And now there’ve suddenly been enough mathematics-themed comics for a fresh collection of the things. If there’s any theme this time around it’s to mathematics I remember filtering into popular culture in the 80s: the Drake Equation (which I, at least, first saw in Carl Sagan’s Cosmos and found haunting), and the Rubik’s Cube, which pop mathematics writers in the early 80s latched onto with an eagerness matched only by how they liked polyominoes in the mid-70s, and the Mandelbrot Set, which I think of as a mid-to-late 80s thing because that’s when it started covering science-oriented magazine covers and the screens of IBM PS/2’s being used by the kids in the math and science magnet programs.
Incidentally, this time around I’ve tried to include the Between Friends that I talk about, because I’m not convinced the link to its Comics Kingdom home site will last indefinitely. Gocomics.com seems to keep links from expiring, even for non-subscribers, but I’m curious whether it would be better-liked if I included images of the strips I talk about? I’m fairly confident that this is fair use, as I talk about mathematical subjects inspired by the strips, but I don’t know whether people care much about saving a click before reading my attempts to say something, anything, about a kid given a word problem about airplanes that he answers in a flippant manner.
Wulff and Morgenthaler’s WuMo (May 15) features Professor Rubik, “five minutes after” inventing what he’s famous for. Ernö Rubik really is a Professor (of architecture, at the Budapest College of Applied Arts when he invented his famous cube), and was interested in the relationships of things in space and of objects moving in space. The Rubik’s Cube is of interest mathematically because it offers a great excuse to introduce group theory to the average person. Group theory is, among other things, a way of studying structures that look like arithmetic but that aren’t necessarily on numbers. Rotations work very much like the addition of numbers, at least, the modular addition (where if a result is less than zero, or greater than some upper bound, you add or subtract that upper bound until the result is back in range), and the Rubik’s Cube offers several interacting sets of things to rotate, so that the groups represented by it are fascinatingly complex.
Though the cube was invented in 1974 it didn’t become an overwhelming phenomenon until 1979, and then much of the early 80s was spent in people making jokes about how frustrating they found it and occasionally buying books that were supposed to tell you how to solve it, but you couldn’t after all. Then there was a Saturday morning cartoon about the cube which I watched because I had horrible, horrible, horrible taste in cartoons as a kid. Anyway, it turns out that if you played it perfectly you could solve any Rubik’s Cube in no more than twenty steps, although this wasn’t proven until 2010. I confess I usually just give up around step 35 and take the cube apart. Don’t watch the cartoon.
Eric the Circle (May 17), this entry by “Designroo”, features Eric in the midst of the Mandelbrot Set. The Mandelbrot Set, basis for two-thirds of all the posters on the walls in the mathematics department from 1986 through 2002, was discovered by Benoît Mandelbrot in one of those triumphs of numerical computing. It’s not hard to describe how to make it — it’s only a little more advanced than the pastime of hitting a square root or a square button on a calculator and watching numbers dwindle to zero or grow infinitely large — but the number of calculations that need to be done to see it mean it’d never have been discovered before there were computers to do the hard work, of calculation and of visualization.
Among the neat things about the Mandelbrot set are that it does have inlets that look like circles, and it has an infinite number of them: if you zoom in closely at any point on the boundary of the Mandelbrot set you’ll find a not-quite-perfect replica of the original set, with the big carotid shape and the budding circles on the edges, over and over, inexhaustibly.
Bill Amend’s FoxTrot (May 17, rerun) asks why there aren’t geometry books on tape. It’s not quite an absurd question: in principle, geometry is a matter of deductive logic, and is about the relationship between ideas we call “points” and “lines” and “angles” and the like. Pictures are nice to have, as appeals to intuition, but our intuition can be wrong, and pictures can lead us astray, as any optical illusion will prove. And yet it’s so very hard to do away with that intuition. We may not know a compelling reason why the things we draw on sheets of paper should correspond to the results of logical, deductive reasoning that ought to be true whether drawn or not and whether, for that matter, a universe existed or not, but seeing representations of the relationships of geometric objects seems to help nearly everyone understand them better than simply knowing the reasons they should have those relationships.
The notion of learning geometry without drawings takes one fairly close to the Bourbaki project, the famous/infamous early 20th century French mathematical collective that tried to work out the logical structure of all mathematics on a purely formal, reasoned basis without any appeals to diagrams or physical intuition at all. It was an ambitious, controversial, and fruitful program that got permanently tainted because following from it was the “New Math”, an attempt at mathematics educational reform of the 60s and 70s which crashed hard against the problem that parents will only support educational reform that doesn’t involve teaching a thing in ways different from how they learned it.
T Lewis and Michael Fry’s Over The Hedge (May 18) showcases the Drake Equation, a wonderful bit of reasoning that tries to answer the question of “how many species capable of interstellar communication are there”, considering that we only have evidence for at most one. It’s a wonderful bit of word-problem-type reasoning: given what we do know, which amounts mostly to how many stars there are, how can we work out what we would like to know? Frank Drake, astronomer, and co-designer of the plaque on Pioneers 10 and 11, made some estimates of what factors are relevant in going from what we do know to what we would like to know, and how they might relate. When Drake first published the equation only the number of stars could be reasonably estimated; today we can also add a good estimate of how likely a star is to have planets, and a fair estimate of how likely a planet is to be livable. The other steps are harder to estimate. But the process Drake used, of evaluating what he would need to know in order to give an answer, is still strong: there may be things about the equation which are wrong — factors that interact in ways not previously considered, for example — but it divides a huge problem into a series of smaller ones that can, hopefully, be studied and understood in pieces and through this process be turned into knowledge.
And finally, Jeff Harris’s Shortcuts (May 18), a kid’s activity/information panel, spends a half-a-comics-page talking about numbers and numerals. It’s a pretty respectable short guide to numbers and their representations, including some of the more famous number-representation schemes.