The last full week of 2017 was also a slow one for mathematically-themed comic strips. You can tell by how many bits of marginally relevant stuff I include. In this case, it also includes a couple that just mention the current or the upcoming year. So you’ve been warned.
John Graziano’s Ripley’s Believe It Or Not for the 24th mentions the bit of recreational group theory that normal people know, the Rubik’s Cube. The group theory comes in from rotations: you can take rows or columns on the cube and turn them, a quarter or a half or a three-quarters turn. Which rows you turn, and which ways you turn them, form a group. So it’s a toy that inspires deep questions. Who wouldn’t like to know in how few moves a cube could be solved? We know there are at least some puzzles that take 18 moves to solve. (You can calculate the number of different cube arrangements there are, and how many arrangements you could make by shuffling a cube around with 17 moves. There’s more possible arrangements than there are ones you can get to in 17 moves; therefore, there must be at least one arrangement that takes 18 moves to solve.) A 2010 computer-assisted proof by Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge showed that at most 20 face turns are needed for every possible cube to be solved. I don’t know if there’s been any success figuring out whether 19 or even 18 is necessarily enough.
Bill Griffith’s Zippy the Pinhead for the 26th just mentions algebra as a thing that Griffith can’t really remember, even in one of his frequent nostalgic fugues. I don’t know that Zippy’s line about the fifth dimension is meant to refer to geometry. It might refer to the band, but that would be a bit odd. Yes, I know, Zippy the Pinhead always speaks oddly, but in these nostalgic fugue strips he usually provides some narrative counterpoint.
Larry Wright’s Motley Classics for the 26th originally ran in 1986. I mention this because it makes the odd dialogue of getting “a new math program” a touch less odd. I confess I’m not sure what the kid even got. An educational game? Something for numerical computing? The coal-fired, gear-driven version of Mathematica that existed in the 1980s? It’s a mystery, it is.
Jef Mallett’s Frazz for the 29th is similarly a calendar joke. It does play on 2017 being a prime number, a fact that doesn’t really mean much besides reassuring us that it’s not a leap year. I’m not sure just what’s meant by saying it won’t repeat for another 2017 years, at least that wouldn’t be just as true for (say) 2015 or 2019. But as Frazz points out, we do cling to anything that floats in times like these.
Can’t say this was too fast or too slow a week for mathematically-themed comic strips. A bunch of the strips were panel comics, so that’ll do for my theme.
Norm Feuti’s Retail for the 21st mentions every (not that) algebra teacher’s favorite vague introduction to group theory, the Rubik’s Cube. Well, the ways you can rotate the various sides of the cube do form a group, which is something that acts like arithmetic without necessarily being numbers. And it gets into value judgements. There exist algorithms to solve Rubik’s cubes. Is it a show of intelligence that someone can learn an algorithm and solve any cube? — But then, how is solving a Rubik’s cube, with or without the help of an algorithm, a show of intelligence? At least of any intelligence more than the bit of spatial recognition that’s good for rotating cubes around?
I don’t see that learning an algorithm for a problem is a lack of intelligence. No more than using a photo reference shows a lack of drawing skill. It’s still something you need to learn, and to apply, and to adapt to the cube as you have it to deal with. Anyway, I never learned any techniques for solving it either. Would just play for the joy of it. Here’s a page with one approach to solving the cube, if you’d like to give it a try yourself. Good luck.
Nate Fakes’s Break of Day for the 24th features the traditional whiteboard full of mathematics scrawls as a sign of intelligence. The scrawl on the whiteboard looks almost meaningful. The integral, particularly, looks like it might have been copied from a legitimate problem in polar or cylindrical coordinates. I say “almost” because while I think that some of the r symbols there are r’ I’m not positive those aren’t just stray marks. If they are r’ symbols, it’s the sort of integral that comes up when you look at surfaces of spheres. It would be the electric field of a conductive metal ball given some charge, or the gravitational field of a shell. These are tedious integrals to solve, but fortunately after you do them in a couple of introductory physics-for-majors classes you can just look up the answers instead.
In today’s installment of Reading The Comics, mathematics gets name-dropped a bunch in strips that aren’t really about my favorite subject other than my love. Also, I reveal the big lie we’ve been fed about who drew the Henry comic strip attributed to Carl Anderson. Finally, I get a question from Queen Victoria. I feel like this should be the start of a podcast.
Patrick Roberts’ Todd the Dinosaur for the 6th of April just name-drops mathematics. The flash cards suggest it. They’re almost iconic for learning arithmetic. I’ve seen flash cards for other subjects. But apart from learning the words of other languages I’ve never been able to make myself believe they’d work. On the other hand, I haven’t used flash cards to learn (or teach) things myself.
Joe Martin’s Boffo for the 7th of April is a solid giggle. (I have a pretty watery giggle myself.) There are unknowable, or at least unprovable, things in mathematics. Any logic system with enough rules to be interesting has ideas which would make sense, and which might be true, but which can’t be proven. Arithmetic is such a system. But just fractions and long division by itself? No, I think we need something more abstract for that.
Carl Anderson’s Henry for the 7th of April is, of course, a rerun. It’s also a rerun that gives away that the “Carl Anderson” credit is a lie. Anderson turned over drawing the comic strip in 1942 to John Liney, for weekday strips, and Don Trachte for Sundays. There is no possible way the phrase “New Math” appeared on the cover of a textbook Carl Anderson drew. Liney retired in 1979, and Jack Tippit took over until 1983. Then Dick Hodgins, Jr, drew the strip until 1990. So depending on how quickly word of the New Math penetrated Comic Strip Master Command, this was drawn by either Liney, Tippit, or possibly Hodgins. (Peanuts made New Math jokes in the 60s, but it does seem the older the comic strip the longer it takes to mention new stuff.) I don’t know when these reruns date from. I also don’t know why Comics Kingdom is fibbing about the artist. But then they went and cancelled The Katzenjammer Kids without telling anyone either.
Eric the Circle for the 8th, this one by “lolz”, declares that Eric doesn’t like being graphed. This is your traditional sort of graph, one in which points with coordinates x and y are on the plot if their values make some equation true. For a circle, that equation’s something like (x – a)2 + (y – b)2 = r2. Here (a, b) are the coordinates for the point that’s the center of the circle, and r is the radius of the circle. This looks a lot like Eric is centered on the origin, the point with coordinates (0, 0). It’s a popular choice. Any center is as good. Another would just have equations that take longer to work with.
Richard Thompson’s Cul de Sac rerun for the 10th is so much fun to look at that I’m including it even though it just name-drops mathematics. The joke would be the same if it were something besides fractions. Although see Boffo.
Norm Feuti’s Gil rerun for the 10th takes on mathematics’ favorite group theory application, the Rubik’s Cube. It’s the way I solved them best. This approach falls outside the bounds of normal group theory, though.
Mac King and Bill King’s Magic in a Minute for the 10th shows off a magic trick. It’s also a non-Rubik’s-cube problem in group theory. One of the groups that a mathematics major learns, after integers-mod-four and the like, is the permutation group. In this, the act of swapping two (or more) things is a thing. This puzzle restricts the allowed permutations down to swapping one item with the thing next to it. And thanks to that, an astounding result emerges. It’s worth figuring out why the trick would work. If you can figure out the reason the first set of switches have to leave a penny on the far right then you’ve got the gimmick solved.
Bill Amend’s FoxTrot Classics for the 28th of November (originally run in 2004) depicts a “Christmas Card For Smart People”. It uses the familiar motif of “ability to do arithmetic” as denoting smartness. The key to the first word is remembering that mathematicians use the symbol ‘e’ to represent a number that’s just a little over 2.71828. We call the number ‘e’, or something ‘the base of the natural logarithm’. It turns up all over the place. If you have almost any quantity that grows or that shrinks at a speed proportional to how much there is, and describe how much of stuff there is over time, you’ll find an ‘e’. Leonhard Euler, who’s renowned for major advances in every field of mathematics, is also renowned for major advances in notation in physics, and he gave us ‘e’ for that number.
The key to the second word there is remembering from physics that force equals mass times acceleration. Therefore the force divided by the acceleration is …
And so that inspires this essay’s edition title. There are several comics in this selection that are about the symbols or the representations of mathematics, and that touch on the subject as a visual art.
Matt Janz’s Out of the Gene Pool for the 28th of November first ran the 26th of October, 2002. It would make for a good word problem, too, with a couple of levels: given the constraints of (a slightly looser) budget, how do they get the greatest number of cookies? Or if some cookies are better than others, how do they get the most enjoyment from their cookie purchase? Working out the greatest amount of enjoyment within a given cookie budget, with different qualities of cookies, can be a good introduction to optimization problems and how subtle they can be.
Bill Holbrook’s On The Fastrack for the 29th of November speaks in support of accounting. It’s a worthwhile message. It doesn’t get much respect, not from the general public, and not from typical mathematics department. The general public maybe thinks of accounting as not much more than a way companies nickel-and-dime them. If the mathematics departments I’ve associated with are fair representatives, accounting isn’t even thought of except by the assistant professor doing a seminar on financial mathematics. (And I’m not sure accounting gets mentioned there, since there’s exciting stuff about the Black-Scholes Equation and options markets to think about instead.) This despite that accounting is probably, by volume, the most used part of mathematics. Anyway, Holbrook’s strip probably won’t get the field a better reputation. But it has got some great illustrations of doing things with numbers. The folks in mathematics departments certainly have had days feeling like they’ve done each of these things.
Dave Coverly’s Speed Bump for the 30th of November is a compound interest joke. I admit I’ve told this sort of joke myself, proposing that the hour cut out of the day in spring when Daylight Saving Time starts comes back as a healthy hour and three minutes in autumn when it’s taken out of saving. If I can get the delivery right I might have someone going for that three minutes.
Mikael Wulff and Anders Morgenthaler’s Truth Facts for the 30th of November is a Venn diagram joke for breakfast. I would bet they’re kicking themselves for not making the intersection be the holes in the center.
Mark Anderson’s Andertoons for this week interests me. It uses a figure to try explaining how to relate gallon and quart an pint and other units relate to each other. I like it, but I’m embarrassed to say how long it took in my life to work out the relations between pints, quarts, gallons, and particularly whether the quart or the pint was the larger unit. I blame part of that on my never really having to mix a pint of something with a quart of something else, which ought to have sorted that out. Anyway, let’s always cherish good representations of information. Good representations organize information and relationships in ways that are easy to remember, or easy to reconstruct or extend.
John Graziano’s Ripley’s Believe It or Not for the 2nd of December tries to visualize how many ways there are to arrange a Rubik’s Cube. Counting off permutations of things by how many seconds it’d take to get through them all is a common game. The key to producing a staggering length of time is that it one billion seconds are nearly 32 years, and the number of combinations of things adds up really really fast. There’s over eight billion ways to draw seven letters in a row, after all, if every letter is equally likely and if you don’t limit yourself to real or even imaginable words. Rubik’s Cubes have a lot of potential arrangements. Graziano misspells Rubik, but I have to double-check and make sure I’ve got it right every time myself. I didn’t know that about the pigeons.
Charles Schulz’s Peanuts for the 2nd of December (originally run in 1968) has Peppermint Patty reflecting on the beauty of numbers. I don’t think it’s unusual to find some numbers particularly pleasant and others not. Some numbers are easy to work with; if I’m trying to add up a set of numbers and I have a 3, I look instinctively for a 7 because of how nice 10 is. If I’m trying to multiply numbers, I’d so like to multiply by a 5 or a 25 than by a 7 or an 18. Typically, people find they do better on addition and multiplication with lower numbers like two and three, and get shaky with sevens and eights and such. It may be quirky. My love is a wizard with 7’s, but can’t do a thing with 8. But it’s no more irrational than the way a person might a pyramid attractive but a sphere boring and a stellated icosahedron ugly.
I’ve seen some comments suggesting that Peppermint Patty is talking about numerals, that is, the way we represent numbers. That she might find the shape of the 2 gentle, while 5 looks hostile. (I can imagine turning a 5 into a drawing of a shouting person with a few pencil strokes.) But she doesn’t seem to say one way or another. She might see a page of numbers as visual art; she might see them as wonderful things with which to play.
I knew if I kept going long enough some cartoonist not on Gocomics.com would have to mention mathematics. That finally happened with one from Comics Kingdom, and then one from the slightly freak case of Rick Detorie’s One Big Happy. Detorie’s strip is on Gocomics.com, but a rerun from several years ago. He has a different one that runs on the normal daily pages. This is for sound economic reasons: actual newspapers pay much better than the online groupings of them (considering how cheap Comics Kingdom and Gocomics are for subscribers I’m not surprised) so he doesn’t want his current strips run on Gocomics.com. As for why his current strips do appear on, for example, the fairly good online comics page of AZcentral.com, that’s a good question, and one that deserves a full answer.
Vic Lee’s Pardon My Planet (November 9), which broke the streak of Comics Kingdom not making it into these pages, builds around a quote from Einstein I never heard of before but which sounds like the sort of vaguely inspirational message that naturally attaches to famous names. The patient talks about the difficulty of finding something in “the middle of four-dimensional curved space-time”, although properly speaking it could be tricky finding anything within a bounded space, whether it’s curved or not. The generic mathematics problem you’d build from this would be to have some function whose maximum in a region you want to find (if you want the minimum, just multiply your function by minus one and then find the maximum of that), and there’s multiple ways to do that. One obvious way is the mathematical equivalent of getting to the top of a hill by starting from wherever you are and walking the steepest way uphill. Another way is to just amble around, picking your next direction at random, always taking directions that get you higher and usually but not always refusing directions that bring you lower. You can probably see some of the obvious problems with either approach, and this is why finding the spot you want can be harder than it sounds, even if it’s easy to get started looking.
Reuben Bolling’s Super Fun-Pak Comix (November 6), which is technically a rerun since the Super Fun-Pak Comix have been a longrunning feature in his Tom The Dancing Bug pages, is primarily a joke about the Heisenberg Uncertainty Principle, that there is a limit to what information one can know about the universe. This limit can be understood mathematically, though. The wave formulation of quantum mechanics describes everything there is to know about a system in terms of a function, called the state function and normally designated Ψ, the value of which can vary with location and time. Determining the location or the momentum or anything about the system is done by a process called “applying an operator to the state function”. An operator is a function that turns one function into another, which sounds like pretty sophisticated stuff until you learn that, like, “multiply this function by minus one” counts.
In quantum mechanics anything that can be observed has its own operator, normally a bit tricker than just “multiply this function by minus one” (although some are not very much harder!), and applying that operator to the state function is the mathematical representation of making that observation. If you want to observe two distinct things, such as location and momentum, that’s a matter of applying the operator for the first thing to your state function, and then taking the result of that and applying the operator for the second thing to it. And here’s where it gets really interesting: it doesn’t have to, but it can depend what order you do this in, so that you get different results applying the first operator and then the second from what you get applying the second operator and then the first. The operators for location and momentum are such a pair, and the result is that we can’t know to arbitrary precision both at once. But there are pairs of operators for which it doesn’t make a difference. You could, for example, know both the momentum and the electrical charge of Scott Baio simultaneously to as great a precision as your Scott-Baio-momentum-and-electrical-charge-determination needs are, and the mathematics will back you up on that.
Ruben Bolling’s Tom The Dancing Bug (November 6), meanwhile, was a rerun from a few years back when it looked like the Large Hadron Collider might never get to working and the glitches started seeming absurd, as if an enormous project involving thousands of people and millions of parts could ever suffer annoying setbacks because not everything was perfectly right the first time around. There was an amusing notion going around, illustrated by Bolling nicely enough, that perhaps the results of the Large Hadron Collider would be so disastrous somehow that the universe would in a fit of teleological outrage prevent its successful completion. It’s still a funny idea, and a good one for science fiction stories: Isaac Asimov used the idea in a short story dubbed “Thiotimoline and the Space Age”, published 1959, which resulted in the attempts to manipulate a compound which dissolves before it adds water might have accidentally sent hurricanes Carol, Edna, and Diane into New England in 1954 and 1955.
Francesco Marciuliano’s Medium Large (November 7) is about two of the fads of the early 80s, those of turning everything into a breakfast cereal somehow and that of playing with Rubik’s Cubes. Rubik’s Cubes have long been loved by a certain streak of mathematicians because they are a nice tangible representation of group theory — the study of things that can do things that look like addition without necessarily being numbers — that’s more interesting than just picking up a square and rotating it one, two, three, or four quarter-turns. I still think it’s easier to just peel the stickers off (and yet, the die-hard Rubik’s Cube Popularizer can point out there’s good questions about polarity you can represent by working out the rules of how to peel off only some stickers and put them back on without being detected).
Rick Detorie’s One Big Happy (November 9), and I’m sorry, readers about a month in the future from now, because that link’s almost certainly expired, is another entry in the subject of word problems resisted because the thing used to make the problem seem less abstract has connotations that the student doesn’t like.
Mac and Bill King’s Magic In A Minute (November 9) shows off a mathematically-based slight-of-hand trick, describing a way to make it look like you’re reading your partner-monkey’s mind. This is probably a nice prealgebra problem to work out just why it works. You could also consider this a toe-step into the problem of encoding messages, finding a way to send information about something in a way that the original information can be recovered, although obviously this particular method isn’t terribly secure for more than a quick bit of stage magic.
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.
I’ve gotten enough comics, I think, to justify a fresh roundup of mathematics appearances in the comic strips. Unfortunately the first mathematics-linked appearance since my most recent entry is also the most badly dated. Pab Sugenis’s The New Adventures of Queen Victoria took (the appropriate) day to celebrate the birthday of Tom Lehrer, but fails to mention his actual greatest contribution to American culture, the “Silent E” song for The Electric Company. He’s also author of the humorous song “Lobachevsky”, which is pretty much the only place to go if you need a mathematics-based song and can’t use They Might Be Giants for some reason. (I regard Lehrer’s “New Math” song as not having a strong enough melody to count.)