I’m surprised to discover it’s been over a month since I had a roster of mathematics-themed comic strips to share, but that’s how things happen to happen. It’s also been a month with repeated references to “finding square roots”, I suppose because that sounds like a really math-y thing to do. It’s certainly computationally challenging; the task of finding such is even a (very minor) moment in Isaac Asimov’s magnificent short story about arithmetic, “The Feeling Of Power”. I remember reading the procedure for finding them when I was a kid, and finding that with considerable effort, I was able to, though I’d probably refuse to do more than give a rough estimate of such a root nowadays.
Bill Watterson’s Calvin and Hobbes (June 4, rerun) is another entry in the long string of jokes about “why bother studying mathematics”, but Watterson’s craft lifts it above average. Admire that fourth panel: that’s every resistant student in one pose.
Tony Rubino and Gary Markstein’s Daddy’s Home (June 7) references mathematics, and particularly the problem of finding square roots, although that’s just to serve the purpose of being some other class to be on the kid’s mind. Trying to work out what Hamlet had in mind would serve the setup just as well, and get the strip taped to a different set of teachers’ doors.
Scott Hilburn’s The Argyle Sweater (June 9) finally gets back in with a collection of anthropomorphized numbers gags, and Hilburn goes back to the subject on the 10th, possibly from the fear I’d stop paying attention to him. The 10th is perhaps the most slyly written given that its wordplay.
Allison Barrows’s Preeteena (June 14, rerun) does a students-misunderstanding-the-word-problem gag, about how can there be “square” feet in a rectangle. But past the obvious follow-up question — isn’t a square a kind of rectangle? — there are some good conceptual questions about area that could be inspired by this.
For example, it’s often an easy way to think of the area of something to imagine putting a grid of squares of uniform size over it, and finding how many of the squares are within the shape. It works with grid paper; it works with the tiles on the kitchen floor (probably the easiest metaphor to use); it works with the grid lines turned on for your drawing software. But do you have to use squares? Couldn’t you cover your floor with hexagons of the same size, or triangles, or something else as your basic unit? How many triangles of the same size could you fit inside on the kitchen floor? How about circles?
Well, you couldn’t tile a normal kitchen floor with circular tiles, not with them all the same size and without overlapping, the way you could with squares or triangles or hexagons, at least if you’re willing to chop the tiles up. But what if you had circular tiles of a given size that was, say, one square foot, and then other circular tiles with smaller areas, and smaller ones yet … could you tile the floor that way? In reality, no, since you’d get tiles too small to deal with and they’d be lost in the grout anyway, but, if the realistic obstacles of working with infinitely many impossibly tiny tiles could be waved away, could you do it? And would the area of those circular tiles, added together, necessarily be the same as the area of the square tiles or the triangular tiles you might have chosen otherwise?
I won’t say, which may lead you to think you know the answer already. But here’s another neat thing one can do: starting from a rectangle as described in PreTeena’s word problem, and using the classic straightedge-and-compass of ancient Greek geometry, it’s easy to create a square that has exactly the same area as the original rectangle. Historically this work of creating squares with the same area as something you were interested in was known as “quadrature”, and finding ways to do this within the rules of straightedge-and-compass work created the problem of “squaring the circle”, which can’t be done, but required several thousand years of mathematical development to show couldn’t be done. The term “quadrature” has fallen out of common use, at least in American English, except for some reason as the term “numerical quadrature”, which is setting up computers to solve differential equations. This turns out to be the same sort of work as finding these squares of equal area; it just doesn’t look like it, at first, and I remember the skeptical looks of my students in Numerical Methods courses when I explained that was why this slight fossil of a name was in their textbooks.
On June 15, PreTeena was still crashing against mathematical homework, with the proposal to work on “finding square roots”, which has apparently been the designated mathematical thing for the past month.
Mark Tatulli’s Lio (June 18) draws his own comic for the day, and a pun which annoyed a number of moderately cranky commenters on GoComics.com. I have read etymologies of “airplane” which claim the “plane” in that sense derives from the same root as “planet” — that is, “wanderer” — while the “plane” in geometry derives from “planum”, as in “a wide flat surface” like, er, plains (fruited or not), so the joke depends on one of those cases where words of different origin happen to have merged spelling and sound. I suppose that’s near every pun, though.
Paul Trap’s Thatababy (June 30) is maybe a reach for this column, but, what the heck; it shows the sort of diagram a physics student draws to figure what it takes to keep something moving in a circle. Just add gravity.
Bill Hinds’s Tank McNamara (July 2) plays around the ambiguity of what it means for something to be “100 percent”, which is one of those things that often nags at me when percentage changes are discussed. I’m among those who find nothing offensive in the sports cliche (not used in this strip) of “giving 110 percent”, for about the same reason I didn’t find anything absurd in the Space Shuttle Main Engines operating, for much of the ascent, at “104 percent” of full thrust (for, as ever, historical factors which are interesting if you’re interested in the detailed evolution of a technology). There’s also a touch of Spinal Tap to the whole concept, of course.
Morrie Turner’s Wee Pals (July 2) gives us the bookish Oliver trying to find the square root of a pretty large number. This can’t possibly be homework, even putting aside that it’s a strip dated July (long-running comic strips can become pretty sloppy about matching the time of year sensibly): 8,765,510, which is a whole number, isn’t a perfect square — a whole number times itself — so its square root is an irrational number; Oliver will have to go on forever or else give up when he’s satisfied he’s worked out enough digits. I could grant a textbook giving a problem of finding the square root of a perfect square — say, 8,767,521 — by hand, but not this. (I also kind of expect Morrie Turner’s kids to still be taking the New Math, for which something as computationally useless as finding square roots by hand would probably be discouraged or at least made into a neat puzzle game, but that would still call for a perfect square instead.)
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (July 5) does a little teasing of the fourth wall along the way to name-dropping the “Weingarten surface”, which I admit is a classification I don’t remember encountering, and poking about for images of such on the web hasn’t turned up anything too inspirational. An explanatory page by Rafael López Camino, of the University of Granada, has a couple explanes of “generating curves”. Imagine taking a printout of the pictures on Camino’s page, and rotating it around a horizontal line; the space swept out by the darker curves would be one of these surfaces, some of which are pretty straightforward.
The MacTutor biography of Julius Weingarten credits him with considerable work in the study of surfaces, but notes that he had one of those frustrating careers of moving from one ill-fitting academic post to another while doing his best work. His crowning achievement was finding a way to turn a particular geometric problem into a problem of solving a certain kind of partial differential equation, which is the sort of unification of geometry and analysis that’s served so many mathematicians so very well over the centuries.