## Reading the Comics, February 11, 2014: Running Out Pi Edition

I’d figured I had enough mathematics comic strips for another of these entries, and discovered during the writing that I had much more to say about one than I had anticipated. So, although it’s no longer quite the 11th, or close to it, I’m going to exile the comics from after that date to the next of these entries.

Melissa DeJesus and Ed Power’s My Cage (February 6, rerun) makes another reference to the infinite-monkeys-with-typewriters scenario, which, since it takes place in a furry universe allows access to the punchline you might expect. I’ve written about that before, as the infinite monkeys problem sits at a wonderful intersection of important mathematics and captivating metaphors.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (starting February 10) (and when am I going to make a macro for that credit and title?) has Cynthia given a slightly baffling homework lesson: to calculate the first ten digits of pi. The story continues through the 11th, the 12th, the 13th, finally resolving on the the 14th, in the way such stories must. I admit I’m not sure why exactly calculating the digits of π would be a suitable homework assignment; I can see working out division problems until the numbers start repeating, or doing a square root or something by hand until you’ve found enough digits.

π, though … well, there’s the question of why it’d be an assignment to start with, but also, what formula for generating π could be plausibly appropriate for an elementary school class. The one that seems obvious to me — π is equal to four times (1/1 minus 1/3 plus 1/5 minus 1/7 plus 1/9 minus 1/11 and so on and so on) — also takes way too long to work. If a little bit of coding is right, it takes something like 160 terms to get just the first two digits of π correct and that isn’t even stable. (The first 160 terms add to 3.135; the first 161 terms to 3.147.) Getting it to ten digits would take —

Well, I thought it might be as few was 10,000 terms, because it turns out the sum of the first ten thousand terms in that series is 3.1414926536, which looks dead-on until you notice that π is 3.1415926536. That’s a neat coincidence, though.

Anyway, obviously, that formula wouldn’t do, and we see on the strip of the 14th that Lucretia isn’t using that. There are a great many formulas that generate the value of π, any of which might be used for a project like this; some of them get the digits right quite rapidly, usually at a cost of being very complicated. The formula shown in the strip of the 14th, though, doesn’t seem to be right. Lucretia’s work uses the formula $\pi = \sqrt{12} \cdot \sum_{k = 0}^{\infty} \frac{(-3)^{-k}}{2k + 1}$, which takes only about 21 terms to get to the demanded ten digits of accuracy. I don’t want to guess how many pages of work it would take to get to 13,908 places.

If I don’t miss my guess the formula used here is one by Abraham Sharp, an astronomer and mathematician who worked for the Royal Observatory at Greenwich and set a record by calculating π to 72 decimal digits. He was also an instrument-maker, of rather some skill, and I found a page purporting to show his notes of how to cut some complicated polyhedrons out of a block of wood, so, if my father wants to carve a 120-sided figure, here’s his chance. Sharp seems to have started with Leibniz’s formula (yes, that Leibniz) — that the arctangent of a number x is equal to x minus one-third x cubed plus one-fifth x to the fifth power minus one-seventh x to the seventh power, et cetera — with the knowledge that the arctangent of the square root of one-third is equal to one-sixth π and produced this series that looks a lot like the one we started with, but which gets digits correct so very much more quickly.

Darrin Bell’s Candorville (February 13) is primarily a bit of guys insulting friends, but what do you know and π makes a cameo appearance here.

Shannon Wheeler’s Too Much Coffee Man (February 10) is a Venn Diagram cartoon in the service of arguing that Venn Diagram cartoons aren’t funny. Putting aside the smoke and sparks popping out of the Nomad space probe which Kirk and Spock are rushing to the transporter room, I don’t think it’s quite fair: the ease the Venn diagram gives to grouping together concepts and showing how they relate helps organize one’s understanding of concepts and can be a really efficient way to set up a joke. Granting that, perhaps Wheeler’s seen too many Venn Diagram cartoons that fail, a complaint I’m sympathetic to.

Bill Amend’s FoxTrot (February 11, rerun) was one of those strips trying to be taped to the math teacher’s door, with the pun-based programming for the Math Channel.