## Reading the Comics, January 13, 2019: January 13, 2019 Edition

I admit I’m including a fairly marginal strip in this, just so I can have the fun of another single-day edition. What can I say? I can be easily swayed by silly things. Also, somehow, all four strips today have circumstances where one might mistake them for reruns. Let’s watch.

Bill Amend’s FoxTrot for the 13th is wordplay, mashing up ‘cell division’ with ‘long division’. As you might expect from Bill Amend — who loves sneaking legitimate mathematics and physics in where it’s not needed — Paige’s long cell division is a legitimate one. If you’d like a bit of recreational mathematics fun, you can figure out which microscopic organisms correspond to which numerals. The answer is also the Featured Comment on the page, at least as I write this. So if you need an answer, or you want to avoid having the answer spoiled, know what’s there.

Greg Evans’s Luann Againn for the 13th is the strip of most marginal relevance here. Part of Luann’s awful ay is a mathematics test. The given problems are nothing particularly meaningful. There is the sequence ‘mc2’ in the problem, although written as $m^c 2$. There’s also a mention of ‘googleplex’, which when the strip was first published in 1991 was nothing more than a misspelling of the quite large number. (‘Googol’ is the number; ‘Google’ a curious misspelling. Or perhaps a reversion. The name was coined in 1938 by Milton Sirotta. Sirotta was seven years old at the time. I accept that it is at least possible Sirotta was thinking of the then-very-popular serial-comic strip Barney Google, and that his uncle Edward Kasner, who brought the name to mathematics, wrote it down wrong.) And that carries with it the connotation that big numbers are harder than small numbers. This is … kind of true. At least, long numbers are more tedious than short numbers. But you don’t really do different work, dividing 1428 by 7, than you do dividing 147 by 7. It’s just longer. “Hard” is a flexible idea.

Mac King and Bill King’s Magic in a Minute for the 13th felt like a rerun to me. It took a bit of work to find, but yeah, it was. The strip itself, as presented, is new. But the same neat little modular-arithmetic coincidence was used the 31st of July, 2016.

Mathematics on clock faces is often used as a way to introduce modular arithmetic, a variation on arithmetic with only finitely many integers. This can help, if you’re familiar with clock faces. Like regular arithmetic, modular arithmetic can form a group and a ring. Clock faces won’t give you a group or ring, not unless you replace the number before ‘1’ with a ‘0’. To be a group, you need a collection of items, and a binary operation on the items. This operation we often think of as either addition or multiplication, depending on what makes sense for the problem. To be a ring, you need two binary operations, which interact by a distributive law. So the operations are often matched to addition and multiplication. Modular arithmetic is fun, yes. It’s also useful, not just as a way to do something like arithmetic that’s different. Many schemes for setting up checksums, quick and easy tests against data entry errors, rely on modular arithmetic on the data. And many schemes for generating ‘random’ numbers are built on finding multiplicative inverses in modular arithmetic. This isn’t truly random, of course. But you can look at a string of digits and not see any clear patterns. This is often as close to random as you need.

Rick DeTorie’s One Big Happy for the 13th is mostly a bunch of complaints the old always have against the young. Well, the complaint about parallel parking I haven’t seen before. But the rest are common enough. Featured in it is a complaint that the young can’t do arithmetic. I’m not sure there was ever a time that the older generation thought the young were well-trained in arithmetic. Nor that there was ever a time that the current educational vogue wasn’t blamed for destroying a generation’s ability to calculate. I’m sure there are better and worse ways to teach calculation. But I suspect any teaching method will fall short of addressing a couple issues. One is that people over-rate their own competence and under-rate other’s competence. So the older generation will see itself as having got the best possible arithmetic education and anything that’s different is a falling away. And another is that people get worse at stuff they don’t think is enjoyable or don’t have to do a lot. If you haven’t got a use for the fact, or an appreciation for the beauty in it, three times six is a bit of trivia, and not one that inspires much conversation when shared.

There’s more comics with something of a mathematical theme that got published last week. When I get to them the essays should be at this link.

## Reading the Comics, January 12, 2019: A Edition

As I said Sunday, last week was a slow one for mathematically-themed comic strips. Here’s the second half of them. They’re not tightly on point. But that’s all right. They all have titles starting with ‘A’. I mean if you ignore the article ‘the’, the way we usually do when alphabetizing titles.

Tony Cochran’s Agnes for the 11th is basically a name-drop of mathematics. The joke would be unchanged if the teacher asked Agnes to circle all the adjectives in a sentence, or something like that. But there are historically links between religious thinking and mathematics. The Pythagoreans, for example, always a great and incredible starting point for any mathematical topic or just some preposterous jokes that might have nothing to do with their reality, were at least as much a religious and philosophical cult. For a long while in the Western tradition, the people with the time and training to do advanced mathematics work were often working for the church. Even as people were more able to specialize, a mystic streak remained. It’s easy to understand why. Mathematics promises to speak about things that are universally true. It encourages thinking about the infinite. It encourages thinking about the infinitely tiny. It courts paradoxes as difficult as any religious Mystery. It’s easy to snark at someone who takes numerology seriously. But I’m not sure the impulse that sees magic in arithmetic is different to the one that sees something supernatural in a “transfinite” item.

Scott Hilburn’s The Argyle Sweater for the 11th is another mistimed Pi Day joke. π is, famously, an irrational number. But so is every number, except for a handful of strange ones that we’ve happened to find interesting. That π should go on and on follows from what an irrational number means. It’s a bit surprising the 4 didn’t know all this before they married.

I appreciate the secondary joke that the marriage counselor is a “Hugh Jripov”, and the counselor’s being a ripoff is signaled by being a &div; sign. It suggests that maybe successful reconciliation isn’t an option. I’m curious why the letters ‘POV’ are doubled, in the diploma there. In a strip with tighter drafting I’d think it was suggesting the way a glass frame will distort an image. But Hilburn draws much more loosely than that. I don’t know if it means anything.

Mark Anderson’s Andertoons for the 12th is the Mark Anderson’s Andertoons for the essay. I’m so relieved to have a regular stream of these again. The teacher thinks Wavehead doesn’t need to annotate his work. And maybe so. But writing down thoughts about a problem is often good practice. If you don’t know what to do, or you aren’t sure how to do what you want? Absolutely write down notes. List the things you’d want to do. Or things you’d want to know. Ways you could check your answer. Ways that you might work similar problems. Easier problems that resemble the one you want to do. You find answers by thinking about what you know, and the implications of what you know. Writing these thoughts out encourages you to find interesting true things.

And this was too marginal a mention of mathematics even for me, even on a slow week. But Georgia Dunn’s Breaking Cat News for the 12th has a cat having a nightmare about mathematics class. And it’s a fun comic strip that I’d like people to notice more.

And that’s as many comics as I have to talk about from last week. Sunday, I should have another Reading the Comics post and it’ll be at this link.

## Reading the Comics, January 9, 2018: I Go On About Johnny Appleseed Edition

This was a slow week for mathematically-themed comic strips. Such things happen. I put together a half-dozen that see on-topic enough to talk about, but I stretched to do it. You’ll see.

Mark Anderson’s Andertoons for the 6th mentions addition as one of the things you learn in an average day of elementary school. I can’t help noticing also the mention of Johnny Appleseed, who’s got a weird place in my heart as he and I share a birthday. He got to it first. Although Johnny Appleseed — John Champan — is legendary for scattering apple seeds, that’s not what he mostly did. He would more often grow apple-tree nurseries, from which settlers could buy plants and demonstrate they were “improving” their plots. He was also committed to spreading the word of Emanuel Swedenborg’s New Church, one of those religious movements that you somehow don’t hear about. But there was this like 200-year-long stretch where a particular kind of idiosyncratic thinker was Swedenborgian, or at least influenced by that. I don’t know offhand of any important Swedenborgian mathematicians, I admit, but I’m glad to hear if someone has news.

Justin Thompson’s MythTickle rerun for the 9th mentions “algebra” as something so dreadful that even being middle-aged is preferable. Everyone has their own tastes, yes, although it would be the same joke if it were “gym class” or something. (I suppose that’s not one word. “Dodgeball” would do, but I never remember playing it. It exists just as a legendarily feared activity, to me.) Granting, though, that I had a terrible time with the introduction to algebra class I had in middle school.

Tom Wilson’s Ziggy for the 9th is a very early Pi Day joke, so, there’s that. There’s not much reason a take-a-number dispenser couldn’t give out π, or other non-integer numbers. What the numbers are doesn’t matter. It’s just that the dispensed numbers need to be in order. It should be helpful if there’s a clear idea how uniformly spaced the numbers are, so there’s some idea how long a wait to expect between the currently-serving number and whatever number you’ve got. But that only helps if you have a fair idea of how long an order should on average take.

I’ll close out last week’s comics soon. The next Reading the Comics post, like all the earlier ones, should be at this link.

## Reading the Comics, January 5, 2019: Start of the Year Edition

With me wrapping up the mathematically-themed comic strips that ran the first of the year, you can see how far behind I’m falling keeping everything current. In my defense, Monday was busier than I hoped it would be, so everything ran late. Next week is looking quite slow for comics, so maybe I can catch up then. I will never catch up on anything the rest of my life, ever.

Scott Hilburn’s The Argyle Sweater for the 2nd is a bit of wordplay about regular and irregular polygons. Many mathematical constructs, in geometry and elsewhere, come in “regular” and “irregular” forms. The regular form usually has symmetries that make it stand out. For polygons, this is each side having the same length, and each interior angle being congruent. Irregular is everything else. The symmetries which constrain the regular version of anything often mean we can prove things we otherwise can’t. But most of anything is the irregular. We might know fewer interesting things about them, or have a harder time proving them.

I’m not sure what the teacher would be asking for in how to “make an irregular polygon regular”. I mean if we pretend that it’s not setting up the laxative joke. I can think of two alternatives that would make sense. One is to draw a polygon with the same number of sides and the same perimeter as the original. The other is to draw a polygon with the same number of sides and the same area as the original. I’m not sure of the point of either. I suppose polygons of the same area have some connection to quadrature, that is, integration. But that seems like it’s higher-level stuff than this class should be doing. I hate to question the reality of a comic strip but that’s what I’m forced to do.

Bud Fisher’s Mutt and Jeff rerun for the 4th is a gambler’s fallacy joke. Superficially the gambler’s fallacy seems to make perfect sense: the chance of twelve bad things in a row has to be less than the chance of eleven bad things in a row. So after eleven bad things, the twelfth has to come up good, right? But there’s two ways this can go wrong.

Suppose each attempted thing is independent. In this case, what if each patient is equally likely to live or die, regardless of what’s come before? And in that case, the eleven deaths don’t make it more likely that the next will live.

Suppose each attempted thing is not independent, though. This is easy to imagine. Each surgery, for example, is a chance for the surgeon to learn what to do, or not do. He could be getting better, that is, more likely to succeed, each operation. Or the failures could reflect the surgeon’s skills declining, perhaps from overwork or age or a loss of confidence. Impossible to say without more data. Eleven deaths on what context suggests are low-risk operations suggest a poor chances of surviving any given surgery, though. I’m on Jeff’s side here.

Mark Anderson’s Andertoons for the 5th is a welcome return of Wavehead. It’s about ratios. My impression is that ratios don’t get much attention in themselves anymore, except to dunk on stupid Twitter comments. It’s too easy to jump right into fractions, and division. Ratios underlie this, at least historically. It’s even in the name, ‘rational numbers’.

Wavehead’s got a point in literally comparing apples and oranges. It’s at least weird to compare directly different kinds of things. This is one of those conceptual gaps between ancient mathematics and modern mathematics. We’re comfortable stripping the units off of numbers, and working with them as abstract entities. But that does mean we can calculate things that don’t make sense. This produces the occasional bit of fun on social media where we see something like Google trying to estimate a movie’s box office per square inch of land in Australia. Just because numbers can be combined doesn’t mean they should be.

Larry Wright’s Motley rerun for the 5th has the form of a story problem. And one timely to the strip’s original appearance in 1987, during the National Football League players strike. The setup, talking about the difference in weekly pay between the real players and the scabs, seems like it’s about the payroll difference. The punchline jumps to another bit of mathematics, the point spread. Which is an estimate of the expected difference in scoring between teams. I don’t know for a fact, but would imagine the scab teams had nearly meaningless point spreads. The teams were thrown together extremely quickly, without much training time. The tools to forecast what a team might do wouldn’t have the data to rely on.

The at-least-weekly appearances of Reading the Comics in these pages are at this link.

## Reading the Comics, January 1, 2019: New Year’s Day Edition

It’s a new year. That doesn’t mean I’m not going to keep up some of my old habits. One of them is reading the comics for the mathematics bits. For example …

Johnny Hart’s Back To BC for the 30th presents some curious use of mathematics. At least the grammar of mathematics. It’s a bunch of statements that are supposed to, taken together, overload … I’m going to say BC’s … brain. (I’m shaky on which of the characters is Peter and which is BC. Their difference in hair isn’t much of a visual hook.) Certainly mathematics inspires that feeling that one’s overloaded one’s brain. The long strings of reasoning and (ideally) precise definitions are hard to consider. And the proofs mathematicians find the most fun are, often, built cleverly. That is, going about their business demonstrating things that don’t seem relevant, and at the end tying them together. It’s hard to think.

But then … Peter … isn’t giving a real mathematical argument. He’s giving nonsense. And obvious nonsense, rather than nonsense because the writer wanted something that sounded complicated without caring what was said. Talking about a “four-sided triangle” or a “rectangular circle” has to be Peter trying to mess with BC’s head. Confidently-spoken nonsense can sound as if it’s deeper wisdom than the listener has. Which, fair enough: how can you tell whether an argument is nonsense or just cleverer than you are? Consider the kind of mathematics proof I mentioned above, where the structure might almost be a shaggy dog joke. If you can’t follow the logic, is it because the argument is wrong or because you haven’t worked out why it is right?

I believe that … Peter … is just giving nonsense and trusting that … BC … won’t know the difference, but will wear himself out trying to understand. Pranks.

Tim Lachowski’s Get A Life for the 31st just has some talk about percentages and depreciation and such. It’s meant to be funny that we might think of a brain depreciating, as if anatomy could use the same language as finance. Still, one of the virtues of statistics is the ability to understand a complicated reality with some manageable set of numbers. If we accept the convention that some number can represent the value of a business, why not the convention that some number could represent the health of a brain? So, it’s silly, but I can imagine a non-silly framing for it.

Tony Cochran’s Agnes for the 1st is about calendars. The history of calendars is tied up with mathematics in deep and sometimes peculiar ways. One might imagine that a simple ever-increasing index from some convenient reference starting time would do. And somehow that doesn’t. Also, the deeper you go into calendars the more you wonder if anyone involved in the project knew how to count. If you ever need to feel your head snapping, try following closely just how the ancient Roman calendar worked. Especially from the era when they would occasionally just drop an extra month in to the late-middle of February.

The Julian and Gregorian calendars have a year number that got assigned proleptically, that is, with the year 1 given to a set of dates that nobody present at the time called the year 1. Which seems fair enough; not many people in the year 1 had any idea that something noteworthy was under way. Calendar epochs dated to more clear events, like the reign of a new emperor or the revolution that took care of that whole emperor problem, will more reliably start with people aware of the new numbers. Proleptic dating has some neat side effects, though. If you ever need to not impress someone, you can point out that the dates from the 1st of March, 200 to the 28th of February, 300 both the Julian and the Gregorian calendar dates exactly matched.

Niklas Eriksson’s Carpe Diem for the 2nd is a dad joke about mathematics. And uses fractions as emblematic of mathematics, fairly enough. Introducing them and working with them are the sorts of thing that frustrate and confuse. I notice also the appearance of “37” here. Christopher Miller’s fascinating American Cornball: A Laffopedic Guide to the Formerly Funny identifies 37 as the current “funniest number”, displacing the early 20th century’s preferred 23 (as in skidoo). Among other things, odd numbers have a connotation of seeming more random than even numbers; ask someone to pick a whole number from 1 to 50 and you’ll see 37’s and 33’s more than you’ll see, oh, 48’s. Why? Good question. It’s among the mysteries of psychology. There’s likely no really deep reason. Maybe a sense that odd numbers are, well, odd as in peculiar, and that a bunch of peculiarities will be funny. Now let’s watch the next decade make a food of me and decide the funniest number is 64.

I’m glad to be back on schedule publishing Reading the Comics posts. I should have another one this week. It’ll be at this link when it’s ready. Thanks for reading.

## Reading the Comics, December 28, 2018: More Christmas Break Edition

I apologize for running quite so late. Comic Strip Master Command tried to make it easy for me, by issuing few comic strips that had any mathematical content to speak of. I was just busier than all that, and even now, I can’t say quite how. Well, living, I suppose. But I’ve done plenty of things now and can settle back to the usual, if anyone knows just what that was.

Also I am drawing down on the number of cancelled, in-eternal-reruns comic strips on my daily feed. So that should reduce the number of times I feature a comic strip and realize I’ve described it four times already and haven’t got anything new to say. It’s hard for me, since most of these comics have some charms, or at least pleasant weirdness. But clearly just making a note to myself that I’ve said everything there is to say about Randolph Itch, 2 am, isn’t enough. I’m sorry, Randolph.

Bill Holbrook’s On The Fastrack for the 28th is an example of the cartoonist’s habit of drawing metaphors literally. Dethany does ask the auditor Fi about “accepting his numbers”. In this context the numbers aren’t intersting as numbers. They’re interesting as representations for a narrative. If the numbers are consistent with a believable story? If it’s more believable that they represent a truth than that they’re a hoax? We call that “accepting the numbers”, but what we’re accepting is the story they’re given as evidence for.

Auditing, and any critical thinking about numbers, involves some subtle uses of Bayesian probability. We’re working out the probability that this story is something we should believe. Each piece of evidence makes us think this probability is greater or lesser. With experience and skill one learns of patterns which suggest the story is false. Benford’s Law, for example, is often useful. Honestly-taken samples show tendencies, for example, in what leading digits appear. A discrepancy between what’s expected and what appears, if it can’t be explained, can be a sign of forgery.

Johnny Hart’s Back To BC rerun for the 27th is built on estimating the grains of sand on a beach. This is, as fits the setting, a very old query. Archimedes wrote The Sand Reckoner which estimated how many grains of sand could fit in the universe. Estimating the number of grains of sand on a beach, or in a universe, is a fun mathematical problem. Perhaps not a practical one, not directly. The answer is after all “lots”, and there is no way to verify the number.

But it can still be indirectly practical. To work with enormous but finite numbers of things is hard. We do well working with small numbers like ‘six’ and ‘fourteen’ and some of us are even good at around ‘thirty’. We don’t have a good intuition for how a number like 480,000,000,000,000,000 should work. And that’s important; if we try adding six and fourteen and get thirty, we realize there’s something not quite right before we’ve done too much more work. With enormous numbers we can go on not noticing the mistake’s there. We need to find ways to understand these inconvenient numbers using the skills and intuitions we already have. Aristotle had to develop new terminology for numbers to get the Ancient Greek numerals system to handle the problem coherently. Peter’s invention of a gillion is — I’ll go ahead and say — a sly reenactment of that.

Mark Pett’s Lucky Cow rerun for the 27th I do intend to make this enjoyable but cancelled strip’s last appearance here. It’s a Rubik’s Cube joke. It’s one about using a solution outside the rules of the problem. And as marginal as this one, I couldn’t quite bring myself to write a paragraph about the Todd the Dinosaur strip of the 29th, which also features the Rubik’s Cube.

Ryan Pagelow’s Buni for the 28th I’ll list as the anthropomorphic-numerals joke for the week, since it did turn out to be that slow a week here. I’m a bit curious what the now-9 is figuring to do next year. I suppose that one’s easy; it’s going to be going from 3 to 4 in a couple years that’s a real problem.

The various Reading the Comics posts should all be at this link. I like to think I’ll be back to having a post this coming Sunday, and maybe a second one next week if there are enough comic strips near enough to on-topic. Thanks for reading.

## Reading the Comics, December 22, 2018: Christmas Break Edition

There were just enough mathematically-themed comic strips last week for me to make two posts out of it. This current week? Is looking much slower, at least as of Wednesday night. But that’s a problem for me to worry about on Sunday.

Eric the Circle for the 20th, this one by Griffinetsabine, mentions a couple of shapes. That’s enough for me, at least on a slow comics week. There is a fictional tradition of X marking the spot. It can be particularly credited to Robert Louis Stevenson’s Treasure Island. Any symbol could be used to note a special place on maps, certainly. Many maps are loaded with a host of different symbols to convey different information. Circles and crosses have the advantage of being easy to draw and difficult to confuse for one another. Squares, triangles, and stars are good too.

Bill Whitehead’s Free Range for the 22nd spoofs Wheel of Fortune with “theoretical mathematics”. Making a game out of filling in parts of a mathematical expression isn’t ridiculous, although it is rather niche. I don’t see how the revealed string of mathematical expressions build to a coherent piece, but perhaps a few further pieces would help.

The parts shown are all legitimate enough expressions. Well, like $a^2 + b^2 = (a + b)$ is only true for some specific numbers ‘a’ and ‘b’, but you can find solutions. $-b \pm \sqrt{b^2 - x^2y^2}$ is just an expression, not picking out any particular values of ‘b’ or ‘x’ or ‘y’ as interesting. But in conjunction with $a^2 + b^2 = (a + b)$ or other expressions there might be something useful. On the second row is a graph, highlighting a region underneath a curve (and above the x-axis) between two vertical lines. This is often the sort of thing looked at in calculus. It also turns up in probability, as the area under a curve like this can show the chance that an experiment will turn up something in a range of values. And $\frac{dy}{dx} = x^4 - \left(1 - x^2\right)^4$ is a straightforward differential equation. Its solution is a family of similar-looking polynomials.

Mark Pett’s Lucky Cow for the 22nd has run before. I’ve even made it the title strip for a Reading the Comics post back in 2014. So it’s probably time to drop this from my regular Reading the Comics reporting. The physicists comes running in with the left half of the time-dependent Schrödinger Equation. This is all over quantum mechanics. In this form, quantum mechanics contains information about how a system behaves by putting it into a function named $\psi$. Its value depends on space (‘x’). It can also depend on time (‘t’). The physicists pretends to not be able to complete this. Neil arranges to give the answer.

Schrödinger’s Equation looks very much like a diffusion problem. Normal diffusion problems don’t have that $\imath$ which appears in the part of Neil’s answer. But this form of equation turns up a lot. If you have something that acts like a fluid — and heat counts — then a diffusion problem is likely important in understanding it.

And, yes, the setup reminds me of a mathematical joke that I only encounter in lists of mathematics jokes. That one I told the last time this strip came up in the rotation. You might chuckle, or at least be convinced that it is a correctly formed joke.

Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

## Reading the Comics, December 19, 2018: Andertoons Is Back Edition

I had not wanted to mention, for fear of setting off a panic. But Mark Anderson’s Andertoons, which I think of as being in every Reading the Comics post, hasn’t been around lately. If I’m not missing something, it hasn’t made an appearance in three months now. I don’t know why, and I’ve been trying not to look too worried by it. Mostly I’ve been forgetting to mention the strange absence. This even though I would think any given Tuesday or Friday that I should talk about the strip not having anything for me to write about. Fretting about it would make a great running theme. But I have never spotted a running theme before it’s finished. In any event the good news is that the long drought has ended, and Andertoons reappears this week. Yes, I’m hoping that it won’t be going to long between appearances this time.

Jef Mallett’s Frazz for the 16th talks about probabilities. This in the context of assessing risks. People are really bad at estimating probabilities. We’re notoriously worse at assessing risks, especially when it’s a matter of balancing a present cost like “fifteen minutes waiting while the pharmacy figures out whether insurance will pay for the flu shot” versus a nebulous benefit like “lessened chance of getting influenza, or at least having a less severe influenza”. And it’s asymmetric, too. We view improbable but potentially enormous losses differently from the way we view improbable but potentially enormous gains. And it’s hard to make the rationally-correct choice reliably, not when there are so many choices of this kind every day.

Tak Bui’s PC and Pixel for the 16th features a wall full of mathematical symbols, used to represent deep thought about a topic. The symbols are gibberish, yes. I’m not sure that an actual “escape probability” could be done in a legible way, though. Or even what precisely Professor Phillip might be calculating. I imagine it would be an estimate of the various ways he might try to escape, and what things might affect that. This might be for the purpose of figuring out what he might do to maximize his chances of a successful escape. Although I wouldn’t put it past the professor to just be quite curious what the odds are. There’s a thrill in having a problem solved, even if you don’t use the answer for anything.

Ruben Bolling’s Super-Fun-Pak Comix for the 18th has a trivia-panel-spoof dubbed Amazing Yet Tautological. One could make an argument that most mathematics trivia fits into this category. At least anything about something that’s been proven. Anyway, whether this is a tautological strip depends on what the strip means by “average” in the phrase “average serving”. There’s about four jillion things dubbed “average” and each of them has a context in which they make sense. The thing intended here, and the thing meant if nobody says anything otherwise, is the “arithmetic mean”. That’s what you get from adding up everything in a sample (here, the amount of egg salad each person in America eats per year) and dividing it by the size of the sample (the number of people in America that year). Another “average” which would make sense, but would break this strip, would be the median. That would be the amount of egg salad that half of all Americans eat more than, and half eat less than. But whether every American could have that big a serving really depends on what that median is. The “mode”, the most common serving, would also be a reasonable “average” to expect someone to talk about.

Mark Anderson’s Andertoons for the 19th is that strip’s much-awaited return to my column here. It features solid geometry, which is both an important part of geometry and also a part that doesn’t get nearly as much attention as plane geometry. It’s reductive to suppose the problem is that it’s harder to draw solids than planar figures. I suspect that’s a fair part of the problem, though. Mathematicians don’t get much art training, not anymore. And while geometry is supposed to be able to rely on pure reasoning, a good picture still helps. And a bad picture will lead us into trouble.

Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

## Reading the Comics, December 15, 2018: Early Holiday Edition

So then this happened: Comic Strip Master Command didn’t have much they wanted me to write about this week. I made out three strips as being relevant enough to discuss at all. And even they don’t have topics that I felt I could really dig into. Coincidence, surely, although I like to think they were trying to help me get ahead of deadline on my A To Z essays for this last week of the run. It’s a noble thought, but doomed. I haven’t been more than one essay ahead of deadline the last three months. I know in past years I’ve gotten three or even four essays ahead of time and I don’t know why it hasn’t worked this time. I am going ahead and blaming that this these essays have been way longer than previous years’. So anyway, I thank Comic Strip Master Command for trying to make my Monday and my Thursday this week be less packed. It won’t help.

Darrin Bell and Theron Heir’s Rudy Park for the 10th uses mathematics as shorthand for a deep, thought-out theory of something. In this case, Randy’s theory of how to interest women. (He has rather a large number of romantic events around him.) It’s easy to suppose that people can be modeled mathematically. Even a crude model, one supposing that people have things they like and dislike, can give us good interesting results. This gets into psychology and sociology though. And probably requires computer modeling to get slightly useful results.

Randy’s blackboard has a good number of legitimate equations on it. They’re maybe not so useful to his problem of modeling people, though. The lower left corner, for example, are three of Maxwell’s Equations, describing electromagnetism. I’m not sure about all of these, in part because I think some might be transcribed incorrectly. The second equation in the upper left, for example, looks like it’s getting at the curl of a conserved force field being zero, but it’s idiosyncratic to write that with a ‘d’ to start with. The symbols all over the right with both subscripts and superscripts look to me like tensor work. This turns up in electromagnetism, certainly. Tensors turn up anytime something, such as electrical conductivity, is different in different directions. But I’ve never worked deeply in those fields so all I can confidently say is that they look like they parse.

Lincoln Pierce’s Big Nate for the 14th is part of a bit where Nate’s trying to write a gruesome detective mystery for kids. I’m not sure that’s a ridiculous idea, at least if the gore could be done at a level that wouldn’t be too visceral. Anyway, Nate has here got the idea of merging some educational value into the whole affair. It’s not presented as a story problem, just as characters explaining stuff to one another. There probably would be some room for an actual problem where Barky and Winky wanted to know something and had to work out how to find it from what they knew, though.

Mel Henze’s Gentle Creatures for the 14th uses a story problem to stand in for science fictional calculations. The strip’s in reruns and I’ve included it here at least four times, I discover, so that’s probably enough for the comic until it gets out of reruns.

And since it was a low-volume week, let me mention strips I didn’t decide fit. Ray Kassinger asked about Tim Rickard’s Brewster Rockit for the 12th. Might it be a play on Schrödinger’s Cat, the famous thought-experiment about how to understand the mathematics of quantum mechanics? It’s possible, but I think it’s more likely just that cats like sitting in boxes. Thaves’s Frank and Ernest for the 13th looks like it should be an anthropomorphic numerals joke. But it’s playing on the idiom about three being a crowd, and the whole of the mathematical content is that three is a number. John Zakour and Scott Roberts’s Maria’s Day for the 15th mentions mathematics. Particularly, Maria wishing they weren’t studying it. It’s a cameo appearance; it could be any subject whose value a student doesn’t see. That’s all I can make of it.

This and my other Reading the Comics posts should all be available at this link. And please check back in Tuesday to see whether I make deadline for the letter ‘Y’ in my Fall 2018 Mathematics A To Z glossary.

## Reading the Comics, December 8, 2018: Sam and Son Edition

That there were twelve comic strips making my cut as mention-worthy this week should have let me do three essays of four comics each. But the desire to include all the comics from the same day in one essay leaves me one short here. So be it. Three of the four cartoonists featured here have a name of Sansom or Samson, so, that’s an edition title for you. No, Sam and Silo do not appear here.

Art Sansom and Chip Sansom’s Born Loser for the 6th uses arithmetic as a test of deference. Will someone deny a true thing in order to demonstrate loyalty? Arithmetic is full of things that are inarguably true. If we take the ordinary meanings of one, plus, equals, and three, it can’t be that one plus one equals three. Most fields of human endeavor are vulnerable to personal taste, or can get lost in definitions and technicalities. Or the advance of knowledge: my love and I were talking last night how we remembered hearing, as kids, the trivia that panda bears were not really bears, but a kind of raccoon. (Genetic evidence has us now put giant pandas with the bears, and red pandas as part of the same superfamily as raccoons, but barely.) Or even be subject to sarcasm. Arithmetic has a harder time of that. Mathematical ideas do evolve in time, certainly. But basic arithmetic is pretty stable. Logic is also a reliable source of things we can be confident are true. But arithmetic is more familiar than most logical propositions.

Samson’s Dark Side of the Horse for the 8th is the Roman Numerals joke for the week. It’s also a bit of a wordplay joke, although the music wordplay rather tha mathematics. Me, I still haven’t heard a clear reason why ‘MIC’ wouldn’t be a legitimate Roman numeral representation of 1099. I’m not sure whether ‘MIC’ would step on or augment the final joke, though.

Pab Sungenis’s New Adventures of Queen Victoria for the 8th has a comedia dell’arte-based structure for its joke. (The strip does that, now and then.) The comic uses a story problem, with the calculated answer rejected for the nonsense it would be. I suppose it must be possible for someone to eat eighty apples over a long enough time that it’s not distressing, and yet another twenty apples wouldn’t spoil. I wouldn’t try it, though.

This and my other Reading the Comics posts should all be available at this link.

## Reading the Comics, December 5, 2018: December 5, 2018 Edition

And then I noticed there were a bunch of comic strips with some kind of mathematical theme on the same day. Always fun when that happens.

Bill Holbrook’s On The Fastrack uses one of Holbrook’s common motifs. That’s the depicting as literal some common metaphor. in this case it’s “massaging the numbers”, which might seem not strictly mathematics. But while numbers are interesting, they’re also useful. To be useful they must connect to something we want to know. They need context. That context is always something of human judgement. If the context seems inappropriate to the listener, she thinks the presenter is massaging the numbers. If the context seems fine, we trust the numbers as showing something truth.

Scott Hilburn’s The Argyle Sweater is a seasonal pun that couldn’t wait for a day closer to Christmas. I’m a little curious why not. It would be the same joke with any subject, certainly. The strip did make me wonder if Ebeneezer Scrooge, in-universe, might have taken calculus. This lead me to see that it’s a bit vague what, precisely, Scrooge, or Scrooge-and-Marley, did. The movies are glad to position him as having a warehouse, and importing and exporting things, and making and collecting on loans and whatnot. These are all trades that mathematicians would like to think benefit from knowing advanced mathematics. The logic of making loans implies attention be paid to compounding interest, risks, and expectation values, as well as projecting cash-flow a fair bit into the future. But in the original text he doesn’t make any stated loans, and the only warehouse anyone enters is Fezziwig’s. Well, the Scrooge and Marley sign stands “above the warehouse door”, but we only ever go in to the counting-house. And yes, what Scrooge does besides gather money and misery is irrelevant to the setting of the story.

Teresa Burritt’s Dadaist strip Frog Applause uses knowledge of mathematics as an emblem of intelligence. “Multivariate analysis” is a term of art from statistics. It’s about measuring how one variable changes depending on two or more other variables. The goal is obvious: we know there are many things that influence anything of interest. Can we find what things have the strongest effects? The weakest effects? There are several ways we might mean “strongest” effect, too. It might mean that a small change in the independent variable produces a big change in the dependent one. Or it might mean that there’s very little noise, that a change in the independent variable produces a reliable change in the dependent one. Or we might have several variables that are difficult to measure precisely on their own, but with a combination that’s noticeable. The basic calculations for this look a lot like those for single-variable analysis. But there’s much more calculation. It’s more tedious, at least. My reading suggests that multivariate analysis didn’t develop much until there were computers cheap enough to do the calculations. Might be coincidence, though. Many machine-learning techniques can be described as multivariate analysis problems.

Greg Evans’s Luann Againn is a Pi Day joke from before the time when Pi Day was a thing. Brad’s magazine flipping like that is an unusual bit of throwaway background humor for the comic strip.

Doug Savage’s Savage Chickens is a bunch of shape jokes. Since I was talking about tiling the plane so recently the rhombus seemed on-point enough. I’m think the irregular heptagon shown here won’t tile the plane. But given how much it turns out I didn’t know, I wouldn’t want to commit to that.

I’m working hard on a latter ‘X’ essay for my Fall 2018 Mathematics A To Z glossary. That should appear on Friday. And there should be another Reading the Comics post later this week, at this link.

## Reading the Comics, December 4, 2018: Christmas Specials Edition

This installment took longer to write than you’d figure, because it’s the time of year we’re watching a lot of mostly Rankin/Bass Christmas specials around here. So I have to squeeze words out in-between baffling moments of animation and, like, arguing whether there’s any possibility that Jack Frost was not meant to be a Groundhog Day special that got rewritten to Christmas because the networks weren’t having it otherwise.

Graham Nolan’s Sunshine State for the 3rd is a misplaced Pi Day strip. I did check the copyright to see if it might be a rerun from when it was more seasonal.

Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 3rd is the anthropomorphic numerals joke for the week. … You know, I’ve always wondered in this sort of setting, what are two-digit numbers like? I mean, what’s the difference between a twelve and a one-and-two just standing near one another? How do people recognize a solitary number? This is a darned silly thing to wonder so there’s probably a good web comic about it.

John Hambrock’s The Brilliant Mind of Edison Lee for the 4th has Edison forecast the outcome of a basketball game. I can’t imagine anyone really believing in forecasting the outcome, though. The elements of forecasting a sporting event are plausible enough. We can suppose a game to be a string of events. Each of them has possible outcomes. Some of them score points. Some block the other team’s score. Some cause control of the ball (or whatever makes scoring possible) to change teams. Some take a player out, for a while or for the rest of the game. So it’s possible to run through a simulated game. If you know well enough how the people playing do various things? How they’re likely to respond to different states of things? You could certainly simulate that.

But all sorts of crazy things will happen, one game or another. Run the same simulation again, with different random numbers. The final score will likely be different. The course of action certainly will. Run the same simulation many times over. Vary it a little; what happens if the best player is a little worse than average? A little better? What if the referees make a lot of mistakes? What if the weather affects the outcome? What if the weather is a little different? So each possible outcome of the sporting event has some chance. We have a distribution of the possible results. We can judge an expected value, and what the range of likely outcomes is. This demands a lot of data about the players, though. Edison Lee can have it, I suppose. The premise of the strip is that he’s a genius of unlimited competence. It would be more likely to expect for college and professional teams.

Brian Basset’s Red and Rover for the 4th uses arithmetic as the homework to get torn up. I’m not sure it’s just a cameo appearance. It makes a difference to the joke as told that there’s division and long division, after all. But it could really be any subject.

I’m figuring to get to the letter ‘W’ in my Fall 2018 Mathematics A To Z glossary for Tuesday. Reading the Comics posts this week. And I also figure there should be two more When posted, they’ll be at this link.

## Reading the Comics, November 29, 2018: Closing Out November Edition

Today, I get to wrap up November’s suggested discussion topics as prepared by Comic Strip Master Command.

Mark Tatulli’s Lio for the 28th features a cameo for mathematics. At least mathematics class. It’s painted as the most tedious part of the school day. I’m not sure this is quite right for Lio as a character. He’s clever in a way that I think harmonizes well with how mathematics brings out universal truths. But there is a difference between mathematics and mathematics class, of course.

Tom Toles’s Randolph Itch, 2am for the 28th shows how well my resolution to drop the strip from my rotation here has gone. I don’t seem to have found it worthy of mention before, though. It plays on the difference between a note of money, the number of units of currency that note represents, and between “zero” and “nothing”. Also I’m enchanted now by the idea that maybe some government might publish a zero-dollar bill. At least for the sake of movie and television productions that need realistic-looking cash.

In the footer joke Randolph mentions how you can never have enough zeroes. Yes, but I’d say that’s true of twenties, too. There is a neat sense in which this is true for working mathematicians, though. At least for those doing analysis. One of the reliable tricks that we learn to do in analysis is to “add zero” to a quantity. This is, literally, going from some expression that might be, say, “a – b” to “a + 0 – b”, which of course has the same value. The point of doing that is that we know other things equal to zero. For example, for any number L, “-L + L” is zero. So we get the original expression from “a + 0 – b” over to “a – L + L – b”. And that becomes useful is you picked L so that you know something about “a – L” and about “L – b”. Because then it tells you something about “a – b” that you didn’t know before. Picking that L, and showing something true about “a – L” and “L – b”, is the tricky part.

Dan Collins’s Looks Good On Paper for the 29th is back with another Möbius Strip comic strip. Last time it was presented as the “Möbius Trip”, a looping journey. This time it’s a comic strip proper. If this particular Looks Good On Paper has run before I don’t seem to have mentioned it. Unlike the “Möbius Trip” comic, this one looks more clearly like it actually is a Möbius strip.

The Dumpties in the comic strip are presented as getting nauseated at the strange curling around. It’s good sense for the comic-in-the-comic, which just has to have something happen and doesn’t really need to make sense. But there is no real way to answer where a Möbius strip wraps around itself. I mean, we can declare it’s at the left and right ends of the strip as we hold it, sure. But this is an ad hoc placement. We can roll the belt along a little bit, not changing its shape, but changing the points where we think of the strip as turning over.

But suppose you were a flat creature, wandering a Möbius strip. Would you have any way to tell that you weren’t on the plane? You could, but it takes some subtle work. Like, you could try drawing shapes. These let you count a thing called the Euler Characteristic, which relates the numer of vertices, edges, and faces of a polyhedron. The Euler Characteristic for a Möbius strip is the same as that for a Klein bottle, a cylinder, or a torus. You could try drawing regions, and coloring them in, calling on the four-color map theorem. (Here I want just to mention the five-color map theorem, which is as these things go easy to prove.) A map on the plane needs at most four colors to have no neighboring territories share a color along an edge. (Territories here are contiguous, and we don’t count territories meeting at only a point as sharing an edge.) Same for a sphere, which is good for we folks who have the job of coloring in both globes and atlases. It’s also the same for a cylinder. On a Möbius strip, this number is six. On a torus, it’s seven. So we could tell, if we were on a Möbius strip, that we were. It can be subtle to prove, is all.

All of my regular Reading the Comics posts should all be at this link. The next in my Fall 2018 Mathematics A To Z glossary should be posted Tuesday. I’m glad for it if you do come around and read again.

## Reading the Comics, November 27, 2018: Multiplication Edition

Last week Comic Strip Master Command sent out just enough on-theme comics for two essays, the way I do them these days. The first half has some multiplication in two of the strips. So that’s enough to count as a theme for me.

Aaron Neathery’s Endtown for the 26th depicts a dreary, boring school day by using arithmetic. A lot of times tables. There is some credible in-universe reason to be drilling on multiplication like this. The setting is one where the characters can’t expect to have computers available. That granted, I’m not sure there’s a point to going up to memorizing four times 27. Going up to twelve-times seems like enough for common uses. For multiplying two- and longer-digit numbers together we usually break the problem up into a string of single-digit multiplications.

There are a handful of bigger multiplications that can make your life easier to know, like how four times 25 is 100. Or three times 33 is pretty near 100. But otherwise? … Of course, the story needs the class to do something dull and seemingly pointless. Going deep into multiplication tables communicates that to the reader quickly.

Thaves’s Frank and Ernest for the 26th is a spot of wordplay. Also a shout-out to my friends who record mathematics videos for YouTube. It is built on the conflation between the ideas of something multiplying and the amount of something growing. It’s easy to see where the idea comes from; just keep hitting ‘x 2’ on a calculator and the numbers grow excitingly fast. You get even more exciting results with ‘x 3’ or ‘x π’. But multiplying by 1 is still multiplication. As is multiplying by a number smaller than 1. Including negative numbers. That doesn’t hurt the joke any. That multiplying two things together doesn’t necessarily give you something larger is a consideration when you’re thinking rigorously about what multiplication can do. It doesn’t have to be part of normal speech.

John Hambrock’s The Brilliant Mind of Edison Lee for the 27th uses the form of a word problem to show off Edison’s gluttony. Edison tries to present it as teaching. We all have rationalizations for giving in to our appetites.

Nate Frakes’s Break of Day for the 27th is the anthropomorphic numerals joke for the week. I don’t know that there’s anything in the other numerals being odds rather than evens, or a mixture of odds and evens. It might just be that they needed to be anything but 1.

All of my regular Reading the Comics posts should all be at this link. The next in my Fall 2018 Mathematics A To Z glossary should be posted Tuesday. I’m glad for it if you do come around and read again.

## Reading the Comics, November 24, 2018: Origins Edition

I’m not sure there is a theme to the back half of last week’s mathematically-based comic strips. If there is, it’s about showing some origins of things. I’ll go with that title, then.

Bill Holbrook’s On The Fastrack for the 21st is another in the curious thread of strips about Fi talking about mathematics. She’s presented as doing a good job inspiring kids to appreciate mathematics as a fun, exciting, interesting thing to think about. It’s good work. And I hope this does not sound like I am envious of a more successful, if fictional, mathematics popularizer. But I don’t see much in the strip of her doing this side job well. That is, of making the case that mathematics is worth the time spent on it. That’s a lot to ask given the confines of a syndicated daily newspaper comic strip, yes. What we can expect is some hint of what the actual good argument would look like. But this particular day’s strip rings false to me, for example. I don’t see how “here’s some pizza — but first, here’s a pop quiz” makes mathematics look as something other than a chore.

Pizza area offers many ways into mathematical ideas. How the area depends on the size of the pizza, for example. How the area depends on the shape, even independently of the size. How to slice a pizza fairly, especially if it’s not to be between four or six or eight people. What is the strangest shape you could make that would give people equal areas? Just the way slices intersect at angles inspires neat little geometry problems. How you might arrange toppings opens up symmetries and tilings, which are surprisingly big areas of mathematics. Setting problems on a pizza gives them a tangibility that could help capture young minds, surely. But I can’t make myself believe that this is a conversation to have when the pizza is entering the room.

Mike Peters’s Mother Goose and Grimm for the 22nd is a lottery joke. So if we suppose this was written about the last time the Powerball jackpot reached a half-billion dollars we can work out how far ahead of publication Mike Peters is working. One solid argument against ever buying a lottery ticket is, as Grimm notes, that you have zero chance of winning. (I’m open to an argument based on expectation value. And even more, I don’t object to people spending a reasonable bit of disposable income “foolishly”.) Mother Goose argues that her chances are vastly worse if she doesn’t buy a ticket. This is true. Are her chances “astronomically” worse? … That depends. A one in three hundred million chance (to use, roughly, the Powerball odds) is so small that it won’t happen to you. Is that any different than a zero in three hundred million chance [*]? Or than a six in three hundred million chance? In any case it won’t happen to you.

[*] Do you actually have zero chance of winning if you don’t have a ticket? I say no, you don’t. Someone might give you a winning ticket. Maybe you find one as a bookmark in a library book. Maybe you find it on the street and figure, what the heck, I’ll check. Unlikely? Sure. But impossible? Hardly.

Johnny Hart’s Back to BC for the 22nd has the form of the world’s oldest story problem. It could also be a joke about the discovery of the concept of zero and the struggle to understand it as a number. Given that clams are used as currency in the BC setting it also shows how finance has driven mathematical development. So the strip actually packs a fair bit of stuff into two panels. … And I’ll admit I’m not quite sure the joke parses, but if you read it quickly it looks like a good enough joke.

Johnny Hart’s Back to BC for the 24th is a more obvious joke. And it’s built on the learning abilities of animals, and the number sense of animals. A large animal stomping a foot evokes, to me at least, Clever Hans. This is a horse presented in the early 20th century as being able to actually do arithmetic. The horse would be given a question and would stop his hoof enough times to get to the right answer. However good the horse’s number sense might be, he had quite good behavioral sense. It turned out — after brilliant and pioneering work in animal cognition — that Hans was observing his trainer’s body language. When Wilhelm von Osten was satisfied that there’d been the right number of stomps, the horse stopped. This is sometimes presented as Hans merely’ taking subconscious cues from his trainer. But consider how carefully the horse must be observing an animal with a very different body, and how it must have understood cues of satisfaction. I can’t call that mere’. And the work of tracking down a signal that von Osten himself did not know he was sending (and, apparently, never accepted that he did) is also amazing. It serves as a reminder how hard biologists and zoologists have to work.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th gives a bit of Dad History about perspective. And, particularly, why artists didn’t seem to use it much before the 16th century. It gets more blatantly tied to mathematics by pointing out how it took ten thousand years of civilization to get Cartesian coordinates. We can argue about how many years civilization has been around. But it does seem strange that we went along for certainly the majority of that time without Cartesian coordinates. They seem so obvious it’s almost hard to not think of them. Many good ideas have such a legacy.

It’s easy to say why older pictures didn’t use perspective, though. For the most part, artists didn’t think perspective gave them something they wanted to show. Ancient peoples knew of perspective. It’s not as if ancient peoples were any dumber than we are, or any less able to look at square tiles held at different angles and at different distances. But we can convey information about the importance of things, or the flow of action of things, using position and relative size. That can be more important than showing that yes, an artist is aware that a square building far away looks small.

I’m less sure what I know about the history of coordinate systems, though, and particularly why it took until René Descartes to describe them. We have a legend of Descartes laying in bed, watching a fly on the tiled ceiling, and realizing he could describe where the fly was by what row and column of tile it was on. (In the past I have written this as though it happened. In writing this essay I went looking for a primary source and found nobody seems to have one. I shall try not to pass it on again without being very clear that it is just a legend.) But there have been tiled floors and walls and ceilings for a very long time. There have been flies even longer. Why didn’t anyone notice this?

One answer may be that they did. We just haven’t heard about it, because it was found by someone who didn’t catch the interest of a mathematical community. There’s likely a lot of such lost mathematics out there. But still, why not? Wouldn’t anyone with a mathematical inclination see that this is plainly a great discovery? And maybe not. What made Cartesian coordinates great was the realization that arithmetic and geometry, previously seen as separate liberal arts, were duals. A problem in one had an expression as a problem in the other. If you don’t make that connection, then Cartesian coordinates don’t solve any problems you have. They’re just a new way to index things you didn’t need indexed. So that would slow down using them any.

All of my regular Reading the Comics posts should all be at this link. Tomorrow should see the posting of my next my Fall 2018 Mathematics A To Z essay. And there’s still time to put in requests for the last half-dozen letters of the alphabet.

## Reading the Comics, November 20, 2018: What Mathematics Is For Edition

The first half of last week’s comics offered another bunch of chances to think about what mathematics is for. Before I do get into all that, though, may I mention the most recent update of Gregory Taylor’s serial:

It does conclude with a vote about the next direction to take. So it’s a good chance for people who like to see authors twisting to their audience’s demands.

Mort Walker and Dik Browne’s Hi and Lois for the 23rd of May, 1961 builds off a major use of arithmetic. Budgeting doesn’t get much attention from mathematicians. I suppose it seems to us like all the basic problems are solved: adding? Subtracting? Multiplication? All familiar things. Especially now with decimal currency. There are great unsolved problems in mathematics, but they get into specialized areas of financial mathematics and just don’t matter for ordinary household budgeting.

Hi comes across a bit harsh here. I’m going to suppose he was taken so by surprise by Lois’s problem that he spoke without thinking.

Scott Hilburn’s The Argyle Sweater for the 19th is the anthropomorphic numerals strip for the week. With the title of “improper fractions” it’s wordplay on the common meaning for a mathematical term. Two times over, come to it. That negative refers to a class of numbers as well as disapproval of something is ordinary enough. I’ve mentioned it, I estimate, 840 times this month alone.

Jokes about the technical and common meanings of “improper” are rarer. In a proper fraction, the numerator is a smaller number than the denominator. In an improper fraction, we don’t count on that. I remember a modest bit of time in elementary and middle school working on converting improper fractions into mixed fractions — a whole number plus a proper fraction. And also don’t remember anyone caring about that after calculus. In most arithmetic work, there’s not much that’s easier about “1 + 1/2” than about “3/2”. The one major convenience “1 + 1/2” has is that it’s easy to tell at a glance how big the number is. It’s not mysterious how big a number 3/2 is, but that’s because of long familiarity. If I asked you whether 54/17 or 46/13 was the larger number, you’d be fairly stumped and maybe cranky. So there’s not much reason to worry about improper fractions while you’re doing work. For the final presentation of an answer, proper or mixed fractions may well be better.

Whoever colored that minus symbol before the 5 screwed up and confused the joke. Syndicated cartoonists give precise coloring instructions for Sunday strips. But many of them don’t, or aren’t able to, give coloring instructions for weekday strips like this. And mistakes like that are the unfortunate result.

Pascal Wyse and Joe Berger’s Berger and Wyse for the 19th features a sudoku appearance. It’s labelled a diversion, and so it is, as many mathematics and logic puzzles will be. The lone commenter at GoComics claims to have solved the puzzle, so I will suppose they’re being honest about this.

Brian Fies’s Mom’s Cancer for the 19th I have mentioned before, although not since I started including images for all mentioned comics. It’s set a moment when treatment for Mom’s cancer has been declared a great success.

The trouble is, as Feis lays out, volume is three-dimensional. We are pretty good at measuring the length, or at least the greatest width of something. You might call that the “characteristic length”. A linear dimension. But volume scales as the cube of this characteristic length. And the sad thing is that 0.8 times 0.8 times 0.8 is, roughly, 0.5. This means that the characteristic length dropping by 20% drops the volume by 50%. Or, as Feis is disappointed to see in this strip and its successor, the great news of a 50% reduction in the turmor’s mass is that it’s just 20% less big in every direction. It doesn’t look like enough.

Bill Holbrook’s On The Fastrack for the 20th presents one of Fi’s seminars about why mathematics is a good thing. The offscreen student’s question about why one should learn mathematics goes unanswered. As often happens the question is presented as though it’s too absurd to deserve answering. The questioner is conflating “mathematics” with “calculating arithmetic”, yes. And a computer will be better at these calculations. A related question, sometimes asked (and rarely on-topic for my essays here), is why one needs to learn any specific facts when a computer is so much better at finding them.

Knowing facts is not understanding them, no. But it is hard to understand a thing without knowing facts. More, without loving the knowing of facts. If we don’t need to be good at calculating, we do still need to know what to have calculated. And why to calculate that instead of something else. In calculating we can learn things of great beauty. And some of us do go on to mathematics which cannot be calculated. There is software that will do very well at computing, say, the indefinite integral of functions. I don’t know of any that will even start on a problem like “find the kernel of this ring”. But these are problems we see, and think interesting, because our experience in arithmetic trains us to notice them. Perhaps there is new interesting mathematics that we would notice if we didn’t have preconceptions set by times tables and long division. But it is hard to believe that we can’t find it because we’re not ignorant enough. I wouldn’t risk it.

This and more Reading the Comics posts should all be at this link.

And for the rest of the calendar year my Fall 2018 Mathematics A To Z should continue posting new essays. I’m still looking for topics for the last half-dozen letters of the alphabet. Give your mathematics term a try.

## Reading the Comics, November 16, 2018: The Rest Of The Week Edition

After that busy start last Sunday, Comic Strip Master Command left only a few things for the rest of the week. Here’s everything that seemed worthy of some comment to me:

Alex Hallatt’s Arctic Circle for the 12th is an arithmetic cameo. It’s used as the sort of thing that can be tested, with the straightforward joke about animal testing to follow. It’s not a surprise that machines should be able to do arithmetic. We’ve built machines for centuries to do arithmetic. Literally; Wilhelm Gottfried Leibniz designed and built a calculating machine able to add, subtract, multiply, and divide. This accomplishment from one of the founders of integral calculus is a potent reminder of how much we can accomplish if we’re supposed to be writing instead. (That link is to Robert Benchley’s classic essay “How To Get Things Done”. It is well worth reading, both because it is funny and because it’s actually good, useful advice.)

But it’s also true that animals do know arithmetic. At least a bit. Not — so far as we know — to the point they ponder square roots and such. But certainly to count, to understand addition and subtraction roughly, to have some instinct for calculations. Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics is a fascinating book about this. I’m only wary about going deeper into the topic since I don’t know a second (and, better, third) pop book touching on how animals understand mathematics. I feel more comfortable with anything if I’ve encountered it from several different authors. Anyway it does imply the possibility of testing a polar bear’s abilities at arithmetic, only in the real world.

Berkeley Breathed’s Bloom County rerun for the 13th has another mathematics cameo. Geometry’s a subject worthy of stoking Binkley’s anxieties, though. It has a lot of definitions that have to be carefully observed. And while geometry reflects the understanding we have of things from moving around in space, it demands a precision that we don’t really have an instinct for. It’s a lot to worry about.

Terry Border’s Bent Objects for the 15th is our Venn Diagram joke for the week. I like this better than I think the joke deserves, probably because it is done in real materials. (Which is the Bent Objects schtick; it’s always photographs of objects arranged to make the joke.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th is a joke on knowing how far to travel but not what direction. Normal human conversations carry contextually reasonable suppositions. Told something is two miles away, it’s probably along the major road you’re on, or immediately nearby. I’d still ask for clarification told something was “two miles away”. Two blocks, I’d let slide, on the grounds that it’s no big deal to correct a mistake.

Still, mathematicians carry defaults with them too. They might be open to a weird, general case, certainly. But we have expectations. There’s usually some obvious preferred coordinate system, or directions. If it’s important that we be ready for alternatives we highlight that. We specify the coordinate system we want. Perhaps we specify we’re taking that choice “without loss of generality”, that is, without supposing some other choice would be wrong.

I noticed the mathematician’s customized plate too. “EIPI1” is surely a reference to the expression $e^{\imath \pi} + 1$. That sum, it turns out, equals zero. It reflects this curious connection between exponentiation, complex-valued numbers, and the trigonometric functions. It’s a weird thing to know is true, and it’s highly regarded in certain nerd circles for that weirdness.

Hilary Price’s Rhymes With Orange for the 16th features a what-are-the-odds sort of joke, this one about being struck by a bolt from the sky. Lightning’s the iconic bolt to strike someone, and be surprising about it. Fabric would be no less surprising, though. And there’s no end of stories of weird things falling from the skies. It’s easier to get stuff into the sky than you might think, and there are only a few options once that’s happened.

And as ever, all my Reading the Comics posts should all be at this link.

Through the end of December my Fall 2018 Mathematics A To Z continues. I’m still open for topics to discuss from the last half-dozen letters of the alphabet. Even if someone’s already given a word for some letter, suggest something anyway. You might inspire me in good ways.

## Reading the Comics, November 11, 2018: November 11, 2018 Edition

There were just enough mathematically-themed comic strips last week to make two editions for this coming week. All going well I’ll run the other half on either Wednesday or Thursday. There is a point that isn’t quite well, which is that one of the comics is in dubious taste. I’ll put that at the end, behind a more specific content warning. In the meanwhile, you can find this and hundreds of other Reading the Comics posts at this link.

Thaves’s Frank and Ernest for the 11th is wordplay, built on the conflation of “negative” as in numbers and “negative” as in bad. I’m not sure the two meanings are unrelated. The word ‘negative’ itself derives from the Latin word meaning to deny, which sounds bad. It’s easy to see why the term would attach to what we call negative numbers. A number plus its negation leaves us zero, a nothing. But it does make the negative numbers sound like bad things to have around, or to have to deal with. The convention that a negative number is less than zero implies that the default choice for a number is one greater than zero. And the default choice is usually seen as the good one, with everything else a falling-away. Still, -7 is as legitimate a number as 7 is; it’s we who think one is better than another.

J C Duffy’s Lug Nuts for the 11th has the Dadaist panel present prime numbers as a way to communicate. I suspect Duffy’s drawing from speculations about how to contact alien intelligences. One problem with communicating with the truly alien is how to recognize there is a message being sent. A message too regular will look like a natural process, one conveying no more intelligence than the brightness which comes to most places at dawn and darkness coming at sunset. A message too information-packed, peculiarly, looks like random noise. We need an intermediate level. A signal that it’s easy to receive, and that is too hard to produce by natural processes.

Prime numbers seem like a good compromise. An intelligence that understands arithmetic will surely notice prime numbers, or at least work out quickly what’s novel about this set of numbers once given them. And it’s hard to imagine an intelligence capable of sending or receiving interplanetary signals that doesn’t understand arithmetic. (Admitting that yes, we might be ruling out conversational partners by doing this.) We can imagine a natural process that sends out (say) three pulses and then rests, or five pulses and rests. Or even draws out longer cycles: two pulses and a rest, three pulses and a rest five pulses and a rest, and then a big rest before restarting the cycle. But the longer the string of prime numbers, the harder it is to figure a natural process that happens to hit them and not other numbers.

We think, anyway. Until we contact aliens we won’t really know what it’s likely alien contact would be like. Prime numbers seem good to us, but — even if we stick to numbers — there’s no reason triangular numbers, square numbers, or perfect numbers might not be as good. (Well, maybe not perfect numbers; there aren’t many of them, and they grow very large very fast.) But we have to look for something particular, and this seems like a plausible particularity.

Charles Schulz’s Peanuts Begins for the 11th is an early strip, from the days when Lucy would look to Charlie Brown for information. And it’s a joke built on conflating ‘zero’ with ‘nothing’. Lucy’s right that zero times zero has to be something. That’s how multiplication works. That the number zero is something? That’s a tricky concept. I think being mathematically adept can blind one to how weird that is. If you’re used to how zero is the amount of a thing you have to have nothing of that thing, then we start to see what’s weird about it.

But I’m not sure the strip quite sets that up well. I think if Charlie Brown had answered that zero times zero was “nothing” it would have been right (or right enough) and Lucy’s exasperation would have flowed more naturally. As it is? She must know that zero is “nothing”; but then why would she figure “nothing times nothing” has to be something? Maybe not; it would have left Charlie Brown less reason to feel exasperated or for the reader to feel on Charlie Brown’s side. Young Lucy’s leap to “three” needs to be at least a bit illogical to make any sense.

Now to the last strip and the one I wanted to warn about. It alludes to gun violence and school shootings. If you don’t want to deal with that, you’re right. There’s other comic strips to read out there. And this for a comic that ran on the centennial of Armistice Day, which has to just be an oversight in scheduling the (non-plot-dependent) comic.

## Reading the Comics, November 9, 2018: Standing For Things Edition

There was something in common in two of the last five comic strips worth attention from last week. That’s good enough to give the essay its name.

Greg Cravens’s The Buckets for the 8th showcases Toby discovering the point of letters in algebra. It’s easy to laugh at him being ignorant. But the use of letters this way is something it’s easy to miss. You need first to realize that we don’t need to have a single way to represent a number. Which is implicit in learning, say, that you can write ‘7’ as the Roman numeral ‘VII’ or so, but I’m not sure that’s always clear. And realizing that you could use any symbol to write out ‘7’ if you agree that’s what the symbol means? That’s an abstraction tossed onto people who often aren’t really up for that kind of abstraction. And that we can have a symbol for “a number whose identity we don’t yet know”? Or even “a number whose identity we don’t care about”? Don’t blame someone for rearing back in confusion at this.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th talks about vectors and scalars. And about the little ways that instructors in one subject can sabotage one another. In grad school I was witness to the mathematics department feeling quite put-upon by the engineering departments, who thought we were giving their students inadequate calculus training. Meanwhile we couldn’t figure out what they were telling students about calculus except that it was screwing up their understanding.

To a physicist, a vector is a size and a direction together. (At least until they get seriously into mathematical physics when they need a more abstract idea.) A scalar is a number. Like, a real-valued number such as ‘4’. Maybe a complex-valued number such as ‘4 + 6i’. Vectors are great because a lot of physics problems become easier when thought of in terms of directions and amounts in that direction.

A mathematician would start out with vectors and scalars like that. But then she’d move into a more abstract idea. A vector, to a mathematician, is a thing you can add to another vector and get a vector out. A scalar is something that’s not a vector but that, multiplied by a vector, gets you a vector out. This sounds circular. But by defining ‘vector’ and ‘scalar’ in how they interact with each other we get a really sweet flexibility. We can use the same reasoning — and the same proofs — for lots of things. Directions, yes. But also matrices, and continuous functions, and probabilities of events, and more. That’s a bit much to give the engineering student who’s trying to work out some problem about … I don’t know. Whatever they do over in that department. Truss bridges or electrical circuits or something.

Mark Leiknes’s Cow and Boy for the 9th is really about misheard song lyrics, a subject that will never die now that we don’t have the space to print lyrics in the album lining anymore, or album linings. But it has a joke resonant with that of The Buckets, in supposing that algebra is just some bunch of letters mixed up with numbers. And Cow and Boy was always a strip I loved, as baffling as it might be to a casual reader. It had a staggering number of running jokes, although not in this installment.

Greg Evans’s Luann Againn for the 9th shows Brad happy to work out arithmetic when it’s for something he’d like to know. The figure Luan gives is ridiculously high, though. If he needs 500 hairs, and one new hair grows in each week, then that’s a little under ten years’ worth of growth. Nine years and a bit over seven months to be exact. If a moustache hair needs to be a half-inch long, and it grows at 1/8th of an inch per month, then it takes four months to be sufficiently long. So in the slowest possible state it’d be nine years, eleven months. I can chalk Luann’s answer up to being snidely pessimistic about his hair growth. But his calculator seems to agree and that suggests something went wrong along the way.

John Zakour and Scott Roberts’s Maria’s Day for the 9th is a story problem joke. It looks to me like a reasonable story problem, too: the distance travelled and the speed are reasonable, and give sensible numbers. The two stops add a bit of complication that doesn’t seem out of line. And the kid’s confusion is fair enough. It takes some experience to realize that the problem splits into an easy part, a hard part, and an easy part. The first easy part is how long the stops take all together. That’s 25 minutes. The hard part is realizing that if you want to know the total travel time it doesn’t matter when the stops are. You can find the total travel time by adding together the time spent stopped with the time spent driving. And the other easy part is working out how long it takes to go 80 miles if you travel at 55 miles per hour. That’s just a division. So find that and add to it the 25 minutes spent at the two stops.

The various Reading the Comics posts should all be at this link. Essays which discuss The Buckets are at this link. The incredibly many essays mentioning Saturday Morning Breakfast Cereal are at this link. Essays which mention Cow and Boy are at this link. Essays inspired in part by Luann, both the current-day and the vintage 1990 run, are at this link. The credibly many essays mentioning Maria’s Day are at this link.

And through the end of December my Fall 2018 Mathematics A-To-Z should have two new posts a week. You might like some of them.

## Reading the Comics, November 7, 2018: Shorthand and Reruns Edition

There’s two types of comics for the second of last week’s review. There’s some strips that are reruns. There’s some that just use mathematics as a shorthand for something else. There’s four strips in all.

John Deering’s Strange Brew for the 6th uses mathematics as shorthand for demonstrating intelligence. There’s no making particular sense out of the symbols, of course. And I’d think it dangerous that Lucky seems to be using both capital X and lowercase x in the same formula. There’s often times one does use the capital and lowercase versions of a letter in a formula. This is usually something like “x is one element of the set X, which is all the possible candidates for some thing”. In that case, you might get the case wrong, but context would make it clear what you meant. But, yes, sometimes there’s no sensible alternative and then you have to be careful.

Randy Glasbergen’s Glasbergen Cartoons for the 6th uses mathematics as shorthand for a hard subject. It’s certainly an economical notation. Alas, you don’t just learn from your mistakes. You learn from comparing your mistakes to a correct answer. And thinking about why you made the mistakes you did, and how to minimize or avoid those mistakes again.

So how would I do this problem? Well, carrying out the process isn’t too hard. But what do I expect the answer to be, roughly? To me, I look at this and reason: 473 is about 500. So 473 x 17 is about 500 x 17. 500 x 17 is 1000 times eight-and-a-half. So start with “about 8500”. That’s too high, obviously. I can do better. 8500 minus some correction. What correction? Well, 473 is roughly 500 minus 25. So I’ll subtract 25 times 17. Which isn’t hard, because 25 times 4 is 100. So 25 times 17? That’s 25 times 16 plus 25 times 1. 25 times 16 is 100 times 4. So 25 times 17 is 425. 8500 minus 425 is 8075. I’m still a bit high, by 2 times 17. 2 times 17 is 34. So subtract 34 from 8075: it should be about 8041.

John Zakour and Scott Roberts’s Maria’s Day for the 7th is a joke built on jargon. Every field has its jargon. Some of it will be safely original terms: people’s names (“Bessel function”) or synthetic words (“isomorphism”) that can’t be easily confused with everyday language. But some of it will be common terms given special meaning. “Right” angles and “right” triangles. “Normal” numbers. “Group”. “Right” as a description for angles and triangles goes back a long way, at least to — well, Merriam-Webster.com says 15th century. But EtymologyOnline says late 14th century. Neither offers their manuscripts. I’ll chalk it up to differences in how they interpret the texts. And possibly differences in whether they would count, say, a reference to “a right angle” written in French or German rather than in English directly.

Richard Thompson’s Richard’s Poor Almanac for the 7th has been run before. It references the Infinite Monkey Theorem. The monkeys this time around write up a treasury of Western Literature, not merely the canon of Shakespeare. That’s at least as impressive a feat. Also, while this is a rerun — sad to say Richard Thompson died in 2016, and was forced to retire from drawing before that — his work was fantastic and deserves attention.

This and every Reading the Comics post should be at this link. Essays discussing topics raised by Strange Brew are at this link. The essays discussing Glasbergen Cartoons are at this link. Essays which mention Maria’s Day, are at this link. And essays featuring Richard’s Poor Almanac are at this link.

My Fall 2018 Mathematics A-To-Z averages two new posts a week, through the end of December. Thanks again for reading.