This is the slightly belated close of last week’s topics suggested by Comic Strip Master Command. For the week we’ve had, I am doing very well.
Werner Wejp-Olsen’s Inspector Danger’s Crime Quiz for the 25th of May sees another mathematician killed, and “identifying” his killer in a dying utterance. Inspector Danger has followed killer mathematicians several times before: the 9th of July, 2012, for instance. Or the 4th of July, 2016, for a case so similar that it’s almost a Slylock Fox six-differences puzzle. Apparently realtors and marine biologists are out for mathematicians’ blood. I’m not surprised by the realtors, but hey, marine biology, what’s the deal? The same gimmick got used the 15th of May, 2017, too. (And in fairness to the late Wejp-Olsen, who could possibly care that similar names are being used in small puzzles used years apart? It only stands out because I’m picking out things that no reasonable person would notice.)
Jim Meddick’s Monty for the 25th has the title character inspired by the legend of genius work done during plague years. A great disruption in life is a great time to build new habits, and if Covid-19 has given you the excuse to break bad old habits, or develop good new ones, great! Congratulations! If it has not, though? That’s great too. You’re surviving the most stressful months of the 21st century, I hope, not taking a holiday.
Anyway, the legend mentioned here includes Newton inventing Calculus while in hiding from the plague. The actual history is more complicated, and ambiguous. (You will not go wrong supposing that the actual history of a thing is more complicated and ambiguous than you imagine.) The Renaissance Mathematicus describes, with greater authority and specificity than I could, what Newton’s work was more like. And some of how we have this legend. This is not to say that the 1660s were not astounding times for Newton, nor to deny that he worked with a rare genius. It’s more that we are lying to imagine that Newton looked around, saw London was even more a deathtrap than usual, and decided to go off to the country and toss out a new and unique understanding of the infinitesimal and the continuum.
Mark Anderson’s Andertoons for the 27th is the Mark Anderson’s Andertoons for the week. One of the students — not Wavehead — worries that a geometric ray, going on forever, could endanger people. There’s some neat business going on here. Geometry, like much mathematics, works on abstractions that we take to be universally true. But it also seems to have a great correspondence to ordinary real-world stuff. We wouldn’t study it if it didn’t. So how does that idealization interact with the reality? If the ray represented by those marks on the board goes on to do something, do we have to take care in how it’s used?
Olivia Jaimes’s Nancy for the 8th has Nancy and Sluggo avoiding mathematics homework. Or, “practice”, anyway. There’s more, though; Nancy and Sluggo are doing some analysis of viewing angles. That’s actual mathematics, certainly. Computer-generated imagery depends on it, just like you’d imagine. There are even fun abstract questions that can give surprising insights into numbers. For example: imagine that space were studded, at a regular square grid spacing, with perfectly reflective marbles of uniform size. Is there, then, a line of sight between any two points outside any marbles? Even if it requires tens of millions of reflections; we’re interested in what perfect reflections would give us.
Using playing cards as a makeshift protractor is a creative bit of making do with what you have. The cards spread in a fanfold easily enough and there’s marks on the cards that you can use to keep your measurements reasonably uniform. Creating ad hoc measurement tools like this isn’t mathematics per se. But making a rough tool is a first step to making a precise tool. And you can use reason to improve your estimates.
It’s not on-point, but I did want to share the most wondrous ad hoc tool I know of: You can use an analog clock hand, and the sun, as a compass. You don’t even need a real clock; you can draw the time on a sheet of paper and use that. It’s not a precise measure, of course. But if you need some help, here you go. You’ve got it.
I’m writing this on Thursday, because I’m expecting to be busy Friday and Saturday. It might be a good policy if I planned the deadline for all these Reading the Comics posts to be a couple days before publishing. But it’ll probably ever come to that. I am not yet begun resisting treating this blog like a professional would. Well, what’s been interesting this week so far have been comic strips presenting or about story problems. That’s enough for a theme.
Olivia Jaimes’s Nancy for the 13th makes the familiar complaint that story problems aren’t “useful”. Perhaps not; she can cut the Gordion knot for most common setups. Well, students aren’t likely to get problems for which there’s no other way to find a solution. I suppose what’s happening is that many mathematical puzzles come from questions like, what’s the least amount of information you need to deduce something? Or what’s an indirect way to find that? Mathematicians are often drawn to questions like this. At least Nancy has found there are problems she’s legitimately interested in, questions about how to do a thing she finds important.
Mort Walker and Dik Browne’s vintage Hi and Lois for the 17th has Chip’s new arithmetic book trying to be more relevant. Chip’s still bored by the problem, which chooses to be about foreign aid. (In the 50s and 60s comics discovered it was very funny that the United States would just give money to other countries and not get anything back out of it except maybe their economies staying stable or their countries not going to war too much.)
Norm Feuti’s Gil did not last long enough in syndication. This is a shame. The characters were great, the humor in a mode I like, and young Gil’s fascination with shows about the paranormal was eerily close to my own young self. But it didn’t last; my understanding is newspapers were reluctant to bring in a comic strip starring an impoverished family. This is a many-faceted shame, not least because the eternal tension between Gil’s fantasy life and his reality made it one of the few strips to reproduce the most vital element of Calvin and Hobbes. But Feuti decided to resume drawing Sunday strips, and I choose to include that in my Reading the Comics reading, because this is my blog and I can make the rules here, at least.
So here’s Norm Feuti’s Gil for the 15th. A couple days ago I saw someone amazed at finally learning where sunflower seeds come from. They’re the black part in the center of a sunflower, the part that makes the big yellow flower stand out in such contrast. People were giving the poster a hard time, asking, where did he think they came from? And the answer is just, he hadn’t thought about it. Why would he? It’s quite reasonable to go through life never encountering a sunflower seed except as a snack or as part of bird or squirrel food. Where on the sunflower plant it’d even be just doesn’t come up. If you want to make this a dire commentary on society losing its sense of where things come from, all right, I won’t stop you. But I think it’s more that there are a billion things to notice in the world, and so many things have names that are fanciful or allusive or ironic, that it’s normal not to realize that a phrase might literally represent its content.
So Gil having so associated a quarter with 25 cents, rather than one-fourth of a something, makes sense to me. (Especially given, as noted, that he and his mother are poor, and so he grows up attentive to cash.)
Isaac Asimov, prolific writer of cozy mysteries, had one short story built on the idea that a person might misremember 5:50, seen on a digital clock, as half-past five. I mention this to show how the difference between a quarter of a hundred of things, and the quarter of sixty things, will get mixed together.
Greg Evans’s Luann Againn for the 15th sees Luann struggling with algebra. And thinking of ways to at least get the answers. One advantage mathematics instructors have which many other subjects don’t is that you can create more problems easily. If for some reason isn’t usable anymore, you can make it and still be testing the same skills. But if you want to (as is reasonable) stick to what’s in a published text, yeah, you’re vulnerable to this.
And you can’t always just change a problem arbitrarily. For example, the expression in the second panel of the top row — — I notice factors into . I don’t know the objective of Luann’s homework, but it would probably be messed up if the problem were just changed to . Not that this couldn’t be worked, but that the work would involve annoying and complicated expressions instead of nice whole numbers or reasonable fractions.
Paul Trap’s Thatababy for the 15th presents Thatabay’s first counting-exponentially book, with the number of rabbits doubling every time. I admire the work Trap put in to drawing — in what we see here — 255 bunnies. I’m trusting there’s 128 in the last bunny panel; I’m not counting. At any rate he drew enough bunnies to not make it obvious to me where he repeats figures.
The traditional ever-increasing bunny spiral is the Fibonacci series. But in that, each panel would on average have only about three-fifths more bunnies than the one before it. That’s good, but it isn’t going to overwhelm as fast as the promise of 256 bunnies on the next page will.
The “dual” here is a mathematical term. Many mathematical things have duals. Polyhedrons have a commonly defined dual shape, though. Start with a polyhedron like, oh, the cube. The dual is a new polyhedron. The vertices of the dual are at the centers of the faces of the original polyhedron. And if two faces of the original polyhedron meet at an edge, then there’s an edge connecting the vertices at the centers of those faces. If several faces meet at a vertex in the original polyhedron, then in the dual there’s a face connecting the vertices dual to the original faces. Work all this out and you get, as you might expect, that the shape that’s dual to a cube is the octahedron we’re told just walked into the bar. The dual to the octahedron, meanwhile … well, that is a cube, which is nice and orderly. You might get a bit of a smile working out what the dual to a tetrahedron is.
Duals are useful, generically, because usually if you can prove something about a dual then you can prove it about the original thing. And we may find that something is easier to prove for the dual than for the original. This isn’t guaranteed, especially for geometric shapes like this, where it’s hard to say that either shape is harder to work with than the other. But it’s one of the tools we have to try sliding between the problem we need to do and the problem we can do.
Olivia Jaimes’s Nancy for the 17th has claims about the usefulness of arithmetic. And Nancy skeptical of them, as you expect for a kid facing mathematics in a comic strip. I admit I’ve never needed to do much arithmetic when I cooked. The most would be figuring out how to adjust the cooking time when two things need very different temperatures. But I always do that by winging it. Now I’m curious whether there are good references for suggested alternate times.
There were a healthy number of comic strips with at least a bit of mathematical content the past week. Enough that I would maybe be able to split them across three essays in all. This conflicts with my plans to post two A-To-Z essays, and two short pieces bringing archived things back to some attention, when you consider the other thing I need to post this week. Well, I’ll work out something, this week at least. But if Comic Strip Master Command ever sends me a really busy week I’m going to be in trouble.
Bud Blake’s Tiger rerun for the 7th has Punkinhead ask one of those questions so basic it ends up being good and deep. What is arithmetic, exactly? Other than that it’s the mathematics you learn in elementary school that isn’t geometry? — an answer that’s maybe not satisfying but at least has historical roots. The quadrivium, four of the seven liberal arts of old, were arithmetic, geometry, astronomy, and music. Each of these has a fair claim on being a mathematics study, though I’d agree that music is a small part of mathematics these days. (I first wrote a “minor” piece, and didn’t want people to think I was making a pun, but you’ll notice I’m sharing it anyway.) I can’t say what people who study music learn about mathematics these days. Still, I’m not sure I can give a punchy answer to the question.
Mathworld offers the not-quite-precise definition that arithmetic is the field of mathematics dealing with integers or, more generally, numerical computation. But then it also offers a mnemonic for the spelling of arithmetic, which I wouldn’t have put in the fourth sentence of an article on the subject. I’m also not confident in that limitation to integers. Arithmetic certainly is about things we do on the integers, like addition and subtraction, multiplication and division, powers, roots, and factoring. So, yes, adding five and two is certainly arithmetic. But would we say that adding one-fifth and two is not arithmetic? Most other definitions I find allow that it can be about the rational numbers, or the real numbers. Some even accept the complex-valued numbers. The core is addition and subtraction, multiplication and division.
Arithmetic blends almost seamlessly into more complicated fields. One is number theory, which is the posing of problems that anyone can understand and that nobody can solve. If you ever run across a mathematical conjecture that’s over 200 years old and that nobody’s made much progress on besides checking that it’s true for all the whole numbers below 21,000,000,000 – 1, it’s probably number theory. Another is group theory, in which we think about structures that look like arithmetic without necessarily having all its fancy features like, oh, multiplication or the ability to factor elements. And it weaves into computing. Most computers rely on some kind of floating-point arithmetic, which approximates a wide range of the rational numbers that we’d expect to actually need.
So arithmetic is one of those things so fundamental and universal that it’s hard to take a chunk and say that this is it.
John Zakour and Scott Roberts’s Maria’s Day for the 8th has Maria fretting over what division means for emotions. I was getting ready to worry about Maria having the idea division means getting less of something. Five divided by one-half is not less than either five or one-half. My understanding is this unsettles a great many people learning division. But she does explicitly say, divide two, which I’m reading as “divide by two”. (I mean to be charitable in my reading of comic strips. It’s only fair.)
Still, even division into two things does not necessarily make things less. One of the fascinating and baffling discoveries of the 20th century was the Banach-Tarski Paradox. It’s a paradox only in that it defies intuition. According to it, one ball can be divided into as few as five pieces, and the pieces reassembled to make two whole balls. I would not expect Maria’s Dad to understand this well enough to explain.
There were naturally comic strips with too marginal a mention of mathematics to rate paragraphs. Among them the past week were these.
Stephen Bentley’s Herb and Jamaal rerun for the 11th portrays the aftermath of realizing a mathematics problem is easier than it seemed. Realizing this after a lot of work should feel good, as discovering a clever way around tedious work is great. But the lost time can still hurt.
Each week Comic Strip Master Command sends out some comics that mention mathematics, but that aren’t substantial enough to write miniature essays about. This past week, too. Here are the comics that just mention mathematics. You may like them; there’s just not more to explain is all.
Dan Collins’s Looks Good On Paper rerun for the 27th uses a blackboard of mathematics — geometry-related formulas — to stand in for all classwork. This strip also ran in 2017 and in 2015. I haven’t checked 2013. I know the strip is still in original production, as it’ll include strips referring to current events, so I’ll keep reading it a while yet.
Ernie Bushmiller’s Nancy Classics for the 29th, which originally ran the 23rd of November, 1949, is a basic cheating-in-class joke. It works for mathematics in a way it wouldn’t for, say, history. Mathematics has enough symbols that don’t appear in ordinary writing that you could copy them upside-down without knowing that you transcribe something meaningless. Well, not realizing an upside-down 4 isn’t anything is a bit odd, but anyone can get pretty lost in symbols.
Ernie Bushmiller’s Nancy Classics for the 27th uses arithmetic as an economical way to demonstrate intelligence. At least, the ability to do arithmetic is used as proof of intelligence. Which shouldn’t surprise. The conventional appreciation for Ernie Bushmiller is of his skill at efficiently communicating the ideas needed for a joke. That said, it’s a bit surprising Sluggo asks the dog “six times six divided by two”; if it were just showing any ability at arithmetic “one plus one” or “two plus two” would do. But “six times six divided by two” has the advantage of being a bit complicated. That is, it’s reasonable Sluggo wouldn’t know it right away, and would see it as something only the brainiest would. But it’s not so complicated that Sluggo wouldn’t plausibly know the question.
Eric the Circle for the 28th, this one by AusAGirl, uses “Non-Euclidean” as a way to express weirdness in shape. My first impulse was to say that this wouldn’t really be a non-Euclidean circle. A non-Euclidean geometry has space that’s different from what we’re approximating with sheets of paper or with boxes put in a room. There are some that are familiar, or roughly familiar, such as the geometry of the surface of a planet. But you can draw circles on the surface of a globe. They don’t look like this mooshy T-circle. They look like … circles. Their weirdness comes in other ways, like how the circumference is not π times the diameter.
On reflection, I’m being too harsh. What makes a space non-Euclidean is … well, many things. One that’s easy to understand is to imagine that the space uses some novel definition for the distance between points. Distance is a great idea. It turns out to be useful, in geometry and in analysis, to use a flexible idea of of what distance is. We can define the distance between things in ways that look just like the Euclidean idea of distance. Or we can define it in other, weirder ways. We can, whatever the distance, define a “circle” as the set of points that are all exactly some distance from a chosen center point. And the appearance of those “circles” can differ.
There are literally infinitely many possible distance functions. But there is a family of them which we use all the time. And the “circles” in those look like … well, at the most extreme, they look like squares. Others will look like rounded squares, or like slightly diamond-shaped circles. I don’t know of any distance function that’s useful that would give us a circle like this picture of Eric. But there surely is one that exists and that’s enough for the joke to be certified factually correct. And that is what’s truly important in a comic strip.
Sandra Bell-Lundy’s Between Friends for the 29th is the Venn Diagram joke for the week. Formally, you have to read this diagram charitably for it to parse. If we take the “what” that Maeve says, or doesn’t say, to be particular sentences, then the intersection has to be empty. You can’t both say and not-say a sentence. But it seems to me that any conversation of importance has the things which we choose to say and the things which we choose not to say. And it is so difficult to get the blend of things said and things unsaid correct. And I realize that the last time Between Friends came up here I was similarly defending the comic’s Venn Diagram use. I’m a sympathetic reader, at least to most comic strips.
And that was the conclusion of comic strips through the 29th of June which mentioned mathematics enough for me to write much about. There were a couple other comics that brought up something or other, though. Wulff and Morgenthaler’s WuMo for the 27th of June has a Rubik’s Cube joke. The traditional Rubik’s Cube has three rows, columns, and layers of cubes. But there’s no reason there can’t be more rows and columns and layers. Back in the 80s there were enough four-by-four-by-four cubes sold that I even had one. Wikipedia tells me the officially licensed cubes have gotten only up to five-by-five-by-five. But that there was a 17-by-17-by-17 cube sold, with prototypes for 22-by-22-by-22 and 33-by-33-by-33 cubes. This seems to me like a great many stickers to peel off and reattach.
I concede I am late in wrapping up last week’s mathematically-themed comics. But please understand there were important reasons for my not having posted this earlier, like, I didn’t get it written in time. I hope you understand and agree with me about this.
Bill Griffith’s Zippy the Pinhead for the 9th brings up mathematics in a discussion about perfection. The debate of perfection versus “messiness” begs some important questions. What I’m marginally competent to discuss is the idea of mathematics as this perfect thing. Mathematics seems to have many traits that are easy to think of as perfect. That everything in it should follow from clearly stated axioms, precise definitions, and deductive logic, for example. This makes mathematics seem orderly and universal and fair in a way that the real world never is. If we allow that this is a kind of perfection then … does mathematics reach it?
Even the idea of a “precise definition” is perilous. If it weren’t there wouldn’t be so many pop mathematics articles about why 1 isn’t a prime number. It’s difficult to prove that any particular set of axioms that give us interesting results are also logically consistent. If they’re not consistent, then we can prove absolutely anything, including that the axioms are false. That seems imperfect. And few mathematicians even prepare fully complete, step-by-step proofs of anything. It takes ridiculously long to get anything done if you try. The proofs we present tend to show, instead, the reasoning in enough detail that we’re confident we could fill in the omitted parts if we really needed them for some reason. And that’s fine, nearly all the time, but it does leave the potential for mistakes present.
Zippy offers up a perfect parallelogram. Making it geometry is of good symbolic importance. Everyone knows geometric figures, and definitions of some basic ideas like a line or a circle or, maybe, a parallelogram. Nobody’s ever seen one, though. There’s never been a straight line, much less two parallel lines, and even less the pair of parallel lines we’d need for a parallellogram. There can be renderings good enough to fool the eye. But none of the lines are completely straight, not if we examine closely enough. None of the pairs of lines are truly parallel, not if we extend them far enough. The figure isn’t even two-dimensional, not if it’s rendered in three-dimensional things like atoms or waves of light or such. We know things about parallelograms, which don’t exist. They tell us some things about their shadows in the real world, at least.
Mark Litzler’s Joe Vanilla for the 9th is a play on the old joke about “a billion dollars here, a billion dollars there, soon you’re talking about real money”. As we hear more about larger numbers they seem familiar and accessible to us, to the point that they stop seeming so big. A trillion is still a massive number, at least for most purposes. If you aren’t doing combinatorics, anyway; just yesterday I was doing a little toy problem and realized it implied 470,184,984,576 configurations. Which still falls short of a trillion, but had I made one arbitrary choice differently I could’ve blasted well past a trillion.
Ruben Bolling’s Super-Fun-Pak Comix for the 9th is another monkeys-at-typewriters joke, that great thought experiment about probability and infinity. I should add it to my essay about the Infinite Monkey Theorem. Part of the joke is that the monkey is thinking about the content of the writing. This doesn’t destroy the prospect that a monkey given enough time would write any of the works of William Shakespeare. It makes the simple estimates of how unlikely that is, and how long it would take to do, invalid. But the event might yet happen. Suppose this monkey decided there was no credible way to delay Hamlet’s revenge to Act V, and tried to write accordingly. Mightn’t the monkey make a mistake? It’s easy to type a letter you don’t mean to. Or a word you don’t mean to. Why not a sentence you don’t mean to? Why not a whole act you don’t mean to? Impossible? No, just improbable. And the monkeys have enough time to let the improbable happen.
Eric the Circle for the 10th, this one by Kingsnake, declares itself set in “the 20th dimension, where shape has no meaning”. This plays on a pop-cultural idea of dimensions as a kind of fairyland, subject to strange and alternate rules. A mathematician wouldn’t think of dimensions that way. 20-dimensional spaces — and even higher-dimensional spaces — follow rules just as two- and three-dimensional spaces do. They’re harder to draw, certainly, and mathematicians are not selected for — or trained in — drawing, at least not in United States schools. So attempts at rendering a high-dimensional space tend to be sort of weird blobby lumps, maybe with a label “N-dimensional”.
And a projection of a high-dimensional shape into lower dimensions will be weird. I used to have around here a web site with a rotatable tesseract, which would draw a flat-screen rendition of what its projection in three-dimensional space would be. But I can’t find it now and probably it ran as a Java applet that you just can’t get to work anymore. Anyway, non-interactive videos of this sort of thing are common enough; here’s one that goes through some of the dimensions of a tesseract, one at a time. It’ll give some idea how something that “should” just be a set of cubes will not look so much like that.
Steve Kelly and Jeff Parker’s Dustin for the 11th is a variation on the “why do I have to learn this” protest. This one is about long division and the question of why one needs to know it when there’s cheap, easily-available tools that do the job better. It’s a fair question and Hayden’s answer is a hard one to refute. I think arithmetic’s worth knowing how to do, but I’ll also admit, if I need to divide something by 23 I’m probably letting the computer do it.
I do not know that the Ziggy printed here is a rerun. I don’t seem to have mentioned it in previous Reading the Comics posts, but that isn’t definite. How much mathematical content a comic strip needs to rate a mention depends on many things, and a strip that seems too slight one week might inspire me another. I’ll explain why I’ve started to get suspicious of the quite humanoid figure.
Tom II Wilson’s Ziggy for the 12th is framed around weather forecasts. It’s the probability question people encounter most often, unless they’re trying to outsmart the contestants on Let’s Make A Deal. (And many games on The Price Is Right, too.) Many people have complained about not knowing the meaning of a “50% chance of rain” for a day. If I understand it rightly, it means, when conditions have been like this in the recorded past, it’s rained about 50% of the time. I’m open to correction from meteorologists and it just occurred to me I know one. Mm.
Few people ask about the probability a forecast is correct. In some ways it’s an unanswerable question. To say there is a one-in-six chance a fairly thrown die will turn up a ‘1’ is not wrong just because it’s rolled a ‘1’ eight times out of the last ten. But it does seem like a forecast such as this should include a sense of confidence, how sure the forecaster is that the current weather is all that much like earlier times.
I’m not sure how much of the joke is meant to be the repetition of “50% chance”. The joke might be meant to say that if he’s got a 50% chance of being wrong, then, isn’t the 50% chance of rain “correctly” a 50% chance of not-rain … which is the same chance of rain? The logic doesn’t hold up, if you pay attention, but it sounds like it should make sense, and having the “wrong” version of something be the same as the original is a valid comic construction.
So now for the promised Ziggy rerun scandal. To the best of my knowledge Ziggy is presented as being in new run. It’s done by the son of the comic strip’s creator, but that’s common enough for long-running comic strips. This Monday, though, ran a Ziggy-at-the-psychiatrist joke that was, apart from coloring, exactly the comic run the 2nd of March, barely two weeks before. (Compare the scribbles in the psychiatrist’s diploma.) It wouldn’t be that weird if a comic were accidentally repeated; production mistakes happen, after all. It’s slightly weird that the daily, black-and-white, original got colored in two different ways, but I can imagine this happening by accident.
Still, that got me primed to look for Ziggy repeats. I couldn’t find this one having an earlier appearance. But I did find that the 9th of January this year was a reprint of the Ziggy from the 11th of January, 2017. I wrote about both appearances, without noticing they were reruns. Here’s the 2017 essay, and over here is the 2019 essay, from before I was very good at remembering what the year was. Mercifully I didn’t say anything contradictory on the two appearances. I’m more interested in how I said things differently in the two appearances. Anyway this earlier year seems to have been part of a week’s worth of reruns, noticeable by the copyright date. I can’t begrudge a cartoonist their vacation. The psychiatrist strip doesn’t seem to be part of that, though, and its repetition is some as-yet-unexplained event.
Tony Rubino and Gary Markstein’s Daddy’s Home for the 13th has a much more casual and non-controversial bit of mathematics. Pete tosses out a calculate-the-square-root problem as a test of Peggy’s omniscience. One of the commenters points out that the square root of 532 is closer to 23.06512519 than it is Peggy’s 23.06512818. It suggests the writers found the square root by something that gave plenty of digits. For example, the macOS Calculator program offers me “23.065 125 189 341 592”. But then they chopped off, rather than rounding off, digits when the panel space ran out.
Olivia Jaimes’s Nancy for the 13th has Nancy dividing up mathematics problems along the equals sign. That’s cute and fanciful enough. One could imagine working out expressions on either side of the equals sign in the hopes of getting them to match. That wouldn’t work for these algebra problems, but, that’s something.
This isn’t what Nancy might do, unless she flashed forward to college and became a mathematics or physics major. But one great trick in differential equations is called the separation of variables. Differential equations describe how quantities change. They’re great. They’re hard. A lot of solving differential equations amounts to rewriting them as simpler differential equations.
Separation is a trick usable when there’s two quantities whose variation affect each other. If you can rewrite the differential equation so that one variable only appears on the left side, and the other variable only appears on the right? Then you can split this equation into two simpler equations. Both sides of the equation have to be some fixed number. So you can separate the differential equations of two variables into two differential equations, each with one variable. One with the first variable, one with the other. And, usually, a differential equation of one variable is easier than a differential equation with two variables. So Nancy and Esther could work each half by themselves. But the work would have to be put together at the end, too.
I thought I had a flood of mathematically-themed comic strips last week. On reflection, many of them were slight enough not to need further context. You’ll see in the paragraph of not-discussed strips at the end of this. What did rate discussion turned out to get more interesting to me the more I wrote about them.
Stephen Beals’s Adult Children for the 6th uses mathematics as icon of things that are indisputably true. Two plus two equals four is a good example of such. If we take the ordinary meanings of ‘two’ and ‘plus’ and ‘equals’ and ‘four’ there’s no disputing it. The result follows from some uncontroversial-seeming axioms and a lot of deduction. By the rules of logic, the conclusion has to be true, whoever makes it. Even, for that matter, if nobody makes it. It’s difficult to imagine a universe in which nobody ever notices two plus two equals four. But we can imagine that there are mathematical truths that will never be noticed by anyone. (Here’s one. There is some largest finite whole number that any human-created project will ever use in any context. Consider the equation represented by “that number plus two equals (even bigger number)”.)
But you see cards palmed there. What do we mean by ‘two’? Have we got a good definition? Might there be a different definition that’s more useful? Probably not, for ‘two’ anyway. But a part of mathematics, especially as a field develops, is working out what are the important concepts, and what their definitions should be. What a ‘function’ is, for example, went through a lot of debate and change over the 19th century. There is an elusiveness to facts, even in mathematics, where you’d think epistemology would be simpler.
Frank Page’s Bob the Squirrel for the 6th continues the SAT prep questions from earlier in the week. There’s two more problems in shuffling around algebraic expressions here. The first one, problem 5, is probably easiest to do by eliminating wrong answers. is a tedious mess. But look at just the terms: they have to add up to , so, the answer has to be either c or d. So next look at the terms and oh, that’s nice. They add up to zero. The answer has to be c. If you feel like checking the terms, go ahead; that’ll offer some reassurance, if you do the addition correctly.
The second one, problem 8, is probably easier to just think out. If then there’s a lot of places to go. What stands out to me is that has the reciprocal of in it. So, the reciprocal of has to equal the reciprocal of . So . And is, well, four times , so, four times one-half, or two. There’s other ways to go about this. In honestly, what I did when I looked at the problem was multiply both sides of by . But it’s harder to explain why that struck me as an obviously right thing to do. It’s got shortcuts I grew into from being comfortable with the more methodical approach. Someone who does a lot of problems like these will discover shortcuts.
Rick Detorie’s One Big Happy for the 6th asks one of those questions you need to be a genius or a child to ponder. Why don’t the numbers eleven and twelve follow the pattern of the other teens, or for that matter of twenty-one and thirty-two, and the like? And the short answer is that they kind of do. At least, “eleven” and “twelve”, etymologists agree, derive from the Proto-Germanic “ainlif” and “twalif”. If you squint your mouth you can get from “ain” to “one” (it’s probably easier if you go through the German “ein” along the way). Getting from “twa” to “two” is less hard. If my understanding is correct, etymologists aren’t fully agreed on the “lif” part. But they are settled on it means the part above ten. Like, “ainlif” would be “one left above ten”. So it parses as one-and-ten, putting it in form with the old London-English preference for one-and-twenty or two-and-thirty as word constructions.
It’s not hard to figure how “twalif” might over centuries mutate to “twelve”. We could ask why “thirteen” didn’t stay something more Old Germanic. My suspicion is that it amounts to just, well, it worked out like that. It worked out the same way in German, which switches to “-zehn” endings from 13 on. Lithuanian has all the teens end with “-lika”; Polish, similarly, but with “-ście”. Spanish — not a Germanic language — has “custom” words for the numbers up to 15, and then switches to “diecis-” as a prefix to the numbers 6 through 9. French doesn’t switch to a systematic pattern until 17. (And no I am not going to talk about France’s 80s and 90s.) My supposition is that different peoples came to different conclusions about whether they needed ten, or twelve, or fifteen, or sixteen, unique names for numbers before they had to resort to systemic names.
Doug Savage’s Savage Chickens for the 6th is a percentages comic. It makes reference to an old series of (American, at least) advertisements in which four out of five dentists would agree that chewing sugarless gum is a good thing. Shifting the four-out-of-five into 80% riffs is not just fun with tautologies. Percentages have this connotation of technical precision; 80% sounds like a more rigorously known number than “four out of five”. It doesn’t sound as scientific as “0.80”, quite. But when applied to populations a percentage seems less bizarre than a decimal.
There are some subjects that seem to come up all the time in these Reading the Comics posts. Lotteries. Roman numerals. Venn Diagrams. The New Math. Kids not doing arithmetic well, or not understanding when they do it. This is the slate of comics for today’s discussion.
Olivia Jaimes’s Nancy for the 27th is the Roman Numerals joke for the week. I am not certain there is a strong consensus about the origins of Roman numerals. It’s hard to suppose that the first several numerals, though, are all that far from tally marks. Adding serifs just makes the numerals probably easier to read, if harder to write. I’ll go along with Nancy’s excuse of using the weights to represent work with a lesser weight.
Rick DeTorie’s One Big Happy for the 28th has the kid, Joe, impressed by something that he ought to have already expected. Grandpa uses this to take a crack at “that new, new math”, as though there were a time people weren’t amazed by what they should have deduced. Or a level of person who’s not surprised by the implications. One of Richard Feynman’s memoirs recounts him pranking people who have taken calculus by pointing out how whatever way you hold a French curve, the lowest point on it has a horizontal slope. This is true of the drafting instrument; but it’s also true of any curve that hasn’t got a corner or discontinuity.
There aren’t comments (so far as I’m aware) on Creators.com, which hosted this strip. So there weren’t any cracks about Common Core. But I am curious whether DeTorie wrote Grandpa as mentioning the New Math because the character would, plausibly, have seen that educational reform movement come and go. Or did DeTorie just riff on the New Math because that’s been a reliable punching bag since the mid-60s?
The first two comics for this essay have titles of the form Name’s Thing, so, that’s why this edition title. That’s good enough, isn’t it? And besides this series there was a Perry Bible Fellowship which at least depicted mathematical symbols. It’s a rerun, though, even among those shown on GoComics.com. It was rerun recently enough that I featured it around here back in June. It’s a bit risque. But the strip was rerun the 12th. Maybe I also need to drop Perry Bible Fellowship from the roster of comics I read for this.
On to the comics I haven’t dropped.
Tony Buino and Gary Markstein’s Daddy’s Home for the 11th tries using specific examples to teach mathematics. There’s strangeness to arithmetic. It’s about these abstract things like “thirty” and “addition” and such. But these things match very well the behaviors of discrete objects, ones that don’t blend together or shatter by themselves. So we can use the intuition we have for specific things to get comfortable working with the abstract. This doesn’t stop, either. Mathematicians like to work on general, abstract questions; they let us answer big swaths of questions all at once. But working out a specific case is usually easier, both to prove and to understand. I don’t know what’s the most advanced mathematics that could be usefully practiced by thinking about cupcakes. Probably something in group theory, in studying the rotations of objects that are perfectly, or nearly, rotationally symmetric.
John Zakour and Scott Roberts’s Maria’s Day for the 11th is a follow-up to a strip featured last week. Maria’s been getting help on her mathematics from one of her closet monsters. And includes the usual joke about Common Core being such a horrible thing that it must come from monsters. I don’t know whether in the comic strip’s universe the monster is supposed to be imaginary. (Usually, in a comic strip, the question of whether a character is imaginary-or-real is pointless. I think Richard Thompson’s Cul de Sac is the only one to have done something good with it.) But if the closet monster is in Maria’s imagination, it’s quite in line for her to think that teaching comes from some malevolent and inscrutable force.
Olivia Jaimes’s Nancy for the 12th features one of the first interesting mathematics questions you do in physics. This is often done with calculus. Not much, but more than Nancy and Esther could realistically have. It could be worked out experimentally, and that’s likely what the teacher was hoping for. Calculus isn’t really necessary, although it does show skeptical students there’s some value in all this d-dx business they’ve been working through. You can find the same answers by dimensional analysis, which is less intimidating. But you’d still need to know some trigonometry functions. That’s beyond whatever Nancy’s grade level is too. In any case, Nancy is an expert at identifying unstated assumptions, and working out loopholes in them. I’m curious whether the teacher would respect Nancy’s skill here. (The way the writing’s been going, I think she would.)
Francesco Marciuliano and Jim Keefe’s Sally Forth for the 13th is about new-friend Jenny trying to work out her relationship with Hilary-Faye-and-Nona. It’s a good bit of character work, but that is outside my subject here. In the last panel Nona admits she’s been talking, or at least thinking about τ versus π. This references a minor nerd-squabble that’s been going on a couple years. π is an incredibly well-known, useful number. It’s the only transcendental number you can expect a normal person to have ever heard of. Humans noticed it, historically, because the length of the circumference of a circle is π times the length of its diameter. Going between “the distance across” and “the distance around” turns out to be useful.
The thing is, many mathematical and physics formulas find it more convenient to write things in terms of the radius of a circle or sphere. And this makes 2π show up in formulas. A lot. Even in things that don’t obviously have circles in them. For example, the Gaussian distribution, which describes how much a sample looks like the population it’s sampled from, has 2π in it. So, the τ argument does, why write out 2π all these places? Why not decide that that’s the useful number to think about, give it the catchy name τ, and use that instead? All the interesting questions about π have exact, obvious parallel questions about τ. Any answers about one give us answers about the other. So why not make this switch and then … pocket the savings in having shorter formulas?
You may sense in me a certain skepticism. I don’t see where changing over gets us anything worth the bother. But there are fashions in mathematics as with everything else. Perhaps τ has some ability to clarify things in ways we’ll come to better appreciate.
Ryan North’s Dinosaur Comics for the 18th is based on Hilbert’s Hotel. This is a construct very familiar to eager young mathematicians. It’s an almost unavoidable pop-mathematics introduction to infinitely large sets. It’s a great introduction because the model is so mundane as to be easily imagined. But you can imagine experiments with intuition-challenging results. T-Rex describes one of the classic examples in the third through fifth panels.
The strip made me wonder about the origins of Hilbert’s Hotel. Everyone doing pop mathematics uses the example, but who created it? And the startling result is, David Hilbert, kind of. My reference here is Helge Kragh’s paper The True (?) Story of Hilbert’s Infinite Hotel. Apparently in a 1924-25 lecture series in Göttingen, Hilbert encouraged people to think of a hotel with infinitely many rooms. He apparently did not use it for so many examples as pop mathematicians would. He just used the question of how to accommodate a single new guest after the infinitely many rooms were first filled. And then went to imagine an infinite dance party. I don’t remember ever seeing the dance party in the wild; perhaps it’s a casualty of modern rave culture.
Hilbert’s Hotel seems to have next seen print in George Gamow’s One, Two Three … Infinity. Gamow summoned the hotel back from the realms of forgotten pop mathematics with a casual, jokey tone that fooled Kragh into thinking he’d invented the model and whimsically credited Hilbert with it. (Gamow was prone to this sort of lighthearted touch.) He came back to it in The Creation Of The Universe, less to make readers consider the modern understanding of infinitely large sets than to argue for a universe having infinitely many things in it.
And then it disappeared again, except for cameo appearances trying to argue that the steady-state universe would be more bizarre than what we actually see. The philosopher Pamela Huby seems to have made Hilbert’s Hotel a thing to talk about again, as part of a debate about whether a universe could be infinite in extent. William Lane Craig furthered using the hotel, as part of the theological debate about whether there could be an infinite temporal regress of events. Rudy Rucker and Eli Maor wrote descriptions of the idea in the 1980s, with vague ideas about whether Hilbert actually had anything to do with the place. And since then it’s stayed, a famous fictional hotel.
David Hilbert was born in 1862; T-Rex misspoke.
Ernie Bushmiller’s Nancy Classics for the 20th gets me out of my Olivia Jaimes rut. We could probably get a good discussion going about whether giving an example of a sphere is an adequate description of a sphere. Granted that a bubble-gum bubble won’t be perfectly spherical; neither will any example that exists in reality. We always trust that we can generalize to an ideal example of this thing.
I did get to wondering, in Sluggo’s description of the octagon, why the specification of eight sides and eight angles. I suspect it’s meant to avoid calling an octagon something that, say, crosses over itself, thus having more angles than sides. Not sure, though. It might be a phrasing intended to make sure one remembers that there are sides and there are angles and the polygon can be interesting for both sets of component parts.
John Atkinson’s Wrong Hands for the 20th is the Venn Diagram joke for the week. The half-week anyway. Also a bunch of other graph jokes for the week. Nice compilation of things. I love the paradoxical labelling of the sections of the Venn Diagram.
Tom II Wilson’s Ziggy for the 20th is a plaintive cry for help from a despairing soul. Who’s adding up four- and five-digit numbers by hand for some reason. Ziggy’s got his projects, I guess is what’s going on here.
Glenn McCoy and Gary McCoy’s The Duplex for the 21st is set up as an I-hate-word-problems joke. The cop does ask something people would generally like to know, though: how much longer would it take, going 60 miles per hour rather than 70? It turns out it’s easy to estimate what a small change in speed does to arrival time. Roughly speaking, reducing the speed one percent increases the travel time one percent. Similarly, increasing speed one percent decreases travel time one percent. Going about five percent slower should make the travel time a little more than five percent longer. Going from 70 to 60 miles per hour reduces the speed about fifteen percent. So travel time is going to be a bit more than 15 percent longer. If it was going to be an hour to get there, now it’ll be an hour and ten minutes. Roughly. The quality of this approximation gets worse the bigger the change is. Cutting the speed 50 percent increases the travel time rather more than 50 percent. But for small changes, we have it easier.
There are a couple ways to look at this. One is as an infinite series. Suppose you’re travelling a distance ‘d’, and had been doing it at the speed ‘v’, but now you have to decelerate by a small amount, ‘s’. Then this is something true about your travel time ‘t’, and I ask you to take my word for it because it has been a very long week and I haven’t the strength to argue the proposition:
‘d’ divided by ‘v’ is how long your travel took at the original speed. And, now, — the fraction of how much you’ve changed your speed — is, by assumption, small. The speed only changed a little bit. So is tiny. And is impossibly tiny. And is ridiculously tiny. You make an error in dropping these squared and cubed and forth-power and higher terms. But you don’t make much of one, not if s is small enough compared to v. And that means your estimate of the new travel time is:
Or, that is, if you reduce the speed by (say) five percent of what you started with, you increase the travel time by five percent. Varying one important quantity by a small amount we know as “perturbations”. Working out the approximate change in one quantity based on a perturbation is a key part of a lot of calculus, and a lot of mathematical modeling. It can feel illicit; after a lifetime of learning how mathematics is precise and exact, it’s hard to deliberately throw away stuff you know is not zero. It gets you to good places, though, and fast.
Morrie Turner’s Wee Pals for the 21st shows Wellington having trouble with partitions. We can divide any counting number up into the sum of other counting numbers in, usually, many ways. I can kind of see his point; there is something strange that we can express a single idea in so many different-looking ways. I’m not sure how to get Wellington where he needs to be. I suspect that some examples with dimes, quarters, and nickels would help.
Some of the comics last week don’t leave me much to talk about. Well, there should be another half-dozen comics under review later in the week. You’ll stick around, won’t you please?
Anthony Blades’s Bewley for the 16th is a rerun, and an old friend. It’s appeared the 14th of August, 2016, and in April 2015 and in May 2013. Maybe it’s time I dropped the strip from my reading. The scheme by which the kids got the right answer out of their father is a variation on the Clever Hans trick. Clever Hans was a famous example of animal perception: the horse appeared to be able to do arithmetic, tapping his hoof to signal a number. Brilliant experimental design found what was going on. Not that the horse was clever enough to tell (to make up an example) 18 divided by 3. But that the horse was clever enough to recognize the slight change in his trainer’s expression when he had counted off six. Animals (besides humans) do have some sense of numbers, but not that great a sense.
Jeff Stahler’s Moderately Confused for the 16th is the old joke told about accountants and lawyers when they encounter mathematics, recast to star the future disgraced former president. The way we normally define ‘two’ and ‘plus’ and ‘two’ and ‘equals’ and ‘four’ there’s not room for quibbling about their relationship. Not without just lying, anyway. Thus this satisfies the rules of joke formation.
Olivia Jaimes’s Nancy for the 16th is, I think, the point that Jaimes’s Nancy has appeared in my essays more than Guy Gilchrist’s ever did. Well, different artists have different interests. This one depicts Nancy getting the motivation she needed to excel in arithmetic. I’m not convinced of the pedagogical soundness of the Nancy comic strip. But it’s not as though people won’t practice things for rewards.
The title doesn’t mean anything. My laptop’s random-draw of pictures pulled up one from the county fair last year is all. I’m just working too close to deadline to have a good one. Pet rabbit has surgery scheduled and we are hoping that turns out well for everyone involved.
Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 12th has the blackboard of mathematical symbols. Familiar old shorthand of conflating mathematics ability with genius, or at least intelligence. The blackboard isn’t particularly full of expressions, possibly because Caulfield and Rouillard’s art might not be able to render too much detail clearly. It’s also got a sort-of appearance of Einstein’s most famous equation. Although with perhaps an extra joke to it. Suppose we’re to take ‘E’ and ‘M’ and ‘C’ to mean what they do in Einstein’s use. Then has to equal zero. And there are many things you can safely do with zero. Dividing by it, though, isn’t one. I shan’t guess whether Caulfield and Rouillard were being that sly, though.
Marty Links’s Emmy Lou rerun for the 13th tries to be a paradox. How can one like mathematics without liking figures? But arithmetic is just one part of mathematics. Surely the most-used part, if we go by real-world utility. But not everything. Arithmetic is often useful, yes. But you can do good work in (say) logic or knot theory or geometry with only a slight ability to add or subtract or multiply. There’s not enough emphasis put on that in early education. I suppose it reflects the reasonable feeling that people do need to be competent at arithmetic, which is useful. But it gives one a distorted view of what mathematics can be.
Mark Parisi’s Off The Markfor the 13th is the anthropomorphic numerals joke for the week. And it presents being multiplied by zero as a terrifying fate for other numbers. This seems to reflect the idea that being multiplied by zero is equivalent to being made into nothing. That it’s being killed. Zero enjoys this dual meaning, culturally, representing both a number and the concept of a thing that doesn’t exist and the concept of non-existence. If being turned from one number to another is a numeral murder, then a 2 sneaking in with a + sign would be at least as horrifying. But that joke wouldn’t work, and I know that too.
Olivia Jaimes’s Nancy for the 14th is another recreational-mathematics puzzle. I know nothing of Jaimes’s background but apparently it involves a keen interest in that kind of play that either makes someone love or hate mathematics. (Myself, I’m only slightly interested in these kinds of puzzles, most of the time.) This one — add one line to ‘fix’ the equation 5 + 5 + 5 + 5 = 555 — I hadn’t encountered before. Took some fuming to work it out. The obvious answer, of course, is to add a slash across the = sign so that it means “does not equal”.
But that answer’s dull. What mathematicians like are statements that are true and interesting. There are many things that 5 + 5 + 5 + 5 does not equal. Why single out 555 from that set? So negating the equals sign meets the specifications of the problem, slightly better than Nancy does herself. It doesn’t have the surprise of the answer Nancy’s teacher wants.
If you don’t get how to do it, highlight over the paragraph below for a hint.
There are actually three ways to add the stroke to make this equation true. The three ways are equivalent, though. Notice that the symbols on the board comprise strokes and curves and consider that the meaning of the symbol can be changed by altering the composition of those strokes and curves.
Ted Shearer’s Quincy for the 21st of May, 1979, and rerun the 14th is a joke about making mathematics problems relevant. And, yeah, I’ll give Mrs Glover credit for making problems that reflect stuff students know they’re going to have to deal with. Also that they may have already dealt with and so have some feeling for what plausible answers will be. It’s tough to find many problems like that which don’t repeat themselves too much. (“If your pants need a new patch every two months how many would you have in three years?”).
I apologize for not having a more robust introduction here. My week’s been chopped up by concern with the health of the older of our rabbits. Today’s proved to be less alarming than we had feared, but it’s still a lot to deal with. I appreciate your kind thoughts. Thank you.
Meanwhile the comics from last week have led me to discover something really weird going on with the Mutt and Jeff reruns.
Charles Schulz’s Peanuts Classics for the 6th has the not-quite-fully-formed Lucy trying to count the vast. She’d spend a while trying to count the stars and it never went well. It does inspire the question of how to count things when doing a simple tally is too complicated. There are many mathematical approaches. Most of them are some kind of sampling. Take a small enough part that you can tally it, and estimate the whole based on what your sample is. This can require ingenuity. For example, when estimating our goldfish population, it was impossible to get a good sample at one time. When tallying the number of visible stars in the sky, we have the problem that the Galaxy has a shape, and there are more stars in some directions than in others. This is why we need statisticians.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th looks initially like it’s meant for a philosophy blog’s Reading the Comics post. It’s often fruitful in the study of ethics to ponder doing something that is initially horrible, but would likely have good consequences. Or something initially good, but that has bad effects. These questions challenge our ideas about what it is to do good or bad things, and whether transient or permanent effects are more important, and whether it is better to be responsible for something (or to allow something) by action or inaction.
It comes to mathematics in the caption, though, and with an assist from the economics department. Utilitarianism seems to offer an answer to many ethical problems. It posits that we need to select a primary good of society, and then act so as to maximize that good. This does have an appeal, I suspect even to people who don’t thrill of the idea of finding the formula that describes society. After all, if we know the primary good of society, why should we settle for anything but the greatest value of that good? It might be difficult in practice, say, to discount the joy a musician would bring over her lifetime with her performances fairly against the misery created by making her practice the flute after school when she’d rather be playing. But we can imagine working with a rough approximation, at least. Then the skilled thinkers point out even worse problems and we see why utilitarianism didn’t settle all the big ethical questions, even in principle.
The mathematics, though. As Weinersmith’s caption puts it, we can phrase moral dilemmas as problems of maximizing evil. Typically we pose them as ones of maximizing good. Or at least of minimizing evil. But if we have the mechanism in place to find where evil is maximized, don’t we have the tools to find where good is? If we can find the set of social parameters x, y, and z which make E(x, y, z) as big as possible, can’t we find where -E(x, y, z) is as big, too? And isn’t that then where E(x, y, z) has to be smallest?
And, sure. As long as the maximum exists, or the minimum exists. Maybe we can tell whether or not there is one. But this is why when you look at the mathematics of finding maximums you realize you’re also doing minimums, or vice-versa. Pretty soon you either start referring to what you find as extremums. Or you stop worrying about the difference between a maximum and a minimum, at least unless you need to check just what you have found. Or unless someone who isn’t mathematically expert looks at you wondering if you know the difference between positive and negative numbers.
Bud Fisher’s Mutt and Jeff for the 7th has run here before. Except that was before they redid the lettering; it was a roast beef in earlier iterations. I was thinking to drop Mutt and Jeff from my Reading the Comics routine before all these mysteries in the lettering turned up. Anyway. The strip’s joke starts with a work-rate problems. Given how long some people take to do a thing, how long does it take a different number of people to do a thing? These are problems that demand paying attention to units, to the dimensions of a thing. That seems to be out of fashion these days, which is probably why these questions get to be baffling. But if eating a ham takes 25 person-minutes to do, and you have ten persons eating, you can see almost right away how long to expect it to take. If the ham’s the same size, anyway.
Olivia Jaimes’s Nancy for the 7th is built on a spot of recreational mathematics. Also on the frustration one can have when a problem looks like it’s harmless innocent fun and turns out to take just forever and you’re never sure you have the answers just right. The commenters on GoComics.com have settled on 18. I’m content with that answer.
Although the hyperbolic cosine is interesting and I could go on about it.
Eric the Circle for the 18th of June is a bit of geometric wordplay for the week. A secant is — well, many things. One of the important things is it’s a line that cuts across a circle. It intersects the circle in two points. This is as opposed to a tangent, which touch it in one. Or missing it altogether, which I think hasn’t got any special name. “Secant” also appears as one of the six common trig functions out there.
In value the secant of an angle is just the reciprocal of the cosine of that angle. Where the cosine is never smaller than -1 nor larger than 1, the secant is always either greater than 1 or smaller than -1. It’s a useful function to have by name. We can write “the secant of angle θ” as . The otherwise sensible-looking is unavailable, because we use that to mean “the angle whose cosine is θ”. We need to express that idea, the “arc-cosine” or “inverse cosine”, quite a bit too. And would look like we wanted the cosine of one divided by θ. Ultimately, we have a lot of ideas we’d like to write down, and only so many convenient quick shorthand ways to write them. And by using secant as its own function we can let the arc-cosine have a convenient shorthand symbol. These symbols are a point where you see the messy, human, evolutionary nature of mathematical symbols at work.
We can understand the cosine of an angle θ by imagining a right triangle with hypotenuse of length 1. Set that so the hypotenuse makes angle θ with respect to the x-axis. Then the opposite leg of that right triangle will be the cosine of θ away from the origin. The secant, now, that works differently. Again here imagine a right triangle, but this time one of the legs has length 1. And that leg is at an angle θ with respect to the x-axis. Then the far leg of that right triangle is going to cross the x-axis. And it’ll do that at a point that’s the secant of θ away from the origin.
Larry Wright’s Motley Classics for the 19th speaks of algebra as the way to explain any sufficiently complicated thing. Algebra’s probably not the right tool to analyze a soap opera, or any drama really. The interactions of characters are probably more a matter for graph theory. That’s the field that studies groups of things and the links between them. Occasionally you’ll see analyses of, say, which characters on some complicated science fiction show spend time with each other and which ones don’t. I’m not aware of any that were done on soap operas proper. I suspect most mathematics-oriented nerds view the soaps as beneath their study. But most soap operas do produce a lot of show to watch, and to summarize; I can’t blame them for taking a smaller, easier-to-summarize data set to study. (Also I’m not sure any of these graphs reveal anything more enlightening than, “Huh, really thought The Doctor met Winston Churchill more often than that”.)
Dave Whamond’s Reality Check for the 19th is our Venn Diagram strip for the week. I say the real punch line is the squirrel’s, though. Properly, yes, the Venn Diagram with the two having nothing in common should still have them overlap in space. There should be a signifier inside that there’s nothing in common, such as the null symbol or an x’d out intersection. But not overlapping at all is so commonly used that it might as well be standard.
Teresa Bullitt’s Frog Applause for the 21st uses a thought balloon full of mathematical symbols as icon for far too much deep thinking to understand. I would like to give my opinion about the meaningfulness of the expressions. But they’re too small for me to make out, and GoComics doesn’t allow for zooming in on their comics anymore. I looks like it’s drawn from some real problem, based on the orderliness of it all. But I have no good reason to believe that.
I have another Reading the Comics post with a title that’s got nothing to do with the post. It has got something to do with how I spent my weekend. Not sure if I’ll ever get around to explaining that since there’s not much mathematical content to that weekend. I’m not sure whether the nonsense titles are any better than trying to find a theme in what Comic Strip Master Command has sent the past week. It takes time to pick something when anything would do, after all.
Scott Hilburn’s The Argyle Sweater for the 10th is the anthropomorphic numerals strip for the week. Also arithmetic symbols. The ÷ sign is known as “the division symbol”, although now and then people try to promote it as the “obelus”. They’re not wrong to call it that, although they are being the kind of person who tries to call the # sign the “octothorp”. Sometimes social media pass around the false discovery that the ÷ sign is a representation of a fraction, , with the numbers replaced by dots. It’s a good mnemonic for linking fractions and division. But it’s wrong to say that’s what the symbol means. ÷ started being used for division in Western Europe in the mid-17th century, in competition with many symbols, including / (still in common use), : (used in talking about ratios or odds), – (not used in this context anymore, and just confusing if you do try to use it so). And ÷ was used in northern Europe to mean “subtraction” for several centuries after this.
Tom Toles’s Randolph Itch, 2am for the 11th is a repeat; the too-short-lived strip has run through several cycles since I started doing these summaries. But it is also one of the great pie chart jokes ever and I have no intention of not telling people to enjoy it.
Pie charts, and the also-mentioned bar charts, come to us originally from the economist William Playfair, who in the late 1700s and early 1800s devised nearly all the good ways to visualize data. But we know them thanks to Florence Nightingale. Among her other works, she recognized in these charts good ways to represent her studies about Crimean War medicine and about sanitation in India. Nightingale was in 1859 named the first woman in the Royal Statistical Society, and was named an honorary member of the American Statistical Association in 1874.
Olivia Jaimes’s Nancy for the 12th uses arithmetic as iconic for classwork nobody wants to do. Algebra, too; I understand the reluctance to start. Simultaneous solutions; the challenge is to find sets of values ‘x’ and ‘y’ that make both equations true together. That second equation is a good break, though. makes it easy to write what ‘y’ has to be in terms of ‘x’. Then you can replace the ‘y’ in the first equation with its expression in terms of ‘x’. In a slightly tedious moment, it’s going to turn out there’s multiple sets of answers. Four sets, if I haven’t missed something. But they’ll be clearly related to each other. Even attractively arranged.
is an equation that’s true if the numbers ‘x’ and ‘y’ are coordinates of the points on a circle. This is if the coordinates are using the Cartesian coordinate system for the plane, which is such a common thing to do that mathematicians can forget they’re doing that. The circle has radius . So you can look at the first equation and draw a circle and write down a note that its radius is and you’ve got it. looks like an equation that’s true if the numbers ‘x’ and ‘y’ are coordinates of the points on a hyperbola. Again in the Cartesian coordinate system. But I have to feel a little uncomfortable saying this. If the equation were (say) then it’d certainly be a hyperbola, which mostly looks like a mirror-symmetric pair of arcs. But equalling zero? That’s called a “degenerate hyperbola”, which makes it sound like the hyperbola is doing something wrong. Unfortunate word, but one we’re stuck with.
The description just reflects that the hyperbola is boring in some way. In this case, it’s boring because the ‘x’ and ‘y’ that make the equation true are just the points on a pair of straight lines that go through the origin, the point with coordinates (0, 0). And they’re going to be mirror-images of each other around the x- and the y-axis. So it seems like a waste to use the form of a hyperbola when we could do just as well using the forms of straight lines to describe the same points. This hyperbola will look like an X, although it might be a pretty squat ‘x’ or a pretty narrow one or something. Depends on the exact equation.
So. The solutions for ‘x’ and ‘y’ are going to be on the points that are on both a circle centered around the origin and on an X centered around the origin. This is a way to see why I would expect four solutions. Also they they would look about the same. There’d be an answer with positive ‘x’ and positive ‘y’, and then three more answers. One answer has ‘x’ with the same size but a minus sign. One answer has ‘y’ with the same size but a minus sign. One has both ‘x’ and ‘y’ with the same values but minus signs.
Sorry I wasn’t there to partner with.
Dave Coverly’s Speed Bump for the 12th is a Rubik’s Cube joke. Here it merges the idea with the struggles of scheduling anything anymore. I’m not sure that the group-theory operations of lining up a Rubik’s cube can be reinterpreted as the optimization problems of scheduling stuff. But there are all sorts of astounding and surprising links between mathematical problems. So I wouldn’t rule it out.
John Allen’s Nest Heads for the 13th is a lotteries joke. I’m less dogmatic than are many mathematicians about the logic of participating in a lottery. At least in the ones as run by states and regional authorities the chance of a major payout are, yes, millions to one against. There can be jackpots large enough that the expectation value of playing becomes positive. In this case the reward for that unlikely outcome is so vast that it covers the hundreds of millions of times you play and lose. But even then, you have the question of whether doing something that just won’t pay out is worth it. My taste is to say that I shall do much more foolish things with my disposable income than buying a couple tickets each year. And while I would like to win the half-billion-dollar jackpot that would resolve all my financial woes and allow me to crush those who had me imprisoned in the Château d’If, I’d also be coming out ahead if I won, like, one of the petty $10,000 prizes.
And now, closer to deadline than I like, let me wrap up last week’s mathematically-themed comic strips. I had a lot happening, that’s all I can say.
Glenn McCoy and Gary McCoy’s The Flying McCoys for the 10th is another tragic moment in the mathematics department. I’m amused that white lab coats are taken to read as “mathematician”. There are mathematicians who work in laboratories, naturally. Many interesting problems are about real-world things that can be modelled and tested and played with. It’s hardly the mathematics-department uniform, but then, I’m not sure mathematicians have a uniform. We just look like academics is all.
It also shows off that motif of mathematicians as doing anything with numbers in a more complicated way than necessary. I can’t imagine anyone in an emergency trying to evoke 9-1-1 by solving any kind of puzzle. But comic strip characters are expected to do things at least a bit ridiculously. I suppose.
Mark Litzler’s Joe Vanilla for the 11th is about random numbers. We need random numbers; they do so much good. Getting them is hard. People are pretty lousy at picking random numbers in their head. We can say what “lousy” random numbers look like. They look wrong. There’s digits that don’t get used as much as the others do. There’s strings of digits that don’t get used as much as other strings of the same length do. There are patterns, and they can be subtle ones, that just don’t look right.
And yet we have a terrible time trying to say what good random numbers look like. Suppose we want to have a string of random zeroes and ones: is 101010 better or worse than 110101? Or 000111? Well, for a string of digits that short there’s no telling. It’s in big batches that we should expect to see no big patterns. … Except that occasionally randomness should produce patterns. How often should we expect patterns, and of what size? This seems to depend on what patterns we’ve found interesting enough to look for. But how can the cultural quirks that make something seem interesting be a substantial mathematical property?
Olivia Jaimes’s Nancy for the 11th uses mathematics-assessment tests for its joke. It’s of marginal relevance, yes, but it does give me a decent pretext to include the new artist’s work here. I don’t know how long the Internet is going to be interested in Nancy. I have to get what attention I can while it lasts.
Well, that’s got me thinking. Obviously all the sides of a triangle can be rational, and so its perimeter can be too. But … the area of an equilateral triangle is times the square of the length of any side. It can have a rational side and an irrational area, or vice-versa. Just as the circle has. If it’s not an equilateral triangle?
Can you have a triangle that has three rational sides and a rational area? And yes, you can. Take the right triangle that has sides of length 5, 12, and 13. Or any scaling of that, larger or smaller. There is indeed a whole family of triangles, the Heronian Triangles. All their sides are integers, and their areas are integers too. (Sides and areas rational are just as good as sides and areas integers. If you don’t see why, now you see why.) So there’s that at least. The name derives from Heron/Hero, the ancient Greek mathematician whom we credit with that snappy formula that tells us, based on the lengths of the three sides, what the area of the triangle is. Not the Pythagorean formula, although you can get the Pythagorean formula from it.
Still, I’m going to bet that there’s some key measure of even a Heronian Triangle that ends up being irrational. Interior angles, most likely. And there are many ways to measure triangles; they can’t all end up being rational at once. There are over two thousand ways to define a “center” of a triangle, for example. The odds of hitting a rational number on all of them at once? (Granted, most of these triangle centers are unknown except to the center’s discoverer/definer and that discoverer’s proud but baffled parents.)
Carla Ventresca and Henry Beckett’s On A Claire Day for the 12th mentions taking classes in probability and statistics. They’re the classes nobody doubts are useful in the real world. It’s easy to figure probability is more likely to be needed than functional analysis on some ordinary day outside the university. I can’t even compose that last sentence without the language of probability.
I’d kind of agree with calling the courses intense, though. Well, “intense” might not be the right word. But challenging. Not that you’re asked to prove anything deep. The opposite, really. An introductory course in either provides a lot of tools. Many of them require no harder arithmetic work than multiplication, division, and the occasional square root. But you do need to learn which tool to use in which scenario. And there’s often not the sorts of proofs that make it easy to understand which tool does what. Doing the proofs would require too much fussing around. Many of them demand settling finicky little technical points that take you far from the original questions. But that leaves the course as this archipelago of small subjects, each easy in themselves. But the connections between them are obscured. Is that better or worse? It must depend on the person hoping to learn.
It was a slow week for mathematically-themed comic strips. What I have are meager examples. Small topics to discuss. The end of the week didn’t have anything even under loose standards of being on-topic. Which is fine, since I lost an afternoon of prep time to thunderstorms that rolled through town and knocked out power for hours. Who saw that coming? … If I had, I’d have written more the day before.
Mac King and Bill King’s Magic in a Minute for the 29th of October looks like a word problem. Well, it is a word problem. It looks like a problem about extrapolating a thing (price) from another thing (quantity). Well, it is an extrapolation problem. The fun is in figuring out what quantities are relevant. Now I’ve spoiled the puzzle by explaining it all so.
Olivia Walch’s Imogen Quest for the 30th doesn’t say it’s about a mathematics textbook. But it’s got to be. What other kind of textbook will have at least 28 questions in a section and only give answers to the odd-numbered problems in back? You never see that in your social studies text.
Eric the Circle for the 30th, this one by Dennill, tests how slow a week this was. I guess there’s a geometry joke in Jane Austen? I’ll trust my literate readers to tell me. My doing the world’s most casual search suggests there’s no mention of triangles in Pride and Prejudice. The previous might be the most ridiculously mathematics-nerdy thing I have written in a long while.
Tony Murphy’s It’s All About You for the 31st does some advanced-mathematics name-dropping. In so doing, it’s earned a spot taped to the door of two people in any mathematics department with more than 24 professors across the country. Or will, when they hear there was a gap unification theory joke in the comics. I’m not sure whether Murphy was thinking of anything particular in naming the subject “gap unification theory”. It sounds like a field of mathematical study. But as far as I can tell there’s just one (1) paper written that even says “gap unification theory”. It’s in partition theory. Partition theory is a rich and developed field, which seems surprising considering it’s about breaking up sets of the counting numbers into smaller sets. It seems like a time-waster game. But the game sneaks into everything, so the field turns out to be important. Gap unification, in the paper I can find, is about studying the gaps between these smaller sets.
There’s also a “band-gap unification” problem. I could accept this name being shortened to “gap unification” by people who have to say its name a lot. It’s about the physics of semiconductors, or the chemistry of semiconductors, as you like. The physics or chemistry of them is governed by the energies that electrons can have. Some of these energies are precise levels. Some of these energies are bands, continuums of possible values. When will bands converge? When will they not? Ask a materials science person. Going to say that’s not mathematics? Don’t go looking at the papers.
Whether partition theory or materials since it seems like a weird topic. Maybe Murphy just put together words that sounded mathematical. Maybe he has a friend in the field.
Bill Amend’s FoxTrot Classics for the 1st of November is aiming to be taped up to the high school teacher’s door. It’s easy to show how the square root of two is irrational. Takes a bit longer to show the square root of three is. Turns out all the counting numbers are either perfect squares — 1, 4, 9, 16, and so on — or else have irrational square roots. There’s no whole number with a square root of, like, something-and-three-quarters or something-and-85-117ths. You can show that, easily if tediously, for any particular whole number. What’s it look like to show for all the whole numbers that aren’t perfect squares already? (This strip originally ran the 8th of November, 2006.)