I haven’t forgotten about the comic strips. It happens that last week’s were mostly quite casual mentions, strips that don’t open themselves up to deep discussions. I write this before I see what I actually have to write about the strips. But here’s the first half of the past week’s. I’ll catch up on things soon.
Bill Amend’s FoxTrot for the 22nd, a new strip, has Jason and Marcus using arithmetic problems to signal pitches. At heart, the signals between a pitcher and catcher are just an index. They’re numbers because that’s an easy thing to signal given that one only has fingers and that they should be visually concealed. I would worry, in a pattern as complicated as these two would work out, about error correction. If one signal is mis-read — as will happen — how do they recognize it, and how do they fix it? This may seem like a lot of work to put to a trivial problem, but to conceal a message is important, whatever the message is.
James Beutel’s Banana Triangle for the 23rd has a character trying to convince himself of his intelligence. And doing so by muttering mathematics terms, mostly geometry. It’s a common shorthand to represent deep thinking.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is a joke about orders of magnitude. The order of magnitude is, roughly, how big the number is. Often the first step of a physics problem is to try to get a calculation that’s of the right order of magnitude. Or at least close to the order of magnitude. This may seem pretty lax. If we want to find out something with value, say, 231, it seems weird to claim victory that our model says “it will be a three-digit number”. But getting the size of the number right is a first step. For many problems, particularly in cosmology or astrophysics, we’re intersted in things whose functioning is obscure. And relies on quantities we can measure very poorly. This is why we can see getting the order magnitude about right as an accomplishment.
And here’s the other half of last week’s comic strips that name-dropped mathematics in such a way that I couldn’t expand it to a full paragraph. We’ll likely be back to something more normal next week.
David Malki’s Wondermark for the 20th is built on the common idiom of giving more than 100%. I’m firmly on the side of allowing “more than 100%” in both literal and figurative uses of percent, so there’s not much more to say.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses Big Numbers as the sort of thing that need a down-to-earth explanation. The strip is about explanations that don’t add clarity. It shows my sense of humor that I love explanations that are true but explain nothing. The more relevant and true without helping the better. Right up until it’s about something I could be explaining instead.
Tom Batiuk’s vintage Funky Winkerbean for the 21st is part of a week of strips from the perspective of a school desk. It includes a joke about football players working mathematics problems. The strip originally ran the 8th of February, 1974, looks like.
The second half of last week’s mathematically-themed comic strips had an interesting range of topics. Two of them seemed to circle around the making of models. So that’s my name for this installment.
Ryan North’s Dinosaur Comics for the 26th has T-Rex trying to build a model. In this case, it’s to project how often we should expect to see a real-life Batman. T-Rex is building a simple model, which is fine. Simple models, first, are usually easier to calculate with. How they differ from reality can give a guide to how to make a more complex model. Or they can indicate the things that have to be learned in order to make a more complex model. The difference between a model’s representation and the observed reality (or plausibly expected reality) can point out problems in one’s assumptions, too.
For example, T-Rex supposes that a Batman needs to have billionaire parents. This makes for a tiny number of available parents. But surely what’s important is that a Batman be wealthy enough he doesn’t have to show up to any appointments he doesn’t want to make. Having a half-billion dollars, or a “mere” hundred million, would allow that. Even a Batman who had “only” ten million dollars would be about as free to be a superhero. Similarly, consider the restriction to Olympic athletes. Astronaut Ed White, who on Gemini IV became the first American to walk in space, was not an Olympic athlete; but he certainly could have been. He missed by a split-second in the 400 meter hurdles race. Surely someone as physically fit as Ed White would be fit enough for a Batman. Not to say that “Olympic athletes or NASA astronauts” is a much bigger population than “Olympic athletes”. (And White was unusually fit even for NASA astronauts.) But it does suggest that merely counting Olympic athletes is too restrictive.
But that’s quibbling over the exact numbers. The process is a good rough model. List all the factors, suppose that all the factors are independent of one another, and multiply how likely it is each step happens by the population it could happen to. It’s hard to imagine a simpler model, but it’s a place to start.
Greg Wallace’s Nothing Is Not Something for the 26th is a bit of a geometry joke. It’s built on the idiom of the love triangle, expanding it into more-sided shapes. Relationships between groups of people like this can be well-represented in graph theory, with each person a vertex, and each pair of involved people an edge. There are even “directed graphs”, where each edge contains a direction. This lets one represent the difference between requited and unrequited interests.
Brian Anderson’s Dog Eat Doug for the 27th has Sophie the dog encounter some squirrels trying to disprove a flat Earth. They’re not proposing a round Earth either; they’ve gone in for a rhomboid. Sophie’s right to point out that drilling is a really hard way to get through the Earth. That’s a practical matter, though.
Is it possible to tell something about the shape of a whole thing from a small spot? In the terminology, what kind of global knowledge can we get from local information? We can do some things. For example, we can draw a triangle on the surface of the Earth and measure the interior angles to see what they sum to. If this could be done perfectly, finding that the interior angles add up to more than 180 degrees would show the triangle’s on a spherical surface. But that also has practical limitations. Like, if we find that locally the planet is curved then we can rule out it being entirely flat. But it’s imaginable that we’d be on the one dome of an otherwise flat planet. At some point you have to either assume you’re in a typical spot, or work out ways to find what’s atypical. In the Conspiracy Squirrels’ case, that would be the edge between two faces of the rhomboid Earth. Then it becomes something susceptible to reason.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th has the mathematician making another model. And this is one of the other uses of a model: to show a thing can’t happen, show that it would have results contrary to reason. But then you have to validate the model, showing that its premises do represent reality so well that its conclusion should be believed. This can be hard. There’s some nice symbol-writing on the chalkboard here, although I don’t see that they parse. Particularly, the bit on the right edge of the panel, where the writing has a rotated-by-180-degrees ‘E’ followed by an ‘x’, a rotated-by-180-degrees ‘A’, and then a ‘z’, is hard to fit inside an equation like this. The string of symbols mean “there exists some x for which, for all z, (something) is true”. This fits at the start of a proof, or before an equation starts. It doesn’t make grammatical sense in the middle of an equation. But, in the heat of writing out an idea, mathematicians will write out ungrammatical things. As with plain-text writing, it’s valuable to get an idea down, and edit it into good form later.
Tom Batiuk’s Funky Winkerbean Vintage for the 28th sees the school’s Computer explaining the nature of its existence, and how it works. Here the Computer claims to just be filled with thousands of toes to count on. It’s silly, but it is the case that there’s no operation a computer does that isn’t something a human can do, manually. If you had the paper and the time you could do all the steps of a Facebook group chat, a game of SimCity, or a rocket guidance computer’s calculations. The results might just be impractically slow.
I’d meant to get back into discussing continuous functions this week, and then didn’t have the time. I hope nobody was too worried.
Bill Amend’s FoxTrot for the 19th is set up as geometry or trigonometry homework. There are a couple of angles that we use all the time, and they do correspond to some common unit fractions of a circle: a quarter, a sixth, an eighth, a twelfth. These map nicely to common cuts of circular pies, at least. Well, it’s a bit of a freak move to cut a pie into twelve pieces, but it’s not totally out there. If someone cuts a pie into 24 pieces, flee.
Tom Batiuk’s vintage Funky Winkerbean for the 19th of May is a real vintage piece, showing off the days when pocket electronic calculators were new. The sales clerk describes the calculator as having “a floating decimal”. And here I must admit: I’m poorly read on early-70s consumer electronics. So I can’t say that this wasn’t a thing. But I suspect that Batiuk either misunderstood “floating-point decimal”, which would be a selling point, or shortened the phrase in order to make the dialogue less needlessly long. Which is fine, and his right as an author. The technical detail does its work, for the setup, by existing. It does not have to be an actual sales brochure. Reducing “floating point decimal” to “floating decimal” is a useful artistic shorthand. It’s the dialogue equivalent to the implausibly few, but easy to understand, buttons on the calculator in the title panel.
Floating point is one of the ways to represent numbers electronically. The storage scheme is much like scientific notation. That is, rather than think of 2,038, think of 2.038 times 103. In the computer’s memory are stored the 2.038 and the 3, with the “times ten to the” part implicit in the storage scheme. The advantage of this is the range of numbers one can use now. There are different ways to implement this scheme; a common one will let one represent numbers as tiny as 10-308 or as large as 10308, which is enough for most people’s needs.
The disadvantage is that floating point numbers aren’t perfect. They have only around (commonly) sixteen digits of significance. That is, the first sixteen or so nonzero numbers in the number you represent mean anything; everything after that is garbage. Most of the time, that trailing garbage doesn’t hurt. But most is not always. Trying to add, for example, a tiny number, like 10-20, to a huge number, like 1020 won’t get the right answer. And there are numbers that can’t be represented correctly anyway, including such exotic and novel numbers as . A lot of numerical mathematics is about finding ways to compute that avoid these problems.
Back when I was a grad student I did have one casual friend who proclaimed that no real mathematician ever worked with floating point numbers, because of the limitations they impose. I could not get him to accept that no, in fact, mathematicians are fine with these limitations. Every scheme for representing numbers on a computer has limitations, and floating point numbers work quite well. At some point, you have to suspect some people would rather fight for a mistaken idea they already have than accept something new.
Mac King and Bill King’s Magic in a Minute for the 19th does a bit of stage magic supported by arithmetic: forecasting the sum of three numbers. The trick is that all eight possible choices someone would make have the same sum. There’s a nice bit of group theory hidden in the “Howdydoit?” panel, about how to do the trick a second time. Rotating the square of numbers makes what looks, casually, like a different square. It’s hard for human to memorize a string of digits that don’t have any obvious meaning, and the longer the string the worse people are at it. If you’ve had a person — as directed — black out the rows or columns they didn’t pick, then it’s harder to notice the reused pattern.
The different directions that you could write the digits down in represent symmetries of the square. That is, geometric operations that would replace a square with something that looks like the original. This includes rotations, by 90 or 180 or 270 degrees clockwise. Mac King and Bill King don’t mention it, but reflections would also work: if the top row were 4, 9, 2, for example, and the middle 3, 5, 7, and the bottom 8, 1, 6. Combining rotations and reflections also works.
If you do the trick a second time, your mark might notice it’s odd that the sum came up 15 again. Do it a third time, even with a different rotation or reflection, and they’ll know something’s up. There are things you could do to disguise that further. Just double each number in the square, for example: a square of 4/18/8, 14/10/6, 12/2/16 will have each row or column or diagonal add up to 30. But this loses the beauty of doing this with the digits 1 through 9, and your mark might grow suspicious anyway. The same happens if, say, you add one to each number in the square, and forecast a sum of 18. Even mathematical magic tricks are best not repeated too often, not unless you have good stage patter.
Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead’s marveling at what seems at first like an asymmetry, about squares all being rhombuses yet rhombuses not all being squares. There are similar results with squares and rectangles. Still, it makes me notice something. Nobody would write a strip where the kid marvelled that all squares were polygons but not all polygons were squares. It seems that the rhombus connotes something different. This might just be familiarity. Polygons are … well, if not a common term, at least something anyone might feel familiar. Rhombus is a more technical term. It maybe never quite gets familiar, not in the ways polygons do. And the defining feature of a rhombus — all four sides the same length — seems like the same thing that makes a square a square.
Three of the five comic strips I review today are reruns. I think that I’ve only mentioned two of them before, though. But let me preface all this with a plea I’ve posted before: I’m hosting the Playful Mathematics Blog Carnival the last week in September. Have you run across something mathematical that was educational, or informative, or playful, or just made you glad to know about? Please share it with me, and we can share it with the world. It can be for any level of mathematical background knowledge. Thank you.
Tom Batiuk’s Funky Winkerbean vintage rerun for the 10th is part of an early storyline of Funky attempting to tutor football jock Bull Bushka. Mathematics — geometry, particularly — gets called on as a subject Bull struggles to understand. Geometry’s also well-suited for the joke because it has visual appeal, in a way that English or History wouldn’t. And, you know, I’ll take “pretty” as a first impression to geometry. There are a lot of diagrams whose beauty is obvious even if their reasons or points or importance are obscure.
Dan Collins’s Looks Good on Paper for the 10th is about everyone’s favorite non-orientable surface. The first time this strip appeared I noted that the road as presented isn’t a Möbius strip. The opossums and the car are on different surfaces. Unless there’s a very sudden ‘twist’ in the road in the part obscured from the viewer, anyway. If I’d drawn this in class I would try to save face by saying that’s where the ‘twist’ is, but none of my students would be convinced. But we’d like to have it that the car would, if it kept driving, go over all the pavement.
Bud Fisher’s Mutt and Jeff for the 10th is a joke about story problems. The setup suggests that there’s enough information in what Jeff has to say about the cop’s age to work out what it must be. Mutt isn’t crazy to suppose there is some solution possible. The point of this kind of challenge is realizing there are constraints on possible ages which are not explicit in the original statements. But in this case there’s just nothing. We would call the cop’s age “underdetermined”. The information we have allows for many different answers. We’d like to have just enough information to rule out all but one of them.
John Rose’s Barney Google and Snuffy Smith for the 11th is here by popular request. Jughead hopes that a complicated process of dubious relevance will make his report card look not so bad. Loweezey makes a New Math joke about it. This serves as a shocking reminder that, as most comic strip characters are fixed in age, my cohort is now older than Snuffy and Loweezey Smith. At least is plausibly older than them.
Anyway it’s also a nice example of the lasting cultural reference of the New Math. It might not have lasted long as an attempt to teach mathematics in ways more like mathematicians do. But it’s still, nearly fifty years on, got an unshakable and overblown reputation for turning mathematics into doubletalk and impossibly complicated rules. I imagine it’s the name; “New Math” is a nice, short, punchy name. But the name also looks like what you’d give something that was being ruined, under the guise of improvement. It looks like that terrible moment of something familiar being ruined even if you don’t know that the New Math was an educational reform movement. Common Core’s done well in attracting a reputation for doing problems the complicated way. But I don’t think its name is going to have the cultural legacy of the New Math.
Mark Anderson’s Andertoons for the 11th is another kid-resisting-the-problem joke. Wavehead’s obfuscation does hit on something that I have wondered, though. When we describe things, we aren’t just saying what we think of them. We’re describing what we think our audience should think of them. This struck me back around 1990 when I observed to a friend that then-current jokes about how hard VCRs were to use failed for me. Everyone in my family, after all, had no trouble at all setting the VCR to record something. My friend pointed out that I talked about setting the VCR. Other people talk about programming the VCR. Setting is what you do to clocks and to pots on a stove and little things like that; an obviously easy chore. Programming is what you do to a computer, an arcane process filled with poor documentation and mysterious problems. We framed our thinking about the task as a simple, accessible thing, and we all found it simple and accessible. Mathematics does tend to look at “problems”, and we do, especially in teaching, look at “finding solutions”. Finding solutions sounds nice and positive. But then we just go back to new problems. And the most interesting problems don’t have solutions, at least not ones that we know about. What’s enjoyable about facing these new problems?
So today I am trying out including images for all the mathematically-themed comic strips here. This is because of my discovery that some links even on GoComics.com vanish without warning. I’m curious how long I can keep doing this. Not for legal reasons. Including comics for the purpose of an educational essay about topics raised by the strips is almost the most fair use imaginable. Just because it’s a hassle copying the images and putting them up on WordPress.com and that’s even before I think about how much image space I have there. We’ll see. I might try to figure out a better scheme.
Also in this batch of comics are the various Pi Day strips. There was a healthy number of mathematically-themed comics on the 14th of March. Many of those were just coincidence, though, with no Pi content. I’ll group the Pi Day strips together.
Tom Batiuk’s Funky Winkerbean for the 2nd of April, 1972 is, I think, the first appearance of Funky Winkerbean around here. Comics Kingdom just started running the strip, as well as Bud Blake’s Tiger and Bill Hoest’s Lockhorns, from the beginning as part of its Vintage Comics roster. And this strip really belonged in Sunday’s essay, but I noticed the vintage comics only after that installment went to press. Anyway, this strip — possibly the first Sunday Funky Winkerbean — plays off a then-contemporary fear of people being reduced to numbers in the face of a computerized society. If you can imagine people ever worrying about something like that. The early 1970s were a time in American society when people first paid attention to the existence of, like, credit reporting agencies. Just what they did and how they did it drew a lot of critical examination. Josh Lauer’s recently published Creditworthy: a History of Consumer Surveillance and Financial Identity in America gets into this.
Bob Scott’s Bear With Me for the 14th sees Molly struggling with failure on a mathematics test. Could be any subject and the story would go as well, but I suppose mathematics gets a connotation of the subject everybody has to study for, even the geniuses. (The strip used to be called Molly and the Bear. In either name this seems to be the first time I’ve tagged it, although I only started tagging strips by name recently.)
Bud Fisher’s Mutt and Jeff rerun for the 14th is a rerun from sometime in 1952. I’m tickled by the problem of figuring out how many times Fisher and his uncredited assistants drew Mutt and Jeff. Mutt saying that the boss “drew us 14,436 times” is the number of days in 45 years, so that makes sense if he’s counting the number of strips drawn. The number of times that Mutt and Jeff were drawn is … probably impossible to calculate. There’s so many panels each strip, especially going back to earlier and earlier times. And how many panels don’t have Mutt or don’t have Jeff or don’t have either in them? Jeff didn’t appear in the strip until March of 1908, for example, four months after the comic began. (With a different title, so the comic wasn’t just dangling loose all that while.)
Scott Hilburn’s The Argyle Sweater for the 14th starts the Pi Day off, of course, with a pun and some extension of what makes 3/14 get its attention. And until Hilburn brought it up I’d never thought about the zodiac sign for someone born the 14th of March, so that’s something.
Mark Parisi’s Off The Mark for the 14th riffs on one of the interesting features of π, that it’s an irrational number. Well, that its decimal representation goes on forever. Rational numbers do that too, yes, but they all end in the infinite repetition of finitely many digits. And for a lot of them, that digit is ‘0’. Irrational numbers keep going on with more complicated patterns. π sure seems like it’s a normal number. So we could expect that any finite string of digits appears somewhere in its decimal expansion. This would include a string of digits that encodes any story you like, The Neverending Story included. This does not mean we might ever find where that string is.
Michael Cavna’s Warped for the 14th combines the two major joke threads for Pi Day. Specifically naming Archimedes is a good choice. One of the many things Archimedes is famous for is finding an approximation for π. He’d worked out that π has to be larger than 310/71 but smaller than 3 1/7. Archimedes used an ingenious approach: we might not know the precise area of a circle given only its radius. But we can know the area of a triangle if we know the lengths of its legs. And we can draw a series of triangles that are enclosed by a circle. The area of the circle has to be larger than the sum of the areas of those triangles. We can draw a series of triangles that enclose a circle. The area of the circle has to be less than the sum of the areas of those triangles. If we use a few triangles these bounds are going to be very loose. If we use a lot of triangles these bounds can be tight. In principle, we could make the bounds as close together as we could possibly need. We can see this, now, as a forerunner to calculus. They didn’t see it as such at the time, though. And it’s a demonstration of what amazing results can be found, even without calculus, but with clever specific reasoning. Here’s a run-through of the process.
John Zakour and Scott Roberts’s Working Daze for the 15th is a response to Dr Stephen Hawking’s death. The coincidence that he did die on the 14th of March made for an irresistibly interesting bit of trivia. Zakour and Roberts could get there first, thanks to working on a web comic and being quick on the draw. (I’m curious whether they replaced a strip that was ready to go for the 15th, or whether they normally work one day ahead of publication. It’s an exciting but dangerous way to go.)
Back on “Pi Day” I shared a terrible way of calculating the digits of π. It’s neat in principle, yes. Drop a needle randomly on a uniformly lined surface. Keep track of how often the needle crosses over a line. From this you can work out the numerical value of π. But it’s a terrible method. To be sure that π is about 3.14, rather than 3.12 or 3.38, you can expect to need to do over three and a third million needle-drops. So I described this as a terrible way to calculate π.
A friend on Twitter asked if it was worse than adding up 4 * (1 – 1/3 + 1/5 – 1/7 + … ). It’s a good question. The answer is yes, it’s far worse than that. But I want to talk about working π out that way.
This isn’t part of the main post. But the comic strip happened to mention π on a day when I’m talking about π so who am I to resist coincidence?