There were more mathematically-themed comic strips last week than I had time to deal with. This is in part because of something Saturday which took several more hours than I had expected. So let me start this week with some of the comics that, last week, mentioned mathematics in a marginal enough way there’s nothing to say about them besides yeah, that’s a comic strip which mentioned mathematics.

Jef Mallett’s Frazz for the 27th has a kid wondering why they need in-person instruction for arithmetic. (I’d agree that rehearsing arithmetic skills is very easy to automate. You can make practice problems pretty near without limit. How much this has to do with mathematics is a point of debate.)

It might be more fair to call this a blackboard edition, as three of the strips worth discussing feature that element. But I think I’ve used that name recently. And two of the strips feature specifically 2 + 2, so I’ll use that instead.

And here’s a possible movie heads-up. Turner Classic Movies, United States feed, is showing Monday at 9:30 am (Eastern/Pacific) All-American Chump. All I know about this 1936 movie is from its Leonard Maltin review:

[ Stuart ] Erwin is funny, in his usual country bumpkin way, as a small-town math whiz known as “the human adding machine” who is exploited by card sharks and hustlers. Fairly diverting double-feature item.

People with great powers of calculation were — and still are — with us. Before calculating machines were common they were, pop mathematicians tell us, in demand for doing the kinds of arithmetic mathematicians and engineers need a lot of. They’d also have value in performing, if they can put together some good patter. And, sure, gambling is just another field that needs calculation done well. I have no idea the quality of the film (it’s rated two and a half stars, but Leonard Maltin rates many things two and a half stars). But it’s there if you’re curious. The film also stars Robert Armstrong. I assume it’s not the guy I know but, you know? We live in a strange world. Now on to the comics.

Glenn McCoy and Gary McCoy’s The Flying McCoys for the 13th uses the image of a blackboard full of mathematics symbols to represent deep thought. The equations on the board are mostly nonsense, although some, like , have obvious meaning. Many of the other symbols have some meaning to them too. In the upper-right corner, for example, is what looks like . This any physics major would recognize: it’s the energy of a photon, which is equal to Planck’s constant (that stuff) times its frequency.

And there are other physics-relevant symbols. In the bottom center is a line that starts . The capital B is commonly used to represent a magnetic field. The arrow above the capital B is a warning that this is a vector, which magnetic fields certainly are. (Mathematicians see vectors as a quite abstract concept. Physicists are more likely to see them as an intensity and direction, like forces, and the fields that make fields.) The symbol comes from vector calculus. It represent an integral taken along a closed loop, a shape that goes out along some path and comes back to where it started without crossing itself. This turns out to be useful all the time in dynamics problems. So the McCoys drew something that doesn’t mean anything, but looks ready to mean things.

“Overthinking this” is a problem common to mathematicians, even at an advanced level. Real problems don’t make clear what their boundaries are, the things that are important and the things that aren’t and the things that are convenient but not essential. Making mistakes picking them out, and working too hard on the wrong matters, will happen.

Graham Harrop’s Ten Cats for the 14th sees the cats pondering the counts of vast things. These are famous problems. Archimedes composed a text, The Sand Reckoner, which tried to estimate how much sand there could be in the universe. To work on the question he had to think of new ways to represent numbers. Grains of sand become numerous by being so tiny. Stars become numerous by the universe being so vast. Comparing the two quantities is a good challenge. For both numbers we have to make estimates. The volume of beaches in the world. The typical size of a grain of sand. The number of galaxies in the universe. The typical number of stars in a galaxy. There’s room to dispute all these numbers; we really have to come up with a range of possible values, with maybe some idea of what seems more likely.

Thaves’s Frank and Ernest for the 15th has the student bringing authority to his answer. The mathematician is called on to prove an answer is “technically” correct. I’m not sure whether the kid is meant to be prefacing the answer he’s about to give, or whether his answer was rewriting the horizontal “2 + 2 = ” in a vertical form.

Brant Parker and Johnny Hart’s The Wizard of Id Classics for the 15th is built around the divisibility of whole numbers, and of relative primes. Setting the fee as some simple integer fraction of the whole has practicality to it. It likely seemed even more practical in the days before currencies decimalized. The common £sd style currency Europeans used before decimals could be subdivided many ways evenly, with one-third of a pound (livre, Reichsgulden, etc) becoming 80 pence (deniers, Pfennig, etc). Unit fractions, and combinations of unit fractions, could offer interesting ways to slice up anything to a desired amount.

Jim Unger’s Herman for the 16th is a student-talking-back-to-the-teacher strip. It also uses the 2 + 2 problem. It’s a common thing for teachers to say they learn from their students. It’s even true, although I son’t know that people ever quite articulate how teachers learn. A good mistake is a great chance to learn. A good mistake shows off a kind of brilliant twist. That the student has understood some but not all of the idea, and has filled in the misunderstood parts with something plausible enough one has to think about why it’s wrong. And why someone would think the wrong idea might be right. There is a kind of mistake that inspires you to think closely about what “right” has to be, and students who know how to make those mistakes are treasures.

I concede I am late in wrapping up last week’s mathematically-themed comics. But please understand there were important reasons for my not having posted this earlier, like, I didn’t get it written in time. I hope you understand and agree with me about this.

Bill Griffith’s Zippy the Pinhead for the 9th brings up mathematics in a discussion about perfection. The debate of perfection versus “messiness” begs some important questions. What I’m marginally competent to discuss is the idea of mathematics as this perfect thing. Mathematics seems to have many traits that are easy to think of as perfect. That everything in it should follow from clearly stated axioms, precise definitions, and deductive logic, for example. This makes mathematics seem orderly and universal and fair in a way that the real world never is. If we allow that this is a kind of perfection then … does mathematics reach it?

Even the idea of a “precise definition” is perilous. If it weren’t there wouldn’t be so many pop mathematics articles about why 1 isn’t a prime number. It’s difficult to prove that any particular set of axioms that give us interesting results are also logically consistent. If they’re not consistent, then we can prove absolutely anything, including that the axioms are false. That seems imperfect. And few mathematicians even prepare fully complete, step-by-step proofs of anything. It takes ridiculously long to get anything done if you try. The proofs we present tend to show, instead, the reasoning in enough detail that we’re confident we could fill in the omitted parts if we really needed them for some reason. And that’s fine, nearly all the time, but it does leave the potential for mistakes present.

Zippy offers up a perfect parallelogram. Making it geometry is of good symbolic importance. Everyone knows geometric figures, and definitions of some basic ideas like a line or a circle or, maybe, a parallelogram. Nobody’s ever seen one, though. There’s never been a straight line, much less two parallel lines, and even less the pair of parallel lines we’d need for a parallellogram. There can be renderings good enough to fool the eye. But none of the lines are completely straight, not if we examine closely enough. None of the pairs of lines are truly parallel, not if we extend them far enough. The figure isn’t even two-dimensional, not if it’s rendered in three-dimensional things like atoms or waves of light or such. We know things about parallelograms, which don’t exist. They tell us some things about their shadows in the real world, at least.

Mark Litzler’s Joe Vanilla for the 9th is a play on the old joke about “a billion dollars here, a billion dollars there, soon you’re talking about real money”. As we hear more about larger numbers they seem familiar and accessible to us, to the point that they stop seeming so big. A trillion is still a massive number, at least for most purposes. If you aren’t doing combinatorics, anyway; just yesterday I was doing a little toy problem and realized it implied 470,184,984,576 configurations. Which still falls short of a trillion, but had I made one arbitrary choice differently I could’ve blasted well past a trillion.

Ruben Bolling’s Super-Fun-Pak Comix for the 9th is another monkeys-at-typewriters joke, that great thought experiment about probability and infinity. I should add it to my essay about the Infinite Monkey Theorem. Part of the joke is that the monkey is thinking about the content of the writing. This doesn’t destroy the prospect that a monkey given enough time would write any of the works of William Shakespeare. It makes the simple estimates of how unlikely that is, and how long it would take to do, invalid. But the event might yet happen. Suppose this monkey decided there was no credible way to delay Hamlet’s revenge to Act V, and tried to write accordingly. Mightn’t the monkey make a mistake? It’s easy to type a letter you don’t mean to. Or a word you don’t mean to. Why not a sentence you don’t mean to? Why not a whole act you don’t mean to? Impossible? No, just improbable. And the monkeys have enough time to let the improbable happen.

Eric the Circle for the 10th, this one by Kingsnake, declares itself set in “the 20th dimension, where shape has no meaning”. This plays on a pop-cultural idea of dimensions as a kind of fairyland, subject to strange and alternate rules. A mathematician wouldn’t think of dimensions that way. 20-dimensional spaces — and even higher-dimensional spaces — follow rules just as two- and three-dimensional spaces do. They’re harder to draw, certainly, and mathematicians are not selected for — or trained in — drawing, at least not in United States schools. So attempts at rendering a high-dimensional space tend to be sort of weird blobby lumps, maybe with a label “N-dimensional”.

And a projection of a high-dimensional shape into lower dimensions will be weird. I used to have around here a web site with a rotatable tesseract, which would draw a flat-screen rendition of what its projection in three-dimensional space would be. But I can’t find it now and probably it ran as a Java applet that you just can’t get to work anymore. Anyway, non-interactive videos of this sort of thing are common enough; here’s one that goes through some of the dimensions of a tesseract, one at a time. It’ll give some idea how something that “should” just be a set of cubes will not look so much like that.

Steve Kelly and Jeff Parker’s Dustin for the 11th is a variation on the “why do I have to learn this” protest. This one is about long division and the question of why one needs to know it when there’s cheap, easily-available tools that do the job better. It’s a fair question and Hayden’s answer is a hard one to refute. I think arithmetic’s worth knowing how to do, but I’ll also admit, if I need to divide something by 23 I’m probably letting the computer do it.

And now, closer to deadline than I like, let me wrap up last week’s mathematically-themed comic strips. I had a lot happening, that’s all I can say.

Glenn McCoy and Gary McCoy’s The Flying McCoys for the 10th is another tragic moment in the mathematics department. I’m amused that white lab coats are taken to read as “mathematician”. There are mathematicians who work in laboratories, naturally. Many interesting problems are about real-world things that can be modelled and tested and played with. It’s hardly the mathematics-department uniform, but then, I’m not sure mathematicians have a uniform. We just look like academics is all.

It also shows off that motif of mathematicians as doing anything with numbers in a more complicated way than necessary. I can’t imagine anyone in an emergency trying to evoke 9-1-1 by solving any kind of puzzle. But comic strip characters are expected to do things at least a bit ridiculously. I suppose.

Mark Litzler’s Joe Vanilla for the 11th is about random numbers. We need random numbers; they do so much good. Getting them is hard. People are pretty lousy at picking random numbers in their head. We can say what “lousy” random numbers look like. They look wrong. There’s digits that don’t get used as much as the others do. There’s strings of digits that don’t get used as much as other strings of the same length do. There are patterns, and they can be subtle ones, that just don’t look right.

And yet we have a terrible time trying to say what good random numbers look like. Suppose we want to have a string of random zeroes and ones: is 101010 better or worse than 110101? Or 000111? Well, for a string of digits that short there’s no telling. It’s in big batches that we should expect to see no big patterns. … Except that occasionally randomness should produce patterns. How often should we expect patterns, and of what size? This seems to depend on what patterns we’ve found interesting enough to look for. But how can the cultural quirks that make something seem interesting be a substantial mathematical property?

Olivia Jaimes’s Nancy for the 11th uses mathematics-assessment tests for its joke. It’s of marginal relevance, yes, but it does give me a decent pretext to include the new artist’s work here. I don’t know how long the Internet is going to be interested in Nancy. I have to get what attention I can while it lasts.

Scott Hilburn’s The Argyle Sweater for the 12th is the anthropomorphic-geometry joke for the week. Unless there was one I already did Sunday that I already forgot. Oh, no, that was anthropomorphic-numerals. It’s easy to see why a circle might be labelled irrational: either its radius or its area has to be. Both can be. The triangle, though …

Well, that’s got me thinking. Obviously all the sides of a triangle can be rational, and so its perimeter can be too. But … the area of an equilateral triangle is times the square of the length of any side. It can have a rational side and an irrational area, or vice-versa. Just as the circle has. If it’s not an equilateral triangle?

Can you have a triangle that has three rational sides and a rational area? And yes, you can. Take the right triangle that has sides of length 5, 12, and 13. Or any scaling of that, larger or smaller. There is indeed a whole family of triangles, the Heronian Triangles. All their sides are integers, and their areas are integers too. (Sides and areas rational are just as good as sides and areas integers. If you don’t see why, now you see why.) So there’s that at least. The name derives from Heron/Hero, the ancient Greek mathematician whom we credit with that snappy formula that tells us, based on the lengths of the three sides, what the area of the triangle is. Not the Pythagorean formula, although you can get the Pythagorean formula from it.

Still, I’m going to bet that there’s some key measure of even a Heronian Triangle that ends up being irrational. Interior angles, most likely. And there are many ways to measure triangles; they can’t all end up being rational at once. There are over two thousand ways to define a “center” of a triangle, for example. The odds of hitting a rational number on all of them at once? (Granted, most of these triangle centers are unknown except to the center’s discoverer/definer and that discoverer’s proud but baffled parents.)

Carla Ventresca and Henry Beckett’s On A Claire Day for the 12th mentions taking classes in probability and statistics. They’re the classes nobody doubts are useful in the real world. It’s easy to figure probability is more likely to be needed than functional analysis on some ordinary day outside the university. I can’t even compose that last sentence without the language of probability.

I’d kind of agree with calling the courses intense, though. Well, “intense” might not be the right word. But challenging. Not that you’re asked to prove anything deep. The opposite, really. An introductory course in either provides a lot of tools. Many of them require no harder arithmetic work than multiplication, division, and the occasional square root. But you do need to learn which tool to use in which scenario. And there’s often not the sorts of proofs that make it easy to understand which tool does what. Doing the proofs would require too much fussing around. Many of them demand settling finicky little technical points that take you far from the original questions. But that leaves the course as this archipelago of small subjects, each easy in themselves. But the connections between them are obscured. Is that better or worse? It must depend on the person hoping to learn.

It’s too many comics to call this a famine edition, after last week’s feast. But there’s not a lot of theme to last week’s mathematically-themed comic strips. There’s a couple that include vintage comic strips from before 1940, though, so let’s run with that as a title.

Glenn McCoy and Gary McCoy’s The Flying McCoys for the 4th of June is your traditional blackboard full of symbols to indicate serious and deep thought on a subject. It’s a silly subject, but that’s fine. The symbols look to me gibberish, but clown research will go along non-traditional paths, I suppose.

Bill Hinds’s Tank McNamara for the 4th is built on mathematics’ successful invasion and colonization of sports management. Analytics, sabermetrics, Moneyball, whatever you want to call it, is built on ideas not far removed from the quality control techniques that changed corporate management so. Look for patterns; look for correlations; look for the things that seem to predict other things. It seems bizarre, almost inhuman, that we might be able to think of football players as being all of a kind, that what we know about (say) one running back will tell us something about another. But if we put roughly similarly capable people through roughly similar training and set them to work in roughly similar conditions, then we start to see why they might perform similarly. Models can help us make better, more rational, choices.

Morrie Turner’s Wee Pals rerun for the 4th is another word-problem resistance joke. I suppose it’s also a reminder about the unspoken assumptions in a problem. It also points out why mathematicians end up speaking in an annoyingly precise manner. It’s an attempt to avoid being shown up like Oliver is.

Which wouldn’t help with Percy Crosby’s Skippy for the 7th of April, 1930, and rerun the 5th. Skippy’s got a smooth line of patter to get out of his mother’s tutoring. You can see where Percy Crosby has the weird trait of drawing comics in 1930 that would make sense today still; few pre-World-War-II comics do.

Niklas Eriksson’s Carpe Diem for the 7th is a joke about mathematics anxiety. I don’t know that it actually explains anything, but, eh. I’m not sure there is a rational explanation for mathematics anxiety; if there were, I suppose it wouldn’t be anxiety.

George Herriman’s Krazy Kat for the 15th of July, 1939, and rerun the 8th, extends that odd little faintly word-problem-setup of the strips I mentioned the other day. I suppose identifying when two things moving at different speeds will intersect will always sound vaguely like a story problem.