As anyone who tries to be funny knows, some words just are funnier than others. Or at least they sound funnier. (This brings us into the problem of whether something is actually funny or whether we just think it is.) Westbury, Shaoul, Moroschan, and Ramscar try testing whether a common measure of how unpredictable something is — the entropy, a cornerstone of information theory — can tell us how funny a word might be.
We’ve encountered entropy in these parts before. I used it in that series earlier this year about how interesting a basketball tournament was. Entropy, in this context, is low if something is predictable. It gets higher the more unpredictable the thing being studied is. You see this at work in auto-completion: if you have typed in ‘th’, it’s likely your next letter is going to be an ‘e’. This reflects the low entropy of ‘the’ as an english word. It’s rather unlikely the next letter will be ‘j’, because English has few contexts that need ‘thj’ to be written out. So it will suggest words that start ‘the’ (and ‘tha’, and maybe even ‘thi’), while giving ‘thj’ and ‘thq’ and ‘thd’ a pass.
Westbury, Shaoul, Moroschan, and Ramscar found that the entropy of a word, how unlikely that collection of letters is to appear in an English word, matches quite well how funny people unfamiliar with it find it. This fits well with one of the more respectable theories of comedy, Arthur Schopenhauer’s theory that humor comes about from violating expectations. That matches well with unpredictability.
Of course it isn’t just entropy that makes words funny. Anyone trying to be funny learns that soon enough, since a string of perfect nonsense is also boring. But this is one of the things that can be measured and that does have an influence.
(I doubt there is any one explanation for why things are funny. My sense is that there are many different kinds of humor, not all of them perfectly compatible. It would be bizarre if any one thing could explain them all. But explanations for pieces of them are plausible enough.)
The coming US summer vacation suggests Comic Strip Master Command will slow down production of mathematics-themed comic strips. But they haven’t quite yet. And this week I also found a couple comics that, while not about mathematics, amused me enough that I want to include them anyway. So those bonus strips I’ll run at the end of my regular business here.
Bill Hinds’s Tank McNamara (June 6) does a pi pun. The pithon mathematical-snake idea is fun enough and I’d be interested in a character design. I think the strip’s unjustifiably snotty about tattoos. But comic strips have a strange tendency to get snotty about other forms of art.
A friend happened to mention one problem with tattoos that require straight lines or regular shapes is that human skin has a non-flat Gaussian curvature. Yes, that’s how the friend talks. Gaussian curvature is, well, a measure of how curved a surface is. That sounds obvious enough, but there are surprises: a circular cylinder, such as the label of a can, has the same curvature as a flat sheet of paper. You can see that by how easy it is to wrap a sheet of paper around a can. But a ball hasn’t, and you see that by how you can’t neatly wrap a sheet of paper around a ball without crumpling or tearing the paper. Human skin is kind of cylindrical in many places, but not perfectly so, and it changes as the body moves. So any design that looks good on paper requires some artistic imagination to adapt to the skin.
Bill Amend’s FoxTrot (June 7) sets Jason and Marcus working on their summer tans. It’s a good strip for adding to the cover of a trigonometry test as part of the cheat-sheet.
Dana Simpson’s Phoebe and her Unicorn (June 8) makes what I think is its first appearance in my Reading the Comics series. The strip, as a web comic, had been named Heavenly Nostrils. Then it got the vanishingly rare chance to run as a syndicated newspaper comic strip. And newspaper comics page editors don’t find the word “nostril” too inherently funny to pass up. Thus the more marketable name. After that interesting background I’m sad to say Simpson delivers a bog-standard “kids not understanding fractions” joke. I can’t say much about that.
Ruben Bolling’s Super Fun-Pak Comix (June 10, rerun) is an installment of everyone’s favorite literary device model of infinite probabilities. A Million Monkeys At A Million Typewriters subverts the model. A monkey thinking about the text destroys the randomness that it depends upon. This one’s my favorite of the mathematics strips this time around.
And now here’s a couple strips that aren’t mathematical but that I just liked too much to ignore. Also this lets Mark Anderson’s Andertoons get back on my page. The June 10th strip is a funny bit of grammar play.
Percy Crosby’s Skippy (June 6, rerun from sometime in 1928) tickles me for its point about what you get at the top and the bottom of the class. Although tutorials and office hours and extracurricular help, and automated teaching tools, do customize things a bit, teaching is ultimately a performance given to an audience. Some will be perfectly in tune with the performance, and some won’t. Audiences are like that.
And now, I think, I’ve got caught up on the mathematics-themed comics that appeared at Comics Kingdom and at Gocomics.com over the past week and a half. I’m sorry to say today’s entries don’t get to be about as rich a set of topics as the previous bunch’s, but on the other hand, there’s a couple Comics Kingdom strips that I feel comfortable using as images, so there’s that. And come to think of it, none of them involve the setup of a teacher asking a student in class a word problem, so that’s different.
Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (February 21) tells the old joke about how much of fractions someone understands. To me the canonical version of the joke was a Sydney Harris panel in which one teacher complains that five-thirds of the class doesn’t understand a word she says about fractions, but it’s all the same gag. I’m a touch amused that three and five turn up in this version of the joke too. That probably reflects writing necessity — especially for this B.C. the numbers have to be a pair that obviously doesn’t give you one-half — and that, somehow, odd numbers seem to read as funnier than even ones.
Bud Fisher’s Mutt and Jeff (February 21) decimates one of the old work-rate problems, this one about how long it takes a group of people to eat a pot roast. It was surely an old joke even when this comic first appeared (and I can’t tell you when it was; Gocomics.com’s reruns have been a mixed bunch of 1940s and 1950s ones, but they don’t say when the original run date was), but the spread across five panels treats the joke well as it’s able to be presented as a fuller stage-ready sketch. Modern comic strips value an efficiently told, minimalist joke, but pacing and minor punch lines (“some men don’t eat as fast as others”) add their charm to a comic.
I know that it’s disappointing to everyone, given that one of the comic strips in today’s roundup of mathematically-themed such gives me such a good excuse to explain what Fourier Transforms are and why they’re interesting and well worth the time learning. But I’m not going to do that today. There’s enough other things to think about and besides you probably aren’t going to need Fourier Transforms in class for a couple more weeks yet. For today, though, no, I’ll go on to other things instead. Sorry to disappoint.
Glen McCoy and Gary McCoy’s The Flying McCoys (October 9) jokes about how one can go through life without ever using algebra. I imagine other departments get this, too, like, “I made it through my whole life without knowing anything about US History!” or “And did any of that time I spent learning Art do anything for me?” I admit a bias here: I like learning stuff even if it isn’t useful because I find it fun to learn stuff. I don’t insist that you share in finding that fun, but I am going to look at you weird if you feel some sense of triumph about not learning stuff.
Tom Thaves’s Frank and Ernest (October 10) does a gag about theoretical physics, and string theory, which is that field where physics merges almost imperceptibly into mathematics and philosophy. The rough idea of string theory is that it’d be nice to understand why the particles we actually observe exist, as opposed to things that we could imagine existing that that don’t seem to — like, why couldn’t there be something that’s just like an electron, but two times as heavy? Why couldn’t there be something with the mass of a proton but three-quarters the electric charge? — by supposing that what we see are the different natural modes of behavior of some more basic construct, these strings. A natural mode is, well, what something will do if it’s got a bunch of energy and is left to do what it will with it.
Probably the most familiar kind of natural mode is how if you strike a glass or a fork or such it’ll vibrate, if we’re lucky at a tone we can hear, and if we’re really lucky, at one that sounds good. Things can have more than one natural mode. String theory hopes to explain all the different kinds of particles, and the different ways in which they interact, as being different modes of a hopefully small and reasonable variety of “strings”. It’s a controversial theory because it’s been very hard to find experiments that proves, or soundly rules out, a particular model of it as representation of reality, and the models require invoking exotic things like more dimensions of space than we notice. This could reflect string theory being an intriguing but ultimately non-physical model of the world; it could reflect that we just haven’t found the right way to go about proving it yet.
Charles Schulz’s Peanuts (October 10, originally run October 13, 1967) has Sally press Charlie Brown into helping her with her times tables. She does a fair bit if guessing, which isn’t by itself a bad approach. For one, if you don’t know the exact answer, but you can pin down a lower and and upper bound, you’re doing work that might be all you really need and you’re doing work that may give you a hint how to get what you really want. And for that matter, guessing at a solution can be the first step to finding one. One of my favorite areas of mathematics, Monte Carlo methods, finds solutions to complicated problems by starting with a wild guess and making incremental refinements. It’s not guaranteed to work, but when it does, it gets extremely good solutions and with a remarkable ease. Granted this, doesn’t really help the times tables much.
On the 11th (originally run October 14, 1967), Sally incidentally shows the hard part of refining guesses about a solution; there has to be some way of telling whether you’re getting warmer. In your typical problem for a Monte Carlo approach, for example, you have some objective function — say, the distance travelled by something going along a path, or the total energy of a system — and can measure whether an attempted change is improving your solution — say, minimizing your distance or reducing the potential energy — or is making it worse. Typically, you take any refinement that makes the provisional answer better, and reject most, but not all, refinements that make the provisional answer worse.
That said, “Overly-Eight” is one of my favorite made-up numbers. A “Quillion” is also a pretty good one.
Jeff Mallet’s Frazz (October 12) isn’t explicitly about mathematics, but it’s about mathematics. “Why do I have to show my work? I got the right answer?” There are good responses on two levels, the first of which is practical, and which blends into the second: if you give me-the-instructor the wrong answer then I can hopefully work out why you got it wrong. Did you get it wrong because you made a minor but ultimately meaningless slip in your calculations, or did you get it wrong because you misunderstood the problem and did not know what kind of calculation to do? Error comes in many forms; some are boring — wrote the wrong number down at the start and never noticed, missed a carry — some are revealing — doesn’t know the order of operations, doesn’t know how the chain rule applies in differentiation — and some are majestic.
These last are the great ones, the errors that I love seeing, even though they’re the hardest to give a fair grade to. Sometimes a student will go off on a tack that doesn’t look anything like what we did in class, or could have reasonably seen in the textbook, but that shows some strange and possibly mad burst of creative energy. Usually this is rubbish and reflects the student flailing around, but, sometimes the student is on to something, might be trying an approach that, all right, doesn’t work here, but which if it were cleaned of its logical flaws might be a new and different way to work out the problem.
And that blends to the second reason: finding answers is nice enough and if you’re good at that, I’m glad, but is it all that important? We have calculators, after all. What’s interesting, and what is really worth learning in mathematics, is how to find answers: what approaches can efficiently be used on this problem, and how do you select one, and how do you do it to get a correct answer? That’s what’s really worth learning, and what is being looked for when the instruction is to show your work. Caulfield had the right answer, great, but is it because he knew a good way to work out the problem, or is it because he noticed the answer was left on the blackboard from the earlier class when this one started, or is it because he guessed and got lucky, or is it because he thought of a clever new way to solve the problem? If he did have a clever new way to do the problem, shouldn’t other people get to see it? Coming up with clever new ways to find answers is the sort of thing that gets you mathematical immortality as a pioneer of some approach that gets mysteriously named for somebody else.
Zach Weinersmith’s Saturday Morning Breakfast Cereal (October 14) makes fun of tenure, the process by which people with a long track record of skill, talent, and drive are rewarded with no longer having to fear being laid off or fired except for cause. (Though I should sometime write about Fourier Transforms, as they’re rather neat.)
Margaret Shulock’s turn at Six Chix (October 14) (the comic strip is shared among six women because … we couldn’t have six different comic strips written and drawn by women all at the same time, I guess?) evokes the classic image of Albert Einstein, the genius, and drawing his famous equation out of the ordinary stuff of daily life. (I snark a little; Shulock is also the writer for Apartment 3-G, to the extent that things can be said to be written in Apartment 3-G.)
Must be the start of school or something. In today’s roundup of mathematically-themed comics there are a couple of strips that I think touch on the question of defining just what the problem is: what are you trying to measure, what are you trying to calculate, what are the rules of this sort of calculation? That’s a lot of what’s really interesting about mathematics, which is how I’m able to say something about a rerun Archie comic. It’s not easy work but that’s why I get that big math-blogger paycheck.
John Hambrock’s The Brilliant Mind of Edison Lee (September 2) talks about the shape of the universe. Measuring the world, or the universe, is certainly one of the older influences on mathematical thought. From a handful of observations and some careful reasoning, for example, one can understand how large the Earth is, and how far away the Moon and the Sun must be, without going past the kinds of reasoning or calculations that a middle school student would probably be able to follow.
There is something deeper to consider about the shape of space, though: the geometry of the universe affects what things can happen in them, and can even be seen in the kinds of physics that happen. A famous, and astounding, result by the mathematical physicist Emmy Noether shows that symmetries in space correspond to conservation laws. That the universe is, apparently, rotationally symmetric — everything would look the same if the whole universe were picked up and rotated (say) 80 degrees along one axis — means that there is such a thing as the conservation of angular momentum. That the universe is time-symmetric — the universe would look the same if it had got started five hours later (please pretend that’s a statement that can have any coherent meaning) — means that energy is conserved. And so on. It may seem, superficially, like a cosmologist is engaged in some almost ancient-Greek-style abstract reasoning to wonder what shapes the universe could have and what it does, but (putting aside that it gets hard to divide mathematics, physics, and philosophy in this kind of field) we can imagine observable, testable consequences of the answer.
Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 5) tells a joke starting with “two perfectly rational perfectly informed individuals walk into a bar”, along the way to a joke about economists. The idea of “perfectly rational perfectly informed” people is part of the mathematical modeling that’s become a popular strain of economic thought in recent decades. It’s a model, and like many models, is properly speaking wrong, but it allows one to describe interesting behavior — in this case, how people will make decisions — without complications you either can’t handle or aren’t interested in. The joke goes on to the idea that one can assign costs and benefits to continuing in the joke. The idea that one can quantify preferences and pleasures and happiness I think of as being made concrete by Jeremy Bentham and the utilitarian philosophers, although trying to find ways to measure things has been a streak in Western thought for close to a thousand years now, and rather fruitfully so. But I wouldn’t have much to do with protagonists who can’t stay around through the whole joke either.
Marc Anderson’s Andertoons (September 6) was probably composed in the spirit of joking, but it does hit something that I understand baffles kids learning it every year: that subtracting a negative number does the same thing as adding a positive number. To be fair to kids who need a couple months to feel quite confident in what they’re doing, mathematicians needed a couple generations to get the hang of it too. We have now a pretty sound set of rules for how to work with negative numbers, that’s nice and logically tested and very successful at representing things we want to know, but there seems to be a strong intuition that says “subtracting a negative three” and “adding a positive three” might just be different somehow, and we won’t really know negative numbers until that sense of something being awry is resolved.
Andertoons pops up again the next day (September 7) with a completely different drawing of a chalkboard and this time a scientist and a rabbit standing in front of it. The rabbit’s shown to be able to do more than multiply and, indeed, the mathematics is correct. Cosines and sines have a rather famous link to exponentiation and to imaginary- and complex-valued numbers, and it can be useful to change an ordinary cosine or sine into this exponentiation of a complex-valued number. Why? Mostly, because exponentiation tends to be pretty nice, analytically: you can multiply and divide terms pretty easily, you can take derivatives and integrals almost effortlessly, and then if you need a cosine or a sine you can get that out at the end again. It’s a good trick to know how to do.
Craig Boldman and Henry Scarpelli’s Archie (September 8, rerun) goes back to one of arithmetic’s traditional comic strip applications, that of working out the tip. Poor Moose is driving himself crazy trying to work out 15 percent of $8.95, probably from a quiz-inspired fear that if he doesn’t get it correct to the penny he’s completely wrong. Being able to do a calculation precisely is useful, certainly, but he’s forgetting that in tis real-world application he gets some flexibility in what has to be calculated. He’d save some effort if he realized the tip for $8.95 is probably close enough to the tip for $9.00 that he could afford the difference, most obviously, and (if his budget allows) that he could just as well work out one-sixth the bill instead of fifteen percent, and give up that workload in exchange for sixteen cents.
Well, I did say we were getting to the end of summer. It’s taken only a couple days to get a fresh batch of enough mathematics-themed comics to include here, although the majority of them are about mathematics in ways that we’ve seen before, sometimes many times. I suppose that’s fair; it’s hard to keep thinking of wholly original mathematics jokes, after all. When you’ve had one killer gag about “537”, it’s tough to move on to “539” and have it still feel fresh.
Tom Toles’s Randolph Itch, 2 am (August 27, rerun) presents Randolph suffering the nightmare of contracting a case of entropy. Entropy might be the 19th-century mathematical concept that’s most achieved popular recognition: everyone knows it as some kind of measure of how disorganized things are, and that it’s going to ever increase, and if pressed there’s maybe something about milk being stirred into coffee that’s linked with it. The mathematical definition of entropy is tied to the probability one will find whatever one is looking at in a given state. Work out the probability of finding a system in a particular state — having particles in these positions, with these speeds, maybe these bits of magnetism, whatever — and multiply that by the logarithm of that probability. Work out that product for all the possible ways the system could possibly be configured, however likely or however improbable, just so long as they’re not impossible states. Then add together all those products over all possible states. (This is when you become grateful for learning calculus, since that makes it imaginable to do all these multiplications and additions.) That’s the entropy of the system. And it applies to things with stunning universality: it can be meaningfully measured for the stirring of milk into coffee, to heat flowing through an engine, to a body falling apart, to messages sent over the Internet, all the way to the outcomes of sports brackets. It isn’t just body parts falling off.
Randy Glasbergen’s The Better Half (August 28) does the old joke about not giving up on algebra someday being useful. Do teachers in other subjects get this? “Don’t worry, someday your knowledge of the Panic of 1819 will be useful to you!” “Never fear, someday they’ll all look up to you for being able to diagram a sentence!” “Keep the faith: you will eventually need to tell someone who only speaks French that the notebook of your uncle is on the table of your aunt!”
Eric the Circle (August 28, by “Gilly” this time) sneaks into my pages again by bringing a famous mathematical symbol into things. I’d like to make a mention of the links between mathematics and music which go back at minimum as far as the Ancient Greeks and the observation that a lyre string twice as long produced the same note one octave lower, but lyres and strings don’t fit the reference Gilly was going for here. Too bad.
Zach Weinersmith’s Saturday Morning Breakfast Cereal (August 28) is another strip to use a “blackboard full of mathematical symbols” as visual shorthand for “is incredibly smart stuff going on”. The symbols look to me like they at least started out as being meaningful — they’re the kinds of symbols I expect in describing the curvature of space, and which you can find by opening up a book about general relativity — though I’m not sure they actually stay sensible. (It’s not the kind of mathematics I’ve really studied.) However, work in progress tends to be sloppy, the rough sketch of an idea which can hopefully be made sound.
Now, there, see? The school year (in the United States) has let out for summer and the rush of mathematics-themed comic strips has subsided; it’s been over two weeks since the last bunch was big enough. Given enough time, though, a handful of comics will assemble that I can do something with, anything, and now’s that time. I hate to admit also that they’re clearly not trying very hard with these mathematics comics as they’re not about very juicy topics. Call it the summer doldroms, as I did.
Mason Mastroianni and Mick Mastroianni’s B.C. (July 6) spends most of its text talking about learning cursive, as part of a joke built around the punch line that gadgets are spoiling students who learn to depend on them instead of their own minds. So it would naturally get around to using calculators (or calculator apps, which is a fair enough substitute) in place of mathematics lessons. I confess I come down on the side that wonders why it’s necessary to do more than rough, approximate arithmetic calculations without a tool, and isn’t sure exactly what’s gained by learning cursive handwriting, but these are subjects that inspire heated and ongoing debates so you’ll never catch me admitting either position in public.
Eric the Circle (July 7), here by “andel”, shows what one commenter correctly identifies as a “pi fight”, which might have made a better caption for the strip, at least for me, because Eric’s string of digits wasn’t one of the approximations to pi that I was familiar with. I still can’t find it, actually, and wonder if andel didn’t just get a digit wrong. (I might just not have found a good web page that lists the digits of various approximations to pi, I admit.) Erica’s approximation is the rather famous 22/7.
Joe Martin’s Mr Boffo (July 18, sorry that I can’t find a truly permanent link) plays with one of Martin’s favorite themes, putting deep domesticity to great inventors and great minds. I suspect but do not know that Martin was aware that Einstein’s first wife, Mileva Maric, was a fellow student with him at the Swiss Federal Polytechnic. She studied mathematics and physics. The extent to which she helped Einstein develop his theories is debatable; as far as I’m aware the evidence only goes so far as to prove she was a bright, outside mind who could intelligently discuss whatever he might be wrangling over. This shouldn’t be minimized: describing a problem is often a key step in working through it, and a person who can ask good follow-up questions about a problem is invaluable even if that person doesn’t do anything further.
I’d like to take a moment — after my most popular month ever, according to the WordPress statistics, one in which my little math blog managed to reach not just the 7,777th viewer but also the 8,000th, with the nice round 8,192nd looking like it’s coming soon — to announce the inauguration of another blog of my hopefully creative writing.
This one, Nebushumor.wordpress.com, is for the writing that I at least intend to be funny, and while I concede the number of people who actually share my sense of humor is perhaps not overwhelming, there’s probably some out there who do, and I’d appreciate your looking and following if you like what you see. Right now I’m still in the stage of starting a new project where I feel hideously embarrassed that I put myself forward like that, all set for public humiliation and whatnot, but that feeling will pass. I hope.
Also, I found a setting that lets me turn on little numeric ratings for all my various articles. I’m hoping that people enjoy this because if there’s one thing I’ve learned about the Internet, it’s that people like getting to click stuff. Plus turning it on satisfied my urge to fiddle with the look of this blog a while, since I can’t find another WordPress theme that I like for it.