Reading the Comics, August 31, 2019: Martin V Edition

And so the Reading the Comics posts have returned to Sunday after a month in exile to Tuesdays. I’m curious whether Sunday is actually the best day to post my signature series of essays, since everybody is usually doing stuff on the weekends. Tuesdays more people are at work and looking for other things to think about. But at least for the duration of the A to Z series there’s not a good time to schedule them besides Sundays. So Sundays it is and I’ll possibly think things over again in December, if all goes well.

Ralph Hagen’s The Barn for the 27th poses a question that’s ridiculous when you look at it. Why should being twenty times as old as your newborn (sic) when you’re twenty years old imply you’d be twenty times as old as the newborn when you’re sixty? Age increases linearly. The ratios between ages, though, those decrease, in a ratio asymptotically approaching 1. So as far as that goes, this strip isn’t much of anything.

But I do like how it captures the way a mathematics puzzle can come from nowhere. Often interesting ones seem to generate themselves. You notice a pattern and wonder whether it reaches some interesting point. If you convince yourself it does, you wonder when it does. If it does not, you wonder why it can’t. This is the fun sort of mathematics, and you create it by looking at the two separate tile patterns in the kitchen or, as here, thinking about the ages of parent and child. Anything that catches the imagination of a bored mind. It’s fun being there.

Rory (the sheep) makes a common enough slip. Saying a twenty-year-old with a newborn is twenty times as old as the newborn is, implicitly, saying the newborn is one year old. This kind of error is so common it’s got a folksy name, the “fencepost error”. It has a more respectable name, for its LinkedIn profile, the “off-by-one error”. But you see the problem. Say that your birthday is the 1st of September. How many times were you alive on the 1st of September by the time you’re ten years old? Eleven times, the first one being the one you were born on, with one more counted up each year you’d lived. This was probably more clear before I explained it.

John Rose’s Barney Google and Snuffy Smith for the 27th has Mis Prunelly complimenting Jughaid’s creativity, but not wanting it in arithmetic. There is creativity in mathematics. And there is great value in calculating something in an original way. There’s value in calculating things wrong, too, if it’s an approximate calculation. Knowing whether your answer is nearer 10 or 20 is of some value, and it might be all that you in fact want. That’s being wrong in a productive way, though.

Harry Bliss and Steve Martin’s Bliss for the 27th uses a string of mathematical symbols as emblem of genius. Most of the symbols look just near enough meaningful that I wonder if Bliss and Martin got a mathematician friend of theirs to give them some scraps. Why I say mathematician rather than, say, physicist is because some of the lines look more mathematician than physicist.

The most distinctive one, to me, is right above Dumbo’s pencil and trunk there: $g^{-1}\cdot g = e$. This is the kind of equation you’ll see all the time in group theory. It’s an important field of mathematics, the one studying sets that work like arithmetic does. This starts with groups, which have a set of things and a binary operation between those things. Think of it as either addition or multiplication. You notice that $g^{-1} \cdot g = e$ already looks like multiplication. ‘g’ and ‘h’ serve, for group theory, the roles that ‘x’ and ‘y’ do in (high school) algebra. ‘x’ and ‘y’ mean some number, whose value we might or might not care about. Similarly, ‘g’ and ‘h’ are some elements, things in the set for our group. We might or might not care which ones they are. $e$ means the identity element, the thing which won’t change the value of the other partner in an operation. The thing that works like zero for addition, or like one for multiplication. And $g^{-1}$ means the inverse of $g$: the thing which, added (or multiplied) to $g$ gives us the identity element. So if we were talking addition and $g$ were 5, then $g^{-1}$ would be -5. This might not sound like very much, but we can make it complicated.

Also distinctive to me: that first line. I’m not perfectly sure I’m transcribing this right. But it looks a good deal to me like the binomial distribution. This is the probability of seeing something happen k times, if you give it n chances to happen, and every chance has the same probability p of it happening. The formula isn’t quite right. It’s missing a power on the (1 – p) term at the end. But it’s wrong in ways that make sense for the need to draw something legible.

Just under Dumbo’s pencil, too, is a line that I had to look up how to render in WordPress’s LaTeX. It’s the one about $\left| X \cup Y \right| = \left| X \right| + \left| Y \right|$. The union symbol, the U there, speaks of set theory. It means to form a new set, one that has all the elements in the set called X or the set called Y or both. The straight vertical lines flanking these set names or descriptions are how we describe taking the norm, finding the size, of a set. This is ordinarily how many things are inside the set. If the sets X and Y have no elements in common, then the size of the union of X and Y will be the size of the set X plus the size of the set Y.

There’s other lines that come near making sense. The line about $f : x \rightarrow xnW$ has the form of the “mapping” way to define a function. I just don’t understand what the rule here means. The final line, $= e \frac{-t^2}{2} !$, first … well, this sort of e-raised-to-the-minus-something-squared form turns up all the time. But second, to end a bit of work with an exclamation point really captures the surprise and joy of having reached a goal. Mathematicians take delight in their work, like you’d expect.

Maria Scrivan’s Half Full for the 29th is a Rubik’s Cube joke. A variation of it ran back in June 2018. I hate that this time I noticed that on the right, the cubelet — with white on top, red on the lower left, and green on the lower right — is inconsistent with the ordered cube. The corresponding cubelet there has blue on top, red on the lower left, and green on the lower right. Well, maybe the cube on the right had its color stickers applied differently. This is a little thing. But it’s close to a problem that turns up all the time in representing geometry. It’s easy to say you have, say, axes going in the x, y, and z directions. But which direction is x? Which is y? Which is z? You can lay all three out so every pair makes a right angle. Whatever way you lay them out will turn out to be, up to a rotation, one of two patterns. Let’s say the x axis points east, and the y axis points north. Then the z axis can point up. Or it can point down. You can pick which one makes sense for your problem. The two choices are mirror images of the other. You get primed to notice this when you do mathematical physics. The Rubik’s Cube on the left is just this kind of representation, with (let’s say) the red face pointing in the x direction, the green face pointing in the y direction, and the blue pointing in the z direction. Which is a lot of thought to put into what was an arbitrary choice, as I’m sure the cartoonist (or whoever did the coloring) just wanted a cube that looked attractive.

There were a surprising number of comics that mentioned mathematics, but not enough for a paragraph. I’ll feature them in another essay run here sometime this week. Also starting this week: the Fall 2019 Mathematics A To Z. It’s still not too late to suggest topics for the letters C through H!

Reading the Comics, March 10, 2018: I Will Get To Pi Day Edition

There were fewer Pi Day comic strips than I had expected for this year. It’s gotten much more public mention than I had expected a pop-mathematics bit of whimsy might. But I’m still working off last week’s strips; I’ll get to this week’s next week. This makes sense to me, which is as good as making sense at all.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 7th is a percentages joke, as applied to hair. Lard doesn’t seem clear whether this would be 10% off the hair by individual strand length or by total volume. Either way, Lard’s right to wonder about the accuracy.

Mark Pett’s Mr Lowe rerun for the 7th is a standardized test joke. Part of the premise of Pett’s strip is that Mister Lowe is a brand-new teacher, which is why he makes mistakes like this problem. (This is touchy to me, as in grad school I hoped to make some spare money selling questions to a standardized testing company. I wasn’t good enough at it, and ultimately didn’t have the time to train up to their needs.) A multiple-choice question needs to clear and concise and to have one clearly best answer. As the given question’s worded, though, I could accept ‘2’ or ’12’ as a correct answer. With a bit of experience Lowe would probably clarify that Tommy and Suzie are getting the same number of apples and that together they should have 20 total.

Then on the 9th Mr Lowe has a joke about cultural bias in standardized tests. It uses an arithmetic problem as the type case. Mathematicians like to think of themselves as working in a universal, culturally independent subject. I suppose it is, but only in ways that aren’t interesting: if you suppose these rules of logic and these axioms and these definitions then these results follow, and it doesn’t matter who does the supposing. But start filtering that by stuff people care about, such as the time it takes for two travelling parties to meet, and you’ve got cultural influence. (Back when this strip was new the idea that a mathematics exam could be culturally biased was a fresh new topic of mockery among people who don’t pay much attention to the problems of teaching but who know what those who do are doing wrong.)

Ralph Hagen’s The Barn for the 8th — a new tag for my comics, by the way — lists a bunch of calculation tools and techniques as “obsolete” items. I’m assuming Rory means that longhand multiplication is obsolete. I’m not sure that it is, but I have an unusual perspective on this.

Thaves’s Frank and Ernest for the 8th is an anthropomorphic-numerals joke. I was annoyed when I first read this because I thought, wait, 97 isn’t a prime number. It is, of course. I have no explanation for my blunder.

Jon Rosenberg’s Scenes from a Multiverse has restarted its run on GoComics. The strip for the 8th is a riff on Venn Diagrams. And, it seems to me, about those logic-bomb problems about sets consisting of sets that don’t contain themselves and the like. You get weird and apparently self-destructive results pondering that stuff. The last time GoComics ran the Scenes from a Multiverse series I did not appreciate right away that there were many continuing stories. There might be follow-ups to this Former Venn Prime Universe story.

Brian Fies’s The Last Mechanical Monster for the 9th has the Mad Scientist, struggling his way into the climax of the story, testing his mind by calculating a Fibonacci Sequence. Whatever keeps you engaged and going. You can build a Fibonacci Sequence from any two starting terms. Each term after the first two is the sum of the previous two. If someone just says “the Fibonacci Sequence” they mean the sequence that starts with 0, 1, or perhaps with 1, 1. (There’s no interesting difference.) Fibonacci Sequences were introduced to the west by Leonardo of Pisa, who did so much to introduce Hindu-Arabic Numerals to a Europe that didn’t know it wanted this stuff. They touch on some fascinating stuff: the probability of not getting two tails in a row of a set number of coin tosses. Chebyshev polynomials. Diophantine equations. They also touch on the Golden Ratio, which isn’t at all important but that people like.

Nicholas Gurewitch’s Perry Bible Fellowship for the 9th just has a blackboard of arithmetic to stand in for schoolwork.