For today’s round of mathematics-themed comic strips a little deeper pattern turns out to have emerged: π, that most popular of the transcendental numbers, turns up quite a bit in the comics that drew my attention the past couple weeks. Let me explain.
Dan Thompson’s Brevity (January 23) returns to the anthropomorphic numbers racket, with the kind of mathematics puns designed to get the strip pasted to the walls of the teacher’s lounge. I wonder how that’s going for him.
Greg Evans’s Luann Againn (January 25, rerun from 1986) has Luann not understanding how to work out an arithmetic problem until it’s shown how to do it: use the calculator. This is a joke that’s probably going to be with us as long as there are practical, personal calculating devices, because it is a good question why someone should bother learning arithmetic when a device will do it faster and better by every reasonable measure. I admit not being sure there is much point to learning arithmetic, other than as a way to practice a particular way of learning how to apply algorithms. I suppose it also stands as a way to get people who are really into mathematics to highlight themselves: someone who memorizes the times tables is probably interested in the kinds of systematic thought that mathematics depends on. But that’s a weak reason to demand it of every student. I suppose arithmetic is very testable, but that’s an even worse reason to make students go through it.
Mind you, I am quite open to the idea that arithmetic drills are useful for students. That I don’t know a particular reason why I should care whether a seventh-grader can divide 391 by 17 by hand doesn’t mean that I don’t think there is one.
Darby Conley’s Get Fuzzy (January 26) has, in a fresh twist for the strip, the cat Bucky and the human Rob get into a fight over something Bucky heard wrong, in this case, a human-interest feature about a kid who had developed a new technique for calculating pi. This is just the sort of thing that would attract human-interest stories: there are many different ways to calculate π, besides that whole dividing a circle’s circumference by its diameter thing. That’s really one of the worse ways to do it, at least if you want any precision.
But you could probably fill a modest book with the different ways people have found to calculate π and the digits of π. Some of them calculate all the digits once you pass the decimal point. Some of them are able to calculate individual digits, without calculating the ones before or after — that is, you can calculate the 2,038th digit of the number without having to work out the 2,037 preceding it. (Kind of; the only formulas I’m aware of calculate the binary or the hexadecimal digits of π, with no formula known for finding a single decimal digit. I don’t know if that just reflects the limits of what we happen to know right now, or whether there’s something about base ten which prohibits a formula for finding a single particular digit of π from existing.)
There are some interesting questions about the digits of π, most of them related to whether it’s a normal number (one in which every possible finite string of digits appears, and appears as often as every other string of the same length), and working out formulas and learning its digits allow some study of that. And working out digits of π makes for a convenient understandable-but-reproducible test of computing power and the like. Mostly, though, π is a really popular, charismatic number that attracts attention the way Mount Everest will attract climbers, although with less personal risk.
Jef Mallet’s Frazz (January 28) has Caulfield try using an idea about how averages work to mess with his teacher’s head, as he offers to turn in one assignment a day early and another a day late. Obviously this approach is useless in meeting a hard deadline, but it’s a normal part of logistics problems to need to plan for processes which take a well-known average time but whose completion times are fairly dispersed. And when you start studying things for which you know the distribution — what the average value is, and how dispersed values are around that average — you start learning statistical mechanics and thermodynamics and quantum mechanics. Not here, though.
Rick Detorie’s One Big Happy (January 28) has the kid, Joey, praying for a miracle, that seven times three equals eighteen. There’s a sound mathematical if not theological problem here: could there be a universe in which seven times three equals eighteen, provided that “seven” and “three” and “eighteen” and “times” and “equals” mean there what they mean here? Could an omnipotent God work out a universe where Joey’s prayer came true? I can’t see how one could, since, given the definitions of seven and three and all that, the fact that seven times three is not eighteen follows from a sound logical deduction, and I can’t see how even an omnipotent God could make true what is deductively false. (If we assume that God isn’t pulling a scam and making people think what’s false is true and all that, I mean.) But this does also feel like it crashes against what the word “omnipotent” means, and here I must plead ignorance. I am certain that philosophers and theologians have given much better thought to what “omnipotence” could mean than I have, and I should learn something of that intellectual heritage before I push my naivete on too many people.
Cory Thomas’s Watch Your Head (January 28) name-drops Srinivasa Ramanujan, which is a quite apt reference. Cory’s demurring on being called a mathematical genius just because he’s able to explain the binomial theorem. Ramanujan is perhaps the iconic mathematical genius: starting from the age of ten, and teaching himself from a couple textbooks, he not just mastered the books’ contents but explored and expanded on them, and proceeded to develop theorems in infinite series, mathematical analysis, and number theory of staggering insight and imagination. Rather fortunately the British mathematician Godfrey Harold Hardy discovered Ramanujan’s work and genius and brought the man to Cambridge, where he could (for too short a time) be part of the mainstream of mathematics. Yes, some of his infinite series theorems are ways to calculate π.
Pab Sungenis’s The New Adventures of Queen Victoria (January 29) starts with a dust bunny, and a dust bunny, and then two dust bunnies, then three, then five, and as Victoria says, “We sense a pattern”, which does continue on the 30th, to 89 dust bunnies and some frantic shouting. This is of course the famous Fibonacci series, introduced to Western mathematics by Leonardo Fibonacci’s 1202 book Liber Abaci and are iconically introduced with the idea of bunnies, who never die and never stop reproducing, piling up into ever-greater numbers. While the sequence gets introduced with the very artificial construct of immortal rabbits breeding to clockwork precision, Fibonacci number-like patterns appear rather often in nature, particularly in the patterns of petals on flowers and of seeds on sunflowers.
I’m surprised also to learn that every positive number can be written as the sum of Fibonacci numbers, with each Fibonacci number being used at most once. That should make for some fun recreational puzzle-writing.
Jerry Van Amerongen’s Ballard Street (February 1) shows Lewis “[looking] for mathematical patterns in the random dumping of toaster crumbs”, which he would, because Ballard Street is all about inscrutable people doing odd things for vague reasons. However, Lewis here actually could do something mathematically meaningful here: random processes like dumping toaster crumbs can show the results of mathematical computations, sometimes more efficiently than actually solving the problem directly would do.
Here’s the simplest example. Suppose you need to find the area of a complicated shape. Put the shape down on top of a plate whose size you do know. Then drop toaster crumbs at random. If, say, three-quarters of the toaster crumbs fall onto the shape, then, the area of the shape should be about three-quarters the area of the plate. You’re probably slapping your head that this is terribly obvious, and so it is, but did you think of it a paragraph ago? This approach gets to be more attractive if you’re trying to figure out the equivalent of the volume of a complicated and multidimensional shape, because the programming that simulates doing this experiment works just as well whatever the shape is and whatever space you’re working in.
Here’s a similar but, I think, more wonderful example. Suppose you have a series of parallel lines, all the same width apart. The floor in my dining room, with long slats of wood, does nicely for this. Imagine dropping toothpicks on the floor, so that where they land and how they’re oriented are randomly distributed. If I count the number of times I drop a toothpick, and count the number of times the toothpick crosses one the edge of one of the wood slats, I can use that to derive the digits of π. (This is about the worst way to calculate π, by the way, taking up way too much time compared to every other method of calculating the digits of π. But it’s possible, and if you program a computer to simulate the dropping and do the counting, then it’s not quite so time-consuming.) And at heart it’s not that different to random dumpings of toaster crumbs.