The rest of last week had more mathematically-themed comic strips than Sunday alone did. As sometimes happens, I noticed an objectively unimportant detail in one of the comics and got to thinking about it. Whether I could solve the equation as posted, or whether at least part of it made sense as a mathematics problem. Well, you’ll see.
Patrick McDonnell’s Mutts for the 25th of September I include because it’s cute and I like when I can feature some comic in these roundups. Maybe there’s some discussion that could be had about what “equals” means in ordinary English versus what it means in mathematics. But I admit that’s a stretch.
Olivia Walch’s Imogen Quest for the 25th uses, and describes, the mathematics of a famous probability problem. This is the surprising result of how few people you need to have a 50 percent chance that some pair of people have a birthday in common. It then goes over to some other probability problems. The examples are silly. But the reasoning is sound. And the approach is useful. To find the chance of something happens it’s often easiest to work out the chance it doesn’t. Which is as good as knowing the chance it does, since a thing can either happen or not happen. At least in probability problems, which define “thing” and “happen” so there’s not ambiguity about whether it happened or not.
Piers Baker’s Ollie and Quentin rerun for the 26th I’m pretty sure I’ve written about before, although back before I included pictures of the Comics Kingdom strips. (The strip moved from Comics Kingdom over to GoComics, which I haven’t caught removing old comics from their pages.) Anyway, it plays on a core piece of probability. It sets out the world as things, “events”, that can have one of multiple outcomes, and which must have one of those outcomes. Coin tossing is taken to mean, by default, an event that has exactly two possible outcomes, each equally likely. And that is near enough true for real-world coin tossing. But there is a little gap between “near enough” and “true”.
Rick Stromoski’s Soup To Nutz for the 27th is your standard sort of Dumb Royboy joke, in this case about him not knowing what percentages are. You could do the same joke about fractions, including with the same breakdown of what part of the mathematics geek population ruins it for the remainder.
Nate Fakes’s Break of Day for the 28th is not quite the anthropomorphic-numerals joke for the week. Anthropomorphic mathematics problems, anyway. The intriguing thing to me is that the difficult, calculus, problem looks almost legitimate to me. On the right-hand-side of the first two lines, for example, the calculation goes from
This is a little sloppy. The first line ought to end in a ‘dt’, and the second ought to have a constant of integration. If you don’t know what these calculus things are let me explain: they’re calculus things. You need to include them to express the work correctly. But if you’re just doing a quick check of something, the mathematical equivalent of a very rough preliminary sketch, it’s common enough to leave that out.
It doesn’t quite parse or mean anything precisely as it is. But it looks like the sort of thing that some context would make meaningful. That there’s repeated appearances of , or , particularly makes me wonder if Frakes used a problem he (or a friend) was doing for some reason.
I don’t actually like it when a split week has so many more comics one day than the next, but I also don’t like splitting across a day if I can avoid it. This week, I had to do a little of both since there were so many comic strips that were relevant enough on the 8th. But they were dominated by the idea of going back to school, yet.
Randy Glasbergen’s Glasbergen Cartoons rerun for the 8th is another back-to-school gag. And it uses arithmetic as the mathematics at its most basic. Arithmetic might not be the most fundamental mathematics, but it does seem to be one of the parts we understand first. It’s probably last to be forgotten even on a long summer break.
Mark Pett’s Mr Lowe rerun for the 8th is built on the familiar old question of why learn arithmetic when there’s computers. Quentin is unconvinced of this as motive for learning long division. I’ll grant the case could be made better. I admit I’m not sure how, though. I think long division is good as a way to teach, especially, the process of estimating and improving estimates of a calculation. There’s a lot of real mathematics in doing that.
Guy Gilchrist’s Nancy for the 8th is another back-to-school strip. Nancy’s faced with “this much math” so close to summer. Her given problem’s a bit of a mess to me. But it’s mostly teaching whether the student’s got the hang of the order of operations. And the instructor clearly hasn’t got the sense right. People can ask whether we should parse “12 divided by 3 times 4” as “(12 divided by 3) times 4” or as “12 divided by (3 times 4)”, and that does make a major difference. Multiplication commutes; you can do it in any order. Division doesn’t. Leaving ambiguous phrasing is the sort of thing you learn, instinctively, to avoid. Nancy would be justified in refusing to do the problem on the grounds that there is no unambiguous way to evaluate it, and that the instructor surely did not mean for her to evaluate it all four different plausible ways.
By the way, I’ve seen going around Normal Person Twitter this week a comment about how they just discovered the division symbol ÷, the obelus, is “just” the fraction bar with dots above and below where the unknown numbers go. I agree this is a great mnemonic for understanding what is being asked for with the symbol. But I see no evidence that this is where the symbol, historically, comes from. We first see ÷ used for division in the writings of Johann Henrich Rahn, in 1659, and the symbol gained popularity particularly when John Pell picked it up nine years later. But it’s not like Rahn invented the symbol out of nowhere; it had been used for subtraction for over 125 years at that point. There were also a good number of writers using : or / or \ for division. There were some people using a center dot before and after a / mark for this, like the % sign fell on its side. That ÷ gained popularity in English and American writing seems to be a quirk of fate, possibly augmented by it being relatively easy to produce on a standard typewriter. (Florian Cajori notes that the National Committee on Mathematical Requirements recommended dropping ÷ altogether in favor of a symbol that actually has use in non-mathematical life, the / mark. The Committee recommended this in 1923, so you see how well the form agenda is doing.)
Mark Leiknes’s Cow and Boy rerun for the 9th only mentions mathematics, and that as a course that Billy would rather be skipping. But I like the comic strip and want to promote its memory as much as possible. It’s a deeply weird thing, because it has something like 400 running jokes, and it’s hard to get into because the first couple times you see a pastoral conversation interrupted by an orca firing a bazooka at a cat-helicopter while a panda brags of blowing up the moon it seems like pure gibberish. If you can get through that, you realize why this is funny.
Dave Blazek’s Loose Parts for the 9th uses chalkboards full of stuff as the sign of a professor doing serious thinking. Mathematics is will-suited for chalkboards, at least in comic strips. It conveys a lot of thought and doesn’t need much preplanning. Although a joke about the difficulties in planning out blackboard use does take that planning. Yes, there is a particular pain that comes from having more stuff to write down in the quick yet easily collaborative medium of the chalkboard than there is board space to write.
Brian Basset’s Red and Rover for the 9th also really only casually mentions mathematics. But it’s another comic strip I like a good deal so would like to talk up. Anyway, it does show Red discovering he doesn’t mind doing mathematics when he sees the use.
It was a busy week at Comic Strip Master Command last week, since they wanted to be sure I was overloaded ahead of the start of the Summer 2017 A To Z project. So here’s the couple of comics I didn’t have time to review on Sunday.
Mort (“Addison”) Walker’s Boner’s Ark for the 7th of September, 1971 was rerun the 27th of July. It mentions mathematics but just as a class someone might need more work on. Could be anything, but mathematics has the connotations of something everybody struggles with, and in an American comic strip needs only four letters to write. Most economical use of word balloon space.
Neil Kohney’s The Other End for the 28th also mentions mathematics without having any real mathematics content. Barry tries to make the argument that mathematics has a timeless and universal quality that makes for good aesthetic value. I support this principle. Art has many roles. One is to make us see things which are true which are not about ourselves. This mathematics does. Whether it’s something as instantly accessible as, say, RobertLovesPi‘s illustrations of geometrical structures, or something as involved as the five-color map theorem mathematics gives us something. This isn’t any excuse to slum, though.
Rob Harrell’s Big Top rerun for the 29th features a word problem. It’s cast in terms of what a lion might find interesting. Cultural expectations are inseparable from the mathematics we do, however much we might find universal truths about them. Word problems make the cultural biases more explicit, though. Also, note that Harrell shows an important lesson for artists in the final panel: whenever possible, draw animals wearing glasses.
Samson’s Dark Side Of The Horse for the 29th is another sheep-counting joke. As Samson will often do this includes different representations of numbers before it all turns to chaos in the end. This is why some of us can’t sleep.
Somehow this is not the title of every Reading The Comics review! But it is for this post and we’ll explore why below.
Piers Baker’s Ollie and Quentin for the 18th is a Zeno’s Paradox-based joke. This uses the most familiar of Zeno’s Paradoxes, about the problem of covering any distance needing infinitely many steps to be done in a finite time. Zeno’s Paradoxes are often dismissed these days (probably were then, too), on the grounds that the Ancient Greeks Just Didn’t Understand about convergence. Hardly; they were as smart as we were. Zeno had a set of paradoxes, built on the questions of whether space and time are infinitely divisible or whether they’re not. Any answer to one paradox implies problems in others. There’s things we still don’t really understand about infinity and infinitesimals and continuity. Someday I should do a proper essay about them.
Dave Coverly’s Speed Bump for the 18th is not exactly an anthropomorphic-numerals joke. It is about making symbols manifest in the real world, at least. The greater-than and less-than signs as we know them were created by the English mathematician Thomas Harriot, and introduced to the world in his posthumous Artis Analyticae Praxis (1631). He also had an idea of putting a . between the numerals of an expression and the letters multiplied by them, for example, “4.x” to mean four times x. We mostly do without that now, taking multiplication as assumed if two meaningful quantities are put next to one another. But we will use, now, a vertically-centered dot to separate terms multiplied together when that helps our organization. The equals sign we trace to the 16th century mathematician Robert Recorde, whose 1557 Whetsone of Witte uses long but recognizable equals signs. The = sign went into hibernation after that, though, until the 17th century and it took some time to quite get well-used. So it often is with symbols.
Ted Shearer’s Quincy for the 25th of April, 1978 and rerun the 19th of June, starts from the history of zero. It’s worth noting there are a couple of threads woven together in the concept of zero. One is the idea of “nothing”, which we’ve had just forever. I mean, the idea that there isn’t something to work with. Another is the idea of the … well, the additive identity, there being some number that’s one less than one and two less than two. That you can add to anything without changing the thing. And then there’s symbols. There’s the placeholder for “there are no examples of this quantity here”. There’s the denotation of … well, the additive identity. All these things are zeroes, and if you listen closely, they are not quite the same thing. Which is not weird. Most words mean a collection of several concepts. We’re lucky the concepts we mean by “zero” are so compatible in meaning. Think of the poor person trying to understand the word “bear”, or “cleave”.
John Deering’s Strange Brew for the 19th is a “New Math” joke, fittingly done with cavemen. Well, numerals were new things once. Amusing to me is that — while I’m not an expert — in quite a few cultures the symbol for “one” was pretty much the same thing, a single slash mark. It’s hard not to suppose that numbers started out with simple tallies, and the first thing to tally might get dressed up a bit with serifs or such but is, at heart, the same thing you’d get jabbing a sharp thing into a soft rock.
Guy Gilchrist’s Today’s Dogg for the 19th I’m sure is a rerun and I think I’ve featured it here before. So be it. It’s silly symbol-play and dog arithmetic. It’s a comic strip about how dogs are cute; embrace it or skip it.
Zach Weinersmith’s Saturday Morning Breakfast Cereal is properly speaking reruns when it appears on GoComics.com. For whatever reason Weinersmith ran a patch of mathematics strips there this past week. So let me bundle all that up. On the 19th he did a joke mathematicians get a lot, about how the only small talk anyone has about mathematics is how they hated mathematics. I’m not sure mathematicians have it any better than any other teachers, though. Have you ever known someone to say, “My high school gym class gave me a greater appreciation of the world”? Or talk about how grade school history opened their eyes to the wonders of the subject? It’s a sad thing. But there are a lot of things keeping teachers from making students feel joy in their subjects.
For the 21st Weinersmith makes a statisticians joke. I can wrangle some actual mathematics out of an otherwise correctly-formed joke. How do we ever know that something is true? Well, we gather evidence. But how do we know the evidence is relevant? Even if the evidence is relevant, how do we know we’ve interpreted it correctly? Even if we have interpreted it correctly, how do we know that it shows what we want to know? Statisticians become very familiar with hypothesis testing, which amounts to the question, “does this evidence indicate that some condition is implausibly unlikely”? And they can do great work with that. But “implausibly unlikely” is not the same thing as “false”. A person knowledgeable enough and honest turns out to have few things that can be said for certain.
The June 23rd strip I’ve seen go around Mathematics Twitter several times, as see above tweet, about the ways in which mathematical literacy would destroy modern society. It’s a cute and flattering portrait of mathematics’ power, probably why mathematicians like passing it back and forth. But … well, how would “logic” keep people from being fooled by scams? What makes a scam work is that the premise seems logical. And real-world problems — as opposed to logic-class problems — are rarely completely resolvable by deductive logic. There have to be the assumptions, the logical gaps, and the room for humbuggery that allow hoaxes and scams to slip through. And does anyone need a logic class to not “buy products that do nothing”? And what is “nothing”? I have more keychains than I have keys to chain, even if we allow for emergencies and reasonable unexpected extra needs. This doesn’t stop my buying keychains as souvenirs. Does a Penn Central-logo keychain “do nothing” merely because it sits on the windowsill rather than hold any sort of key? If so, was my love foolish to buy it as a present? Granted that buying a lottery ticket is a foolish use of money; is my life any worse for buying that than, say, a peanut butter cup that I won’t remember having eaten a week afterwards? As for credit cards — It’s not clear to me that people max out their credit cards because they don’t understand they will have to pay it back with interest. My experience has been people max out their credit cards because they have things they must pay for and no alternative but going further into debt. That people need more money is a problem of society, yes, but it’s not clear to me that a failure to understand differential equations is at the heart of it. (Also, really, differential equations are overkill to understand credit card debt. A calculator with a repeat-the-last-operation feature and ten minutes to play is enough.)
For the first time in ages there aren’t enough mathematically-themed comic strips to justify my cutting the week’s roundup in two. No, I have no idea what I’m going to write about for Thursday. Let’s find out together.
Jenny Campbell’s Flo and Friends for the 19th faintly irritates me. Flo wants to make sure her granddaughter understands that just because it takes people on average 14 minutes to fall asleep doesn’t mean that anyone actually does, by listing all sorts of reasons that a person might need more than fourteen minutes to sleep. It makes me think of a behavior John Allen Paulos notes in Innumeracy, wherein the statistically wise points out that someone has, say, a one-in-a-hundred-million chance of being killed by a terrorist (or whatever) and is answered, “ah, but what if you’re that one?” That is, it’s a response that has the form of wisdom without the substance. I notice Flo doesn’t mention the many reasons someone might fall asleep in less than fourteen minutes.
But there is something wise in there nevertheless. For most stuff, the average is the most common value. By “the average” I mean the arithmetic mean, because that is what anyone means by “the average” unless they’re being difficult. (Mathematicians acknowledge the existence of an average called the mode, which is the most common value (or values), and that’s most common by definition.) But just because something is the most common result does not mean that it must be common. Toss a coin fairly a hundred times and it’s most likely to come up tails 50 times. But you shouldn’t be surprised if it actually turns up tails 51 or 49 or 45 times. This doesn’t make 50 a poor estimate for the average number of times something will happen. It just means that it’s not a guarantee.
Gary Wise and Lance Aldrich’s Real Life Adventures for the 19th shows off an unusually dynamic camera angle. It’s in service for a class of problem you get in freshman calculus: find the longest pole that can fit around a corner. Oh, a box-spring mattress up a stairwell is a little different, what with box-spring mattresses being three-dimensional objects. It’s the same kind of problem. I want to say the most astounding furniture-moving event I’ve ever seen was when I moved a fold-out couch down one and a half flights of stairs single-handed. But that overlooks the caged mouse we had one winter, who moved a Chinese finger-trap full of crinkle paper up the tight curved plastic to his nest by sheer determination. The trap was far longer than could possibly be curved around the tube. We have no idea how he managed it.
J R Faulkner’s Promises, Promises for the 20th jokes that one could use Roman numerals to obscure calculations. So you could. Roman numerals are terrible things for doing arithmetic, at least past addition and subtraction. This is why accountants and mathematicians abandoned them pretty soon after learning there were alternatives.
Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for the week. Probably anything would do for the blackboard problem, but something geometry reads very well.
Jef Mallett’s Frazz for the 21st makes some comedy out of the sort of arithmetic error we all make. It’s so easy to pair up, like, 7 and 3 make 10 and 8 and 2 make 10. It takes a moment, or experience, to realize 78 and 32 will not make 100. Forgive casual mistakes.
Bud Fisher’s Mutt and Jeff rerun for the 22nd is a similar-in-tone joke built on arithmetic errors. It’s got the form of vaudeville-style sketch compressed way down, which is probably why the third panel could be made into a satisfying final panel too.
Bud Blake’s Tiger rerun for the 23rd just name-drops mathematics; it could be any subject. But I need some kind of picture around here, don’t I?
The week started out quite busy and I was expecting I’d have to split my essay again. It didn’t turn out that way; Comic Strip Master Command called a big break on mathematically-themed comics from Tuesday on. And then nobody from Comics Kingdom or from Creators.com needed inclusion either. I just have a bunch of GoComics links and a heap of text here. I bet that changes by next week. Still no new Jumble strips.
Kevin Fagan’s Drabble for the 22nd uses arithmetic as the sort of problem it’s easy to get clearly right or clearly wrong. It’s a more economical use of space than (say) knowing how many moons Saturn’s known to have. (More than we thought there were as long ago as Thursday.) I do like that there’s a decent moral to this on the way to the punch line.
Bill Amend’s FoxTrot for the 22nd has Jason stand up for “torus” as a better name for doughnuts. You know how nerdy people will like putting a complicated word onto an ordinary thing. But there are always complications. A torus ordinarily describes the shape made by rotating a circle around an axis that’s in the plane of the circle. The result is a surface, though, the shell of a doughnut and none of the interior. If we’re being fussy. I don’t know of a particular name for the torus with its interior and suspect that, if pressed, a mathematician would just say “torus” or maybe “doughnut”.
We can talk about toruses in two dimensions; those look just like circles. The doughnut-shell shape is a torus in three dimensions. There’s torus shapes made by rotating spheres, or hyperspheres, in four or more dimensions. I’m not going to draw them. And we can also talk about toruses by the number of holes that go through them. If a normal torus is the shape of a ring-shaped pool toy, a double torus is the shape of a two-seater pool toy, a triple torus something I don’t imagine exists in the real world. A quadruple torus could look, I imagine, like some pool toys Roller Coaster Tycoon allows in its water parks. I’m saying nothing about whether they’re edible.
Mikael Wulff and Anders Morgenthaler’s TruthFacts for the 23rd jokes about mathematics skills versus life. The growth is fine enough; after all, most of us are at, or get to, our best at something while we’re training in it or making regular use of it. So the joke peters out into the usual “I never use mathematics in real life” crack, which, eh. I agree it’s what I feel like my mathematics skills have done ever since I got my degree, at any rate.
Michael Fry’s Committed rerun for the 28th just riffs on the escalation of hyperbole, and what sure looks like an exponential growth of hyperbolic numbers. There’s a bit of scientific notation in the last panel. The “1 x” part isn’t necessary. It doesn’t change the value of the expression “1 x 1026”. But it might be convenient to use the “1 x” anyway. Scientific notation is about separating the size of the number from the interesting digits that the number has. Often when you compare numbers you’re interested in the size or else you’re interested in the important digits. Get into that habit and it’s not worth making an exception just because the interesting digits turn out to be boring in this case.
Comic Strip Master Command decreed that last week should be busy again. So I’m splitting its strips into two essays. It’s a week that feels like it had more anthropomorphic numerals jokes than usual, but see if I actually count these things.
Mike Peters’s Mother Goose and Grimm for the 15th I figured would be the anthropomorphic numerals joke for the week. Shows what I know. It is an easy joke, but I do appreciate the touch of craft involved in picking the numerals. The joke is just faintly dirty if the numbers don’t add to six. If they were a pair of 3’s, there’d be the unwanted connotations of a pair of twins talking about all this. A 6 and a 0 would make at least one character weirdly obsessed. So it has to be a 4 and a 2, or a 5 and a 1. I imagine Peters knew this instinctively, at this point in his career. It’s one of the things you learn in becoming an expert.
Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 15th is mostly physical comedy, with a touch of — I’m not sure what to call this kind of joke. The one where a little arithmetic error results in bodily harm. In this sort of joke it’s almost always something not being carried that’s the error. I suppose that’s a matter of word economy. “Forgot to carry the (number)” is short, and everybody’s done it. And even if they don’t remember making this error, the phrasing clarifies to people that it’s a little arithmetic mistake. I think in practice mistaking a plus for a minus (or vice-versa) is the more common arithmetic error. But it’s harder to describe that clearly and concisely.
Jef Mallett’s Frazz for the 15th puzzled me. I hadn’t heard this thing the kid says about how if you can “spew ten random lines from a classic movie” to convince people you’ve seen it. (I don’t know the kid’s name; it happens.) I suppose that it would be convincing, though. I certainly know a couple lines from movies I haven’t seen, what with living in pop culture and all that. But ten would be taxing for all but the most over-saturated movies, like any of the Indiana Jones films. (There I’m helped by having played the 90s pinball machine a lot.) Anyway, knowing ten random mathematics things isn’t convincing, especially since you can generate new mathematical things at will just by changing a number. But I would probably be convinced that someone who could describe what’s interesting about ten fields of mathematics had a decent understanding of the subject. That requires remembering more stuff, but then, mathematics is a bigger subject than even a long movie is.
In Bill Holbrook’s On The Fastrack for the 16th Fi speaks of tallying the pluses and minuses of her life. Trying to make life into something that can be counted is an old decision-making technique. I think Benjamin Franklin explained how he found it so useful. It’s not a bad approach if a choice is hard. The challenging part is how to weight each consideration. Getting into fractions seems rather fussy to me, but some things are just like that. There is the connotation here that a fraction is a positive number smaller than 1. But the mathematically-trained (such as Fi) would be comfortable with fractions larger than 1. Or also smaller than zero. “Fraction” is no more bounded than “real number”. So, there’s the room for more sweetness here than might appear to the casual reader.
Scott Hilburn’s The Argyle Sweater for the 16th is the next anthropomorphic numerals joke for this week. I’m glad Hilburn want to be in my pages more. 5’s concern about figuring out x might be misplaced. We use variables for several purposes. One of them is as a name to give a number whose value we don’t know but wish to work out, and that’s how we first see them in high school algebra. But a variable might also be a number whose value we don’t particularly care about and will never try to work out. This could be because the variable is a parameter, with a value that’s fixed for a problem but not what we’re interested in. We don’t typically use ‘x’ for that, though; usually parameter are something earlier in the alphabet. That’s merely convention, but it is convention that dates back to René Descartes. Alternatively, we might use ‘x’ as a dummy variable. A dummy variable serves the same role that falsework on a building or a reference for an artistic sketch does. We use dummy variables to organize and carry out work, but we don’t care what its values are and we don’t even see the dummy variable in the final result. A dummy variable can be any name, but ‘x’ and ‘t’ are popular choices.
Terry LaBan and Patty LaBan’s Edge City rerun for the 16th plays on the idea that mathematics people talk in algebra. Funny enough, although, “the opposing defense is a variable of 6”? That’s an idiosyncratic use of “variable”. I’m going to suppose that Charles is just messing with Len’s head because, really, it’s fun doing a bit of that.
So now let me get to the other half of last week’s comics. Also, not to spoil things, but this coming week is looking pretty busy so I may have anothe split-week Reading the Comics coming up. The shocking thing this time is that the Houston Chronicle has announced it’s discontinuing its comics page. I don’t know why; I suppose because they’re fed up with people coming loyally to a daily feature. I will try finding alternate sources for the things I had still been reading there, but don’t know if I’ll make it.
I’m saddened by this. Back in the 90s comics were just coming onto the Internet. The Houston Chronicle was one of a couple newspapers that knew what to do with them. It, and the Philadelphia Inquirer and the San Jose Mercury-News, had exactly what we wanted in comics: you could make a page up of all the strips you wanted to read, and read them on a single page. You could even go backwards day by day in case you missed some. The Philadelphia Inquirer was the only page that let you put the comics in the order you wanted, as opposed to alphabetical order by title. But if you were unafraid of opening up URLs you could reorder the Houston Chronicle page you built too.
And those have all faded away. In the interests of whatever interest is served by web site redesigns all these papers did away with their user-buildable comics pages. The Chronicle was the last holdout, but even they abolished their pages a few years ago, with a promise for a while that they’d have a replacement comics-page scheme up soon. It never came and now, I suppose, never will.
Most of the newspapers’ sites had become redundant anyway. Comics Kingdom and GoComics.com offer user-customizable comics pages, with a subscription model that makes it clear that money ought to be going to the cartoonists. I still had the Chronicle for a few holdouts, like Joe Martin’s strips or the Jumble feature. And from that inertia that attaches to long-running Internet associations.
So among the other things January 2017 takes away from us, it is taking the last, faded echo of the days in the 1990s when newspapers saw comics as awesome things that could be made part of their sites.
Lorie Ransom’s The Daily Drawing for the 11th is almost but not quite the anthropomorphized-numerals joke for this installment. It’s certainly the most numerical duck content I’ve got on record.
Tom II Wilson’s Ziggy for the 11th is an Early Pi Day joke. Cosmically there isn’t any reason we couldn’t use π in take-a-number dispensers, after all. Their purpose is to give us some certain order in which to do things. We could use any set of numbers which can be put in order. So the counting numbers work. So do the integers. And the real numbers. But practicality comes into it. Most people have probably heard that π is a bit bigger than 3 and a fair bit smaller than 4. But pity the two people who drew and figuring out who gets to go first. Still, I won’t be surprised if some mathematics-oriented place uses a gimmick like this, albeit with numbers that couldn’t be confused. At least not confused by people who go to mathematics-oriented places. That would be for fun rather than cake.
I can’t promise that the Jumble for the 11this the last one I’ll ever feature here. I might find where David L Hoyt and Jeff Knurek keep a linkable reference to their strips and point to them. But just in case of the worst here’s an abacus gag for you to work on.
Corey Pandolph, Phil Frank, and Joe Troise’s The Elderberries for the 12th is, I have to point out, a rerun. So if you’re trying to do the puzzle the reference to “the number of the last president” isn’t what you’re thinking of. It is an example of the conflation of intelligence with skill at arithmetic. It’s also an example the conflation of intelligence with a mastery of trivia. But I think it leans on arithmetic more. I am not sure when this strip first appeared. “The last president” might have been Bill Clinton (42) or George W Bush (43). But this means we’re taking the square root of either 33 or 34. And there’s no doing that in your head. The square root of a whole number is either a whole number — the way the square root of 36 is — or else it’s an irrational number. You can work out the square root of a non-perfect-square by hand. But it’s boring and it’s worse than just writing “” or “”. Except in figuring out if that number is larger than or smaller than five or six. It’s good for that.
Dave Blazek’s Loose Parts for the 13th is the actuary joke for this installment. Actuarial studies are built on one of the great wonders of statistics: that it is possible to predict how often things will happen. They can happen to a population, as in forecasts of how many people will be in traffic accidents or fires or will lose their jobs or will move to a new city. We may have no idea to whom any of these will happen, and they may have no way of guessing, but the enormous number of people and great number of things that can combine to make a predictable state of affairs. I suppose it’s imaginable that a group could study its dynamics well enough to identify who screws up the most and most seriously. So they might be able to say what the odds are it is his fault. But I imagine in practice it’s too difficult to define screw-ups or to assign fault consistently enough to get the data needed.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is another multiverse strip, echoing the Dinosaur Comics I featured here Sunday. I’ll echo my comments then. If there is a multiverse — again, there is not evidence for this — then there may be infinitely many versions of every book of the Bible. This suggests, but it does not mandate, that there should be every possible incarnation of the Bible. And a multiverse might be a spendthrift option anyway. Just allow for enough editions, and the chance that any of them will have a misprint at any word or phrase, and we can eventually get infinitely many versions of every book of the Bible. If we wait long enough.
And now I can finish off last week’s mathematically-themed comic strips. There’s a strong theme to them, for a refreshing change. It would almost be what we’d call a Comics Synchronicity, on Usenet group rec.arts.comics.strips, had they all appeared the same day. Some folks claiming to be open-minded would allow a Synchronicity for strips appearing on subsequent days or close enough in publication, but I won’t have any of that unless it suits my needs at the time.
Ernie Bushmiller’s for the 6th would fit thematically better as a Cameo Edition comic. It mentions arithmetic but only because it’s the sort of thing a student might need a cheat sheet on. I can’t fault Sluggo needing help on adding eight or multiplying by six; they’re hard. Not remembering 4 x 2 is unusual. But everybody has their own hangups. The strip originally ran the 6th of December, 1949.
Bill holbrook’s On The Fastrack for the 7th seems like it should be the anthropomorphic numerals joke for this essay. It doesn’t seem to quite fit the definition, but, what the heck.
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers on the 7th starts off the run of E = mc2 jokes for this essay. This one reminds me of Gary Larson’s Far Side classic with the cleaning woman giving Einstein just that little last bit of inspiration about squaring things away. It shouldn’t surprise anyone that E equalling m times c squared isn’t a matter of what makes an attractive-looking formula. There’s good reasons when one thinks what energy and mass are to realize they’re connected like that. Einstein’s famous, deservedly, for recognizing that link and making it clear.
Mark Pett’s Lucky Cow rerun for the 7th has Claire try to use Einstein’s famous quote to look like a genius. The mathematical content is accidental. It could be anything profound yet easy to express, and it’s hard to beat the economy of “E = mc2” for both. I’d agree that it suggests Claire doesn’t know statistics well to suppose she could get a MacArthur “Genius” Grant by being overheard by a grant nominator. On the other hand, does anybody have a better idea how to get their attention?
Harley Schwadron’s 9 to 5 for the 8th completes the “E = mc2” triptych. Calling a tie with the equation on it a power tie elevates the gag for me. I don’t think of “E = mc2” as something that uses powers, even though it literally does. I suppose what gets me is that “c” is a constant number. It’s the speed of light in a vacuum. So “c2” is also a constant number. In form the equation isn’t different from “E = m times seven”, and nobody thinks of seven as a power.
Morrie Turner’s Wee Pals rerun for the 8th is a bit of mathematics wordplay. It’s also got that weird Morrie Turner thing going on where it feels unquestionably earnest and well-intentioned but prejudiced in that way smart 60s comedies would be.
Mort Walker’s Beetle Bailey for the 18th of May, 1960 was reprinted on the 9th. It mentions mathematics — algebra specifically — as the sort of thing intelligent people do. I’m going to take a leap and suppose it’s the sort of algebra done in high school about finding values of ‘x’ rather than the mathematics-major sort of algebra, done with groups and rings and fields. I wonder when holding a mop became the signifier of not just low intelligence but low ambition. It’s subverted in Jef Mallet’s Frazz, the title character of which works as a janitor to support his exercise and music habits. But it is a standard prop to signal something.
Here I’m just closing out last week’s mathematically-themed comics. The new week seems to be bringing some more in at a good pace, too. Should have stuff to talk about come Sunday.
Darrin Bell and Theron Heir’s Rudy Park for the 24th brings out the ancient question, why do people need to do mathematics when we have calculators? As befitting a comic strip (and Sadie’s character) the question goes unanswered. But it shows off the understandable confusion people have between mathematics and calculation. Calculation is a fine and necessary thing. And it’s fun to do, within limits. And someone who doesn’t like to calculate probably won’t be a good mathematician. (Or will become one of those master mathematicians who sees ways to avoid calculations in getting to an answer!) But put aside the obviou that we need mathematics to know what calculations to do, or to tell whether a calculation done makes sense. Much of what’s interesting about mathematics isn’t a calculation. Geometry, for an example that people in primary education will know, doesn’t need more than slight bits of calculation. Group theory swipes a few nice ideas from arithmetic and builds its own structure. Knot theory uses polynomials — everything does — but more as a way of naming structures. There aren’t things to do that a calculator would recognize.
John Atkinson’s Wrong Hands for the 25th is the awaited anthropomorphic-numerals and symbols joke for this past week. I enjoy the first commenter’s suggestion tha they should have stayed in unknown territory.
Rick Kirkman and Jerry Scott’s Baby Blues for the 26th does a little wordplay built on pre-algebra. I’m not sure that Zoe is quite old enough to take pre-algebra. But I also admit not being quite sure what pre-algebra is. The central idea of (primary school) algebra — that you can do calculations with a number without knowing what the number is — certainly can use some preparatory work. It’s a dazzling idea and needs plenty of introduction. But my dim recollection of taking it was that it was a bit of a subject heap, with some arithmetic, some number theory, some variables, some geometry. It’s all stuff you’ll need once algebra starts. But it is hard to say quickly what belongs in pre-algebra and what doesn’t.
Art Sansom and Chip Sansom’s The Born Loser for the 26th uses two ancient staples of jokes, probabilities and weather forecasting. It’s a hard joke not to make. The prediction for something is that it’s very unlikely, and it happens anyway? We all laugh at people being wrong, which might be our whistling past the graveyard of knowing we will be wrong ourselves. It’s hard to prove that a probability is wrong, though. A fairly tossed die may have only one chance in six of turning up a ‘4’. But there’s no reason to think it won’t, and nothing inherently suspicious in it turning up ‘4’ four times in a row.
We could do it, though. If the die turned up ‘4’ four hundred times in a row we would no longer call it fair. (This even if examination proved the die really was fair after all!) Or if it just turned up a ‘4’ significantly more often than it should; if it turned up two hundred times out of four hundred rolls, say. But one or two events won’t tell us much of anything. Even the unlikely happens sometimes.
Even the impossibly unlikely happens if given enough attempts. If we do not understand that instinctively, we realize it when we ponder that someone wins the lottery most weeks. Presumably the comic’s weather forecaster supposed the chance of snow was so small it could be safely rounded down to zero. But even something with literally zero percent chance of happening might.
Imagine tossing a fair coin. Imagine tossing it infinitely many times. Imagine it coming up tails every single one of those infinitely many times. Impossible: the chance that at least one toss of a fair coin will turn up heads, eventually, is 1. 100 percent. The chance heads never comes up is zero. But why could it not happen? What law of physics or logic would it defy? It challenges our understanding of ideas like “zero” and “probability” and “infinity”. But we’re well-served to test those ideas. They hold surprises for us.
The mathematically-themed comic strips of the past week tended to touch on some classic topics and classic motifs. That’s enough for me to declare a title for these comics. Enjoy, won’t you please?
John McPherson’s Close To Home for the 9th uses the classic board full of mathematics to express deep thinking. And it’s deep thinking about sports. Nerds like to dismiss sports as trivial and so we get the punch line out of this. But models of sports have been one of the biggest growth fields in mathematics the past two decades. And they’ve shattered many longstanding traditional understandings of strategy. It’s not proper mathematics on the board, but that’s all right. It’s not proper sabermetrics either.
Vic Lee’s Pardon My Planet for the 10th is your classic joke about putting mathematics in marketable terms. There is an idea that a mathematical idea to be really good must be beautiful. And it’s hard to say exactly what beauty is, but “short” and “simple” seem to be parts of it. That’s a fine idea, as long as you don’t forget how context-laden these are. Whether an idea is short depends on what ideas and what concepts you have as background. Whether it’s simple depends on how much you’ve seen similar ideas before. π looks simple. “The smallest positive root of the solution to the differential equation y”(x) = -y(x) where y(0) = 0 and y'(0) = 1” looks hard, but however much mathematics you know, rhetoric alone tells you those are the same thing.
Scott Hilburn’s The Argyle Sweater for the 10th is your classic anthropomorphic-numerals joke. Well, anthropomorphic-symbols in this case. But it’s the same genre of joke.
Randy Glasbergen’s Glasbergen Cartoons rerun for the 10th is your classic sudoku-and-arithmetic-as-hard-work joke. And it’s neat to see “programming a VCR” used as an example of the difficult-to-impossible task for a comic strip drawn late enough that it’s into the era of flat-screen, flat-bodied desktop computers.
Bill Holbrook’s On The Fastrack for 11th is your classic grumbling-about-how-mathematics-is-understood joke. Well, statistics, but most people consider that part of mathematics. (One could mount a strong argument that statistics is as independent of mathematics as physics or chemistry are.) Statistics offers many chances for intellectual mischief, whether deliberately or just from not thinking matters through. That may be inevitable. Sampling, as in political surveys, must talk about distributions, about ranges of possible results. It’s hard to be flawless about that.
That said I’m not sure I can agree with Fi in her example here. Take her example, a political poll with three-point margin of error. If the poll says one candidate’s ahead by three points, Fi asserts, they’ll say it’s tied when it’s as likely the lead is six. I don’t see that’s quite true, though. When we sample something we estimate the value of something in a population based on what it is in the sample. Obviously we’ll be very lucky if the population and the sample have exactly the same value. But the margin of error gives us a range of how far from the sample value it’s plausible the whole population’s value is, or would be if we could measure it. Usually “plausible” means 95 percent; that is, 95 percent of the time the actual value will be within the margin of error of the sample’s value.
So here’s where I disagree with Fi. Let’s suppose that the first candidate, Kirk, polls at 43 percent. The second candidate, Picard, polls at 40 percent. (Undecided or third-party candidates make up the rest.) I agree that Kirk’s true, whole-population, support is equally likely to be 40 percent or 46 percent. But Picard’s true, whole-population, support is equally likely to be 37 percent or 43 percent. Kirk’s lead is actually six points if his support was under-represented in the sample and Picard’s was over-represented, by the same measures. But suppose Kirk was just as over-represented and Picard just as under-represented as they were in the previous case. This puts Kirk at 40 percent and Picard at 43 percent, a Kirk-lead of minus three percentage points.
So what’s the actual chance these two candidates are tied? Well, you have to say what a tie is. It’s vanishingly impossible they have precisely the same true support and we can’t really calculate that. Don’t blame statisticians. You tell me an election in which one candidate gets three more votes than the other isn’t really tied, if there are more than seven votes cast. We can work on “what’s the chance their support is less than some margin?” And then you’d have all the possible chances where Kirk gets a lower-than-surveyed return while Picard gets a higher-than-surveyed return. I can’t say what that is offhand. We haven’t said what this margin-of-tying is, for one thing.
But it is certainly higher than the chance the lead is actually six; that only happens if the actual vote is different from the poll in one particular way. A tie can happen if the actual vote is different from the poll in many different ways.
Doing a quick and dirty little numerical simulation suggests to me that, if we assume the sampling respects the standard normal distribution, then in this situation Kirk probably is ahead of Picard. Given a three-point lead and a three-point margin for error Kirk would be expected to beat Picard about 92 percent of the time, while Picard would win about 8 percent of the time.
Here I have been making the assumption that Kirk’s and Picard’s support are to an extent independent. That is, a vote might be for Kirk or for Picard or for neither. There’s this bank of voting-for-neither-candidate that either could draw on. If there are no undecided candidates, every voter is either Kirk or Picard, then all of this breaks down: Kirk can be up by six only if Picard is down by six. But I don’t know of surveys that work like that.
Not to keep attacking this particular strip, which doesn’t deserve harsh treatment, but it gives me so much to think about. Assuming by “they” Fi means news anchors — and from what we get on panel, it’s not actually clear she does — I’m not sure they actually do “say the poll is tied”. What I more often remember hearing is that the difference is equal to, or less than, the survey’s margin of error. That might get abbreviated to “a statistical tie”, a usage that I think is fair. But Fi might mean the candidates or their representatives in saying “they”. I can’t fault the campaigns for interpreting data in ways useful for their purposes. The underdog needs to argue that the election can yet be won. The leading candidate needs to argue against complacency. In either case a tie is a viable selling point and a reasonable interpretation of the data.
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 12th is a classic use of Einstein and general relativity to explain human behavior. Everyone’s tempted by this. Usually it’s thermodynamics that inspires thoughts that society could be explained mathematically. There’s good reason for this. Thermodynamics builds great and powerful models of complicated systems by supposing that we never know, or need to know, what any specific particle of gas or fluid is doing. We care only about aggregate data. That statistics shows we can understand much about humanity without knowing fine details reinforces this idea. The Wingartens and Clark probably shifted from thermodynamics to general relativity because Einstein is recognizable to normal people. And we’ve all at least heard of mass warping space and can follow the metaphor to money warping law.
In vintage comics, Dan Barry’s Flash Gordon for the 14th (originally run the 28th of November, 1961) uses the classic idea that sufficient mathematics talent will outwit games of chance. Many believe it. I remember my grandmother’s disappointment that she couldn’t bring the underaged me into the casinos in Atlantic City. This did save her the disappointment of learning I haven’t got any gambling skill besides occasionally buying two lottery tickets if the jackpot is high enough. I admit that an irrational move on my part, but I can spare two dollars for foolishness once or twice a year. The idea of beating a roulette wheel, at least a fair wheel, isn’t absurd. In principle if you knew enough about how the wheel was set up and how the ball was weighted and how it was launched into the spin you could predict where it would land. In practice, good luck. I wouldn’t be surprised if a good roulette wheel weren’t chaotic, or close to it. If it’s chaotic then while the outcome could be predicted if the wheel’s spin and the ball’s initial speed were known well enough, they can’t be measured well enough for a prediction to be meaningful. The comic also uses the classic word balloon full of mathematical symbols to suggest deep reasoning. I spotted Einstein’s famous quote there.
The last week in mathematically themed comics was a pleasant one. By “a pleasant one” I mean Comic Strip Master Command sent enough comics out that I feel comfortable splitting them across two essays. Look for the other half of the past week’s strips in a couple days at a very similar URL.
Mac King and Bill King’s Magic in a Minute feature for the 2nd shows off a bit of number-pattern wonder. Set numbers in order on a four-by-four grid and select four as directed and add them up. You get the same number every time. It’s a cute trick. I would not be surprised if there’s some good group theory questions underlying this, like about what different ways one could arrange the numbers 1 through 16. Or for what other size grids the pattern will work for: 2 by 2? (Obviously.) 3 by 3? 5 by 5? 6 by 6? I’m not saying I actually have been having fun doing this. I just sense there’s fun to be had there.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 2nd is based on one of those weirdnesses of the way computers add. I remember in the 90s being on a Java mailing list. Routinely it would draw questions from people worried that something was very wrong, as adding 0.01 to a running total repeatedly wouldn’t get to exactly 1.00. Java was working correctly, in that it was doing what the specifications said. It’s just the specifications didn’t work quite like new programmers expected.
What’s going on here is the same problem you get if you write down 1/3 as 0.333. You know that 1/3 plus 1/3 plus 1/3 ought to be 1 exactly. But 0.333 plus 0.333 plus 0.333 is 0.999. 1/3 is really a little bit more than 0.333, but we skip that part because it’s convenient to use only a few points past the decimal. Computers normally represent real-valued numbers with a scheme called floating point representation. At heart, that’s representing numbers with a couple of digits. Enough that we don’t normally see the difference between the number we want and the number the computer represents.
Every number base has some rational numbers it can’t represent exactly using finitely many digits. Our normal base ten, for example, has “one-third” and “two-third”. Floating point arithmetic is built on base two, and that has some problems with tenths and hundredths and thousandths. That’s embarrassing but in the main harmless. Programmers learn about these problems and how to handle them. And if they ask the mathematicians we tell them how to write code so as to keep these floating-point errors from growing uncontrollably. If they ask nice.
Random Acts of Nancy for the 3rd is a panel from Ernie Bushmiller’s Nancy. That panel’s from the 23rd of November, 1946. And it just uses mathematics in passing, arithmetic serving the role of most of Nancy’s homework. There’s a bit of spelling (I suppose) in there too, which probably just represents what’s going to read most cleanly. Random Acts is curated by Ernie Bushmiller fans Guy Gilchrist (who draws the current Nancy) and John Lotshaw.
Thom Bluemel’s Birdbrains for the 4th depicts the discovery of a new highest number. When humans discovered ‘1’ is, I would imagine, probably unknowable. Given the number sense that animals have it’s probably something that predates humans, that it’s something we’re evolved to recognize and understand. A single stroke for 1 seems to be a common symbol for the number. I’ve read histories claiming that a culture’s symbol for ‘1’ is often what they use for any kind of tally mark. Obviously nothing in human cultures is truly universal. But when I look at number symbols other than the Arabic and Roman schemes I’m used to, it is usually the symbol for ‘1’ that feels familiar. Then I get to the Thai numeral and shrug at my helplessness.
Bill Amend’s FoxTrot Classics for the 4th is a rerun of the strip from the 11th of October, 2005. And it’s made for mathematics people to clip out and post on the walls. Jason and Marcus are in their traditional nerdly way calling out sequences of numbers. Jason’s is the Fibonacci Sequence, which is as famous as mathematics sequences get. That’s the sequence of numbers in which every number is the sum of the previous two terms. You can start that sequence with 0 and 1, or with 1 and 1, or with 1 and 2. It doesn’t matter.
Marcus calls out the Perrin Sequence, which I neve heard of before either. It’s like the Fibonacci Sequence. Each term in it is the sum of two other terms. Specifically, each term is the sum of the second-previous and the third-previous terms. And it starts with the numbers 3, 0, and 2. The sequence is named for François Perrin, who described it in 1899, and that’s as much as I know about him. The sequence describes some interesting stuff. Take n points and put them in a ‘cycle graph’, which looks to the untrained eye like a polygon with n corners and n sides. You can pick subsets of those points. A maximal independent set is the biggest subset you can make that doesn’t fit into another subset. And the number of these maximal independent sets in a cyclic graph is the n-th number in the Perrin sequence. I admit this seems like a nice but not compelling thing to know. But I’m not a cyclic graph kind of person so what do I know?
I know it seems like when I write these essays I spend the most time on the first comic in the bunch and give the last ones a sentence, maybe two at most. I admit when there’s a lot of comics I have to write up at once my energy will droop. But Comic Strip Master Command apparently wants the juiciest topics sent out earlier in the week. I have to follow their lead.
Stephen Beals’s Adult Children for the 14th uses mathematics to signify deep thinking. In this case Claremont, the dog, is thinking of the Riemann Zeta function. It’s something important in number theory, so longtime readers should know this means it leads right to an unsolved problem. In this case it’s the Riemann Hypothesis. That’s the most popular candidate for “what is the most important unsolved problem in mathematics right now?” So you know Claremont is a deep-thinking dog.
The big Σ ordinary people might recognize as representing “sum”. The notation means to evaluate, for each legitimate value of the thing underneath — here it’s ‘n’ — the value of the expression to the right of the Sigma. Here that’s . Then add up all those terms. It’s not explicit here, but context would make clear, n is positive whole numbers: 1, 2, 3, and so on. s would be a positive number, possibly a whole number.
The big capital Pi is more mysterious. It’s Sigma’s less popular brother. It means “product”. For each legitimate value of the thing underneath it — here it’s “p” — evaluate the expression on the right. Here that’s . Then multiply all that together. In the context of the Riemann Zeta function, “p” here isn’t just any old number, or even any old whole number. It’s only the prime numbers. Hence the “p”. Good notation, right? Yeah.
This particular equation, once shored up with the context the symbols live in, was proved by Leonhard Euler, who proved so much you sometimes wonder if later mathematicians were needed at all. It ties in to how often whole numbers are going to be prime, and what the chances are that some set of numbers are going to have no factors in common. (Other than 1, which is too boring a number to call a factor.) But even if Claremont did know that Euler got there first, it’s almost impossible to do good new work without understanding the old.
Charlos Gary’s Working It Out for the 14th is this essay’s riff on pie charts. Or bar charts. Somewhere around here the past week I read that a French idiom for the pie chart is the “cheese chart”. That’s a good enough bit I don’t want to look more closely and find out whether it’s true. If it turned out to be false I’d be heartbroken.
Ryan North’s Dinosaur Comics for the 15th talks about everyone’s favorite physics term, entropy. Everyone knows that it tends to increase. Few advanced physics concepts feel so important to everyday life. I almost made one expression of this — Boltzmann’s H-Theorem — a Theorem Thursday post. I might do a proper essay on it yet. Utahraptor describes this as one of “the few statistical laws of physics”, which I think is a bit unfair. There’s a lot about physics that is statistical; it’s often easier to deal with averages and distributions than the mass of real messy data.
Utahraptor’s right to point out that it isn’t impossible for entropy to decrease. It can be expected not to, in time. Indeed decent scientists thinking as philosophers have proposed that “increasing entropy” might be the only way to meaningfully define the flow of time. (I do not know how decent the philosophy of this is. This is far outside my expertise.) However: we would expect at least one tails to come up if we simultaneously flipped infinitely many coins fairly. But there is no reason that it couldn’t happen, that infinitely many fairly-tossed coins might all come up heads. The probability of this ever happening is zero. If we try it enough times, it will happen. Such is the intuition-destroying nature of probability and of infinitely large things.
Tony Cochran’s Agnes on the 16th proposes to decode the Voynich Manuscript. Mathematics comes in as something with answers that one can check for comparison. It’s a familiar role. As I seem to write three times a month, this is fair enough to say to an extent. Coming up with an answer to a mathematical question is hard. Checking the answer is typically easier. Well, there are many things we can try to find an answer. To see whether a proposed answer works usually we just need to go through it and see if the logic holds. This might be tedious to do, especially in those enormous brute-force problems where the proof amounts to showing there are a hundred zillion special cases and here’s an answer for each one of them. But it’s usually a much less hard thing to do.
Johnny Hart and Brant Parker’s Wizard of Id Classics for the 17th uses what seems like should be an old joke about bad accountants and nepotism. Well, you all know how important bookkeeping is to the history of mathematics, even if I’m never that specific about it because it never gets mentioned in the histories of mathematics I read. And apparently sometime between the strip’s original appearance (the 20th of August, 1966) and my childhood the Royal Accountant character got forgotten. That seems odd given the comic potential I’d imagine him to have. Sometimes a character’s only good for a short while is all.
Mark Anderson’s Andertoons for the 18th is the Andertoons representative for this essay. Fair enough. The kid speaks of exponents as a kind of repeating oneself. This is how exponents are inevitably introduced: as multiplying a number by itself many times over. That’s a solid way to introduce raising a number to a whole number. It gets a little strained to describe raising a number to a rational number. It’s a confusing mess to describe raising a number to an irrational number. But you can make that logical enough, with effort. And that’s how we do make the idea rigorous. A number raised to (say) the square root of two is something greater than the number raised to 1.4, but less than the number raised to 1.5. More than the number raised to 1.41, less than the number raised to 1.42. More than the number raised to 1.414, less than the number raised to 1.415. This takes work, but it all hangs together. And then we ask about raising numbers to an imaginary or complex-valued number and we wave that off to a higher-level mathematics class.
Lachowski’s Get A Life for the 18th is the sudoku joke for this essay. It’s also a representative of the idea that any mathematical thing is some deep, complicated puzzle at least as challenging as calculating one’s taxes. I feel like this is a rerun, but I don’t see any copyright dates. Sudoku jokes like this feel old, but comic strips have been known to make dated references before.
Samson’s Dark Side Of The Horse for the 19th is this essay’s Dark Side Of The Horse gag. I thought initially this was a counting-sheep in a lab coat. I’m going to stick to that mistaken interpretation because it’s more adorable that way.
This past week was refreshing. The mathematics comics appeared at a regular, none-too-excessive pace. And some old familiar friends reappeared. Some were comic strips that haven’t been around in a while. Some were jokes that haven’t been. Enjoy.
Scott Hilburn’s The Argyle Sweater gets back in my review here for the first time since April, to my amazement. Used to be you couldn’t go two weeks without Hilburn looking for my attention. And here’s the first Roman Numerals joke since … I don’t quite feel up to checking just now. I’m going to go ahead and suppose it’s the first one since the last time Samson’s Dark Side Of The Horse was here.
It’s anachronistic to speak of Ancient Roman students getting ‘C’ grades. Of course it is; it could hardly be otherwise. It’s a joke; how much is that to be worried about? But if I haven’t been mislead the use of letters, A through E-or-F, in student evaluations is an American innovation of the late 19th century. It developed over the 20th century and took over at least American education, in conjunction with the 100-through-0 points evaluation scale. And in parallel to the Grade Point Average, typically with 4.0 as its highest score.
Samson’s Dark Side Of The Horse makes a comfortable visit back here on the 20th. It’s another counting-sheep and number-representation gag. I love the third panel’s artwork.
Mark Anderson’s Andertoons for the 22nd is a joke about motivating mathematics study. I believe I’ve mentioned this before, but there was a lovely bit on The Mary Tyler Moore Show along these lines once. Fantastically stupid newsman Ted Baxter was struggling to do some arithmetic until Murray Slaughter gave him the advice: “put a dollar sign in front of it”. Then he had the answer instantly.
Nat Fakes’s Break of Day for the 22nd brings back mathematics as signifier of the hardest homework a kid can have, or the toughest thing someone can have to think about. Fine enough stuff, although it isn’t really that stunning to think a parent might not understand what the kid’s homework is about. Often the point of an assignment is not to learn how to do something, but to encourage thinking about ways one could do something. That’s a hard assignment to create, and a harder one to do, and a very hard one to help with. As adults we get used to looking at problems as calculations to identify and do as swiftly as possible. That there is value in wandering around the slow routes needs remembering.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 22nd riffs on … I’m not sure exactly. The idea that the sort of meaningless nonsense that makes for good late-night dorm conversations when you’re 20 comes back around to being the cutting edge of theoretical physics, I suppose. It’s funny enough. A complaint often brought against the most cutting edge of theoretical physics is that it’s so abstract that there aren’t any conceivable tests that would say whether a calculation is right or not. In that condition mathematics and theoretical physics merge back into a thread of philosophy and its question of how can we know what it is for something to be true. Once we have a way of discerning whether an idea might happen to be true we’re ejected again from philosophy and into a science. And then the scientist makes a smug, snarky comment about the impossibility of testing philosophical conclusions.
Since the late 19th century much cutting-edge physics has involved counter-intuitive results. Often they have premises that strain intuition, as see relativity, or that seem to violate it altogether, as see quantum mechanics. But they turn out so very right so very often it’s hard not to feel excited and encouraged by this. Who wouldn’t look for a surprising and counter-intuitive explanation for the world as thrilling and maybe even right idea? I don’t blame anyone for looking to a wild idea like “what if the universe is made of math”. I don’t know what that would mean exactly unless we suppose we do live in a universe of Platonic Forms, in which case perfection runs counter-intuitively to me. I do understand being excited by the question. But the answers probably won’t be that much fun.
It’s been a quiet week. There’s not a lot of comic strips telling mathematically-themed jokes. Those that were didn’t give me a lot to talk about. And then on Friday nobody came around to even look at my blog. I exaggerate but only barely; I was down to about a quarter the usual low point of page views. I have no explanation for this and I just hope it doesn’t come up again. That’s the sort of thing that’ll break a mere blogger’s heart.
Mike Baldwin’s Cornered for the 12th uses the traditional blackboard — well, whiteboard — full of mathematics to represent intelligence. The symbols aren’t in enough detail to mean anything,
Jeremy Kaye’s Up and Out for the 13th uses a smaller blackboard (whiteboard) full of mathematics to represent intelligence. Here the symbols are more clearly focused, on Boring High School Algebra. It was looking like this might be the blackboard (well, whiteboard)-themed week.
Dan Piraro’s Bizarro for the 14th I admit I don’t quite get. I get that it’s circling around the invention of mathematics and of architecture and all that. And I expect the need to build stuff efficiently helped inspire people to do mathematics. I’m just not sure how the joke quite fits together here. It happens.
Bill Amend’s Fox Trot Classics for the 17th reruns a storyline in which Jason tries to de-nerdify himself. The use of many digits past the decimal make up a lot of what’s left of Jason’s nerdiness. And since it’s easy to overlook let me point this out: 0.0675 percent is only half of the difference between 99.865 percent and 100 percent. It’s not exactly a classic nerd move to use decimal points when a fraction would be at least as good. Digits have a hypnotic power; many people would think 0.25 a more mathematical thing than “one-quarter”. But it is quite nerdly to speak of 0.0675 percent instead of “half of what’s left”.
For this week’s roundup of mathematically themed comic strips I have a picture again! After a month or so. It’s great to see again. Also there’s several comics I could swear I’ve shown and featured before. But it’s really quite hot here and I don’t feel like going to the effort of looking. If I repeat myself, so I do. I bet you’ve forgotten the last time I did this Robbie and Bobby too.
Carol Lay’s Lay Lines for the 6th implicitly uses mathematics as an example of perfection. The idea of the straight line is in that territory shared by both mathematics and Platonic ideals. We can imagine a straight line and understand many properties of it even though it can’t be manifest in our real world. The Gods, allegedly, would be able to overcome that and offer perfect circles around imperfect lines. I suppose that’s one way to tell there’s a god involved. The strip also take a moment to riff on the ontological problem, although I don’t know if that’s part of Lay’s intent.
Jonathan Lemon’s Rabbits Against Magic for the 6th uses a bit of mathematics to represent having a theory. It’s true enough that mathematics serves this role in many sciences. We can often put a good explanation for phenomena in a set of equations. But that’s so if you have a good idea what quantities to measure, and how they affect one another. Lettuce’s equation just describes how long an arc within a circle is. It’s true, although I don’t think it rates the status of a theory; it just describes one thing we’d like to know in terms of another thing. And it’s all a setup for a π joke anyway.
Bud Blake’s Tiger for the 8th, as a King Features comic, broke my drought of having images to include with Reading the Comics posts! Celebrate! It’s also the one that made me think I was getting reruns in. But it’s more mysterious than that. The Tiger rerun (Blake died several years ago and all Tiger strips are reruns) for the 20th of April, 2015, is the same joke. I featured it in a Reading The Comics post back then. But it’s not the same strip. The art’s completely redrawn. I can’t fault Blake for having reused a setup-and-punchline. Every comic strip creator does this. Sometimes the cartoonist has improved the joke. (Berkeley Breathed did this several times over.) Sometimes the cartoonist probably just forgot it was done before. (There’s several Peanuts strips suggesting this.) I’m just delighted to catch someone at it.
Ryan Pagelow’s Buni for the 8th uses a blackboard full of mathematics as signifier for explaining the Big Questions of life. And features the traditional little error spotted by someone else. The scribbles are gibberish altogether, but they don’t need to be (and in truth couldn’t be) meaningful. I will defend the backwards-capital-sigma in the upper left of the first panel, though. Sigmas are some of those letters that get pretty sloppy treatment. You get swept up in inspiration and penmanship just collapses. Other Greek letters take some shabby treatment too. And there was a stretch of about three years when I would’ve sworn there was a letter ‘ksee’, a sort of topheavy squiggle. It doesn’t exist, but it’s pretty convenient when you need one more easy Greek letter to use.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th features the “scariest equation” in the universe. The board gives a good description of the quantities in the equation and the relationship makes superficial sense. But it does depend on an assumption I’m not sure about, but I will go with. Weinersmith’s argument supposes that a mis-sent text is equally likely to go to any of your contacts. I am not an experienced texter. But it seems to me that a mis-sent text is more likely to go to a contact you’d recently messaged, or one that’s close to the person you meant to contact. Suppose parents are among the people you text often, or whose contact information is stored where it’s easy to pick by accident. Then you likely send them more messages by accident than this expects. On the other hand, suppose you don’t text parents often or you store their information well away from your significant’s. Then the number of mis-sent messages given to them is lower. Without information about how you organize your contacts, we can’t say what’s a better estimate. So in ignorance we may suppose you mis-send texts to every one of your contacts equally often.
Mell Lazarus’s Momma rerun for the 10th uses a word problem to try to quantify love. Marylou decries the result as “differential calculus”, although it’s really just high school algebra. “Differential calculus” is the funnier term, must admit. Differential calculus refers, generally, to the study of how much one quantity depends on another. On average you can expect something to change if one or more of the variables that describe it change. For example, if you make a rectangle a little larger, its area gets larger. What’s the ratio between how much the area changes and how much the lengths of the rectangle change? If you make an angled corridor wider, then a longer straight object can be fit through the corner. How much longer an object can you bring through the corridor if you make the width a tiny bit bigger? And this also tells us where maximums and minimums are. At a maximum or minimum, a quantity doesn’t change appreciably as the variables that describe it change a little bit. So we can find maximums and minimums by the differential calculus.
OK, that’s three weeks in a row in which all my mathematically-themed comic strips are from Gocomics. Maybe I should commission some generic Reading The Comics art from the cartoonists and artists I know. It could make things more exciting on a visually dull week like this.
Mark Anderson’s Andertoons got its entry on the 25th. We draw the name “exponents” from the example of Michael Stifel, a 16th-century German theologian/mathematician. He’d described them as exponents in his influential 1544 book Arithmetica Integra. But I don’t know why he picked the name “exponent” rather than some other word.
Dave Blazek’s Loose Parts for the 25th is not quite the anthropomorphic shapes joke for this week. The word “isosceles” does trace back to Greek, of course. The first part comes from “isos”, meaning equal; you see the same root in terms like “isobar” and “isometric view”. The “sceles” part comes from “skelos”, meaning leg. Say what you will about an isosceles triangle, and you may as they’ve got poor hearing, but they do have two legs with the same length. If you want to say an equilateral triangle, which has three legs the same length, is an isosceles triangle you can do that. You’ll be right. But you will look like you’re trying a little too hard to make a point, the way you do if you point to a square and start off by calling it a rhombus.
Donna A Lewis’s Reply All Lite for the 25th tries doing a joke about doing mathematics by hand being a sign of old age. If we’re talking about arithmetic … I could go along with that, grudgingly. Calculator applications are so reliable and so quick that it’s hard to justify doing arithmetic by hand unless it’s a very simple problem. If you have fun doing that, good.
But if we’re doing real mathematics, the working out of a model and the implications of that, or working out calculus or group theory or graph theory or the like? There are surely some people who can do all this work in their heads and I am impressed by that. But much of real mathematics is working out implications of ideas, and that’s done so very well by hand. I haven’t found a way of typing in strings of expressions which makes it easier for me to think about the mathematics rather than the formatting. And I would believe in a note-taking program that was as sensitive and precise as pen on paper. I haven’t seen one yet, though. (I have small handwriting, and the applications I’ve tried turn all my writing into tiny, disconnected dots and scribbles.)
Ralph Hagen’s The Barn for the 27th is superficially about Olbers’s Paradox. If there’s an infinitely large, infinitely old universe, then how can the night sky by dark? The light of all those stars should come together to make night even more brilliantly blazing than the daytime sun. This is a legitimate calculus problem. The reasoning is sound. The light of a star trillions upon trillions of light-years away may be impossibly faint. But there are so many stars that would be that far away that they would be, on average, about as bright as the sun is. Integral calculus tells us what happens when we have infinitely large numbers of impossibly tiny things added together. In the case of stars, infinitely many impossibly faint stars would come together to an infinitely bright night sky. That night is dark tells us: the universe can’t be infinitely large and infinitely old. There must be limits to how far away anything can be.
It looks like Comic Strip Master Command wanted to give me a nice, easy start of the year. The first group of mathematics-themed comic strips doesn’t get into deep waters and so could be written up with just a few moments. I foiled them by not having even a few moments to write things up, so that I’m behind on 2016 already. I’m sure I kind of win.
Dan Thompson’s Brevity for the 1st of January starts us off with icons of counting and computing. The abacus, of course, is one of the longest-used tools for computing. The calculator was a useful stopgap between the slide rule and the smart phone. The Count infects numerals with such contagious joy. And the whiteboard is where a lot of good mathematics work gets done. And yes, I noticed the sequence of numbers on the board. The prime numbers are often cited as the sort of message an alien entity would recognize. I suppose it’s likely an intelligence alert enough to pick up messages across space would be able to recognize prime numbers. Whether they’re certain to see them as important building blocks to the ways numbers work, the way we do? I don’t know. But I would expect someone to know the sequence, at least.
Ryan Pagelow’s Buni for New Year’s Day qualifies as the anthropomorphic-numerals joke for this essay.
Scott Hilburn’s The Argyle Sweater for the 2nd of January qualifies as the Roman numerals joke for this essay. It does prompt me to wonder whether about the way people who used Roman numerals as a their primary system thought, though. Obviously, “XCIX red balloons” should be pronounced as “ninety-nine red balloons”. But would someone scan it as “ninety-nine” or would it be read as the characters, “x-c-i-x” and then that converted to a number? I’m not sure I’m expressing the thing I wonder.
Steve Moore’s In The Bleachers for the 4th of January shows a basketball player overthinking the problem of getting a ball in the basket. The overthinking includes a bundle of equations which are all relevant to the problem, though. They’re the kinds of things you get in describing an object tossed up and falling without significant air resistance. I had thought I’d featured this strip — a rerun — before, but it seems not. Moore has used the same kind of joke a couple of other times, though, and he does like getting the equations right where possible.
Justin Boyd’s absurdist Invisible Bread for the 4th of January has Mom clean up a messy hard drive by putting all the 1’s together and all the 0’s together. And, yes, that’s not how data works. We say we represent data, on a computer, with 1’s and 0’s, but those are just names. We need to call them something. They’re in truth — oh, they’re positive or negative electric charges, or magnetic fields pointing one way or another, or they’re switches that are closed or open, or whatever. That’s for the person building the computer to worry about. Our description of what a computer does doesn’t care about the physical manifestation of our data. We could be as right if we say we’re representing data with A’s and purples, or with stop signs and empty cups of tea. What’s important is the pattern, and how likely it is that a 1 will follow a 0, or a 0 will follow a 1. If that sounds reminiscent of my information-theory talk about entropy, well, good: it is. Sweeping all the data into homogenous blocks of 1’s and of 0’s, alas, wipes out the interesting stuff. Information is hidden, somehow, in the ways we line up 1’s and 0’s, whatever we call them.
Richard Thompson’s Cul de Sac for the 19th of December (a rerun, alas, from the 18th of December, 2010) gives me a name for this Reading the Comics installment. Just as in a Richard’s Poor Almanac mentioned last time he gives us a Christmas tree occupying a non-Euclidean space. Non-Euclidean spaces do open up the possibility of many wondrous and counterintuitive phenomena. Trees probably aren’t among them, but I don’t know a better shorthand way to describe their mysteries. And if you’re not sure why so many people say this was the greatest comic strip of our still-young century, look at little Pete in the last panel. Both his expression and the composition of the panel are magnificent.
Richard Thompson’s Poor Richard’s Almanac for the 21st of December justifies my theme with its Platonic Fir. The Platonic Ideals of objects are, properly speaking, philosophical constructs. If they are constructs, anyway, and not the things that truly exist, and yes, we must be careful what we mean by ‘exist’ in this context. But Thompson’s diagram shows this Platonic Fir drawn as a mathematical diagram. That’s another common motif. Mathematical constructs, ideas like “triangles” and “circles” and “rotations”, do suggest Platonic Ideals quite closely. We might be a bit pressed to say what the quintessence of chair-ness is, the thing all chairs must be aspects of. But we can be pretty sure we understand what a triangle is, apart from our messy and imperfect real-world approximations of a true triangle. When mathematics enthusiasts speak of the beauty of pure mathematics it does seem like they speak of the beauty of approaching Platonic Ideals.
Don Asmussen’s Bad Reporter panel for the 23rd of December does a joke that depends on the idea of getting to be “more than infinity”. Every kid has run into the problem of trying to understand “infinity plus one”. The way we speak of “infinity” we can’t really talk about getting “more than infinity”. But we are able to think meaningfully of ways to differentiate sizes of infinity. There are some infinitely large sets that, in a sensible way, are bigger than other infinitely large sets. That’s a fun field of mathematics. You can get to interesting questions in it without needing much background or experience. It’s almost ideal for pop-mathematics essays and if you don’t believe me, then look at how many results you get googling for “Cantor’s Diagonalization Argument”. It’s not an infinite number of results, but it’ll get you quite close.
Mark Litzler’s Joe Vanilla for the 23rd of December is built on the idea that it’s absurd to develop an algorithm that could predict earning potential, hairline at 50, and fidelity. It sounds silly at first glance. But if we’ve learned anything from sabermetrics it’s that all kinds of physical traits can be studied, and modeled, and predicted. With a large and reliable enough data set, and with a mindfully developed algorithm, these models can become quite good at predicting things. The underlying property is that on average, people are average. If we know what is typical, and we have reason to think that “typical” is not changing, then we can forecast the future pretty well based on what we already see. Or if we have reason to expect that “typical” is changing in ways we understand, we can still make good forecasts.
I’d hoped that running a slightly-too-soon edition of Reading the Comics would let me have a better-sized edition for later in this week. Then everybody did comics for the 11th of December. I can have a series of awkward-sized essays or just run what I have. I wonder which I’ll do.
Aaron McGruder’s The Boondocks for the 6th of December is a student-resisting-the-problem joke. It ran originally the 24th of September, 2000, if the copyright information is right. The original problem — “what is 24 divided by 4 minus 2” — is a reasonable one for at least some level of elementary school. (I’m vague on just what grade Caesar is supposed to be in. It’s a problem for any strip with wise-beyond-their-years children. Peanuts plays with this by having the kids give book reports on Peter Rabbit and Tess of the d’Urbervilles.) What makes it a challenge is that you know to know the order of operations. Should you divide 24 by 4 first, and subtract 2 from that, or should you take 4 minus 2 and then divide 24 by whatever that number is?
Absent any confounding information, you should always do multiplication and division before you do addition and subtraction. So this suggests 24 divided by 4, giving us 6, and then subtract 2, giving us 4. The only relevant confounding information, though, would be the direction to do something else first. That’s indicated by putting something in parentheses. (Or brackets, if you have so many parentheses the symbols are getting confusing.) A thing in parentheses has higher priority and should be calculated first. But there’s no way to tell parentheses in dialogue. The best the teacher could do is say something like “24 divided by the quantity four minus two”, or even, “24 divided by parenthesis four minus two close parenthesis”. That’s awkward but it is what we resort to even in the mathematics department.
Eric the Circle for the 6th of December, this one by “Scooterpiggy”, is the anthropomorphic-numerals joke this essay. You might fuss that there’s a difference between a circle and zero. The earliest examples of zero seem to have been simple dots. But the circle, or at least elliptical, shape of zero grew pretty fast. Maybe in a couple of centuries. Maybe there’s something in the empty loop that suggests what it stands for.
Tom Thaves’s Frank and Ernest for the 6th of December tosses in a statistics pun for the final panel. The statistics use of “median” is the number that half the data is less than and half the data is greater than. It’s one of several quantities that get called an “average”. In this case it’s average because if you picked a data point at random you’d be as likely to be above as below the median. In data sets that aren’t too weird, that will usually be pretty close to the arithmetic mean. The arithmetic mean is the thing normal people mean by “average”. It’ll also typically be near the most common value. That most common value mathematicians and statisticians call the “mode”.
I don’t know if the use of “median” for the middle strip of a divided road shares an etymology with the statistics use of the word. It might be one use might have inspired the other, perhaps as metaphor. But the similarity between “being in the middle of the data” and “being in the middle of the street” is straightforward for English.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th of December pinpoints a common failure mode of experts. (The strip almost surely ran before, sometime. The only method I have to find out when, though, is to post an incorrect date and make someone correct me. So let me say it originally ran on Singapore National Day, 2009.) Mathematics is especially prone to it. It’s so seductive to teach something the way an expert sees it. This is usually in a rigorously thought-out, open-ended, flexible method. After all, why would you ever teach something that wasn’t exactly right, with “right” being “the ways experts see things”? A teacher knows the answer: the expert understanding of a thing is hard to get to. That’s why having it takes expertise. The comic strip’s explanation of fractions is correct and reasonable. But it brings up why Bertrand Russell and Alfred North Whitehead needed over four hundred pages to establish 1 + 1 equals 2. That’s a lot of intellectual scaffolding for the quality of paint job required. Sometimes it’s easier to start with a quick and dirty explanation, and then go back later and rebuild the understanding if a student needs it.
Rick Stromoski’s Soup to Nutz for the 7th of December puts forth a kind of Zeno’s paradox problem in the guise of compound interest. If doing something increases life expectancy by a certain percentage, then, how much of the extra time one gets do you need to be immortal? I’m amused by this although I can’t imagine modest alcohol consumption increasing lifespan by 20 percent. (I assume 20 percent of the average expected lifespan.) If the effect were anything near that big the actuaries would have noticed and ordered people to drink long ago.
On looking at all this, I think I’ll save the December 11th strips for later. This is enough text for this early in the morning.