Reading the Comics, March 26, 2014: Kitchen Science Department

It turns out that three of the comic strips to be included in this roundup of mathematics-themed strips mentioned things that could reasonably be found in kitchens, so that’s why I’ve added that as a subtitle. I can’t figure a way to contort the other entries to being things that might be in kitchens, but, given that I don’t get to decide what cartoonists write about I think I’m doing well to find any running themes.

Ralph Hagen’s The Barn (March 19) is built around a possibly accurate bit of trivia which tries to stagger the mind by considering the numinous: how many stars are there? This evokes, to me at least, one of the famous bits of ancient Greek calculations (for which they get much less attention than the geometers and logicians did), as Archimedes made an effort to estimate how many grains of sand could fit inside the universe. Archimedes had apparently little fear of enormous numbers, and had to strain the Greek system for representing numbers to get at such enormous quantities. But he was an ingenious reasoner: he was able to estimate, for example, the sizes and distances to the Moon and the Sun based on observing, with the naked eye, the half-moon; and his work on problems like finding the value of pi get surprisingly close to integral calculus and would probably be a better introduction to the subject than pre-calculus courses are. It’s quite easy in considering how big (and how old) the universe is to get to numbers that are really difficult to envision, so, trying to reduce that by imagining stars as grains of salt might help, if you can imagine a ball of salt eight miles across.

Bill Amend’s FoxTrot (March 19, rerun) aims for the walls of math teacher’s offices, although using slices of pizza to explain certain angles and areas and cans of soda for cylinder volumes is probably not a bad approach. And, pizza areas can be a generally useful bit of geometry to consider. Last month NPR got itself recommended by every mathematics Twitter and forwarded to me a couple times too when Quoctrung Bui reported on the price-per-area for various pizzas, if you’re mostly interested in good price-per-area deals.

Chip Dunham’s Overboard (March 20) uses one of the standard ways of teaching mathematics by representing the abstract idea of addition by the specific idea of adding, here, meatballs. It’s not only at the elementary levels that taking a specific, concrete example of a problem can be useful, though. Often a theorem will be about some fairly abstract construct and it won’t be clear how to even start. Attempting to make the proof about something less abstract — say, instead of trying to prove it about all Hilbert spaces, which are fairly abstract things that have some of the properties that ordinary geometry has, try to prove it about one example, such as, ordinary three-dimensional space — where you have a better feeling for how things fit together. If you’re lucky, you can show whatever you’re interested in for this nice and familiar case works for the original problem; if not, then, the work you did might at least be a sketch for how to do what you really want. If all else fails, you might build your intuition up to be better able to handle the original problem.

Mark Anderson’s Andertoons (March 25) shows a Venn diagram included for its aesthetic rather than informative value. Of course, the problem of how to present information so that it communicates well, and the temptation to be entertaining rather than informative, is an important one which everyone trying to enlighten others has to deal with. But what really comes to my mind is that until last month I’d have called that a Venn diagram too.

Tom Thaves’s Frank and Ernest (March 26) puns on the statistical concept of “regression to the mean”. (It’s also called “reversion to the mean”.) The idea of regression to the mean is straightforward enough: start with something measurable, and repeatable, but that varies: your score on a pinball game, the amount of snow gotten over the winter, the number of car accidents you see on the day’s commute, the size of your water bill, whatever you find interesting. These things have a mean value, which we usually take to be the old-fashioned arithmetic average (add up all the recorded values of the thing in the past and divide by the number of recorded values). Now suppose you have a value that’s away from this mean. The farther this value is from the mean, the greater the probability that the next value is going to be less extreme. Put to a real-world application the idea feels obvious: if you’ve just had a pinball game where you scored ten times what you normally do, the chances are, you’re going to score something closer to the average next time. (Of course this assumes that there is a mean, and that it isn’t changing, or at least isn’t changing rapidly.)

This phenomenon is linked psychologically to the gambler’s fallacy, the idea that if something hasn’t happened as often as it ought to, that it’s due: if you flip a fair coin ten times and it comes up tails all but one time, shouldn’t it come up heads much more often in the next ten flips? The fallacious gambler would suppose that because there were fewer heads this time than expected, it’s more likely that there’ll be more heads than expected next. But all we actually know is that it is likely that the coin will behave more nearly fair in the next ten flips. But that “more nearly fair” is a matter of coming up heads two, or three, or four, or five, or six, or seven, or eight times in the next ten flips. Expecting that isn’t expecting very much out of the coin, and you’re just as likely to get two heads as you are to get eight.