## Reading the Comics, August 5, 2022: Catching Up Edition

I’ve had several weeks since my last Reading the Comics post. They’ve been quiet enough weeks. Let me share some of the recent offerings from Comic Strip Master Command that I enjoyed, though. I enjoy many comic strips but not all of them mention something on-point here.

Isabella Bannerman’s Six Chix for the 18th of July is a wordplay joke, naming a dog Isosceles to respect his front and hindlegs being equal in length. As I say, I’m including these because I like them, not because I have deep thoughts about them.

Bill Amend’s FoxTrot for the 24th of July gets a bit more solidly mathematical, as it’s natural to think of this sort of complicated polyhedron as something mathematicians do. Geometers at least. There’s a comfortable bit of work to be done in these sorts of shapes. They sometimes have appealing properties, for instance balancing weight loads well. Building polyhedrons out of toothpicks and gumdrops, or straws and marshmallows, or some other rigid-and-soft material, is a fine enough activity. I think every mathematics department has some dusty display shelf with a couple of these.

There are many shapes that Paige Fox’s construction might be. To my eye Paige Fox seems to be building a truncated icosahedron, that is to say, the soccer-ball shape. It’s an Archimedean solid, one of the family of thirteen shapes made of nonintersecting regular convex polygons. These are the shapes you discover if you go past Platonic solids. The family is named for that Archimedes, although the work in which he discussed them is now lost.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th of July is a different take on the challenge of motivating students to care about mathematics. The “right approach” argument has its appeal, although it’s obviously thinking of “professor” as the only job a mathematician can hold. And even granting that none of the people who understand your work have the power to fire you, that means people who don’t understand your work do have it. Also, people shut down very hard at your promise that they could understand anything about what you do, even when you know how to express it in common-language terms. This gets old fast. I am also skeptical that women are impressed by men at bars who claim to be employed for their intellect.

Anyway, academic jobs are great and more jobs should work by their rules, which, yes, do include extremely loose set hours and built-in seasons where the amount and kind of work you do varies.

Patrick Roberts’s Todd the Dinosaur for the 5th of August is a mathematics-anxiety dream, represented with the sudden challenge to do mental arithmetic. 47 times 342 is an annoying problem, yes, but one can at least approximate it fairly well quickly. 47 is almost 50, which is half a hundred. So 47 times 342 has to be nearly half of a hundred times 342, that is, half of 34,200. This is an easy number to cut in half, though: 17,100. To get this exactly? 47 is three less than fifty, so, subtract three times 342. 342 is about a third of a thousand, we can make a better estimate by subtracting a thousand: 16,100.

If you’re really good you notice that 342 is nine more than 333, so, three times 342 is three times nine more than three times 333. That is, it’s 27 more than 999. So the 16,100 estimate is 26 more than the correct number, 16,074. And I believe if you check, you will find the card in your hand is the ace of clubs. Am I not right, professor?

And that’s a look at comic strips through to a bit over a week ago. I intend to have more soon. All my Reading the Comics posts should be at this link. Thanks for being with me for these reviews. See you again soon.

## Reading the Comics, July 16, 2022: Brevity is the Soul of Wit Edition

Once more Comic Strip Master Command shows great parsimony with its mathematics-themed comic strips. The one with most substance to it is Dan Thompson’s Brevity, and even that’s not one I can talk about at length. Watch me try, anyway.

Dan Thompson’s Brevity for the 11th of July is the anthropomorphic numerals-and-symbols joke for the week. And it correctly identifies + as representing the addition operator. I talked about operators for the 2019 A-to-Z, though my focus there was on functions of functions. Here, we’ve just got a binary operator, taking two integers (or real numbers) and, if we follow its rule, matching this pair with an integer (or real number). Well, we can embed the integers (or real numbers) in the space of functions, if we want.

It happens that “smooth” is also a term of mathematical art. It refers to a function that you can take enough derivatives of. “Enough” here depends on our use. Depending on what our work is, we might need anything from one derivative to infinitely many derivatives. I’m not sure that I would call the ordinary arithmetic-type addition operator “smooth”. But I’m not confident in saying it’s not.

Greg Cravens’s The Buckets for the 12th of July is more wordplay than mathematics play. I’m including it because I like the playful energy of the comic strip and want more people to notice it. I like the idea of calling such an assembled doll a fraction figure, though.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 14th of July is your usual story problem setup. It’s a standard enough insult joke, although it does use the mathematics content. There aren’t a lot of spelling questions that would let you set up an ambiguous “how many are left” joke. Maybe an English class.

This takes us through the middle of July, as these go. All my Reading the Comics posts should be at this link,. Will there be more comics for next week? I have my suspicions.

## Reading the Comics, June 26, 2022: First Doldrums of Summer Edition

I have not kept secret that I’ve had little energy lately. I hope that’s changing but can do little more than hope. I find it strange that my lack of energy seems to be matched by Comic Strip Master Command. Last week saw pretty slim pickings for mathematically-themed comics. Here’s what seems worth the sharing from my reading.

Lincoln Peirce’s Big Nate for the 22nd is a Pi Day joke, displaced to the prank day at the end of Nate’s school year. It’s also got a surprising number of people in the comments complaining that 3.1416 is only an approximation to π. It is, certainly, but so is any representation besides π or a similar mathematical expression. And introducing it with 3.1416 gives the reader the hint that this is about a mathematics expression and not an arbitrary symbol. It’s important to the joke that this be communicated clearly, and it’s hard to think of better ways to do that.

Dave Whamond’s Reality Check for the 24th is another in the line of “why teach algebra instead of something useful” strips. There are several responses. One is that certainly one should learn how to do a household budget; this was, at least back in the day, called home economics, and it was a pretty clear use of mathematics. Another is that a good education is about becoming literate in all the great thinking of humanity: you should come out knowing at least something coherent about mathematics and literature and exercise and biology and music and visual arts and more. Schools often fail to do all of this — how could they not? — but that’s not reason to fault them on parts of the education that they do. And anther is that algebra is about getting comfortable working with numbers before you know just what they are. That is, how to work out ways to describe a thing you want to know, and then to find what number (or range of numbers) that is. Still, these responses hardly matter. Mathematics has always lived in a twin space, of being both very practical and very abstract. People have always and will always complain that students don’t learn how to do the practical well enough. There’s not much changing that.

Charles Schulz’s Peanuts Begins for the 26th sees Violet challenge Charlie Brown to say what a non-perfect circle would be. I suppose this makes the comic more suitable for a philosophy of language blog, but I don’t know any. To be a circle requires meeting a particular definition. None of the things we ever point to and call circles meets that. We don’t generally have trouble connecting our imperfect representations of circles to the “perfect” ideal, though. And Charlie Brown said something meaningful in describing his drawing as being “a perfect circle”. It’s tricky pinning down exactly what it is, though.

And that is as much as last week moved me to write. This and my other Reading the Comics posts should be at this link. We’ll see whether the upcoming week picks up any.

## Reading the Comics, June 18, 2022: Pizza Edition

I’m back with my longest-running regular feature here. As I’ve warned I’m trying not to include every time one of the newspaper comics (that is, mostly, ones running on Comics Kingdom or GoComics) mentions the existence of arithmetic. So, for example, both Frank and Ernest and Rhymes with Orange did jokes about the names of the kinds of triangles. You can clip those at your leisure; I’m looking to discuss deeper subjects.

Scott Hilburn’s The Argyle Sweater is … well, it’s just an anthropomorphic-numerals joke. I have a weakness for The Wizard of Oz, that’s all. Also, I don’t know but somewhere in the nine kerspillion authorized books written since Baum’s death there be at least one with a “wizard of odds” plot.

Bill Amend’s FoxTrot reads almost like a word problem’s setup. There’s a difference in cost between pizzas of different sizes. Jason and Marcus make the supposition that they could buy the difference in sizes. They are asking for something physically unreasonable, but in a way that mathematics problems might do. The ring of pizza they’d be buying would be largely crust, after all. (Some people like crust, but I doubt any are ten-year-olds like Jason and Marcus.) The obvious word problem to spin out of this is extrapolating the costs of 20-inch or 8-inch pizzas, and maybe the base cost of making any pizza however tiny.

You can think of a 16-inch-wide circle as a 12-inch-wide circle with an extra ring around it. (An annulus, we’d say in the trades.) This is often a useful way to look at circles. If you get into calculus you’ll see the extra area you get from a slight increase in the diameter (or, more likely, the radius) all over the place. Also, in three dimensions, the difference in volume you get from an increase in diameter. There are also a good number of theorems with names like Green’s and Stokes’s. These are all about what you can know about the interior of a shape, like a pizza, from what you know about the ring around the edge.

Jim Meddick’s Monty sees Sedgwick, spoiled scion of New Jersey money, preparing for a mathematics test. He’s allowed the use of an abacus, one of the oldest and best-recognized computational aides. The abacus works by letting us turn the operations of basic arithmetic into physical operations. This has several benefits. We (generally) understand things in space pretty well. And the beads and wires serve as aides to memory, always a struggle. Sedgwick also brings out a “hyperbolic abacus”, a tool for more abstract operations like square roots and sines and cosines. I don’t know of anything by that name, but you can design mechanical tools to do particular computations. Slide rules, for example, generally have markings to let one calculate square roots and cube roots easily. Aircraft pilots might use a flight computer, a set of plastic discs to do quick estimates of flight time, fuel consumption, ground speed, and such. (There’s even an episode of the original Star Trek where Spock fiddles with one!)

I have heard, but not seen, that specialized curves were made to let people square circles with something approximating a compass-and-straightedge method. A contraption to calculate sines and cosines would not be hard to imagine. It would need to be a post on a hinge, mostly, with a set of lines to read off sine and cosine values over a range of angles. I don’t know of one that existed, as it’s easy enough to print out a table of trig functions, but it wouldn’t be hard to make.

And that’s enough for this week. This and all my other Reading the Comics posts should be at this link. I hope to get this back to a weekly column, but that does depend on Comic Strip Master Command doing what’s convenient for me. We’ll see how it turns out.

## Reading the Comics, June 3, 2022: Prime Choices Edition

I intended to be more casual about these comics when I resumed reading them for their mathematics content. I feel like Comic Strip Master Command is teasing me, though. There has been an absolute drought of comics with enough mathematics for me to really dig into. You can see that from this essay, which covers nearly a month of the strips I read and has two pieces that amount to “the cartoonist knows what a prime number is”. I must go with what I have, though.

Mark Anderson’s Andertoons for the 12th of May I would have sworn was a repeat. If it is, I don’t seem to have featured it before. It gives us Wavehead — I’ve learned his name is not consistent — learning about division. The first kind of division, at least, with a quotient and a remainder. The novel thing here, with integer division, is that the result is not a single number, but rather an ordered pair. I hadn’t thought about it that way before, I suppose since integer division and ordered pairs are introduced so far apart in one’s education.

We mostly put away this division-with-remainders as soon as we get comfortable with decimals. 19 &div; 4 becoming “4 remainder 3” or “4.75” or “4 $latex\frac{3}{4}$” all impose a roughly equal cognitive load. But this division reappears in (high school) algebra, when we start dividing polynomials. (Almost anything you can do with integers there’s a similar thing you can do with polynomials. This is not just because you can rewrite the integer “4” as the polynomial “f(x) = 0x + 4”.) There may be something easier to understand in turning $\left(x^2 + 3x - 3\right) \div \left(x - 2\right)$ into $\left(x + 1\right)$ remainder $\left(4x - 1\right)$.

A thing happening here is that integer arithmetic is a ring. We study a lot of rings, as it’s not hard to come up with things that look like addition and subtraction and multiplication. Rings we don’t assume to have division that stays in your set. They can turn into pairs, like with integers or with polynomials. Having that division makes the ring into a field, so-called because we don’t have enough things called a “field” already.

Scott Hilburn’s The Argyle Sweater for the 16th of May is one of the prime number strips from this collection. About the only note worth mention is that the indivisibility of 3 depends on supposing we mean the integer 3. If we decided 3 was a real number, we would have every real number other than zero as a divisor. There’s similar results for complex numbers or polynomials. I imagine there’s a good fight one could get going about whether 3-in-integer-arithmetic is the same number as 3-in-real-arithmetic. I’m not ready for that right now, though.

I like the blood bag Dracula’s drinking from. Nice touch.

Dave Coverly’s Speed Bump for the 16th of May names the ways to classify triangles based on common side lengths (or common angles). There is some non-absurdity in the joke’s premise. Not the existence of these particular pennants. But that someone who loves a subject enough to major in it will often be a bit fannish about it? Yes. It’s difficult to imagine going any other way. You need to get to a pretty high leve of mathematics to go seriously into triangles, but the option is there.

Dave Whamond’s Reality Check for the 3rd of June is the other comic strip playing on the definition of “prime”. Here it’s applied to the hassle of package delivery, and the often comical way that items will get boxed in what seems to be no logical pattern. But there is a reason behind that lack of pattern. It is an extremely hard problem to get bunches of things together at once. It gets even harder when those things have to come from many different sources, and get warehoused in many disparate locations. Add to that the shipper’s understandable desire to keep stuff sitting around, waiting, for as little time as possible. So the waste in package and handling and delivery costs seems worth it to send an order in ten boxes than in finding how to send it all in one.

It feels like an obvious offense to reason to use four boxes to send five items. It can be hard to tell whether the cost of organizing things into fewer boxes outweighs the additional cost of transporting, mostly, air. This is not to say that I think the choice is necessarily made correctly. I don’t trust organizations to not decide “I dunno, we always did it this way”. I want instead to note that when you think hard about a question it often becomes harder to say what a good answer would be.

I can give you a good answer, though, if your question is how to read more comic strips alongside me. I try to put all my Reading the Comics posts at this link. You can see something like a decade’s worth of my finding things to write about students not answering word problems. Thank you for reading along with this.

## Reading the Comics, May 7, 2022: Does Comic Strip Master Command Not Do Mathematics Anymore Edition?

I mentioned in my last Reading the Comics post that it seems there are fewer mathematics-themed comic strips than there used to be. I know part of this is I’m trying to be more stringent. You don’t need me to say every time there’s a Roman numerals joke or that blackboards get mathematics symbols put on them. Still, it does feel like there’s fewer candidate strips. Maybe the end of the 2010s was a boom time for comic strips aimed at high school teachers and I only now appreciate that? Only further installments of this feature will let us know.

Jim Benton’s Jim Benton Cartoons for the 18th of April, 2022 suggests an origin for those famous overlapping circle pictures. This did get me curious what’s known about how John Venn came to draw overlapping circles. There’s no reason he couldn’t have used triangles or rectangles or any shape, after all. It looks like the answer is nobody really knows.

Venn, himself, didn’t name the diagrams after himself. Wikipedia credits Charles Dodgson (Lewis Carroll) as describing “Venn’s Method of Diagrams” in 1896. Clarence Irving Lewis, in 1918, seems to be the first person to write “Venn Diagram”. Venn wrote of them as “Eulerian Circles”, referencing the Leonhard Euler who just did everything. Sir William Hamilton — the philosopher, not the quaternions guy — posthumously published the Lectures On Metaphysics and Logic which used circles in these diagrams. Hamilton asserted, correctly, that you could use these to represent logical syllogisms. He wrote that the 1712 logic text Nucleus Logicae Weisianae — predating Euler — used circles, and was right about that. He got the author wrong, crediting Christian Weise instead of the correct author, Johann Christian Lange.

With 1712 the trail seems to end to this lay person doing a short essay’s worth of research. I don’t know what inspired Lange to try circles instead of any other shape. My guess, unburdened by evidence, is that it’s easy to draw circles, especially back in the days when every mathematician had a compass. I assume they weren’t too hard to typeset, at least compared to the many other shapes available. And you don’t need to even think about setting them with a rotation, the way a triangle or a pentagon might demand. But I also would not rule out a notion that circles have some connotation of perfection, in having infinite axes of symmetry and all points on them being equal in distance from the center and such. Might be the reasons fit in the intersection of the ethereal and the mundane.

Daniel Beyer’s Long Story Short for the 29th of April, 2022 puts out a couple of concepts from mathematical physics. These are all about geometry, which we now see as key to understanding physics. Particularly cosmology. The no-boundary proposal is a model constructed by James Hartle and Stephen Hawking. It’s about the first $10^{-43}$ seconds of the universe after the Big Bang. This is an era that was so hot that all our well-tested models of physical law break down. The salient part of the Hartle-Hawking proposal is the idea that in this epoch time becomes indistinguishable from space. If I follow it — do not rely on my understanding for your thesis defense — it’s kind of the way that stepping away from the North Pole first creates the ideas of north and south and east and west. It’s very hard to think of a way to test this which would differentiate it from other hypotheses about the first instances of the universe.

The Weyl Curvature is a less hypothetical construct. It’s a tensor, one of many interesting to physicists. This one represents the tidal forces on a body that’s moving along a geodesic. So, for example, how the moon of a planet gets distorted over its orbit. The Weyl Curvature also offers a way to describe how gravitational waves pass through vacuum. I’m not aware of any serious question of the usefulness or relevance of the thing. But the joke doesn’t work without at least two real physics constructs as setup.

Liniers’ Macanudo for the 5th of May, 2022 has one of the imps who inhabit the comic asserting responsibility for making mathematics work. It’s difficult to imagine what a creature could do to make mathematics work, or to not work. If pressed, we would say mathematics is the set of things we’re confident we could prove according to a small, pretty solid-seeming set of logical laws. And a somewhat larger set of axioms and definitions. (Few of these are proved completely, but that’s because it would involve a lot of fiddly boring steps that nobody doubts we could do if we had to. If this sounds sketchy, consider: do you believe my claim that I could alphabetize the books on the shelf to my right, even though I’ve never done that specific task? Why?) It would be like making a word-search puzzle not work.

The punch line, the blue imp counting seventeen of the orange imp, suggest what this might mean. Mathematics as a set of statements following some rule, is a niche interest. What we like is how so many mathematical things seem to correspond to real-world things. We can imagine mathematics breaking that connection to the real world. The high temperature rising one degree each day this week may tell us something about this weekend, but it’s useless for telling us about November. So I can imagine a magical creature deciding what mathematical models still correspond to the thing they model. Be careful in trying to change their mind.

And that’s as many comic strips from the last several weeks that I think merit discussion. All of my Reading the Comics posts should be at this link, though. And I hope to have a new one again sometime soon. I’ll ask my contacts with the cartoonists. I have about half of a contact.

## Reading the Comics, April 17, 2022: Did I Catch Comic Strip Master Command By Surprise Edition

Part of the thrill of Reading the Comics posts is that the underlying material is wholly outside my control. The subjects discussed, yes, although there are some quite common themes. (Students challenging the word problem; lottery jokes; monkeys at typewriters.) But also quantity. Part of what burned me out on Reading the Comics posts back in 2020 was feeling the need to say something about lots of comic strips . Now?

I mentioned last week seeing only three interesting strips, and one of them, Andertoons, was a repeat I’d already discussed. This week there were only two strips that drew a first note and again, Andertoons was a repeat I’d already discussed. Mark Anderson’s comic for the 17th I covered in enough detail back in August of 2019. I don’t know how many new Andertoons are put into the rotation at GoComics. But the implication is Comic Strip Master Command ordered mathematics-comics production cut down, and they haven’t yet responded to my doing these again. I guess we’ll know for sure if things pick up in a couple weeks, as the lead time allows.

So Rick McKee and Kent Sligh’s Mount Pleasant for the 15th of April is all I have to discuss. It’s part of the long series of students resisting the teacher’s question. The teacher is asking a fair enough question, that of how to do a problem that has several parts. She does ask how we “should” solve the problem of finding what 4 + 4 – 2 equals. The catch is there are several ways to do this, all of them as good. We know this if we’ve accepted subtraction as a kind of addition, and if we’ve accepted addition as commutative.

So the order is our choice. We can add 4 and 4 and then subtract 2. Or subtract 2 from the second 4, and then add that to the first 4. If you want, and can tell the difference, you could subtract 2 from the first 4, and then add the second 4 to that.

For this problem it doesn’t make any difference. But one can imagine similar ones where the order you tackle things in can make calculations easier, or harder. 5 + 7 – 2, for example, I find easier if I work it out as 5 + ( 7 – 2), that is, 5 + 5. So it’s worth taking a moment to consider whether rearranging it can make the calculation more reliable. I don’t know whether the teacher meant to challenge the students to see that there are alternatives, and no uniquely “right” answer. It’s possible McKee and Sligh did not have the teaching plan worked out.

That makes for another week’s worth of comic strips to discuss. All of my Reading the Comics posts should be at this link. Thanks for reading this and I will let you know if Comic Strip Master Command increases production of comics with mathematics themes.

## Reading the Comics, April 10, 2022: Quantum Entanglement Edition

I remember part of why I stopped doing Reading the Comics posts regularly was their volume. I read a lot of comics and it felt like everyone wanted to do a word problem joke. Since I started easing back into these posts it’s seemed like they’ve disappeared. When I put together this week’s collection, I only had three interesting ones. And one was Andertoons for the 10th of April. Andertoons is a stalwart here, but this particular strip was one I already talked about, back in 2019.

Another was the Archie repeat for the 10th of April. And that only lists mathematics as a school subject. It would be the same joke if it were English lit. Saying “differential calculus” gives it the advantage of specificity. It also suggests Archie is at least a good enough student to be taking calculus in high school, which isn’t bad. Differential calculus is where calculus usually starts, with the study of instantaneous changes. A person can, and should, ask how a change can be instantaneous. Part of what makes differential calculus is learning how to find something that matches our intuition about what it should be. And that never requires us to do something appalling like divide zero by zero. Our current definition took a couple centuries of wrangling to find a scheme that makes sense. It’s a bit much to expect high school students to pick it up in two months.

Ripley’s Believe It Or Not for the 10th of April, 2022 was the most interesting piece. This referenced a problem I didn’t remember having heard about, the “36 Officers puzzle” of Leonhard Euler. Euler’s name you know as he did foundational work in every field of mathematics ever. This particular puzzle ates to 1779, according to an article in Quanta Magazine which one of the Ripley’s commenters offered. Six army regiments each have six officers of six different ranks. How can you arrange them in a six-by-six square so that no row or column repeats a rank or regiment?

The problem sounds like it shouldn’t be hard. The two-by-two version of this is easy. So is three-by-three and four-by-four and even five-by-five. Oddly, seven-by-seven is, too. It looks like some form of magic square, and seems not far off being a sudoku problem either. So it seems weird that six-by-six should be particularly hard, but sometimes it happens like that. In fact, this happens to be impossible; a paper by Gaston Terry in 1901 proved there were none.

The solution discussed by Ripley’s is of a slightly different problem. So I’m not saying to not believe it, just, that you need to believe it with reservations. The modified problem casts this as a quantum-entanglement, in which the rank and regiment of an officer in one position is connected to that of their neighbors. I admit I’m not sure I understand this well enough to explain; I’m not confident I can give a clear answer why a solution of the entangled problem can’t be used for the classical problem.

The problem, at this point, isn’t about organizing officers anymore. It never was, since that started as an idle pastime. Legend has it that it started as a challenge about organizing cards; if you look at the paper you’ll see it presenting states as card suits and values. But the problem emerged from idle curiosity into practicality. These turn out to be applicable to quantum error detection codes. I’m not certain I can explain how myself. You might be able to convince yourself of this by thinking how you know that someone who tells you the sum of six odd numbers is itself an odd number made a mistake somewhere, and you can then look for what went wrong.

And that’s as many comics from last week as I feel like discussing. All my Reading the Comics posts should be gathered at this link. Thanks for reading this and I hope to do this again soon.

## Reading the Comics, April 2, 2022: Pi Day Extra Edition

I’m not sure that I will make a habit of this. It’s been a while since I did a regular Reading the Comics post, looking for mathematics topics in syndicated newspaper comic strips. I thought I might dip my toes in those waters again. Since my Pi Day essay there’ve been only a few with anything much to say. One of them was a rerun I’ve discussed before, too, a Bloom County Sunday strip that did an elaborate calculation to conceal the number 1. I’ve written about that strip twice before, in May 2016 and then in October 2016, so that’s too well-explained to need revisiting.

As it happens two of the three strips remaining were repeats, though ones I don’t think I’ve addressed before here.

Bill Amend’s FoxTrot Classics for the 18th of March looks like a Pi Day strip. It’s not, though: it originally ran the 16th of March, 2001. We didn’t have Pi Day back then.

What Peter Fox is doing is drawing a unit circle — a circle of radius 1 — and dividing it into a couple common angles. Trigonometry students are expected to know the sines and cosines and tangents of a handful of angles. If they don’t know them, they can work these out from first principles. Draw a line from the center of the unit circle at an angle measured counterclockwise from the positive x-axis. Find where that line you’ve just drawn intersects the unit circle. The x-coordinate of that point has the same value as the cosine of that angle. The y-coordinate of that point has the same value as the sine of that angle. And for a handful of angles — the ones Peter marks off in the second panel — you can work them out by reason alone.

These angles we know as, like, 45 degrees or 120 degrees or 135 degrees. Peter writes them as $\frac{\pi}{4}$ or $\frac{2}{3}\pi$ or $\frac{9}{8}\pi$, because these are radian measure rather than degree measure. It’s a different scale, one that’s more convenient for calculus. And for some ordinary uses too: an angle of (say) $\frac{3}{4}\pi$ radians sweeps out an arc of length $\frac{3}{4}\pi$ on the unit circle. You can see where that’s easier to keep straight than how long an arc of 135 degrees might be.

Drawing this circle is a good way to work out or remember sines and cosines for the angles you’re expected to know, which is why you’d get them on a trig test.

Scott Hilburn’s The Argyle Sweater for the 27th of March summons every humorist’s favorite piece of topology, the Möbius strip. Unfortunately the line work makes it look to me like Hilburn’s drawn a simple loop of a steak. Follow the white strip along the upper edge. Could be the restaurant does the best it can with a challenging presentation.

August Ferdinand Möbius by the way was an astronomer, working most of his career at the Observatory at Leipzig. (His work as a professor was not particularly successful; he was too poor a lecturer to keep students.) His father was a dancing teacher, and his mother was a descendant of Martin Luther, although I imagine she did other things too.

Rina Piccolo’s Tina’s Groove for the 2nd of April makes its first appearance in a Reading the Comics post in almost a decade. The strip ended in 2017 and only recently has Comics Kingdom started showing reprints. The strip is about the numerical coincidence between 3.14 of a thing and the digits of π. It originally ran at the end of March, 2007, which like the vintage FoxTrot reminds us how recent a thing Pi Day is to observe.

3.14 hours is three hours, 8.4 minutes, which implies that she clocked in at about 9:56.

And that’s this installment. All my Reading the Comics posts should be at this link. I don’t know when I’ll publish a next one, but it should be there, too. Thanks for reading.

## Reading the Comics, March 14, 2022: Pi Day Edition

As promised I have the Pi Day comic strips from my reading here. I read nearly all the comics run on Comics Kingdom and on GoComics, no matter how hard their web sites try to avoid showing comics. (They have some server optimization thing that makes the comics sometimes just not load.) (By server optimization I mean “tracking for advertising purposes”.)

Pi Day in the comics this year saw the event almost wholly given over to the phonetic coincidence that π sounds, in English, like pie. So this is not the deepest bench of mathematical topics to discuss. My love, who is not as fond of wordplay as I am, notes that the ancient Greeks likely pronounced the name of π about the same way we pronounce the letter “p”. This may be etymologically sound, but that’s not how we do it in English, and even if we switched over, that would not make things better.

Scott Hilburn’s The Argyle Sweater is one of the few strips not to be about food. It is set in the world of anthropomorphized numerals, the other common theme to the day.

John Hambrook’s The Brilliant Mind of Edison Lee leads off with the food jokes, in this case cookies rather than pie. The change adds a bit of Abbott-and-Costello energy to the action.

Mick Mastroianni and Mason Mastroianni’s Dogs of C Kennel gets our first pie proper, this time tossed in the face. One of the commenters observes that the middle of a pecan pie can really hold heat, “Ouch”. Will’s holding it in his bare paw, though, so it can’t be that bad.

Jules Rivera’s Mark Trail makes the most casual Pi Day reference. If the narrator hadn’t interrupted in the final panel no one would have reason to think this referenced anything.

Mark Parisi’s Off The Mark is the other anthropomorphic numerals joke for the day. It’s built on the familiar fact that the digits of π go on forever. This is true for any integer base. In base π, of course, the representation of π is just “10”. But who uses that? And in base π, the number six would be something with infinitely many digits. There’s no fitting that in a one-panel comic, though.

Doug Savage’s Savage Chickens is the one strip that wasn’t about food or anthropomorphized numerals. There is no practical reason to memorize digits of π, other than that you’re calculating something by hand and don’t want to waste time looking them up. In that case there’s not much call go to past 3.14. If you need more than about 3.14159, get a calculator to do it. But memorizing digits can be fun, and I will not underestimate the value of fun in getting someone interested in mathematics.

For my part, I memorized π out to 3.1415926535787932, so that’s sixteen digits past the decimal. Always felt I could do more and I don’t know why I didn’t. The next couple digits are 8462, which has a nice descending-fifths cadence to it. The 626 following is a neat coda. My describing it this way may give you some idea to how I visualize the digits of π. They might help you, if you figure for some reason you need to do this. You do not, but if you enjoy it, enjoy it.

Bianca Xunise’s Six Chix for the 15th ran a day late; Xunise only gets the comic on Tuesdays and the occasional Sunday. It returns to the food theme.

And this brings me to the end of this year’s Pi Day comic strips. All of my Reading the Comics posts, past and someday future, should be at this link. And my various Pi Day essays should be here. Thank you for reading.

## Here Are Past Years’ Pi Day Comic Strips

I haven’t yet read today’s comics; it takes a while to get through them. But I hope to summarize what Comic Strip Master Command has sent out for the syndicated comics for today. In the meanwhile, here’s Pi Day strips of past years.

And I have to offer a warning. GoComics.Com has discontinued a lot of comics in the past couple years. They’ve been brutal about removing the archives of strips they’ve discontinued. Comics Kingdom is similarly ruthless in removing strips not in production. And a recent and, to the user, bad code update broke a lot of what had been non-expiring links. But my discussions of the themes in the comic are still there. And, as I got more into the Reading the Comics project I got more likely to include the original comic. So that’s some compensation.

Here’s the past several years in comics from on or around the 14th of March:

• 2015, featuring The Argyle Sweater, Baldo, The Chuckle Brothers, Dog Eat Doug, FoxTrot Classics, Herb and Jamaal, Long Story Short, The New Adventures of Queen Victoria, Off The Mark, and Working Daze.
• 2016, featuring The Argyle Sweater, B.C., Brewster Rockit, The Brilliant Mind of Edison Lee, Curtis, Dog Eat Doug, F Minus, Free Range, and Holiday Doodles.
• 2017, featuring 2 Cows and a Chicken, Archie, The Argyle Sweater, Arlo and Janis, Lard’s World Peace Tips, Loose Parts, Off The Mark, Saturday Morning Breakfast Cereal, TruthFacts, and Working Daze.
• 2018, featuring The Argyle Sweater, Bear With Me, Funky Winterbean Classic, Mutt and Jeff, Off The Mark, Savage Chickens, Warped, and Working Daze.
• 2019, featuring The Brilliant Mind of Edison Lee, Liz Climo’s Cartoons, The Grizzwells, Off The Mark, and Working Daze.
• 2020, featuring Baldo, Calvin and Hobbes, Off The Mark, Real Life Adventures, Reality Check, and Warped.
• 2021, featuring Agnes, The Argyle Sweater, Between Friends, Breaking Cat News, FoxTrot, Frazz, Get Fuzzy, Heart of the City, Reality Check, and Studio Jantze.

As mentioned, I have yet to read today’s comics. I’m looking forward to it, at least to learn what Funky Winkerbean character I’m going to be most annoyed with this week. It will be Les Moore. I was also going to look forward to seeing if there would ever be a Pi Day strips roundup without The Argyle Sweater or Reality Check. It turns out there was one in 2019. Weird how you can get the impression something is always there even when it’s not.

## Some Progress on the Infinitude of Monkeys

I have been reading Pierre-Simon LaPlace, 1749 – 1827, A Life In Exact Science, by Charles Coulson Gillispie with Robert Fox and Ivor Grattan-Guinness. It’s less of a biography than I expected and more a discussion of LaPlace’s considerable body of work. Part of LaPlace’s work was in giving probability a logically coherent, rigorous meaning. Laplace discusses the gambler’s fallacy and the tendency to assign causes to random events. That, for example, if we came across letters from a printer’s font reading out ‘INFINITESIMAL’ we would think that deliberate. We wouldn’t think that for a string of letters in no recognized language. And that brings up this neat quote from Gillispie:

The example may in all probability be adapted from the chapter in the Port-Royal La Logique (1662) on judgement of future events, where Arnauld points out that it would be stupid to bet twenty sous against ten thousand livres that a child playing with printer’s type would arrange the letters to compose the first twenty lines of Virgil’s Aenid.

The reference here is to a book by Antoine Arnauld and Pierre Nicole that I haven’t read or heard of before. But it makes a neat forerunner to the Infinite Monkey Theorem. That’s the study of what probability means when put to infinitely great or long processes. Émile Borel’s use of monkeys at a typewriter echoes this idea of children playing beyond their understanding. I don’t know whether Borel knew of Arnauld and Nicole’s example. But I did not want my readers to miss a neat bit of infinite-monkey trivia. Or to miss today’s Bizarro, offering yet another comic on the subject.

## In Which I Feel A Little Picked On

This is not a proper Reading the Comics post, since there’s nothing mathematical about this. But it does reflect a project I’ve been letting linger for months and that I intend to finish before starting the abbreviated Mathematics A-to-Z for this year.

In the meanwhile. I have a person dear to me who’s learning college algebra. For no reason clear to me this put me in mind of last year’s essay about Extraneous Solutions. These are fun and infuriating friends. They’re created when you follow the rules about how you can rewrite a mathematical expression without changing its value. And yet sometimes you do these rewritings correctly and get a would-be solution that isn’t actually one. So I’d shared some thoughts about why they appear, and what tedious work keeps them from showing up.

## Reading the Comics, May 25, 2021: Hilbert’s Hotel Edition

I have only a couple strips this time, and from this week. I’m not sure when I’ll return to full-time comics reading, but I do want to share strips that inspire something.

Carol Lay’s Lay Lines for the 24th of May riffs on Hilbert’s Hotel. This is a metaphor often used in pop mathematics treatments of infinity. So often, in fact, a friend snarked that he wished for any YouTube mathematics channel that didn’t do the same three math theorems. Hilbert’s Hotel was among them. I think I’ve never written a piece specifically about Hilbert’s Hotel. In part because every pop mathematics blog has one, so there are better expositions available. I have a similar restraint against a detailed exploration of the different sizes of infinity, or of the Monty Hall Problem.

Hilbert’s Hotel is named for David Hilbert, of Hilbert problems fame. It’s a thought experiment to explore weird consequences of our modern understanding of infinite sets. It presents various cases about matching elements of a set to the whole numbers, by making it about guests in hotel rooms. And then translates things we accept in set theory, like combining two infinitely large sets, into material terms. In material terms, the operations seem ridiculous. So the set of thought experiments get labelled “paradoxes”. This is not in the logician sense of being things both true and false, but in the ordinary sense that we are asked to reconcile our logic with our intuition.

So the Hotel serves a curious role. It doesn’t make a complex idea understandable, the way many demonstrations do. It instead draws attention to the weirdness in something a mathematics student might otherwise nod through. It does serve some role, or it wouldn’t be so popular now.

It hasn’t always been popular, though. Hilbert introduced the idea in 1924, though per a paper by Helge Kragh, only to address one question. A modern pop mathematician would have a half-dozen problems. George Gamow’s 1947 book One Two Three … Infinity brought it up again, but it didn’t stay in the public eye. It wasn’t until the 1980s that it got a secure place in pop mathematics culture, and that by way of philosophers and theologians. If you aren’t up to reading the whole of Kragh’s paper, I did summarize it a bit more completely in this 2018 Reading the Comics essay.

Anyway, Carol Lay does an great job making a story of it.

Leigh Rubin’s Rubes for the 25th of May I’ll toss in here too. It’s a riff on the art convention of a blackboard equation being meaningless. Normally, of course, the content of the equation doesn’t matter. So it gets simplified and abstracted, for the same reason one draws a brick wall as four separate patches of two or three bricks together. It sometimes happens that a cartoonist makes the equation meaningful. That’s because they’re a recovering physics major like Bill Amend of FoxTrot. Or it’s because the content of the blackboard supports the joke. Which, in this case, it does.

The essays I write about comic strips I tag so they appear at this link. You may enjoy some more pieces there.

So this is not a mathematics-themed comic update, not really. It’s just a bit of startling news about frequent Reading the Comics subject Andertoons. A comic strip back in December revealed that Wavehead had a specific name. According to the strip from the 3rd of December, the student most often challenging the word problem or the definition on the blackboard is named Tommy.

And then last week we got this bombshell:

So, also, it turns out I should have already known this since the strip ran in 2018 also. All I can say is I have a hard enough time reading nearly every comic strip in the world. I can’t be expected to understand them too.

So as not to leave things too despairing let me share a mathematics-mentioning Andertoons from yesterday and also from July 2018.

I don’t know if it’s run before that.

## Reading the Comics, December 20, 1948: What is Barnaby’s friend’s name Edition?

Have a special one today. I’ve been reading a compilation of Crockett Johnson’s 1940s comic Barnaby. The title character, an almost too gentle child, follows his fairy godfather Mr O’Malley into various shenanigans. Many (the best ones, I’d say) involve the magical world. The steady complication is that Mr O’Malley boasts abilities beyond his demonstrated competence. (Although most of the magic characters are shown to be not all that good at their business.) It’s a gentle strip and everything works out all right, if farcically.

This particular strip comes from a late 1948 storyline. Mr O’Malley’s gone missing, coincidentally to a fairy cop come to arrest the pixie, who is a con artist at heart. So this sees the entry of Atlas, the Mental Giant, who’s got some pleasant gimmicks. One of them is his requiring mnemonics built on mathematical formulas to work out names. And this is a charming one, with a great little puzzle: how do you get A-T-L-A-S out of the formula Atlas has remembered?

I’m sorry the solution requires a bit of abusing notation, so please forgive it. But it’s a fun puzzle, especially as the joke would not be funnier if the formula didn’t work. I’m always impressed when a comic strip goes to that extra effort.

Johnson, who also wrote the Harold and the Purple Crayon books, painted over a hundred canvasses with theorem-based pictures. There’s a selection of them at the Smithsonian Institute’s web site, here.

## Reading the Comics Follow-up: Where Else Is A Tetrahedron’s Centroid Edition

A Reading the Comics post a couple weeks back inspired me to find the centroid of a regular tetrahedron. A regular tetrahedron, also known as “a tetrahedron”, is the four-sided die shape. A pyramid with triangular base. Or a cone with a triangle base, if you prefer. If one asks a person to draw a tetrahedron, and they comply, they’ll likely draw this shape. The centroid, the center of mass of the tetrahedron, is at a point easy enough to find. It’s on the perpendicular between any of the four faces — the equilateral triangles — and the vertex not on that face. Particularly, it’s one-quarter the distance from the face towards the other vertex. We can reason that out purely geometrically, without calculating, and I did in that earlier post.

But most tetrahedrons are not regular. They have centroids too; where are they?

Thing is I know the correct answer going in. It’s at the “average” of the vertices of the tetrahedron. Start with the Cartesian coordinates of the four vertices. The x-coordinate of the centroid is the arithmetic mean of the x-coordinates of the four vertices. The y-coordinate of the centroid is the mean of the y-coordinates of the vertices. The z-coordinate of the centroid is the mean of the z-coordinates of the vertices. Easy to calculate; but, is there a way to see that this is right?

What’s got me is I can think of an argument that convinces me. So in this sense, I have an easy proof of it. But I also see where this argument leaves a lot unaddressed. So it may not prove things to anyone else. Let me lay it out, though.

So start with a tetrahedron of your own design. This will be less confusing if I have labels for the four vertices. I’m going to call them A, B, C, and D. I don’t like those labels, not just for being trite, but because I so want ‘C’ to be the name for the centroid. I can’t find a way to do that, though, and not have the four tetrahedron vertices be some weird set of letters. So let me use ‘P’ as the name for the centroid.

Where is P, relative to the points A, B, C, and D?

And here’s where I give a part of an answer. Start out by putting the tetrahedron somewhere convenient. That would be the floor. Set the tetrahedron so that the face with triangle ABC is in the xy plane. That is, points A, B, and C all have the z-coordinate of 0. The point D has a z-coordinate that is not zero. Let me call that coordinate h. I don’t care what the x- and y-coordinates for any of these points are. What I care about is what the z-coordinate for the centroid P is.

The property of the centroid that was useful last time around was that it split the regular tetrahedron into four smaller, irregular, tetrahedrons, each with the same volume. Each with one-quarter the volume of the original. The centroid P does that for the tetrahedron too. So, how far does the point P have to be from the triangle ABC to make a tetrahedron with one-quarter the volume of the original?

The answer comes from the same trick used last time. The volume of a cone is one-third the area of the base times its altitude. The volume of the tetrahedron ABCD, for example, is one-third times the area of triangle ABC times how far point D is from the triangle. That number I’d labelled h. The volume of the tetrahedron ABCP, meanwhile, is one-third times the area of triangle ABC times how far point P is from the triangle. So the point P has to be one-quarter as far from triangle ABC as the point D is. It’s got a z-coordinate of one-quarter h.

Notice, by the way, that while I don’t know anything about the x- and y- coordinates of any of these points, I do know the z-coordinates. A, B, and C all have z-coordinate of 0. D has a z-coordinate of h. And P has a z-coordinate of one-quarter h. One-quarter h sure looks like the arithmetic mean of 0, 0, 0, and h.

At this point, I’m convinced. The coordinates of the centroid have to be the mean of the coordinates of the vertices. But you also see how much is not addressed. You’d probably grant that I have the z-coordinate coordinate worked out when three vertices have the same z-coordinate. Or where three vertices have the same y-coordinate or the same x-coordinate. You might allow that if I can rotate a tetrahedron, I can get three points to the same z-coordinate (or y- or x- if you like). But this still only gets one coordinate of the centroid P.

I’m sure a bit of algebra would wrap this up. But I would like to avoid that, if I can. I suspect the way to argue this geometrically depends on knowing the line from vertex D to tetrahedron centroid P, if extended, passes through the centroid of triangle ABC. And something similar applies for vertexes A, B, and C. I also suspect there’s a link between the vector which points the direction from D to P and the sum of the three vectors that point the directions from D to A, B, and C. I haven’t quite got there, though.

I will let you know if I get closer.

## Reading the Comics, April 1, 2021: Why Is Gunther Taking Algebraic Topology Edition

I’m not yet looking to discuss every comic strip with any mathematics mention. But something gnawed at me in this installment of Greg Evans and Karen Evans’s Luann. It’s about the classes Gunther says he’s taking.

The main characters in Luann are in that vaguely-defined early-adult era. They’re almost all attending a local university. They’re at least sophomores, since they haven’t been doing stories about the trauma and liberation of going off to school. How far they’ve gotten has been completely undefined. So here’s what gets me.

Gunther taking vector calculus? That makes sense. Vector calculus is a standard course if you’re taking any mathematics-dependent major. It might be listed as Multivariable Calculus or Advanced Calculus or Calculus III. It’s where you learn partial derivatives, integrals along a path, integrals over a surface or volume. I don’t know Gunther’s major, but if it’s any kind of science, yeah, he’s taking vector calculus.

Algebraic topology, though. That I don’t get. Topology at all is usually an upper-level course. It’s for mathematics majors, maybe physics majors.  Not every mathematics major takes topology.   Algebraic topology is a deeper specialization of the subject. I’ve only seen courses listed as algebraic topology as graduate courses. It’s possible for an undergraduate to take a graduate-level course, yes. And it may be that Gunther is taking a regular topology course, and the instructor prefers to focus on algebraic topology.

But even a regular topology course relies on abstract algebra. Which, again, is something you’ll get as an undergraduate. If you’re a mathematics major you’ll get at least two years of algebra. And, if my experience is typical, still feel not too sure about the subject. Thing is that Intro to Abstract Algebra is something you’d plausibly take at the same time as Vector Calculus.  Then you’d get Abstract Algebra and then, if you wished, Topology.

So you see the trouble. I don’t remember anything in algebra-to-topology that would demand knowing vector calculus. So it wouldn’t mean Gunther took courses without taking the prerequisites. But it’s odd to take an advanced mathematics course at the same time as a basic mathematics course. Unless Gunther’s taking an advanced vector calculus course, which might be. Although since he wants to emphasize that he’s taking difficult courses, it’s odd to not say “advanced”. Especially if he is tossing in “algebraic” before topology.

And, yes, I’m aware of the Doylist explanation for this. The Evanses wanted courses that sound impressive and hard. And that’s all the scene demands. The joke would not be more successful if they picked two classes from my actual Junior year schedule. None of the characters have a course of study that could be taken literally. They’ve been university students full-time since 2013 and aren’t in their senior year yet. It would be fun, is all, to find a way this makes sense.

This and my other essays discussing something from the comic strips are at this link.

## Reading the Comics, March 16, 2021: Where Is A Tetrahedron’s Centroid Edition

Comic Strip Master Command has not, to appearances, been distressed by my Reading the Comics hiatus. There are still mathematically-themed comic strips. Many of them are about story problems and kids not doing them. Some get into a mathematical concept. One that ran last week caught my imagination so I’ll give it some time here. This and other Reading the Comics essays I have at this link, and I figure to resume posting them, at least sometimes.

Ben Zaehringer’s In The Bleachers for the 16th of March, 2021 is an anthropomorphized-geometry joke. Here the centroid stands in for “the waist”, the height below which boxers may not punch.

The centroid is good geometry, something which turns up in plane and solid shapes. It’s a center of the shape: the arithmetic mean of all the points in the shape. (There are other things that can, with reason, be called a center too. Mathworld mentions the existence of 2,001 things that can be called the “center” of a triangle. It must be only a lack of interest that’s kept people from identifying even more centers for solid shapes.) It’s the center of mass, if the shape is a homogenous block. Balance the shape from below this centroid and it stays balanced.

For a complicated shape, finding the centroid is a challenge worthy of calculus. For these shapes, though? The sphere, the cube, the regular tetrahedron? We can work those out by reason. And, along the way, work out whether this rule gives an advantage to either boxer.

The sphere first. That’s the easiest. The centroid has to be the center of the sphere. Like, the point that the surface of the sphere is a fixed radius from. This is so obvious it takes a moment to think why it’s obvious. “Why” is a treacherous question for mathematics facts; why should 4 divide 8? But sometimes we can find answers that give us insight into other questions.

Here, the “why” I like is symmetry. Look at a sphere. Suppose it lacks markings. There’s none of the referee’s face or bow tie here. Imagine then rotating the sphere some amount. Can you see any difference? You shouldn’t be able to. So, in doing that rotation, the centroid can’t have moved. If it had moved, you’d be able to tell the difference. The rotated sphere would be off-balance. The only place inside the sphere that doesn’t move when the sphere is rotated is the center.

This symmetry consideration helps answer where the cube’s centroid is. That also has to be the center of the cube. That is, halfway between the top and bottom, halfway between the front and back, halfway between the left and right. Symmetry again. Take the cube and stand it upside-down; does it look any different? No, so, the centroid can’t be any closer to the top than it can the bottom. Similarly, rotate it 180 degrees without taking it off the mat. The rotation leaves the cube looking the same. So this rules out the centroid being closer to the front than to the back. It also rules out the centroid being closer to the left end than to the right. It has to be dead center in the cube.

Now to the regular tetrahedron. Obviously the centroid is … all right, now we have issues. Dead center is … where? We can tell when the regular tetrahedron’s turned upside-down. Also when it’s turned 90 or 180 degrees.

Symmetry will guide us. We can say some things about it. Each face of the regular tetrahedron is an equilateral triangle. The centroid has to be along the altitude. That is, the vertical line connecting the point on top of the pyramid with the equilateral triangle base, down on the mat. Imagine looking down on the shape from above, and rotating the shape 120 or 240 degrees if you’re still not convinced.

And! We can tip the regular tetrahedron over, and put another of its faces down on the mat. The shape looks the same once we’ve done that. So the centroid has to be along the altitude between the new highest point and the equilateral triangle that’s now the base, down on the mat. We can do that for each of the four sides. That tells us the centroid has to be at the intersection of these four altitudes. More, that the centroid has to be exactly the same distance to each of the four vertices of the regular tetrahedron. Or, if you feel a little fancier, that it’s exactly the same distance to the centers of each of the four faces.

It would be nice to know where along this altitude this intersection is, though. We can work it out by algebra. It’s no challenge to figure out the Cartesian coordinates for a good regular tetrahedron. Then finding the point that’s got the right distance is easy. (Set the base triangle in the xy plane. Center it, so the coordinates of the highest point are (0, 0, h) for some number h. Set one of the other vertices so it’s in the xz plane, that is, at coordinates (0, b, 0) for some b. Then find the c so that (0, 0, c) is exactly as far from (0, 0, h) as it is from (0, b, 0).) But algebra is such a mass of calculation. Can we do it by reason instead?

That I ask the question answers it. That I preceded the question with talk about symmetry answers how to reason it. The trick is that we can divide the regular tetrahedron into four smaller tetrahedrons. These smaller tetrahedrons aren’t regular; they’re not the Platonic solid. But they are still tetrahedrons. The little tetrahedron has as its base one of the equilateral triangles that’s the bigger shape’s face. The little tetrahedron has as its fourth vertex the centroid of the bigger shape. Draw in the edges, and the faces, like you’d imagine. Three edges, each connecting one of the base triangle’s vertices to the centroid. The faces have two of these new edges plus one of the base triangle’s edges.

The four little tetrahedrons have to all be congruent. Symmetry again; tip the big tetrahedron onto a different face and you can’t see a difference. So we’ll know, for example, all four little tetrahedrons have the same volume. The same altitude, too. The centroid is the same distance to each of the regular tetrahedron’s faces. And the four little tetrahedrons, together, have the same volume as the original regular tetrahedron.

What is the volume of a tetrahedron?

If we remember dimensional analysis we may expect the volume should be a constant times the area of the base of the shape times the altitude of the shape. We might also dimly remember there is some formula for the volume of any conical shape. A conical shape here is something that’s got a simple, closed shape in a plane as its base. And some point P, above the base, that connects by straight lines to every point on the base shape. This sounds like we’re talking about circular cones, but it can be any shape at the base, including polygons.

So we double-check that formula. The volume of a conical shape is one-third times the area of the base shape times the altitude. That’s the perpendicular distance between P and the plane that the base shape is in. And, hey, one-third times the area of the face times the altitude is exactly what we’d expect.

So. The original regular tetrahedron has a base — has all its faces — with area A. It has an altitude h. That h must relate in some way to the area; I don’t care how. The volume of the regular tetrahedron has to be $\frac{1}{3} A h$.

The volume of the little tetrahedrons is — well, they have the same base as the original regular tetrahedron. So a little tetrahedron’s base is A. The altitude of the little tetrahedron is the height of the original tetrahedron’s centroid above the base. Call that $h_c$. How can the volume of the little tetrahedron, $\frac{1}{3} A h_c$, be one-quarter the volume of the original tetrahedron, $\frac{1}{3} A h$? Only if $h_c$ is one-quarter $h$.

This pins down where the centroid of the regular tetrahedron has to be. It’s on the altitude underneath the top point of the tetrahedron. It’s one-quarter of the way up from the equilateral-triangle face.

(And I’m glad, checking this out, that I got to the right answer after all.)

So, if the cube and the tetrahedron have the same height, then the cube has an advantage. The cube’s centroid is higher up, so the tetrahedron has a narrower range to punch. Problem solved.

I do figure to talk about comic strips, and mathematics problems they bring up, more. I’m not sure how writing about one single strip turned into 1300 words. But that’s what happens every time I try to do something simpler. You know how it goes.

## Reading the Comics, March 14, 2021: Pi Day Edition

I was embarrassed, on looking at old Pi Day Reading the Comics posts, to see how often I observed there were fewer Pi Day comics than I expected. There was not a shortage this year. This even though if Pi Day has any value it’s as an educational event, and there should be no in-person educational events while the pandemic is still on. Of course one can still do educational stuff remotely, mathematics especially. But after a year of watching teaching on screens and sometimes doing projects at home, it’s hard for me to imagine a bit more of that being all that fun.

But Pi Day being a Sunday did give cartoonists more space to explain what they’re talking about. This is valuable. It’s easy for the dreadfully online, like me, to forget that most people haven’t heard of Pi Day. Most people don’t have any idea why that should be a thing or what it should be about. This seems to have freed up many people to write about it. But — to write what? Let’s take a quick tour of my daily comics reading.

Tony Cochran’s Agnes starts with some talk about Daylight Saving Time. Agnes and Trout don’t quite understand how it works, and get from there to Pi Day. Or as Agnes says, Pie Day, missing the mathematics altogether in favor of the food.

Scott Hilburn’s The Argyle Sweater is an anthropomorphic-numerals joke. It’s a bit risqué compared to the sort of thing you expect to see around here. The reflection of the numerals is correct, but it bothered me too.

Georgia Dunn’s Breaking Cat News is a delightful cute comic strip. It doesn’t mention mathematics much. Here the cat reporters do a fine job explaining what Pi Day is and why everybody spent Sunday showing pictures of pies. This could almost be the standard reference for all the Pi Day strips.

Bill Amend’s FoxTrot is one of the handful that don’t mention pie at all. It focuses on representing the decimal digits of π. At least within the confines of something someone might write in the American dating system. The logic of it is a bit rough but if we’ve accepted 3-14 to represent 3.14, we can accept 1:59 as standing in for the 0.00159 of the original number. But represent 0.0015926 (etc) of a day however you like. If we accept that time is continuous, then there’s some moment on the 14th of March which matches that perfectly.

Jef Mallett’s Frazz talks about the eliding between π and pie for the 14th of March. The strip wonders a bit what kind of joke it is exactly. It’s a nerd pun, or at least nerd wordplay. If I had to cast a vote I’d call it a language gag. If they celebrated Pi Day in Germany, there would not be any comic strips calling it Tortentag.

Steenz’s Heart of the City is another of the pi-pie comics. I do feel for Heart’s bewilderment at hearing π explained at length. Also Kat’s desire to explain mathematics overwhelming her audience. It’s a feeling I struggle with too. The thing is it’s a lot of fun to explain things. It’s so much fun you can lose track whether you’re still communicating. If you set off one of these knowledge-floods from a friend? Try to hold on and look interested and remember any single piece anywhere of it. You are doing so much good for your friend. And if you realize you’re knowledge-flooding someone? Yeah, try not to overload them, but think about the things that are exciting about this. Your enthusiasm will communicate when your words do not.

Dave Whamond’s Reality Check is a pi-pie joke that doesn’t rely on actual pie. Well, there’s a small slice in the corner. It relies on the infinite length of the decimal representation of π. (Or its representation in any integer base.)

Michael Jantze’s Studio Jantze ran on Monday instead, although the caption suggests it was intended for Pi Day. So I’m including it here. And it’s the last of the strips sliding the day over to pie.

But there were a couple of comic strips with some mathematics mention that were not about Pi Day. It may have been coincidence.

Sandra Bell-Lundy’s Between Friends is of the “word problem in real life” kind. It’s a fair enough word problem, though, asking about how long something would take. From the premises, it takes a hair seven weeks to grow one-quarter inch, and it gets trimmed one quarter-inch every six weeks. It’s making progress, but it might be easier to pull out the entire grey hair. This won’t help things though.

Darby Conley’s Get Fuzzy is a rerun, as all Get Fuzzy strips are. It first (I think) ran the 13th of September, 2009. And it’s another Infinite Monkeys comic strip, built on how a random process should be able to create specific outcomes. As often happens when joking about monkeys writing Shakespeare, some piece of pop culture is treated as being easier. But for these problems the meaning of the content doesn’t count. Only the length counts. A monkey typing “let it be written in eight and eight” is as improbable as a monkey typing “yrg vg or jevggra va rvtug naq rvtug”. It’s on us that we find one of those more impressive than the other.

And this wraps up my Pi Day comic strips. I don’t promise that I’m back to reading the comics for their mathematics content regularly. But I have done a lot of it, and figure to do it again. All my Reading the Comics posts appear at this link. Thank you for reading and I hope you had some good pie.

I don’t know how Andertoons didn’t get an appearance here.

## Breaking Andertoons News: Wavehead has a name

I will be late with this week’s A-to-Z essay. I’ve had more demands on my time and my ability to organize thoughts than I could manage and something had to yield. I’m sorry for that but figure to post on Friday something for the letter ‘Y’.

But there is some exciting news in one of my regular Reading the Comics features. It’s about the kid who shows up often in Mark Anderson’s Andertoons. At the nomination of — I want to say Ray Kassinger? — I’ve been calling him “Wavehead”. Last week, though, the strip gave his name. I don’t know if this is the first time we’ve seen it. It is the first time I’ve noticed. He turns out to be Tommy.

And what about my Reading the Comics posts, which have been on suspension since the 2020 A-to-Z started? I’m not sure. I figure to resume them after the new year. I don’t know that it’ll be quite the same, though. A lot of mathematics mentions in comic strips are about the same couple themes. It is exhausting to write about the same thing every time. But I have, I trust, a rotating readership. Someone may not know, or know how to find, a decent 200-word piece about lotteries published four months in the past. I need to better balance not repeating myself.

Also a factor is lightening my overhead. Most of my strips come from Comics Kingdom or GoComics. Both of them also cull strips from their archives occasionally, leaving me with dead links. (GoComics particularly is dropping a lot of strips by the end of 2020. I understand them dumping, say, The Sunshine Club, which has been in reruns since 2007. But Dave Kellett’s Sheldon?)

The only way to make sure a strip I write about remains visible to my readers is to include it here. But to make my including the strip fair use requires that I offer meaningful commentary. I have to write something substantial, and something that’s worsened without the strip to look at. You see how this builds to a workload spiral, especially for strips where all there is to say is it’s a funny story problem. (If any cartoonists are up for me being another, unofficial archive for their mathematics-themed strips? Drop me a comment, Bill Amend, we can work something out if it doesn’t involve me sending more money than I’m taking in.)

So I don’t know how I’ll resolve all this. Key will be remembering that I can just not do the stuff I find tedious here. I will not, in fact, remember that.

## Sally Brown knows some imaginary numbers too

I had remembered this comic strip, and I hoped to use it for yesterday’s A-to-Z essay about Imaginary Numbers. But I wasn’t able to find it before publishing deadline. I figured I could go back and add this to the essay once I found it, and I likely will anyway. (The essay is quite long and any kind of visual appeal helps.)

But I also wanted folks to have the chance to notice it, and an after-the-fact addition doesn’t give that chance.

It is almost certain that Bill Watterson read this strip, and long before his own comic with eleventeen and thirty-twelve and such. Watterson has spoken of Schulz’s influence. That isn’t to say that he copied the joke. “Gibberish number-like words” is not a unique idea, and it’s certainly not original to Schulz. I’d imagine a bit of effort could find prior examples even within comic strips. (I’m reminded in Pogo of Howland Owl describing the Groundhog Child’s gibberish as first-rate algebra.) It’s just fun to see great creative minds working out similar ideas, and how they use those ideas for different jokes.

## Reading the Comics, June 7, 2020: Hiatus Edition

I think of myself as not a prescriptivist blogger. Here and on my humor blog I do what I feel like, and if that seems to work, I do more of it if I can. If I do enough of it, I try to think of a title, give up and use the first four words that kind of fit, and then ask Thomas K Dye for header art. If it doesn’t work, I drop it without mention. Apart from appealing for A-to-Z topics I don’t usually declare what I intend to do.

This feels different. One of the first things I fell into here, and the oldest hook in my blogging, is Reading the Comics. It’s mostly fun. But it is also work. 2020 is not a year when I am capable of expanding my writing work without bounds. Something has to yield, and my employers would rather it not be my day job. So, at least through the completion of the All 2020 Mathematics A-to-Z, I’ll just be reading the comics. Not Reading the Comics for posting here.

And this is likely a good time for a hiatus. There is much that’s fun about Reading the Comics. First is the comic strips, a lifelong love. Second is that they solve the problem of what to blog about. During the golden age of Atlantic City, there was a Boardwalk performer whose gimmick was to drag a trap along the seabed, haul it up, and identify every bit of sea life caught up in that. My schtick is of a similar thrill, with less harm required of the sea life.

But I have felt bored by this the last several months. Boredom is not a bad thing, of course. And if you are to be a writer, you must be able to write something competent and fresh about a topic you are tired of. Admitting that: I do not have one more sentence in me about kids not buying into the story problem. Or observing that yes, that is a blackboard full of mathematics symbols. Or that lotteries exist and if you play them infinitely many times strange conclusions seem to follow. An exercise that is tiring can be good; an exercise that is painful is not. I will put the painful away and see what I feel like later.

For the time being I figure to write only the A-to-Z essays. And, since I have them, to post references back to old A-to-Z essays. These recaps seemed to be received well enough last year. So why not repeat something that was fine when it was just one of many things?

And after all, the A-to-Z theme is still at heart hauling up buckets of sea life and naming everything in it. It’s just something that I can write farther ahead of deadline, but will not.

The Boardwalk performer would, if stumped, make up stuff. What patron was going to care if they went away ill-informed? It was a show. The performer just needed a confident air.

## Reading the Comics, June 6, 2020: Wrapping Up The Week Edition

Let’s see if I can’t close out the first week of June’s comics. I’d rather have published this either Tuesday or Thursday, but I didn’t have the time to write my statistics post for May, not yet. I’ll get there.

One of Gary Larson’s The Far Side reprints for the 4th is one I don’t remember seeing before. The thing to notice is the patient has a huge right brain and a tiny left one. The joke is about the supposed division between left-brained and right-brained people. There are areas of specialization in the brain, so that the damage or destruction of part can take away specific abilities. The popular imagination has latched onto the idea that people can be dominated by specialties of the either side of the brain. I’m not well-versed in neurology. I will hazard the guess that neurologists see “left-brain” and “right-brain” as amusing stuff not to be taken seriously. (My understanding is the division of people into “type A” and “type B” personalities is also entirely bunk unsupported by any psychological research.)

Samson’s Dark Side of the Horse for the 5th is wordplay. It builds on the use of “problem” to mean both “something to overcome” and “something we study”. The mathematics puzzle book is a fanciful creation. The name Lucien Kastner is a Monty Python reference. (I thank the commenters for spotting that.)

Dan Collins’s Looks Good on Paper for the 5th is some wordplay on the term “Möbius Strip”, here applied to a particular profession.

Bud Blake’s Tiger rerun for the 6th has Tiger complaining about his arithmetic homework. And does it in pretty nice form, really, doing some arithmetic along the way. It does imply that he’s starting his homework at 1 pm, though, so I guess it’s a weekend afternoon. It seems like rather a lot of homework for that age. Maybe he’s been slacking off on daily work and trying to make up for it.

John McPherson’s Close To Home for the 6th has a cheat sheet skywritten. It’s for a geometry exam. Any subject would do, but geometry lets cues be written out in very little space. The formulas are disappointingly off, though. We typically use ‘r’ to mean the radius of a circle or sphere, but then would use C for its circumference. That would be $c = 2\pi r$. The area of a circle, represented with A, would be $\pi r^2$. I’m not sure what ‘Vol.C’ would mean, although ‘Volume of a cylinder’ would make sense … if the next line didn’t start “Vol.Cyl”. The volume of a circular cylinder is $\pi r^2 h$, where r is the radius and h the height. For a non-circular cylinder, it’s the area of a cross-section times the height. So that last line may be right, if it extends out of frame.

Granted, though, a cheat sheet does not necessarily make literal sense. It needs to prompt one to remember what one needs. Notes that are incomplete, or even misleading, may be all that one needs.

And this wraps up the comics. This and other Reading the Comics posts are gathered at this link. Next week, I’ll get the All 2020 A-to-Z under way. Thanks once again for all your reading.

## Reading the Comics, June 3, 2020: Subjective Opinions Edition

Thanks for being here for the last week before my All-2020 Mathematics A to Z starts. By the time this posts I should have decided on the A-topic, but I’m still up for B or C topics, if you’d be so kind as to suggest things.

Bob Weber Jr’s Slylock Fox for the 1st of June sees Reeky Rat busted for speeding on the grounds of his average speed. It does make the case that Reeky Rat must have travelled faster than 20 miles per hour at some point. There’s no information about when he did it, just the proof that there must have been some time when he drove faster than the speed limit. One can find loopholes in the reasoning, but, it’s a daily comic strip panel for kids. It would be unfair to demand things like proof there’s no shorter route from the diner and that the speed limit was 20 miles per hour the whole way.

Ted Shearer’s Quincy for the 1st originally ran the 7th of April, 1981. Quincy and his friend ponder this being the computer age, and whether they can let computers handle mathematics.

Jef Mallett’s Frazz for the 2nd has the characters talk about how mathematics offers answers that are just right or wrong. Something without “subjective grading”. It enjoys that reputation. But it’s not so, and that’s obvious when you imagine grading. How would you grade an answer that has the right approach, but makes a small careless error? Or how would you grade an approach that doesn’t work, but that plausibly could?

And how do you know that the approach wouldn’t work? Even in non-graded mathematics, we have subjectivity. Much of mathematics is a search for convincing arguments about some question. What we hope to be convinced of is that there is a sound logical argument making the same conclusions. Whether the argument is convincing is necessarily subjective.

Yes, in principle, we could create a full deductive argument. It will take forever to justify every step from some axiom or definition or rule of inference. And even then, how do we know a particular step is justified? It’s because we think we understand what the step does, and how it conforms to one (or more) rule. That’s again a judgement call.

(The grading of essays is also less subjective than you might think if you haven’t been a grader. The difference between an essay worth 83 points and one worth 85 points may be trivial, yes. But you will rarely see an essay that reads as an A-grade one day and a C-grade the next. This is not to say that essay grading is not subject to biases. Some of these are innocent, such as the way the grader’s mood will affect the grade. Or how the first several papers, or the last couple, will be less consistently graded than the ones done in the middle of the project. Some are pernicious, such as under-rating the work done by ethnic minority students. But these biases affect the way one would grade, say, the partial credit for an imperfectly done algebra problem too.)

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. I could also swear that I’ve featured it here before. I can’t find it, if I have discussed this strip before. I may not have. Wavehead’s observing the difference between zero as an additive identity and its role in multiplication.

Ryan Pagelow’s Buni for the 3rd fits into the anthropomorphic-numerals category of joke. It’s really more of a representation of the year as the four horsemen of the Apocalypse.

Dan Collins’s Looks Good on Paper for the 3rd has a cook grilling a “Möbius Strip Steak”. It’s a good joke for putting on a mathematics instructor’s door.

Doug Savage’s Savage Chickens for the 3rd has, as part of animal facts, the assertion that “llamas have basic math skills”. I don’t know of any specific research on llama mathematics skills. But animals do have mathematics skills. Often counting. Some amount of reasoning. Social animals often have an understanding of transitivity, as well, especially if the social groups have a pecking order.

And this wraps up half of the past week’s mathematically-themed comic strips. I hope to have the rest in a Reading the Comics post at this link in a few days. Thanks for reading.

## Reading the Comics, May 29, 2020: Slipping Into Summer More Edition

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 29th is set in a (virtual) arithmetic class. It builds on the conflation between “nothing” and “zero”.

And that wraps up my week in comic strips. I keep all my Reading the Comics posts at this link. I am also hoping to start my All 2020 Mathematics A-to-Z shortly, and am open for nominations for topics for the first couple letters. Thank you for reading.

## Reading the Comics, May 25, 2020: Slipping into Summer Edition

Comic Strip Master Command wanted to give me a break as I ready for the All 2020 A-to-Z. I appreciate the gesture, especially given the real-world events of the past week. I get to spend this week mostly just listing appearances, even if they don’t inspire deeper thought.

Gordon Bess’s vintage Redeye for the 24th has one of his Cartoon Indians being lousy at counting. Talking about his failures at arithmetic, with how he doesn’t count six shots off well. There’s a modest number of things that people are, typically, able to perceive at once. Six can be done, although it’s easy for a momentary loss of focus to throw you off. This especially for things that have to be processed in sequence, rather than perceived all together.

Wulff and Morgenthaler’s WuMo for the 24th shows a parent struggling with mathematics, billed as part of “the terrible result of homeschooling your kids”. It’s a cameo appearance. It’d be the same if Mom were struggling with history or English. This is just quick for the comic strip reader to understand.

Andrés J. Colmenares’s Wawawiwa for the 25th sets several plants in a classroom. They’re doing arithmetic. This, too, could be any course; it just happens to be mathematics.

Sam Hurt’s Eyebeam for the 25th is built on cosmology. The subject is a blend of mathematics, observation, and metaphysics. The blackboard full of mathematical symbols gets used as shorthand for describing the whole field, not unfairly. The symbols as expressed don’t come together to mean anything. I don’t feel confident saying they don’t mean anything, though.

This is enough for today. I keep all my Reading the Comics posts at this link, and should have another one later this week. And I am trying to get my All 2020 Mathematics A-to-Z ready, with nominations open for the first several letters of the alphabet already. Thank you for reading.

## Reading the Comics, May 23, 2020: Parents Can’t Do Math Edition

This was a week of few mathematically-themed comic strips. I don’t mind. If there was a recurring motif, it was about parents not doing mathematics well, or maybe at all. That’s not a very deep observation, though. Let’s look at what is here.

Liniers’s Macanudo for the 18th puts forth 2020 as “the year most kids realized their parents can’t do math”. Which may be so; if you haven’t had cause to do (say) long division in a while then remembering just how to do it is a chore. This trouble is not unique to mathematics, though. Several decades out of regular practice they likely also have trouble remembering what the 11th Amendment to the US Constitution is for, or what the rule is about using “lie” versus “lay”. Some regular practice would correct that, though. In most cases anyway; my experience suggests I cannot possibly learn the rule about “lie” versus “lay”. I’m also shaky on “set” as a verb.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th shows a mathematician talking, in the jargon of first and second derivatives, to support the claim there’ll never be a mathematician president. Yes, Weinersmith is aware that James Garfield, 20th President of the United States, is famous in trivia circles for having an original proof of the Pythagorean theorem. It would be a stretch to declare Garfield a mathematician, though, except in the way that anyone capable of reason can be a mathematician. Raymond Poincaré, President of France for most of the 1910s and prime minister before and after that, was not a mathematician. He was cousin to Henri Poincaré, who founded so much of our understanding of dynamical systems and of modern geometry. I do not offhand know what presidents (or prime ministers) of other countries have been like.

Weinersmith’s mathematician uses the jargon of the profession. Specifically that of calculus. It’s unlikely to communicate well with the population. The message is an ordinary one, though. The first derivative of something with respect to time means the rate at which things are changing. The first derivative of a thing, with respect to time being positive means that the quantity of the thing is growing. So, that first half means “things are getting more bad”.

The second derivative of a thing with respect to time, though … this is interesting. The second derivative is the same thing as the first derivative with respect to time of “the first derivative with respect to time”. It’s what the change is in the rate-of-change. If that second derivative is negative, then the first derivative will, in time, change from being positive to being negative. So the rate of increase of the original thing will, in time, go from a positive to a negative number. And so the quantity will eventually decline.

So the mathematician is making a this-is-the-end-of-the-beginning speech. The point at which the the second derivative of a quantity changes sign is known as the “inflection point”. Reaching that is often seen as the first important step in, for example, disease epidemics. It is usually the first good news, the promise that there will be a limit to the badness. It’s also sometimes mentioned in economic crises or sometimes demographic trends. “Inflection point” is likely as technical a term as one can expect the general public to tolerate, though. Even that may be pushing things.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 19th has a father who can’t help his son do mathematics. In this case, finding square roots. There are many ways to find square roots by hand. Some are iterative, in which you start with an estimate and do a calculation that (typically) gets you a better estimate of the square root you want. And then repeat the calculation, starting from that improved estimate. Some use tables of things one can expect to have calculated, such as exponentials and logarithms. Or trigonometric tables, if you know someone who’s worked out lots of cosines and sines already.

Henry Scarpelli and Craig Boldman’s Archie rerun for the 20th mentions romantic triangles. And Moose’s relief that there’s only two people in his love triangle. So that’s our geometry wordplay for the week.

Bill Watterson’s Calvin and Hobbes repeat for the 20th has Calvin escaping mathematics class.

Julie Larson’s The Dinette Set rerun for the 21st fusses around words. Along the way Burl mentions his having learned that two negatives can make a positive, in mathematics. Here it’s (most likely) the way that multiplying or dividing two negative numbers will produce a positive number.

This covers the week. My next Reading the Comics post should appear at this tag, when it’s written. Thanks for reading.

## Reading the Comics, May 15, 2020: Squared Away Edition

The end of last week offered just a few more comic strips, and some pretty casual mathematics content. Let me wrap that up.

Daniel Beyer’s Long Story Short for the 13th has the “math department lavatory” represented as a door labelled $1 \pm 2$. It’s an interesting joke in that it reads successfully, but doesn’t make sense. To match the references to the commonly excreted substances they’d want $\frac32 \pm \frac12$.

On funny labels, though, I did once visit a mathematics building in which the dry riser had the label N Bourbaki. Nicholas Bourbaki was not a member of that college’s mathematics department, of course. This is why the joke was correctly formed and therefore funny.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 13th features the rounding-up-sheep joke.

Gary Larson’s The Far Side strips for the 14th includes the famous one of Albert Einstein coming so close to working out $E = mc^2$. The usual derivations for $E = mc^2$ don’t start with that and then explore whether it makes sense, which is what Einstein seems to be doing here. Instead they start from some uncontroversial premises and find that they imply this $E = mc^2$ business. Dimensional analysis would also let you know that, if c is involved, it’s probably to the second power rather than anything else.

But that doesn’t mean we can’t imagine Einstein assuming there must be a relationship between energy and mass, finding one that makes sense, and then finding a reason it’s that rather than something else. That’s a common enough pattern of mathematical discovery. Also, a detail I hadn’t noticed before, is that Einstein tried out $E = mc^3$, rejected it, and then tried it again. This is also a common pattern of discovery.

Mark Litzler’s Joe Vanilla for the 14th has a vague recollection of the Pythagorean Theorem be all that someone says he remembers of mathematics.

Niklas Eriksson’s Carpe Diem for the 15th depicts a couple ancient Greek deep-thinkers. A bit of mathematics, specifically geometry, is used as representative of that deep thinking.

This wraps up the past week’s mathematically-themed comics. Read this and next week’s comic strips at this link. Thank you.

## Reading the Comics, May 12, 2020: Little Oop Counts For More Edition

The past week had a fair number of comic strips mentioning some aspect of mathematics. One of them is, really, fairly slight. But it extends a thread in the comic strip that I like and so that I will feature here.

Jonathan Lemon and Joey Alison Sayers’s Little Oop for the 10th continues the thread of young Alley Oop’s time discovering numbers. (This in a storyline that’s seen him brought to the modern day.) The Moo researchers of the time have found numbers larger than three. As I’d mentioned when this joke was first done, that Oop might not have had a word for “seven” until recently doesn’t mean he wouldn’t have understood that seven of a thing was more than five of a thing, or less than twelve of a thing. At least if he could compare them.

Sam Hurt’s Eyebeam for the 11th uses heaps of mathematical expressions, graphs, charts, and Venn diagrams to represent the concept of “data”. It’s spilled all over to represent “sloppy data”. Usually by the term we mean data that we feel is unreliable. Measurements that are imprecise, or that are unlikely to be reliable. Precision is, roughly, how many significant digits your measurement has. Reliability is, roughly, if you repeated the measurement would you get about the same number?

Nate Fakes’s Break of Day for the 12th is the anthropomorphic numerals joke for the week.

Ryan North’s Dinosaur Comics for the 12th talks about immortality. And what the probability of events means when there are infinitely many opportunities for a thing to happen.

We’re accustomed in probability to thinking of the expectation value. This is the chance that something will happen, given some number N opportunities to happen, if at each opportunity it has the probability p of happening. Let me assume the probability is always the same number. If it’s not, our work gets harder, although it’s basically the same kind of work. But, then, the expectation value, the number of times we’d expect to see the thing happen, is N times p. Which, as Utahraptor points out, we can expect has to be at least 1 for any event, however unlikely, given enough chances. So it should be.

But, then, to take Utahraptor’s example: what is the probability that an immortal being never trips down the stairs? At least not badly enough to do harm? Why should we think that’s zero? It’s not as if there’s a physical law that compels someone to go to stairs and then to fall down them to their death. And, if there’s any nonzero chance of someone not dying this way? Then, if there are enough immortals, there’s someone who will go forever without falling down stairs.

That covers just the one way to die, of course. But the same reasoning holds for every possible way to die. If there’s enough immortals, there’s someone who would not die from falling down stairs and from never being struck by a meteor. And someone who’d never fall down stairs and never be struck by a meteor and never fall off a cliff trying to drop an anvil on a roadrunner. And so on. If there are infinitely many people, there’s at least one who’d avoid all possible accidental causes of death.

More. If there’s infinitely many immortals, then there are going to be a second and a third — indeed, an infinite number — of people who happen to be lucky enough to never die from anything. Infinitely many immortals die of accidents, sure, but somehow not all of them. We can’t even say that more immortals die of accidents than don’t.

My point is that probability gets really weird when you try putting infinities into it. Proceed with extreme caution. But the results of basic, incautious, thinking can be quite heady.

Bill Amend’s FoxTrot Classics for the 12th has Paige cramming for a geometry exam. Don’t cram for exams; it really doesn’t work. It’s regular steady relaxed studying that you need. That and rest. There is nothing you do that you do better for being sleep-deprived.

Bob Weber Jr and Jay Stephens’s Oh Brother for the 12th has Lily tease her brother with a story problem. I believe the strip’s a rerun, but it had been gone altogether for more than a year. It’s nice to see it returned anyway.

And while I don’t regularly cover web-only comics here, Norm Feuti has carried on his Gil as a Sunday-only web comic. The strip for the 10th of May has Gil using a calculator for mathematics homework, with a teacher who didn’t say he couldn’t. I’m surprised she hadn’t set a guideline.

This carries me through half a week. I’ll have more mathematically-themed comic strips at this link soon. Thanks for reading.

## Reading the Comics, May 9, 2020: Knowing the Angles Edition

There were a couple more comic strips in the block of time I want to write about. Only one’s got some deeper content and, I admit, I had to work to find it.

Bob Scott’s Bear With me for the 7th has Bear offering the answer from mathematics class, late.

Jerry Bittle’s Shirley and Sons Classic rerun for the 7th has Louis struggling on an arithmetic test.

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.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 9th has Elliot avoiding doing his mathematics homework.

And that’s got the last week covered. Some more comic strips should follow at a link here, soon. And I hope to have some other stuff to announce here, soon.

## Reading the Comics, May 7, 2020: Getting to Golf Edition

Last week saw a modest number of mathematically-themed comic strips. Then it threw in a bunch of them all on Thursday. I’m splitting the week partway through that, since it gives me some theme to this collection.

Tim Rickard’s Brewster Rockit for the 3rd of May is a dictionary joke, with Brewster naming each kind of chart and making a quick joke about it. The comic may help people who’ve had trouble remembering the names of different kinds of graphs. I doubt people are likely to confuse a pie chart with a bar chart, admittedly. But I could imagine thinking a ‘line graph’ is what we call a bar chart, especially if the bars are laid out horizontally as in the second panel here.

The point of all these graphs is to understand data geometrically. We have fair intuitions about relatives lengths and areas. Bar charts represent relative magnitudes in lengths. Pie charts and bubble charts represent magnitudes in area. We have okay skills in noticing structures in complex shapes. Line graphs and scatter plots use that skill. So these pictures can help us understand some abstraction or something we can’t sense using a sense we do have. It’s not necessarily great; note that I said our intuitions were ‘fair’ and ‘okay’. But we hope to use reason helped by intuition to better understand what we are doing.

Jef Mallett’s Frazz for the 3rd is a resisting-the-story-problem joke. It’s built not just on wondering the point of story problems at all, but of these story problems during the pandemic. (Which Mallett on the 27th of April, would be taking “some liberties” with the real world. It’s a respectable decision.)

And, yes, in the greater scheme of things, any homework or classwork problem is trivial. It’s meant to teach how to calculate things we would like to know. The framing of the story is meant to give us a reason to want to know a thing. But they are practice, and meant to be practice. One practices on something of no consequence, where errors in one’s technique can be corrected without breaking anything.

It happens a round of story problems broke out among my family. My sister’s house has some very large trees. There turns out to be a poorly-organized process for estimating the age of these trees from their circumference. This past week saw a lot of chatter and disagreement about what the ages of these trees might be.

Jason Poland’s Robbie and Bobby for the 4th riffs on the difference between rectangles and trapezoids. It’s also a repeat, featured here just five years ago. Amazing how time slips on like that.

Samson’s Dark Side of the Horse for the 4th is another counting-sheep joke. It features one of those shorthands for large numbers which often makes them more manageable.

Michael Fry’s Committed rerun for the 7th finally gets us to golf. The Lazy Parent tries to pass off watching golf as educational, with working out the distance to the pin as a story problem. Structurally this is just fine, though: a golfer would be interested to know how far the ball has yet to go. All the information needed is given. It’s the question of whether anyone but syndicated cartoonists cares about golf that’s a mystery.

Bill Amend’s FoxTrot Classics for the 7th is another golf and mathematics joke. Jason has taken the homonym of ‘fore’ for ‘four’, and then represented ‘four’ in a needlessly complicated way. Amend does understand how nerd minds work. The strip originally ran the 21st of May, 1998.

That’s enough comics for me for today. I should have the rest of last week’s in a post at this link soon. Thank you.

## Reading the Comics, May 2, 2020: What Is The Cosine Of Six Edition

The past week was a light one for mathematically-themed comic strips. So let’s see if I can’t review what’s interesting about them before the end of this genially dumb movie (1940’s Hullabaloo, starring Frank Morgan and featuring Billie Burke in a small part). It’ll be tough; they’re reaching a point where the characters start acting like they care about the plot either, which is usually the sign they’re in the last reel.

Patrick Roberts’s Todd the Dinosaur for the 26th of April presents mathematics homework as the most dreadful kind of homework.

Jenny Campbell’s Flo and Friends for the 26th is a joke about fumbling a bit of practical mathematics, in this case, cutting a recipe down. When I look into arguments about the metric system, I will sometimes see the claim that English traditional units are advantageous for cutting down a recipe: it’s quite easy to say that half of “one cup” is a half cup, for example. I doubt that this is much more difficult than working out what half of 500 ml is, and my casual inquiries suggest that nobody has the faintest idea what half of a pint would be. And anyway none of this would help Ruthie’s problem, which is taking two-fifths of a recipe meant for 15 people. … Honestly, I would have just cut it in half and wonder who’s publishing recipes that serve 15.

Ed Bickford and Aaron Walther’s American Chop Suey for the 28th uses a panel of (gibberish) equations to represent deep thinking. It’s in part of a story about an origami competition. This interests me because there is serious mathematics to be done in origami. Most of these are geometry problems, as you might expect. The kinds of things you can understand about distance and angles from folding a square may surprise. For example, it’s easy to trisect an arbitrary angle using folded squares. The problem is, famously, impossible for compass-and-straightedge geometry.

Origami offers useful mathematical problems too, though. (In practice, if we need to trisect an angle, we use a protractor.) It’s good to know how to take a flat, or nearly flat, thing and unfold it into a more interesting shape. It’s useful whenever you have something that needs to be transported in as few pieces as possible, but that on site needs to not be flat. And this connects to questions with pleasant and ordinary-seeming names like the map-folding problem: can you fold a large sheet into a small package that’s still easy to open? Often you can. So, the mathematics of origami is a growing field, and one that’s about an accessible subject.

Nate Fakes’s Break of Day for the 29th is the anthropomorphic-symbols joke for the week, with an x talking about its day job in equations and its free time in games like tic-tac-toe.

Bill Holbrook’s On The Fastrack for the 2nd of May also talks about the use of x as a symbol. Curt takes eagerly to the notion that a symbol can represent any number, whether we know what it is or not. And, also, that the choice of symbol is arbitrary; we could use whatever symbol communicates. I remember getting problems to work in which, say, 3 plus a box equals 8 and working out what number in the box would make the equation true. This is exactly the same work as solving 3 + x = 8. Using an empty box made the problem less intimidating, somehow.

Dave Whamond’s Reality Check for the 2nd is, really, a bit baffling. It has a student asking Siri for the cosine of 174 degrees. But it’s not like anyone knows the cosine of 174 degrees off the top of their heads. If the cosine of 174 degrees wasn’t provided in a table for the students, then they’d have to look it up. Well, more likely they’d be provided the cosine of 6 degrees; the cosine of an angle is equal to minus one times the cosine of 180 degrees minus that same angle. This allows table-makers to reduce how much stuff they have to print. Still, it’s not really a joke that a student would look up something that students would be expected to look up.

… That said …

If you know anything about trigonometry, you know the sine and cosine of a 30-degree angle. If you know a bit about trigonometry, and are willing to put in a bit of work, you can start from a regular pentagon and work out the sine and cosine of a 36-degree angle. And, again if you know anything about trigonometry, you know that there are angle-addition and angle-subtraction formulas. That is, if you know the cosine of two angles, you can work out the cosine of the difference between them.

So, in principle, you could start from scratch and work out the cosine of 6 degrees without using a calculator. And the cosine of 174 degrees is minus one times the cosine of 6 degrees. So it could be a legitimate question to work out the cosine of 174 degrees without using a calculator. I can believe in a mathematics class which has that as a problem. But that requires such an ornate setup that I can’t believe Whamond intended that. Who in the readership would think the cosine of 174 something to work out by hand? If I hadn’t read a book about spherical trigonometry last month I wouldn’t have thought the cosine of 6 a thing someone could reasonably work out by hand.

I didn’t finish writing before the end of the movie, even though it took about eighteen hours to wrap up ten minutes of story. My love came home from a walk and we were talking. Anyway, this is plenty of comic strips for the week. When there are more to write about, I’ll try to have them in an essay at this link. Thanks for reading.

## Reading the Comics, April 25, 2020: Off Brand Edition

Comic Strip Master Command decided I should have a week to catch up on things, and maybe force me to write something original. Of all the things I read there were only four strips that had some mathematics content. And three of them are such glancing mentions that I don’t feel it proper to include the strip. So let me take care of this.

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead apparently wants to know whether $\frac{3}{4}$ or $\frac{6}{8}$ is the better of these equivalent forms. I understand the impulse. Rarely in real life do we see two things that are truly equivalent; there’s usually some way in which one is better than the other. There may be two ways to get home for example, both taking about the same time to travel. One might have better scenery, though, or involve fewer difficult turns or less traffic this time of day. This is different, though: $\frac{3}{4}$ or $\frac{6}{8}$ are two ways to describe the same number. Which one is “better”?

The only answer is, better for what? What do you figure to do with this number afterwards? I admit, and suppose most people have, a preference for $\frac{3}{4}$. But that’s trained into us, in large part, by homework set to reduce fractions to “lowest terms”. There’s honest enough reasons behind that. It seems wasteful to have a factor in the numerator that’s immediately divided out by the denominator.

If this were 25 years ago, I could ask how many of you have written out a check for twenty-two and 3/4 dollars, then, rather than twenty-two and 75/100 dollars? The example is dated but the reason to prefer an equivalent form is not. If I know that I need the number represented by $\frac{3}{4}$, and will soon be multiplying it by eight, then $\frac{6}{8}$ may save me the trouble of thinking what three times two is. Or if I’ll be adding it to $\frac{5}{8}$, or something like that. If I’m measuring this for a recipe I need to cut in three, because the original will make three dozen cookies and I could certainly eat three dozen cookies, then $\frac{3}{4}$ may be more convenient than $\frac{6}{8}$. What is the better depends on what will clarify the thing I want to do.

A significant running thread throughout all mathematics, not just arithmetic, is finding equivalent forms. Ways to write the same concept, but in a way that makes some other work easier. Or more likely to be done correctly. Or, if the equivalent form is more attractive, more likely to be learned or communicated. It’s of value.

Jan Eliot’s Stone Soup Classics rerun for the 20th is a joke about how one can calculate what one is interested in. In this case, going from the number of days left in school to the number of hours and minutes and even seconds left. Personally, I have never had trouble remembering there are 24 hours in the day, nor that there are 86,400 seconds in the day. That there are 1,440 minutes in the day refuses to stick in my mind. Your experiences may vary.

Thaves’s Frank and Ernest for the 22nd is the Roman Numerals joke for the week, shifting the number ten to the representation “X” to the prefix “ex”.

Harry Bliss’s Bliss for the 23rd speaks of “a truck driver with a PhD in mathematical logic”. It’s an example of signifying intelligence through mathematics credentials. (It’s also a bit classicist, treating an intelligent truck driver as an unlikely thing.)

I’m caught up! This coming Sunday I hope to start discussingthis week’s comics in a post at this link. And for this week? I don’t know; maybe I’ll figure something to write. We’ll see. Thanks for reading.

## Reading the Comics, April 17, 2020: Creating Models Edition

And now let me close out a week ago, in the comics. It was a slow week and it finished on a bunch of casual mentions of mathematical topics.

Gary Larson’s The Far Side compilation “Hands Off My Bunsen Burner” features this panel creating a model of how to get rights out of wrongs. The material is a joke, but trying to find a transformation from one mathematical object to another is a reasonable enough occupation.

Ted Shearer’s Quincy rerun for the 15th is one in the lineage of strips about never using mathematics in later life. Quincy challenges us to think of a time a reporter asks the President how much is 34 times 587.

That’s an unpleasant multiplication to do. But I can figure some angles on it. 34 is just a bit over one-third of 100. 587 is just a bit under 600. So, 34 times 587 has to be tolerably near one-third of 100 times 600. So it should be something around 20,000. To get it more exact: 587 is 13 less than 600. So, 587 times one-third of a hundred will be 600 times one-third of a hundred minus 13 times one-third of a hundred. That’s one-third of 130, which is about 40. So the product has to be something close to 19,960. And the product has be some number which ends in an 8, what with 4 times 7 being 28. So the answer has to be one of 19,948, 19,958, or 19,968. And, indeed, it’s 19,958. I doubt I could do that so well during a press conference, I’ll admit. (If I wanted to be sure about that second digit, I’d have worked out: the tens unit in 34 times the ones in 587 is three times seven which is 21; the ones unit in 34 times the tens unit in 587 is four times eight which is 32; and the 4 times 7 being 28 gives me a 2 in the tens unit. So, 1 plus 2 plus 2 is 5, and there we go.)

Brian Anderson’s Dog Eat Doug for the 15th uses blackboards full of equations to represent deep thinking. I can’t make out what the symbols say. They look quite good, though, and seem to have the form of legitimate expressions.

Terri Liebenson’s The Pajama Diaries for the 17th imagines creating a model for the volume of a laundry pile. The problem may seem trivial, but it reflects an important kind of work. Many processes are about how something that’s always accumulating will be handled. There’s usually a hard limit to the rate at which whatever it is gets handled. And there’s usually very little reserve, in either capacity or time. This will cause, for example, a small increase in traffic in a neighborhood to produce great jams, or how a modest rain can overflow the whole city’s sewer systems. Or how a day of missing the laundry causes there to be a week’s backlog of dirty clothes.

And a little final extra comic strip. I don’t generally mention web comics here, except for those that have fallen in with a syndicator like GoComics.com. (This is not a value judgement against web comics. It’s that I have to stop reading sometime.) But Kat Swenski’s KatRaccoon Comics recently posted this nice sequence with a cat facing her worst fear: a calculus date.

And that’s my comics for a week ago. Later this week I’ll cover the past week’s handful of comics, in an essay at this link. Thanks for reading.