or that I can sort-of talk about. There’s enough for a regular essay just about the comics from the 5th of April alone. So today’s Reading the Comics entry is just the strips up through the 4th of April. That makes for a slightly short collection but what can I do besides schedule these for a consistent day of the week regardless of how many comics there are to talk about?
Dave Whamond’s Reality Check for the 3rd of April mentions the infinite-monkeys tale. And it even does so in iconic form, in talking about writing Shakespeare’s Hamlet. I don’t mean to disparage the comic, especially when it’s put five punch lines into the panel. (I admit I’m a little disappointed when a Sunday strip is the same one- or three-panel format as a regular daily comic, though.) But I’m pretty sure this same premise was done by Fred Allen on the radio sometime around 1940. I don’t think that mentioned the infinite monkeys, though.
Missy Meyer’s Holiday Doodles for the 4th of April mentioned that it was Square Root Day. I am curious whether the comic will mention anything for the 9th of April. I have noticed some people muttering about this Perfect Squares Day. Also I’m surprised that “glases with tape over the bridge” is still a signifier of square-ness.
Brandon Sheffield and Dami Lee’s Hot Comics for Cool People for the 4th titles its installment Perfect Geometry Comics. And it presents, as often will happen, some muddle of algebra and geometry as the way to work out a brilliantly perfect solution. Also, the comic features a dog in safety goggles, which is always good to see.
Graham Nolan’s Sunshine State for the 4th presents a word problem that might be a good introduction to asymptotes. The ratio of two people’s ages will approach without ever quite equalling 1. But it will, if the people last long enough, come as close as one might want. There’s probably also a good lesson to be made by comparing this age problem to the problem of Achilles and the tortoise.
It’s summer, so the rate of mathematically-themed comic strips slowed. That’s fine. I had a very busy, distracting couple of weeks and wasn’t able to keep up reading everything quite so faithfully. As it is you’ll notice I’m posting this without having read all my Sunday comics. They can fit in the next edition, surely.
Carol Lay’s Lay Lines (August 16) is a nicely mind-expanding story about the clash between incredibly unlikely events and an infinitely large universe. This is also my favorite of this week’s strips.
Infinity and probability interact in weird ways. Infinitely many chances for something to happen, for example, seem to imply that everything that is even remotely possible ought to happen. However … it doesn’t, really. For example, it’s imaginable that one might flip a fair coin infinitely many times and yet never see a streak where it comes up tails more than ten times in a row. Is that improbable? Sure. But impossible? And if it’s not impossible, then mustn’t it happen, given enough chances to? Mathematics and philosophy blend into one another. We see this often in logic. But probability is another of the fields that stands insistently with one foot in mathematics’s question of “what can we say about this model” and one foot in philosophy’s question of “what does it mean for something to be true”.
Mark Pett’s Lucky Cow (August 16) is about geometry. To study the real world we use straight lines and perfect circles and right angles and regular polygons and such. None of these things happen in the real world. Even things that could be like them, such as the path of a laser, or the area highlighted by a cone of light falling on the wall, are only close to the line or the ellipse or other shape in our ideal. But the abstractions are such very useful things. Well, Clare can rest assured that stacking the meal trays straight would not, in the end, produce a straight line anyway.
Mike Peters’s Mother Goose and Grimm (August 21) is the oft-made pun about the area of circles. Of course the blackboard with a lot of formulas is used as signifier for “mathematician” or perhaps “really bright person”. The writing on it is basically nonsense, although I’m curious why so many words are written out.
Mind, it is often very useful when setting out to write out, in plain English, what your variables represent. It helps set out what you think you’re doing, and check back that you’re doing things that are sensible. But, for example, “R” means “Radius” so very often that it’s silly to write that down. More useful would be “R = Radius Of [ something ]”. And then why write out circumference, and diameter, and circle twice on this board? … Besides the fact that what’s on the board is meaningless, that is, and we shouldn’t bother reading it. It exists and that is all it need do.
Bill Rechin’s Crock (August 22) is not exactly an anthropomorphized numerals joke. But it’s something in the field of turning numbers into physical objects. It’s cute enough.
I am curious why the first two panels are duplicates, though. (Look at the hatching on the cannon, or whatever the scribbles are at the bottom of his shirt.) Actually, everything about Crock is a bit mysterious. Cartoonist Bill Rechin died in May of 2011, and his family decided to stop drawing new strips in May of 2012. However, for some reason, reruns were to be distributed for three further years. Still, the strips are dated 2015. I don’t remember seeing them before. Of course, I admit I don’t have many Crock strips committed to memory, but I’d have imagined at least some would have struck me as familiar. In short, there’s a lot I don’t understand about this comic strip.
The Twop Twips account on Twitter — I’m not sure how to characterize what it is exactly, but friends retweet it often enough — had the above advice about the infinite monkeys problem, and what seems to me correct advice that turning on autocorrect will get them to write the works of Shakespeare more quickly. And then John Kovaleski’s monkey-featuring comic strip Bo Nanas featured the infinite monkey problem today, so obviously I have to spend more time thinking of it.
It seems fair that monkeys with autocorrect will be more likely to hit a word than a monkey without will be. Let’s try something simpler than Shakespeare and just consider the chance of typing the word “the”, and to keep the numbers friendly let’s imagine that the keyboard has just the letters and a space bar. We’ll not care about punctuation or numbers; that’s what copy editors would be for, if anyone had been employed as a copy editor since 1996, when someone in the budgeting office discovered there was autocorrect.
Anyway, there’s 27 characters on this truncated keyboard, and if the monkeys were equally likely to hit any one of them, then, there’d be 27 times 27 times 27 — that is, 19,683 — different three-character strings they might hit. Exactly one of them is the desired word “the”. So, roughly, we would expect the monkey to get the word right one time in each 19,683 attempts at a three-character string. (We wouldn’t have to wait quite so long if we’ll accept the monkey as writing continuously and pluck out three characters in a row wherever they appear, but that’s more work than I feel like doing, and I doubt it would significantly change the qualitative results, of how much faster it’d be if autocorrect were on.)
But how many tries would be needed to hit a word that gets autocorrected to “the”? And here we get into the mysteries of the English language. I’d be surprised by a spell checker that couldn’t figure out “teh” probably means “the”. Similarly “hte” should get back to “the”. So we can suppose the five other permutations of the letters in “the” will be autocorrected. So there’s six different strings of the 19,683 possibilities that will get fixed to “the”. The monkey has one chance in 3280.5 of getting one of them and so, on average, the monkey can be expected to be right once in every 3281 attempts.
But there’s other typos possible: “thw” is probably just my finger slipping, and “ghe” isn’t too implausible either. At least my spell checker recognizes both as most likely meant to be “the”. Let’s suppose that a spell checker can get to the right word if any one letter is mistaken. This means that there are some 78 other three-character strings that would get fixed to “the”, for a total of 84 possible three-character strings which are either “the” or would get autocorrected to “the”. With that many, there’s one chance in a touch more than 234 that a three-character string will get corrected to “the”, and we have to wait, considering, not very long at all.
It gets better if two-character errors are allowed, but I can’t make myself believe that the spell check will turn “yje” into “the”, and that’s something which might be typed if you just had the right hand on the wrong keys. My checker hasn’t got any idea what “yje” is supposed to be anyway, so, one wrong letter is probably the limit.
Except. “tie” is one character wrong for “the” and no spell checker will protest “tie”. Similarly “she” and “thy” and a couple of other words. And it’d be a bit much to expect “t e” or “ he” to be turned back into “the” even though both are just the one keystroke off. And a spell checker would probably suppose that “tht” is a typo for “that”. It’s hard to guess how many of the one-character-off words will not actually be caught. Let’s say that maybe half the one-character-off words will be corrected to “the”; that’s still a pretty good 39 one-character misspellings, plus five permutations, plus the correct spelling or 45 candidate three-character strings for autocorrect to get. So our monkey has something like one chance in 450 of getting “the” in banging on the keyboard three times.
For four-letter words there are many more combinations — 531,441, if we just list the strings of our 27 allowed characters — but then there are more strings which would get autocorrected. Let’s say we want the string “thus”; there are 23 ways to arrange those letters in addition to the correct one. And there are 104 one-character-off strings; supposing that half of them will get us to “thus”, then, there’s 76 strings that get one to the desired “thus”. That’s a pretty dismal one chance in about 7,000 of typing one of them, unfortunately. Things get a little better if we suppose that some two-character errors are going to be corrected, although I can’t find one which my spell checker will accept right now, and if a single error and a transposition are viable.
With longer words yet there’s more chances for spell checker forgiveness: you can get pretty far off “accommodate” or “aneurysm” and still be saved by the spell checker, which is good for me as I last spelled “accommodate” correctly sometime in 1992, and I thought it looked wrong then.
So the conclusion has to be: you’ll get a bit of an improvement in speed by turning on autocorrect, for the obvious reason that you’re more likely to get one right out of 450 than you are to get one right out of 19,000. But it’s not going to help you very much; the number of ways to spell things so completely wrong that not even spell check can find you just grows far too rapidly to be helped. If I get a little bored I might work out the chance of getting a permutation-or-one-off for strings of different lengths.
And your monkey might be ill-served by autocorrect anyway. When I lived in Singapore I’d occasionally have teh tarik (“pulled tea”), black tea with sugar and milk tossed back and forth until it’s nice and frothy. It’s a fine drink but hard to write back home about because even if you get past the spell checker, the reader assumes the “teh” is a typo and mentally corrects for it. When this came up I’d include a ritual emphasis that I actually meant what I wrote, but you see the problem. Fortunately Shakespeare wrote relatively little about southeast Asian teas, but if you wanted to expand the infinite monkey problem to the problem of guiding tourists through Singapore, you’d have to turn the autocorrect off to have any hope of success.
There’s been a little rash of comics that bring up mathematical themes, now, which is ordinarily pretty good news. But when I went back to look at my notes I realized most of them are pretty much name-drops, mentioning stuff that’s mathematical without giving me much to expand upon. The exceptions are what might well be the greatest gift which early 20th century probability could give humor writers. That’s enough for me.
Mark Anderson’s Andertoons (March 27) plays on the double meaning of “fifth” as representing a term in a sequence and as representing a reciprocal fraction. It also makes me realize that I hadn’t paid attention to the fact that English (at least) lets you get away with using the ordinal number for the part fraction, at least apart from “first” and “second”. I can make some guesses about why English allows that, but would like to avoid unnecessarily creating folk etymologies.
Hector D Cantu and Carlos Castellanos’s Baldo (March 27) has Baldo not do as well as he expected in predictive analytics, which I suppose doesn’t explicitly require mathematics, but would be rather hard to do without. Making predictions is one of mathematics’s great applications, and drives much mathematical work, in the extrapolation of curves and the solving of differential equations most obviously.
Rick Stromoski’s Soup To Nutz (March 29) name-drops the metric system, as Royboy notices his ten fingers and ten toes and concludes that he is indeed metric. The metric system is built around base ten, of course, and the idea that changing units should be as easy as multiplying and dividing by powers of ten, and powers of ten are easy to multiply and divide by because we use base ten for ordinary calculations. And why do we use base ten? Almost certainly because most people have ten fingers and ten toes, and it’s so easy to make the connection between counting fingers, counting objects, and then to the abstract idea of counting. There are cultures that used other numerical bases; for example, the Maya used base 20, but it’s hard not to notice that that’s just using fingers and toes together.
Greg Cravens’s The Buckets (March 30) brings out a perennial mathematics topic, the infinite monkeys. Here Toby figures he could be the greatest playwright by simply getting infinite monkeys and typewriters to match, letting them work, and harvesting the best results. He hopes that he doesn’t have to buy many of them, to spoil the joke, but the remarkable thing about the infinite monkeys problem is that you don’t actually need that many monkeys. You’ll get the same result — that, eventually, all the works of Shakespeare will be typed — with one monkey or with a million or with infinitely many monkeys; with fewer monkeys you just have to wait longer to expect success. Tim Rickard’s Brewster Rockit (April 1) manages with a mere hundred monkeys, although he doesn’t reach Shakespearean levels.
But making do with fewer monkeys is a surprisingly common tradeoff in random processes. You can often get the same results with many agents running for a shorter while, or a few agents running for a longer while. Processes that allow you to do this are called “ergodic”, and being able to prove that a process is ergodic is good news because it means a complicated system can be represented with a simple one. Unfortunately it’s often difficult to prove that something is ergodic, so you might instead just warn that you are assuming the ergodic hypothesis or ergodicity, and if nothing else you can probably get a good fight going about the validity of “ergodicity” next time you play Scrabble or Boggle.
So the subject line references here a mathematics joke that I never have heard anybody ever tell, and only encounter in lists of mathematics jokes. It goes like this: a couple professors are arguing at lunch about whether normal people actually learn anything about calculus. One of them says he’s so sure normal people learn calculus that even their waiter would be able to answer a basic calc question, and they make a bet on that. He goes back and finds their waiter and says, when she comes with the check he’s going to ask her if she knows what the integral of x is, and she should just say, “why, it’s one-half x squared, of course”. She agrees. He goes back and asks her what the integral of x is, and she says of course it’s one-half x squared, and he wins the bet. As he’s paid off, she says, “But excuse me, professor, isn’t it one-half x squared plus C?”
Let me explain why this is an accurately structured joke construct and must therefore be classified as funny. “The integral of x”, as the question puts it, has not just one correct answer but rather a whole collection of correct answers, which are different from one another only by the addition of a constant whole number, by convention denoted C, and the inclusion of that “plus C” denotes that whole collection. The professor was being sloppy in referring to just a single example from that collection instead of the whole set, as the waiter knew to do. You’ll see why this is relevant to today’s collection of mathematics-themed comics.
Jef Mallet’s Frazz (February 22) points out one of the grand things about mathematics, that if you follow the proper steps in a mathematical problem you get to be right, and to be extraordinarily confident in that rightness. And that’s true, although, at least to me a good part of what’s fun in mathematics is working out what the proper steps are: figuring out what the important parts of something you want to study should be, and what follows from your representation of them, and — particularly if you’re trying to represent a complicated real-world phenomenon with a model — whether you’re representing the things you find interesting in the real-world phenomenon well. So, while following the proper steps gets you an answer that is correct within the limits of whatever it is you’re doing, you still get to work out whether you’re working on the right problem, which is the real fun.
Mark Pett’s Lucky Cow (February 23, rerun) uses that ambiguous place between mathematics and physics to represent extreme smartness. The equation the physicist brings to Neil is the (time-dependent) Schrödinger Equation, describing how probability evolves in time, and the answer is correct. If Neil’s coworkers at Lucky Cow were smarter they’d realize the scam, though: while the equation is impressively scary-looking to people not in the know, a particle physicist would have about as much chance of forgetting this as of forgetting the end of “E equals m c … ”.
Hilary Price’s Rhymes With Orange (February 24) builds on the familiar infinite-monkeys metaphor, but misses an important point. Price is right that yes, an infinite number of monkeys already did create the works of Shakespeare, as a result of evolving into a species that could have a Shakespeare. But the infinite monkeys problem is about selecting letters at random, uniformly: the letter following “th” is as likely to be “q” as it is to be “e”. An evolutionary system, however, encourages the more successful combinations in each generation, and discourages the less successful: after writing “th” Shakespeare would be far more likely to put “e” and never “q”, which makes calculating the probability rather less obvious. And Shakespeare was writing with awareness that the words mean things and they must be strings of words which make reasonable sense in context, which the monkeys on typewriters would not. Shakespeare could have followed the line “to be or not to be” with many things, but one of the possibilities would never be “carport licking hammer worbnoggle mrxl 2038 donkey donkey donkey donkey donkey donkey donkey”. The typewriter monkeys are not so selective.
Mark Pett’s Lucky Cow turns up again (February 28, rerun) for the Rubik’s Cube. The tolerably fun puzzle and astoundingly bad Saturday morning cartoon of the 80s can be used to introduce abstract algebra. When you rotate the nine little cubes on the edge of a Rubik’s cube, you’re doing something which is kind of like addition. Think of what you can do with the top row of cubes: you can leave it alone, unchanged; you can rotate it one quarter-turn clockwise; you can rotate it one quarter-turn counterclockwise; you can rotate it two quarter-turns clockwise; you can rotate it two quarter-turns counterclockwise (which will result in something suspiciously similar to the two quarter-turns clockwise); you can rotate it three quarter-turns clockwise; you can rotate it three quarter-turns counterclockwise.
If you rotate the top row one quarter-turn clockwise, and then another one quarter-turn clockwise, you’ve done something equivalent to two quarter-turns clockwise. If you rotate the top row two quarter-turns clockwise, and then one quarter-turn counterclockwise, you’ve done the same as if you’d just turned it one quarter-turn clockwise and walked away. You’re doing something that looks a lot like addition, without being exactly like it. Something odd happens when you get to four quarter-turns either clockwise or counterclockwise, particularly, but it all follows clear rules that become pretty familiar when you notice how much it’s like saying four hours after 10:00 will be 2:00.
Abstract algebra marks one of the things you have to learn as a mathematics major that really changes the way you start looking at mathematics, as it really stops being about trying to solve equations of any kind. You instead start looking at how structures are put together — rotations are seen a lot, probably because they’re familiar enough you still have some physical intuition, while still having significant new aspects — and, following this trail can get for example to the parts of particle physics where you predict some exotic new subatomic particle has to exist because there’s this structure that makes sense if it does.
Jenny Campbell’s Flo and Friends (March 1) is set off with the sort of abstract question that comes to mind when you aren’t thinking about mathematics: how many five-card combinations are there in a deck of (52) cards? Ruthie offers an answer, although — as the commenters get to disputing — whether she’s right depends on what exactly you mean by a “five-card combination”. Would you say that a hand of “2 of hearts, 3 of hearts, 4 of clubs, Jack of diamonds, Queen of diamonds” is a different one to “3 of hearts, Jack of diamonds, 4 of clubs, Queen of diamonds, 2 of hearts”? If you’re playing a game in which the order of the deal doesn’t matter, you probably wouldn’t; but, what if the order does matter? (I admit I don’t offhand know a card game where you’d get five cards and the order would be important, but I don’t know many card games.)
For that matter, if you accept those two hands as the same, would you accept “2 of clubs, 3 of clubs, 4 of diamonds, Jack of spades, Queen of spades” as a different hand? The suits are different, yes, but they’re not differently structured: you’re still three cards away from a flush, and two away from a straight. Granted there are some games in which one suit is worth more than another, in which case it matters whether you had two diamonds or two spades; but if you got the two-of-clubs hand just after getting the two-of-hearts hand you’d probably be struck by how weird it was you got the same hand twice in a row. You can’t give a correct answer to the question until you’ve thought about exactly what you mean when you say two hands of cards are different.
Brian and Ron Boychuk’s The Chuckle Brothers (February 13) does a little wordplay on how “circumference” sounds like it could kind of be a knightly name, which I remember seeing in a minor Bugs Bunny cartoon back in the day. “Circumference” the word derives from the Latin, “circum” meaning around and “fero” meaning “to carry”; and to my mind, the really interesting question is why do we have the words “perimeter” and “circumference” when it seems like either one would do? “Circumference” does have the connotation of referring to just the boundary of a circular or roughly circular form, but why should the perimeter of circular things be so exceptional as to usefully have its own distinct term? But English is just like that, I suppose.
Paul Trapp’s Thatababy (February 13) brings back the infinite-monkey metaphor. The infinite monkeys also appear in John Deering’s Strange Brew (February 20), which is probably just a coincidence based on how successfully tossing in lots of monkeys can produce giggling. Or maybe the last time Comic Strip Master Command issued its orders it sent out a directive, “more infinite monkey comics!”
Ruben Bolling’s Tom The Dancing Bug (February 14) delivers some satirical jabs about Biblical textual inerrancy by pointing out where the Bible makes mathematical errors. I tend to think nitpicking the Bible mostly a waste of good time on everyone’s part, although the handful of arithmetic errors are a fair wedge against the idea that the text can’t have any errors and requires no interpretation or even forgiveness, with the Ezra case the stronger one. The 1 Kings one is about the circumference and the diameter for a vessel being given, and those being incompatible, but it isn’t hard to come up with a rationalization that brings them plausibly in line (you have to suppose that the diameter goes from outer wall to outer wall, while the circumference is that of an inner wall, which may be a bit odd but isn’t actually ruled out by the text), which is why I think it’s the weaker.
Bill Whitehead’s Free Range (February 16) uses a blackboard full of mathematics as a generic “this is something really complicated” signifier. The symbols as written don’t make a lot of sense, although I admit it’s common enough while working out a difficult problem to work out weird bundles of partly-written expressions or abuses of notation (like on the middle left of the board, where a bracket around several equations is shown as being less than a bracket around fewer equations), just because ideas are exploding faster than they can be written out sensibly. Hopefully once the point is proven you’re able to go back and rebuild it all in a form which makes sense, either by going into standard notation or by discovering that you have soem new kind of notation that has to be used. It’s very exciting to come up with some new bit of notation, even if it’s only you and a couple people you work with who ever use it. Developing a good way of writing a concept might be the biggest thrill in mathematics, even better than proving something obscure or surprising.
Jonathan Lemon’s Rabbits Against Magic (February 18) uses a blackboard full of mathematics symbols again to give the impression of someone working on something really hard. The first two lines of equations on 8-Ball’s board are the time-dependent Schrödinger Equations, describing how the probability distribution for something evolves in time. The last line is Euler’s formula, the curious and fascinating relationship between pi, the base of the natural logarithm e, imaginary numbers, one, and zero.
Todd Clark’s Lola (February 20) uses the person-on-an-airplane setup for a word problem, in this case, about armrest squabbling. Interesting to me about this is that the commenters get into a squabble about how airplane speeds aren’t measured in miles per hour but rather in nautical miles, although nobody not involved in air traffic control really sees that. What amuses me about this is that what units you use to measure the speed of the plane don’t matter; the kind of work you’d do for a plane-travelling-at-speed problem is exactly the same whatever the units are. For that matter, none of the unique properties of the airplane, such as that it’s travelling through the air rather than on a highway or a train track, matter at all to the problem. The plane could be swapped out and replaced with any other method of travel without affecting the work — except that airplanes are more likely than trains (let’s say) to have an armrest shortage and so the mock question about armrest fights is one in which it matters that it’s on an airplane.
Bill Watterson’s Calvin and Hobbes (February 21) is one of the all-time classics, with Calvin wondering about just how fast his sledding is going, and being interested right up to the point that Hobbes identifies mathematics as the way to know. There’s a lot of mathematics to be seen in finding how fast they’re going downhill. Measuring the size of the hill and how long it takes to go downhill provides the average speed, certainly. Working out how far one drops, as opposed to how far one travels, is a trigonometry problem. Trying the run multiple times, and seeing how the speed varies, introduces statistics. Trying to answer questions like when are they travelling fastest — at a single instant, rather than over the whole run — introduce differential calculus. Integral calculus could be found from trying to tell what the exact distance travelled is. Working out what the shortest or the fastest possible trips introduce the calculus of variations, which leads in remarkably quick steps to optics, statistical mechanics, and even quantum mechanics. It’s pretty heady stuff, but I admit, yeah, it’s math.
I’d figured I had enough mathematics comic strips for another of these entries, and discovered during the writing that I had much more to say about one than I had anticipated. So, although it’s no longer quite the 11th, or close to it, I’m going to exile the comics from after that date to the next of these entries.
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (starting February 10) (and when am I going to make a macro for that credit and title?) has Cynthia given a slightly baffling homework lesson: to calculate the first ten digits of pi. The story continues through the 11th,the 12th,the 13th, finally resolving on the the 14th, in the way such stories must. I admit I’m not sure why exactly calculating the digits of π would be a suitable homework assignment; I can see working out division problems until the numbers start repeating, or doing a square root or something by hand until you’ve found enough digits.
π, though … well, there’s the question of why it’d be an assignment to start with, but also, what formula for generating π could be plausibly appropriate for an elementary school class. The one that seems obvious to me — π is equal to four times (1/1 minus 1/3 plus 1/5 minus 1/7 plus 1/9 minus 1/11 and so on and so on) — also takes way too long to work. If a little bit of coding is right, it takes something like 160 terms to get just the first two digits of π correct and that isn’t even stable. (The first 160 terms add to 3.135; the first 161 terms to 3.147.) Getting it to ten digits would take —
Well, I thought it might be as few was 10,000 terms, because it turns out the sum of the first ten thousand terms in that series is 3.1414926536, which looks dead-on until you notice that π is 3.1415926536. That’s a neat coincidence, though.
Anyway, obviously, that formula wouldn’t do, and we see on the strip of the 14th that Lucretia isn’t using that. There are a great many formulas that generate the value of π, any of which might be used for a project like this; some of them get the digits right quite rapidly, usually at a cost of being very complicated. The formula shown in the strip of the 14th, though, doesn’t seem to be right. Lucretia’s work uses the formula , which takes only about 21 terms to get to the demanded ten digits of accuracy. I don’t want to guess how many pages of work it would take to get to 13,908 places.
If I don’t miss my guess the formula used here is one by Abraham Sharp, an astronomer and mathematician who worked for the Royal Observatory at Greenwich and set a record by calculating π to 72 decimal digits. He was also an instrument-maker, of rather some skill, and I found a page purporting to show his notes of how to cut some complicated polyhedrons out of a block of wood, so, if my father wants to carve a 120-sided figure, here’s his chance. Sharp seems to have started with Leibniz’s formula (yes, that Leibniz) — that the arctangent of a number x is equal to x minus one-third x cubed plus one-fifth x to the fifth power minus one-seventh x to the seventh power, et cetera — with the knowledge that the arctangent of the square root of one-third is equal to one-sixth π and produced this series that looks a lot like the one we started with, but which gets digits correct so very much more quickly.
Darrin Bell’s Candorville (February 13) is primarily a bit of guys insulting friends, but what do you know and π makes a cameo appearance here.
Shannon Wheeler’s Too Much Coffee Man (February 10) is a Venn Diagram cartoon in the service of arguing that Venn Diagram cartoons aren’t funny. Putting aside the smoke and sparks popping out of the Nomad space probe which Kirk and Spock are rushing to the transporter room, I don’t think it’s quite fair: the ease the Venn diagram gives to grouping together concepts and showing how they relate helps organize one’s understanding of concepts and can be a really efficient way to set up a joke. Granting that, perhaps Wheeler’s seen too many Venn Diagram cartoons that fail, a complaint I’m sympathetic to.
Bill Amend’s FoxTrot (February 11, rerun) was one of those strips trying to be taped to the math teacher’s door, with the pun-based programming for the Math Channel.