Reading the Comics, September 24, 2015: Yes, I Do So Edition
Yes, in this roundup of mathematically-themed comic strips I talk seriously about the educational techniques of the fictional Great Smokey Mountains community where the comic strip Barney Google and Snuffy Smith takes place. I accept the implications of this.
John Rose’s Barney Google And Snuffy Smith for the 23rd of September is your standard snarky-response joke. I’m a bit surprised to see that at whatever class level Jughaid’s in they’re using “x” to stand in for the not-yet-known number. I thought empty boxes or question marks were more common. But I also think Miz Prunelly’s not working most effectively by getting angry at Jughaid for not knowing what x is.
I would suggest trying this: can Jughaid find some possible values of x that are definitely too small? And some possible values that are certainly too big? Then what kinds of numbers are both not-too-small and not-too-big? One standard mathematician’s trick for finding an unknown quantity is to show that it can’t be smaller than some number, giving us a lower bound. And then show it can’t be larger than some number, giving us an upper bound. If the lower bound and the upper bound are the same number, we’re done. If they’re not the same number we might have to go looking, but at least we’ve got a better idea what a correct answer should look like. If the lower bound is a larger number than the upper bound, we have to go back and check whether there actually is an answer, or if we started off in the wrong direction.
Scott Adams’s Dilbert Classics for the 23rd of September (a rerun from the 30th of July, 1992) mentions “conversational geometry”. It’s built on a bit of geometry that somehow escaped into being a common allusion, and that occasionally riles up grammar nerds. The problem is trying to use “turned around 360 degrees” for “turned completely around”. 360 degrees is certainly turning something all the way around, but it leaves the thing back where it started, apparently unchanged. (Well, there are some oddball structures where you can rotate something 360 degrees and have it not back the way it started, but those only occur in abstract mathematical constructions and in some — not all! — subatomic particles. Yes, it’s weird. It’s like that.)
The grammar nerd will insist that what’s meant is to turn something 180 degrees, reversing its direction. Or maybe changed 90 degrees, looking perpendicular to whatever the original situation was. Personally I can’t get upset about a shorthand English phrase not making literal sense, because there are only about six shorthand English phrases that make even the slightest literal sense, and four of those are tapas orders. Eventually you have to stop with the rage and just say something already. And rotating 360 degrees is a different process from rotating not at all. You move, you break your focus, you break your attention. Even if you face the same things again you face them having refreshed your perceptions. You might now see something you had not before.
John Zakour and Scott Roberts’s Maria’s Day for the 23rd of September asserts that mathematics is important so that one can check one’s accountants. This is true, although it’s hardly everything mathematics is enjoyable for. And while I don’t often get to call attention to comic strip artwork, do look at the different papers; there’s some fun there.
Pab Sungenis’s New Adventures of Queen Victoria for the 24th of September — and the days around it — have seen Victoria and Nikola Tesla facing the end result of too much holiday creep: a holiday singularity. By a singularity a mathematician means a point where stuff gets weird: where a function isn’t defined, where a surface breaks off, where several independent solutions suddenly stop being independent, that sort of thing. It’ll often correspond with some measure becoming infinitely large (as a positive or a negative number), though I don’t think it’s safe to say that always happens.
We generally can’t say what’s happening at a singularity. But the existence of a singularity, and what it behaves like, can tell us something about what’s happening away from the singularity. It can happen, for example, that a singularity is removable. That is, if a function is undefined for some values, perhaps we can come up with a logically compelling definition for what it might do at those values. If you can remove a singularity then we call this a “removable singularity”. This serves to show you don’t necessarily need grad school to understand everything mathematicians are saying. Sometimes a singularity can’t be removed, and those are known as “nonremovable singularities” or “essential singularities” or sometimes some other nastier names.
Usually, if one has a singularity in a mathematical construct, then information about one side of the singularity isn’t enough to extrapolate what might be on the other side. This makes the literary use of a “singularity” as “something magical that does whatever the plot requires” justified enough. Tesla here is clearly using the idea of reaching an infinitely vast, or an infinitely dense, holiday concentration as a singularity. I grant that would be singular enough. The strip does make me think of a fun sequence in Walt Kelly’s Pogo where one year the Bun Rabbit decided to get all the holiday-celebrating done first thing in the year, to clear out the rest. He went about banging the drum and listing every holiday ever, which is what made me aware of the New Jersey Big Sea Day.
Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 24th of September includes a sequence identified as the “Catalan Series”. I’d have said “sequence” myself. The Catalan sequence describes (among other things) how many ways you can break down a regular polygon into a particular number of triangles. A square can be broken down into two triangles just two ways (if orientation counts, which for this problem, it does). A pentagon can be broken down into three triangles in five ways. A hexagon can be broken down into four triangles in fourteen ways, and so on. (The key is you break the polygon into a number of triangles that’s two less than the number of sides. So if you had a 9-sided polygon, you’d break it up into 7 triangles. If you had a 20-sided polygon, you’d break it up into 18 triangles.) The sequence describes more stuff than that, but this is an easy-to-understand application. As the name of the sequence suggests, it comes to us from the Belgian-French mathematician Eugène Charles Catalan (1814 – 1894).
Catalan’s name also might be faintly familiar for a conjecture he posed in 1844, which was finally proven true in 2002 by Preda Mihăilescu. His conjecture is based on observing that the number 2 raised to the third power is 8, while the number 3 raised to the second power is 9, quite close together. Catalan conjectured this was the only case of consecutive powers. That is, there’s nothing like 15 to the twentieth power being one less than 12 to the twenty-fourth power or anything like that. I’m afraid I don’t know enough of this kind of mathematics, known as number theory, to say whether that’s of use for anything more than settling curiosity on the point.