Reading The Comics, November 14, 2014: Rectangular States Edition

I have no idea why Comic Strip Master Command decided this week should see everybody do some mathematics-themed comic strips, but, so they did, and here’s my collection of the, I estimate, six hundred comic strips that touched on something recently. Good luck reading it all.

Samsons Dark Side of the Horse (November 10) is another entry on the theme of not answering the word problem.

Scott Adams’s Dilbert Classics (November 10) started a sequence in which Dilbert gets told the big boss was a geometry major, so, what can he say about rectangles? Further rumors indicate he’s more a geography fan, shifting Dilbert’s topic to the “many” rectangular states of the United States. Of course, there’s only two literally rectangular states, but — and Mark Stein’s How The States Got Their Shapes contains a lot of good explanations of this — many of the states are approximately rectangular. After all, when many of the state boundaries were laid out, the federal government had only vague if any idea what the landscapes looked like in detail, and there weren’t many existing indigenous boundaries the white governments cared about. So setting a proposed territory’s bounds to be within particular lines of latitude and longitude, with some modification for rivers or shorelines or mountain ranges known to exist, is easy, and can be done with rather little of the ambiguity or contradictory nonsense that plagued the eastern states (where, say, a colony’s boundary might be defined as where a river intersects a line of latitude that in fact it never touches). And while perfect rectangularity may be achieved only by Colorado and Wyoming, quite a few states — the Dakotas, Washington, Oregon, Missisippi, Alabama, Iowa — are rectangular enough.

Mikael Wulff and Anders Morgenthaler’s WuMo (November 10) shows that their interest in pi isn’t just a casual thing. They think about what those neglected and non-famous numbers get up to.

Sherman does poorly with mathematic problems that look more like short stories. And don't have pictures.
Jim Toomey’s Sherman’s Lagoon for the 11th of November, 2014. He’s got a point about pictures helping with this kind of problem.

Jim Toomey’s Sherman’s Lagoon starts a “struggling with mathematics homework” story on the 11th, with Sherman himself stumped by a problem that “looks more like a short story” than a math problem. By the 14th Megan points out that it’s a problem that really doesn’t make sense when applied to sharks. Such is the natural hazard in writing a perfectly good word problem without considering the audience.

Homework: Megan points out that a problem about walking in the rain with umbrellas doesn't make much sense for sharks or other underwater creatures.
Jim Toomey’s Sherman’s Lagoon for the 14th of November, 2014.

Mike Peters’s Mother Goose and Grimm (November 12) takes one of its (frequent) breaks from the title characters for a panel-strip-style gag about Roman numerals.

A Roman medic calls for an IV; the nurse asks if he means this numeral 4.
Mike Peters’s Mother Goose and Grimm for the 12th of November, 2014.

Darrin Bell’s Candorville (November 12) starts talking about Zeno’s paradox — not the first time this month that a comic strip’s gotten to the apparent problem of covering any distance when distance is infinitely divisible. On November 13th it’s extended to covering stretches of time, which has exactly the same problem. Now it’s worth reminding people, because a stunning number of them don’t seem to understand this, that Zeno was not suggesting that there’s no such thing as motion (or that he couldn’t imagine an infinite convergent sequence; it’s easy to think of a geometric construction that would satisfy any ancient geometer); he was pointing out that there’s things that don’t make perfect sense about it. Either distance (and time) are infinitely divisible into indistinguishable units, or they are not; and either way has implications that seem contrary to the way motion works. Perhaps they can be rationalized; perhaps they can’t; but when you can find a question that’s easy to pose and hard to answer, you’re probably looking at something really worth thinking hard about.

Bill Amend’s FoxTrot Classics (November 12, a rerun) puns on the various meanings of “irrational”. A fun little fact you might want to try proving sometime, though I wouldn’t fault you if you only tried it out for a couple specific numbers and decided the general case too much to do: any whole number — like 2, 3, 4, or so on — has a square root that’s either another whole number, or else has a square root that’s irrational. There’s not a case where, say, the square root is exactly 45.144 or something like that, though it might be close.

Susan tries to figure out either what year she was in Grade Ten or what her age was back then. She admits she was never any good at math, although the real trouble might be she hasn't got a clear idea what she wants to calculate.
Sandra Bell-Lundy’s Between Friends for the 13th of November, 2014.

Sandra Bell-Lundy’sBetween Friends (November 13) shows one of those cases where mental arithmetic really is useful, as Susan tries to work out — actually, staring at it, I’m not precisely sure what she is trying to work out. Her and her coffee partner’s ages in Grade Ten, probably, or else just when Grade Ten was. That’s most likely her real problem: if you don’t know what you’re looking for it’s very difficult to find it. Don’t start calculating before you know what you’re trying to work out.

If I wanted to work out what year was 35 years ago I’d probably just use a hack: 35 years before 2014 is one year before “35 years before 2015”, which is a much easier problem to do. 35 years before 2015 is also 20 years before 2000, which is 1980, so subtract one and you get 1979. (Alternatively, I might remember it was 35 years ago that the Buggles’ “Video Killed The Radio Star” first appeared, which I admit is not a method that would work for everyone, or for all years.) If I wanted to work out my (and my partner’s) age in Grade Ten … well, I’d use a slightly different hack: I remember very well that I was ten years old in Grade Five (seriously, the fact that twice my grade was my age overwhelmed my thinking on my tenth birthday, which is probably why I had to stay in mathematics), so, add five to that and I’d be 15 in Grade Ten.

Bill Whitehead’s Free Range (November 13) brings up one of the most-quoted equations in the world in order to show off how kids will insult each other, which is fair enough.

Rick Detorie’s One Big Happy (November 13), this one a rerun from a couple years ago because that’s how his strip works on Gocomics, goes to one of its regular bits of the kid Ruthie teaching anyone she can get in range, and while there’s a bit more to arithmetic than just adding two numbers to get a bigger number, she is showing off an understanding of a useful sanity check: if you add together two (positive) numbers, you have to get a result that’s bigger than either of the ones you started with. As for the 14th, and counting higher, well, there’s not much she could do about that.

Steve McGarry’s Badlands (November 14) talks about the kind of problem people wish to have: how to win a lottery where nobody else picks the same numbers, so that the prize goes undivided? The answer, of course, is to have a set of numbers that nobody else picked, but is there any way to guarantee that? And this gets into the curious psychology of random numbers: there is absolutely no reason that 1-2-3-4-5-6, or for that matter 7-8-9-10-11-12, would not come up just as often as, say, 11-37-39-51-52-55, but the latter set looks more random. But we see some strings of numbers as obviously a pattern, while others we don’t see, and we tend to confuse “we don’t know the pattern” with “there is no pattern”. I have heard the lore that actually a disproportionate number of people pick such obvious patterns like 1-2-3-4-5-6, or numbers that form neat pictures on a lottery card, no doubt cackling at how much more clever they are than the average person, and guaranteeing that if such a string ever does come out there’ll a large number of very surprised lottery winners. All silliness, really; the thing to do, obviously, is buy two tickets with the exact same set of numbers, so that if you do win, you get twice the share of anyone else, unless they’ve figured out the same trick.


Reading The Comics, November 4, 2014: Will Pictures Ever Reappear Edition

I had assumed that at some point the good folks at Comics Kingdom would let any of their cartoonists do a panel that’s got mathematical content relevant enough for me to chat about, but apparently that’s just not happening. So for a third time in a row here’s a set of Gocomics-only comic strips, with reasonably stable links and images I don’t feel the need to include. Enjoy, please.

Fred Wagner’s Animal Crackers (October 26) presents an old joke — counting the number of animals by counting the number of legs and dividing by four — although it’s only silly because it’s hard to imagine a case where it’s easier to count the legs on a bunch of animals than it is to count the animals themselves. But if it’s the case that every animal has exactly four legs, then, there’s what’s called a one-to-one relationship between the set of animals and the set of animal legs: if you have some number of animals you have exactly four times that number of animal legs, and if you have some number of animal legs you have exactly one-fourth that number of animals, and you can count whatever’s the more convenient for you and use that to get what you’re really interested in. Showing such a one-to-one relationship exists between two interesting things can often be a start to doing more interesting problems, especially if you can show that the relationship also preserves some interesting interactions; if you have two ways to work out a problem, you can do the easier one.

Mark Anderson’s Andertoons (October 27) riffs on the place value for numbers written in the familiar Arabic style. As befitting a really great innovation, place value becomes invisible when you’re familiar with it; it takes a little sympathy and imagination to remember the alienness of the idea that a “2” means different things based on how many digits are to the right (or, if it’s a decimal, to the left) of it.

Anthony Blades’s charming Bewley (October 27) has one of the kids insisting that instinct alone is enough to do maths problems. The work comes out disastrously bad, of course, or there’d not be a comic strip. However, my understanding is that people do have some instinctive understanding even of problems that would seem to have little survival application. One test I’ve seen demonstrating this asks people to give, without thinking, their answer to whether a multiplication problem might be right or wrong. It’s pretty quick for most people to say that “7 times 9 equals 12” has to be wrong; to say that “7 times 9 equals 59” is wrong takes longer, and that seems to reflect an idea that 59 is, if not the right answer, at least pretty close to it. There’s an instinctive plausibility at work there and it’s amazing to think people should have that. Zach Weinersmith’s Saturday Morning Breakfast Cereal for October 31 circles around this idea, with one person having little idea what 1,892,491,287 times 7,798,721,415 divided by 82,493,726,631 might be, but being pretty sure that “4” isn’t it.

Saturday Morning Breakfast Cereal (October 30) also contains a mention of “cross products”, which are an interesting thing people learning vectors trip over. A cross product is defined for a pair of three-dimensional vectors, and the interesting thing is it’s a new vector that’s perpendicular to the two vectors multiplied together. The length of the cross product vector depends on the lengths of the two vectors multiplied together and the angle they make; the closer the two vectors multiplied together are, the smaller the cross product is, to the point that the cross product of two parallel vectors has length zero. The closer the two vectors multiplied together are to perpendicular the longer the cross product vector is.

More mysterious: if you swap the first vector and the second vector being cross-multiplied together, you get a cross product that’s the same size but pointing the opposite direction, pointing (say) down instead of up. Cross products have some areas where they’re particularly useful, especially in describing the movement of charged particles in magnetic fields.

(There’s something that looks a lot like the cross product which exists for seven-dimensional vectors, but I’ve never even heard of anyone who had a use for it, so, you don’t need to do anything about it.)

Eric the Circle (November 2), this one by “dDave”, presents the idea that that the points on a line might themselves be miniature Erics the Circle. What a line is made of is again one of those problems that straddles the lines between mathematics and philosophy. It seems to be one of the problems of infinity that Zeno’s Paradoxes outlined so perfectly thousands of years ago. To shorten it to the point it becomes misleading, is a line made up of things that have some width? If they’re infinitesimals, things with no width, then, how can an aggregate of things with no width come to have some width? But if they’re made up of things which have some width, how can there be infinitely many of them fitting into a finite space?

We can form good logical arguments about the convergence of infinite series — lining up, essentially, circles of ever-dwindling but ever-positive sizes so that the pile has a finite length — but that seems to suggest that space has to be made up of intervals of different widths, which seems silly; why couldn’t all the miniature circles be the same? In short, space is either infinitely divisible into identical things, or it is not, and neither one is completely satisfying.

Guy Gilchrist’s Nancy (November 2) uses math homework appearing in the clouds, although that’s surely because it’s easier to draw a division problem than it is to depict an assignment for social studies or English.

Todd Clark’s Lola (November 4) uses an insult-the-minor-characters variant of what seems to be the standard way of explaining fractions to kids, that of dividing a whole thing into smaller pieces and counting the number of smaller pieces. As physical interpretations of mathematical concepts goes I suppose that’s hard to beat.