## 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, April 6, 2020: My Perennials Edition

As much as everything is still happening, and so much, there’s still comic strips. I’m fortunately able here to focus just on the comics that discuss some mathematical theme, so let’s get started in exploring last week’s reading. Worth deeper discussion are the comics that turn up here all the time.

Lincoln Peirce’s Big Nate for the 5th is a casual mention. Nate wants to get out of having to do his mathematics homework. This really could be any subject as long as it fit the word balloon.

John Hambrock’s The Brilliant Mind of Edison Lee for the 6th is a funny-answers-to-story-problems joke. Edison Lee’s answer disregards the actual wording of the question, which supposes the group is travelling at an average 70 miles per hour. The number of stops doesn’t matter in this case.

Mark Anderson’s Andertoons for the 6th is the Mark Anderson’s Andertoons for the week. In it Wavehead gives the “just use a calculator” answer for geometry problems.

Not much to talk about there. But there is a fascinating thing about perimeters that you learn if you go far enough in Calculus. You have to get into multivariable calculus, something where you integrate a function that has at least two independent variables. When you do this, you can find the integral evaluated over a curve. If it’s a closed curve, something that loops around back to itself, then you can do something magic. Integrating the correct function on the curve around a shape will tell you the enclosed area.

And this is an example of one of the amazing things in multivariable calculus. It tells us that integrals over a boundary can tell us something about the integral within a volume, and vice-versa. It can be worth figuring out whether your integral is better solved by looking at the boundaries or at the interiors.

Heron’s Formula, for the area of a triangle based on the lengths of its sides, is an expression of this calculation. I don’t know of a formula exactly like that for the perimeter of a quadrilateral, but there are similar formulas if you know the lengths of the sides and of the diagonals.

Richard Thompson’s Cul de Sac rerun for the 6th sees Petey working on his mathematics homework. As with the Big Nate strip, it could be any subject.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th depicts, fairly, the sorts of things that excite mathematicians. The number discussed here is about algorithmic complexity. This is the study of how long it takes to do an algorithm. How long always depends on how big a problem you are working on; to sort four items takes less time than sorting four million items. Of interest here is how much the time to do work grows with the size of whatever you’re working on.

The mathematician’s particular example, and I thank dtpimentel in the comments for finding this, is about the Coppersmith–Winograd algorithm. This is a scheme for doing matrix multiplication, a particular kind of multiplication and addition of squares of numbers. The squares have some number N rows and N columns. It’s thought that there exists some way to do matrix multiplication in the order of N2 time, that is, if it takes 10 time units to multiply matrices of three rows and three columns together, we should expect it takes 40 time units to multiply matrices of six rows and six columns together. The matrix multiplication you learn in linear algebra takes on the order of N3 time, so, it would take like 80 time units.

We don’t know the way to do that. The Coppersmith–Winograd algorithm was thought, after Virginia Vassilevska Williams’s work in 2011, to take something like N2.3728642 steps. So that six-rows-six-columns multiplication would take slightly over 51.796 844 time units. In 2014, François le Gall found it was no worse than N2.3728639 steps, so this would take slightly over 51.796 833 time units. The improvement doesn’t seem like much, but on tiny problems it never does. On big problems, the improvement’s worth it. And, sometimes, you make a good chunk of progress at once.

I’ll have some more comic strips to discuss in an essay at this link, sometime later this week. Thanks for reading.

## Reading the Comics, April 4, 2020: Ruling Things Out Edition

This little essay should let me wrap up the rest of the comic strips from the past week. Most of them were casual mentions. At least I thought they were when I gathered them. But let’s see what happens when I actually write my paragraphs about them.

Darrin Bell and Theron Heir’s Rudy Park rerun for the 1st of April uses arithmetic as emblematic of things which we know with certainty to be true.

Thaves’s Frank and Ernest for the 2nd is a bit of wordplay, having Euclid and Galileo talking about parallel universes. I’m not sure that Galileo is the best fit for this, but I’m also not sure there’s another person connected who could be named. It’d have to be a name familiar to an average reader as having something to do with geometry. Pythagoras would seem obvious, but the joke is stronger if it’s two people who definitely did not live at the same time. Did Euclid and Pythagoras live at the same time? I am a mathematics Ph.D. and have been doing pop mathematics blogging for nearly a decade now, and I have not once considered the question until right now. Let me look it up.

It doesn’t make any difference. The comic strip has to read quickly. It might be better grounded to post Euclid meeting Gauss or Lobachevsky or Euler (although the similarity in names would be confusing) but being understood is better than being precise.

Stephan Pastis’s Pearls Before Swine for the 2nd is a strip about the foolhardiness of playing the lottery. And it is foolish to think that even a \$100 purchase of lottery tickets will get one a win. But it is possible to buy enough lottery tickets as to assure a win, even if it is maybe shared with someone else. It’s neat that an action can be foolish if done in a small quantity, but sensible if done in enough bulk.

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. Wavehead has made a bunch of failed attempts at subtracting seven from ten, but claims it’s at least progress that some thing have been ruled out. I’ll go along with him that there is some good in ruling out wrong answers. The tricky part is in how you rule them out. For example, obvious to my eye is that the correct answer can’t be more than ten; the problem is 10 minus a positive number. And it can’t be less than zero; it’s ten minus a number less than ten. It’s got to be a whole number. If I’m feeling confident about five and five making ten, then I’d rule out any answer that isn’t between 1 and 4 right away. I’ve got the answer down to four guesses and all I’ve really needed to know is that 7 is greater than five but less than ten. That it’s an even number minus an odd means the result has to be odd; so, it’s either one or three. Knowing that the next whole number higher than 7 is an 8 says that we can rule out 1 as the answer. So there’s the answer, done wholly by thinking of what we can rule out. Of course, knowing what to rule out takes some experience.

Mark Parisi’s Off The Mark for the 4th is roughly the anthropomorphic numerals joke for the week. It’s a dumb one, but, that’s what sketchbooks are for.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is the Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th for the week. It shows in joking but not wrong fashion a mathematical physicist’s encounters with orbital mechanics. Orbital mechanics are a great first physics problem. It’s obvious what they’re about, and why they might be interesting. And the mathematics of it is challenging in ways that masses on springs or balls shot from cannons aren’t.

A few problems are very easy, like, one thing in circular orbit of another. A few problems are not bad, like, one thing in an elliptical or hyperbolic orbit of another. All our good luck runs out once we suppose the universe has three things in it. You’re left with problems that are doable if you suppose that one of the things moving is so tiny that it barely exists. This is near enough true for, for example, a satellite orbiting a planet. Or by supposing that we have a series of two-thing problems. Which is again near enough true for, for example, a satellite travelling from one planet to another. But these is all work that finds approximate solutions, often after considerable hard work. It feels like much more labor to smaller reward than we get for masses on springs or balls shot from cannons. Walking off to a presumably easier field is understandable. Unfortunately, none of the other fields is actually easier.

Pythagoras died somewhere around 495 BC. Euclid was born sometime around 325 BC. That’s 170 years apart. So Pythagoras was as far in Euclid’s past as, oh, Maria Gaetana Agnesi is to mine.

I did a little series looking into orbital mechanics, not necessarily ones that look like planetary orbits, a couple years ago. You might enjoy that. And I figure to have more mathematically-themed comic strips in the near future. Thanks for reading.

## Reading the Comics, March 25, 2020: Regular Old Mathematics Mentions Edition

I haven’t forgotten about the comic strips. It happens that last week’s were mostly quite casual mentions, strips that don’t open themselves up to deep discussions. I write this before I see what I actually have to write about the strips. But here’s the first half of the past week’s. I’ll catch up on things soon.

Bill Amend’s FoxTrot for the 22nd, a new strip, has Jason and Marcus using arithmetic problems to signal pitches. At heart, the signals between a pitcher and catcher are just an index. They’re numbers because that’s an easy thing to signal given that one only has fingers and that they should be visually concealed. I would worry, in a pattern as complicated as these two would work out, about error correction. If one signal is mis-read — as will happen — how do they recognize it, and how do they fix it? This may seem like a lot of work to put to a trivial problem, but to conceal a message is important, whatever the message is.

Jerry Scott and Jim Borgman’s Zits for the 23rd has Jeremy preparing for a calculus test. Could be any subject.

James Beutel’s Banana Triangle for the 23rd has a character trying to convince himself of his intelligence. And doing so by muttering mathematics terms, mostly geometry. It’s a common shorthand to represent deep thinking.

Tom Batiuk’s Funky Winkerbean Vintage strip for the 24th, originally run the 13th of May, 1974, is wordplay about acute triangles.

Hector D Cantú and Carlos Castellanos’s Baldo for the 25th has Gracie work out a visual joke about plus signs. Roger Price, name-checked here, is renowned for the comic feature Droodles, extremely minimalist comic panels. He also, along with Get Smart’s Leonard Stern, created Mad Libs.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is a joke about orders of magnitude. The order of magnitude is, roughly, how big the number is. Often the first step of a physics problem is to try to get a calculation that’s of the right order of magnitude. Or at least close to the order of magnitude. This may seem pretty lax. If we want to find out something with value, say, 231, it seems weird to claim victory that our model says “it will be a three-digit number”. But getting the size of the number right is a first step. For many problems, particularly in cosmology or astrophysics, we’re intersted in things whose functioning is obscure. And relies on quantities we can measure very poorly. This is why we can see getting the order magnitude about right as an accomplishment.

There’s another half-dozen strips from last week that at least mention mathematics. I’ll at least mention them soon, in an essay at this link. Thank you.

## Reading the Comics, March 21, 2020: Pragmatic Calculations Edition

There were a handful of other comic strips last week. If they have a common theme (and I’ll try to drag one out) it’s that they circle around pragmatism. Not just using mathematics in the real world but the fussy stuff of what you can calculate and what you can use a calculation for.

And, again, I am hosting the Playful Math Education Blog Carnival this month. If you’ve run across any online tool that teaches mathematics, or highlights some delightful feature of mathematics? Please, let me know about it here, and let me know what of your own projects I should feature with it. The goal is to share things about mathematics that helped you understand more of it. Even if you think it’s a slight thing (“who cares if you can tell whether a number’s divisible by 11 by counting the digits right?”) don’t worry. Slight things count. Speaking of which …

Jef Mallett’s Frazz for the 20th has a kid ask about one of those add-the-digits divisibility tests. What happens if the number is too big to add up all the digits? In some sense, the question is meaningless. We can imagine finding the sum of digits no matter how many digits there are. At least if there are finitely many digits.

But there is a serious mathematical question here. We accept the existence of numbers so big no human being could ever know their precise value. At least, we accept they exist in the same way that “4” exists. If a computation can’t actually be finished, then, does it actually mean anything? And if we can’t figure a way to shorten the calculation, the way we can usually turn the infinitely-long sum of a series into a neat little formula?

This gets into some cutting-edge mathematics. For calculations, some. But also, importantly, for proofs. A proof is, really, a convincing argument that something is true. The ideal of this is a completely filled-out string of logical deductions. These will take a long while. But, as long as it takes finitely many steps to complete, we normally accept the proof as done. We can imagine proofs that take more steps to complete than could possibly be thought out, or checked, or confirmed. We, living in the days after Gödel, are aware of the idea that there are statements which are true but unprovable. This is not that. Gödel’s Incompleteness Theorems tell us about statements that a deductive system can’t address. This is different. This is things that could be proven true (or false), if only the universe were more vast than it is.

There are logicians who work on the problem of what too-long-for-the-universe proofs can mean. Or even what infinitely long proofs can mean, if we allow those. And how they challenge our ideas of what “proof” and “knowledge” and “truth” are. I am not among these people, though, and can’t tell you what interesting results they have concluded. I just want to let you know the kid in Frazz is asking a question you can get a spot in a mathematics or philosophy department pondering. I mean so far as it’s possible to get a spot in a mathematics or philosophy department.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a less heady topic. Its speaker is doing an ethical calculation. These sorts of things are easy to spin into awful conclusions. They treat things like suffering with the same tools that we use to address the rates of fluids mixing, or of video game statistics. This often seems to trivialize suffering, which we feel like we shouldn’t do.

This kind of calculation is often done, though. It’s rather a hallmark of utilitarianism to try writing an equation for an ethical question. It blends often more into economics, where the questions can seem less cruel even if they are still about questions of life and death. But as with any model, what you build into the model directs your results. The lecturer here supposes that guilt is diminished by involving more people. (This seems rather true to human psychology, though it’s likely more that the sense of individual responsibility dissolves in a large enough group. There are many other things at work, though, all complicated and interacting in nonlinear ways.) If we supposed that the important measure was responsibility for the killing, we would get that the more people involved in killing, the worse it is, and that a larger war only gets less and less ethical. (This also seems true to human psychology.)

Jeff Corriveau’s Deflocked for the 20th sees Mamet calculating how many days of life he expects to have left. There are roughly 1,100 days in three years, so, Mamet’s figuring on about 40 years of life. These kinds of calculation are often grim to consider. But we all have long-term plans that we would like to do (retirement, and its needed savings, are an important one) and there’s no making a meaningful plan without an idea of what the goals are.

This finally closes out the last week’s comic strips. Please stop in next week as I get to some more mathematics comics and the Playful Math Education Blog Carnival. Thanks for reading.

## Reading the Comics, February 19, 2020: 90s Doonesbury Edition

The weekday Doonesbury has been in reruns for a very long while. Recently it’s been reprinting strips from the 1990s and something that I remember producing Very Worried Editorials, back in the day.

Garry Trudeau’s Doonesbury for the 17th reprints a sequence that starts off with the dread menace and peril of Grade Inflation, the phenomenon in which it turns out students of the generational cohort after yours are allowed to get A’s. (And, to a lesser extent, the phenomenon in which instructors respond to the treatment of education as a market by giving the “customers” the grades they’re “buying”.) The strip does depict an attitude common towards mathematics, though, the idea that it must be a subject immune to Grade Inflation: “aren’t there absolute answers”? If we are careful to say what we mean by an “absolute answer” then, sure.

But grades? Oh, there is so much subjectivity as to what goes into a course. And into what level to teach that course at. How to grade, and how harshly to grade. It may be easier, compared to other subjects, to make mathematics grading more consistent year-to-year. One can make many problems that test the same skill and yet use different numbers, at least until you get into topics like abstract algebra where numbers stop being interesting. But the factors that would allow any course’s grade to inflate are hardly stopped by the department name.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is a strip about using a great wall of equations as emblem of deep, substantial thought. The equations depicted are several meaningful ones. The top row is from general relativity, the Einstein Field Equations. These relate the world-famous Ricci curvature tensor with several other tensors, describing how mass affects the shape of space. The P = NP line describes a problem of computational science with an unknown answer. It’s about whether two different categories of problems are, in fact, equivalent. The line about $L = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu}$ is a tensor-based scheme to describe the electromagnetic field. The next two lines look, to me, like they’re deep in Schrödinger’s Equation, describing quantum mechanics. It’s possible Weinersmith has a specific problem in mind; I haven’t spotted it.

Ruben Bolling’s Super-Fun-Pak Comix for the 18th is one of the Guy Walks Into A Bar line, each of which has a traditional joke setup undermined by a technical point. In this case, it’s the horse counting in base four, in which representation the number 2 + 2 is written as 10. Really, yes, “10 in base four” is the number four. I imagine properly the horse should say “four” aloud. But it is quite hard to read the symbols “10” as anything but ten. It’s not as though anyone looks at the hexadecimal number “4C” and pronounces it “76”, either.

Garry Trudeau’s Doonesbury for the 19th twisted the Grade Inflation peril to something that felt new in the 90s: an attack on mathematics as “Eurocentric”. The joke depends on the reputation of mathematics as finding objectively true things. Many mathematicians accept this idea. After all, once we’ve seen a proof that we can do the quadrature of a lune, it’s true regardless of what anyone thinks of quadratures and lunes, and whether that person is of a European culture or another one.

But there are several points to object to here. The first is, what’s a quadrature? … This is a geometric thing; it’s finding a square that’s the same area as some given shape, using only straightedge and compass constructions. The second is, what’s a lune? It’s a crescent moon-type shape (hence the name) that you can make by removing the overlap from two circles of specific different radiuses arranged in a specific way. It turns out you can find the quadrature for the lune shape, which makes it seem obvious that you should be able to find the quadrature for a half-circle, a way easier (to us) shape. And it turns out you can’t. The third question is, who cares about making squares using straightedge and compass? And the answer is, well, it’s considered a particularly elegant way of constructing shapes. To the Ancient Greeks. And to those of us who’ve grown in a mathematics culture that owes so much to the Ancient Greeks. Other cultures, ones placing more value on rulers and protractors, might not give a fig about quadratures and lunes.

This before we get into deeper questions. For example, if we grant that some mathematical thing is objectively true, independent of the culture which finds it, then what role does the proof play? It can’t make the thing more or less true. It doesn’t eve matter whether the proof is flawed, or whether it convinces anyone. It seems to imply a mathematician isn’t actually needed for their mathematics. This runs contrary to intuition.

Anyway, this gets off the point of the student here, who’s making a bad-faith appeal to multiculturalism to excuse laziness. It’s difficult to imagine a culture that doesn’t count, at least, even if they don’t do much work with numbers like 144. Granted that, it seems likely they would recognize that 12 has some special relationship with 144, even if they don’t think too much of square roots as a thing.

And do please stop in later this Leap Day week. I figure to have one of my favorite little things, a Reading the Comics day that’s all one day. It should be at this link, when posted. Thank you.

## Reading the Comics, February 8, 2020: Delta Edition

With this essay, I finally finish the comic strips from the first full week of February. You know how these things happen. I’ll get to the comics from last week soon enough, at an essay gathered under this link. For now, some pictures with words:

Art Sansom and Chip Sansom’s The Born Loser for the 7th builds on one of the probability questions people often use. That is the probability of an event, in the weather forecast. Predictions for what the weather will do are so common that it takes work to realize there’s something difficult about the concept. The weather is a very complicated fluid-dynamics problem. It’s almost certainly chaotic. A chaotic system is deterministic, but unpredictable, because to get a meaningful prediction requires precision that’s impossible to ever have in the real world. The slight difference between the number π and the number 3.1415926535897932 throws calculations off too quickly. Nevertheless, it implies that the “chance” of snow on the weekend means about the same thing as the “chance” that Valentinte’s Day was on the weekend this year. The way the system is set up implies it will be one or the other. This is a probability distribution, yes, but it’s a weird one.

What we talk about when we say the “chance” of snow or Valentine’s on a weekend day is one of ignorance. It’s about our estimate that the true value of something is one of the properties we find interesting. Here, past knowledge can guide us. If we know that the past hundred times the weather was like this on Friday, snow came on the weekend less than ten times, we have evidence that suggests these conditions don’t often lead to snow. This is backed up, these days, by numerical simulations which are not perfect models of the weather. But they are ones that represent something very like the weather, and that stay reasonably good for several days or a week or so.

And we have the question of whether the forecast is right. Observing this fact is used as the joke here. Still, there must be some measure of confidence in a forecast. Around here, the weather forecast is for a cold but not abnormally cold week ahead. This seems likely. A forecast that it was to jump into the 80s and stay there for the rest of February would be so implausible that we’d ignore it altogether. A forecast that it would be ten degrees (Fahrenheit) below normal, or above, though? We could accept that pretty easily.

Proving a forecast is wrong takes work, though. Mostly it takes evidence. If we look at a hundred times the forecast was for a 10% chance of snow, and it actually snowed 11% of the time, is it implausible that the forecast was right? Not really, not any more than a coin coming up tails 52 times out of 100 would be suspicious. If it actually snowed 20% of the time? That might suggest that the forecast was wrong. If it snowed 80% of the time? That suggests something’s very wrong with the forecasting methods. It’s hard to say one forecast is wrong, but we can have a sense of what forecasters are more often right than others are.

Doug Savage’s Savage Chickens for the 7th is a cute little bit about counting. Counting things out is an interesting process; for some people, hearing numbers said aloud will disrupt their progress. For others, it won’t, but seeing numbers may disrupt it instead.

Niklas Eriksson’s Carpe Diem for the 8th is a bit of silliness about the mathematical sense of animals. Studying how animals understand number is a real science, and it turns up interesting results. It shouldn’t be surprising that animals can do a fair bit of counting and some geometric reasoning, although it’s rougher than even our untrained childhood expertise. We get a good bit of our basic mathematical ability from somewhere, because we’re evolved to notice some things. It’s silly to suppose that dogs would be able to state the Pythagorean Theorem, at least in a form that we recognize. But it is probably someone’s good research problem to work out whether we can test whether dogs understand the implications of the theorem, and whether it helps them go about dog work any.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th speaks of the “Cinnamon Roll Delta Function”. The point is clear enough on its own. So let me spoil a good enough bit of fluff by explaining that it’s a reference to something. There is, lurking in mathematical physics, a concept called the “Dirac delta function”, named for that innovative and imaginative fellow Paul Dirac. It has some weird properties. Its domain is … well, it has many domains. The real numbers. The set of ordered pairs of real numbers, R2. The set of ordered triples of real numbers, R3. Basically any space you like, there’s a Dirac delta function for it. The Dirac delta function is equal to zero everywhere in this domain, except at one point, the “origin”. At that one function, though? There it’s equal to …

Here we step back a moment. We really, really, really want to say that it’s infinitely large at that point, which is what Weinersmith’s graph shows. If we’re being careful, we don’t say that though. Because if we did say that, then we would lose the thing that we use the Dirac delta function for. The Dirac delta function, represented with δ, is a function with the property that for any set D, in the domain, that you choose to integrate over

$\int_D \delta(x) dx = 1$

whenever the origin is inside the interval of integration D. It’s equal to 0 if the origin is not inside the interval of integration. This, whatever the set is. If we use the ordinary definitions for what it means to integrate a function, and say that the delta function is “infinitely big” at the origin, then this won’t happen; the integral will be zero everywhere.

This is one of those cases where physicists worked out new mathematical concepts, and the mathematicians had to come up with a rationalization by which this made sense. This because the function is quite useful. It allows us, mathematically, to turn descriptions of point particles into descriptions of continuous fields. And vice-versa: we can turn continuous fields into point particles. It turns out we like to do this a lot. So if we’re being careful we don’t say just what the Dirac delta function “is” at the origin, only some properties about what it does. And if we’re being further careful we’ll speak of it as a “distribution” rather than a function.

But colloquially, we think of the Dirac delta function as one that’s zero everywhere, except for the one point where it’s somehow “a really big infinity” and we try to not look directly at it.

The sharp-eyed observer may notice that Weinersmith’s graph does not put the great delta spike at the origin, that is, where the x-axis represents zero. This is true. We can create a delta-like function with a singular spot anywhere we like by the process called “translation”. That is, if we would like the function to be zero everywhere except at the point $a$, then we define a function $\delta_a(x) = \delta(x - a)$ and are done. Translation is a simple step, but it turns out to be useful all the time.

Thanks again for reading. See you soon.

## Reading the Comics, January 25, 2020: Comic Strip Master Command Is Making This Hard For Me Edition

Or they’re making it easy for me. But for another week all the comic strips mentioning mathematics have done so in casual ways. Ones that I don’t feel I can write a substantial paragraph about. And so, ones that I don’t feel I can fairly use the images of here. Here’s strips that at least said “math” somewhere in them:

Mark Pett’s Mr Lowe rerun for the 18th had the hapless teacher giving out a quiz about fractions.

Greg Cravens’s The Buckets for the 19th plays on the conflation of “zero” and “nothing”. The concepts are related, and we wouldn’t have a zero if we weren’t trying to worth with the concept of nothing. But there is a difference that’s quite hard to talk about without confusing matters.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 19th has a student accused of cheating on a pre-algebra test.

Liniers’s Macanudo for the 21st has a kid struggling with mathematics while the imaginary friend goes off and plays.

Lincoln Peirce’s Big Nate: First Class for the 21st has Nate struggling with mathematics. The strip is a reprint of the Big Nate from the 23rd of January, 1995.

Greg Curfman’s Meg for the 21st has Meg doing arithmetic homework.

Scott Hilburn’s The Argyle Sweater for the 23rd is a wordplay joke, with a flash card that has an addition problem on it.

One of Gary Larson’s The Far Side reprints for the 24th has a man demanding the answer to one question: the square root of an arbitrary number. It’s a little over 70, and that’s as far as anyone could reasonably expect to answer off the top of their head.

James Beutel’s Banana Triangle for the 24th quotes The Wizard Of Oz’s famous garbled version of the Pythagorean Theorem.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th presents a sinister reading of the fad of “prove you’re human” puzzles that demanded arithmetic expressions be done. All computer programs, including, like, Facebook group messages are arithmetic operations ultimately. The steps could be translated into simple expressions like this and be done by humans. It just takes work which, I suppose, could also be translated into other expressions.

And with that large pile of mentions I finish off the mathematical comic strips for the day. Also for the month: next Sunday gets us already into February. Sometime then I should post at this link a fresh Reading the Comics essay. Thank you for reading this one.

## Reading the Comics, January 18, 2020: Decimals In Fractions Edition

Let me first share the other comic strips from last week which mentioned mathematics, but in a casual way.

Jerry Scott and Jim Borgman’s Zits for the 14th used the phrase “do the math”, and snarked on the younger generation doing mathematics. This was as part of the longrunning comic’s attempt to retcon the parents from being Baby Boomers to being Generation X. Scott and Borgman can do as they like but, I mean, their kids are named Chad and Jeremy. That’s only tenable if they’re Boomers. (I’m not sure Chad has returned from college in the past ten years.) And even then it was marginal.

John Kovaleski’s Bo Nanas rerun for the 14th is a joke about the probability of birthdays.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th features “the Bertrand Russell Drinking Game”, playing on the famous paradox about self-referential statements of logic.

Stephan Pastis’s Pearls Before Swine for the 17th has Rat use a bunch of mathematical jargon to give his declarations authority.

Cy Olson’s Office Hours for the 18th, rerunning a strip from the 9th of November, 1971, is in the line of jokes about parents not understanding their children’s arithmetic. It doesn’t seem to depend on mocking the New Math, which is a slight surprise for a 1971 comic.

So Mark Anderson’s Andertoons for the 12th is the only comic strip of some substance that I noticed last week. You see what a slender month it’s been. It does showcase the unsettling nature of seeing notations for similar things mixed. It’s not that there’s anything which doesn’t parse about having decimals in the numerator or denominator. It just looks weird. And that can be enough to throw someone out of a problem. They might mistake the problem for one that doesn’t have a coherent meaning. Or they might mistake it for one too complicated to do. Learning to not be afraid of a problem that looks complicated is worth doing. As is learning how to tell whether a problem parses at all, even if it looks weird.

And that’s an end to last week in comics. I plan to have a fresh Reading the Comics post on Sunday. Thank you for reading in the meanwhile.

## Reading the Comics, January 4, 2020: Representations Edition

The start of the year brings me comic strips I can discuss in some detail. There are also some that just mention a mathematical topic, and don’t need more than a mention that the strip exists. I’ll get to those later.

Jonathan Lemon’s Rabbits Against Magic for the 2nd is another comic strip built on a very simple model of animal reproduction. We saw one late last year with a rat or mouse making similar calculations. Any calculation like this builds on some outright untrue premises, particularly in supposing that every rabbit that’s born survives, and that the animals breed as much as could do. It also builds on some reasonable simplifications. Things like an average litter size, or an average gestation period, or time it takes infants to start breeding. These sorts of exponential-growth calculations depend a lot on exactly what assumptions you make. I tried reproducing Lemon’s calculation. I didn’t hit 95 billion offspring. But I got near enough to say that Lemon’s right to footnote this as ‘true’. I wouldn’t call them “baby bunnies”, though; after all, some of these offspring are going to be nearly seven years old by the end of this span.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd justifies why “mathematicians are no longer allowed to [sic] sporting events” with mathematicians being difficult. Each of the signs is mean to convey the message “We’re #1”. The notations are just needlessly inaccessible, in that way nerds will do things.

$0.\bar{9}$ first. The bar over over a decimal like this means to repeat what is underneath the bar without limit. So this is the number represented by 0.99999… and this is another way to write the number 1. This sometimes makes people uncomfortable; the proof is to think what the difference is between 1 and the number represented by 0.999999 … . The difference is smaller than any positive number. It’s certainly not negative. So the difference is zero. So the two numbers have to be the same number.

$0^0$ is the controversial one here. The trouble is that there are two standard rules that clash here. One is the rule that any real number raised to the zeroth power is 1. The other is the rule that zero raised to any positive real number is 0. We don’t ask about zero raised to a negative number. These seem to clash. That we only know zero raised to positive real numbers is 0 seems to break the tie, and justify concluding the number-to-the-zero-power rule should win out. This is probably what Weinersmith, or Weinersmith’s mathematician, was thinking. If you forced me to say what I think $0^0$ should be, and didn’t let me refuse to commit to a value, I’d probably pick “1” too. But.

The expression $x^x$ exists for real-valued numbers x, and that’s fine. We can look at $\lim_{x \rightarrow 0 } x^x$ and that number’s 1. But what if x is a complex-valued number? If that’s the case, then this limit isn’t defined. And mathematicians need to work with complex-valued numbers a lot. It would be daft to say “real-valued $0^0$ is 1, but complex-valued $0^0$ isn’t anything”. So we avoid the obvious daftness and normally defer to saying $0^0$ is undefined.

The last expression is $e^{\frac{\pi}{2}} \imath^{\imath}$. This $\imath$ is that famous base of imaginary numbers, one of those numbers for which $\imath^2 = -1$. Complex-valued numbers can be multiplied and divided and raised to powers just like real-valued numbers can. And, remarkably — it surprised me — the number $\imath^{\imath}$ is equal to $e^{-\frac{\pi}{2}}$. That’s the reciprocal of $e^{\frac{\pi}{2}}$.

There are a couple of ways to show this. A straightforward method uses the famous Euler formula, that $e^{\imath x} = \cos(x) + \imath\sin(x)$. This implies that $e^{\imath \frac{\pi}{2}} = \imath$. So $\imath^{\imath}$ has to equal $(e^{\imath \frac{\pi}{2}})^{\imath}$. That’s equal to $e^{\imath^2 \frac{\pi}{2}})$, or $e^{- \frac{\pi}{2}})$. If you find it weird that an imaginary number raised to an imaginary number gives you a real number — it’s a touch less than 0.208 — then, well, you see how weird even the simple things can be.

Gary Larson’s The Far Side for the 4th references Abraham Lincoln’s famous use of “four score and seven” to represent 87. There have been many ways to give names to numbers. As we’ve gotten comfortable with decimalization, though, most of them have faded away. I think only dozens and half-dozens remain in common use; if it weren’t for Lincoln’s style surely nobody today would remember “score” as a way to represent twenty. It probably avoids ambiguities that would otherwise plague words like “hundred”, but it does limit one’s prose style. The talk about carrying the one and taking away three is flavor. There’s nothing in turning eighty-seven into four-score-and-seven that needs this sort of arithmetic.

I hope later this week to list the comic strips which just mentioned some mathematical topic. That essay, and next week’s review of whatever this week is mathematical, should appear at this link. Thanks for reading.

## Reading the Comics, December 21, 2019: My Favorite Kind Of Explanation Edition

And here’s the other half of last week’s comic strips that name-dropped mathematics in such a way that I couldn’t expand it to a full paragraph. We’ll likely be back to something more normal next week.

David Malki’s Wondermark for the 20th is built on the common idiom of giving more than 100%. I’m firmly on the side of allowing “more than 100%” in both literal and figurative uses of percent, so there’s not much more to say.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 20th has a wall full of mathematical scribbles and plays on the phrase “calculating killer”. The strip originally ran the 7th of January, 2011.

Samson’s Dark Side of the Horse for the 19th is wordplay on “the thought that counts”. The joke demands Horace be pondering arithmetic, as we see.

Maria Scrivan’s Half Full for the 20th is the Venn Diagram joke for this week.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses Big Numbers as the sort of thing that need a down-to-earth explanation. The strip is about explanations that don’t add clarity. It shows my sense of humor that I love explanations that are true but explain nothing. The more relevant and true without helping the better. Right up until it’s about something I could be explaining instead.

Tom Batiuk’s vintage Funky Winkerbean for the 21st is part of a week of strips from the perspective of a school desk. It includes a joke about football players working mathematics problems. The strip originally ran the 8th of February, 1974, looks like.

Thaves’s Frank and Ernest for the 21st is the anthropomorphic-numerals (and letters) joke for the week.

And there we go; thank you for looking over a quick list of things. I should be back with more comic strips on Sunday, barring surprises.

## Reading the Comics, December 9, 2019: It’s A Slow Week Edition, Part II

And here’s the rest of last week’s mathematically-themed comic strips. On reflection, none of them are so substantially about the mathematics they mention for me to go into detail. Again, Comic Strip Master Command is helping me rebuild my energies after the A-to-Z wrapped up. I appreciate it, folks, but would like, you know, two or three strips a week I can sink my teeth into.

Charles Schulz’s Peanuts rerun for the 11th sees Sally Brown working out metric system unit conversions. The strip originally ran the 13th of December, 1972, a year when people in the United States briefly thought there might ever be a reason to use the prefix “deci-” for something besides decibels. “centi-” for anything besides “centimeter” is pretty dodgy too.

Rick Detorie’s One Big Happy for the 13th is a strip about percentages, and the question of whether a percentage over 100 can be meaningful. I’m solidly in the camp that says “of course it can be”.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 13th is titled “Do Not Date A Mathematician”. This seems personal. The point here is the mathematician believing her fiancee has “demonstrated a poor understanding of probability” by declaring his belief in soulmates. The joke seems to be missing some key points, though. Just declaring a belief in soulmates doesn’t say anything about his understanding of probability. If we suppose that he believed every person had exactly one soulmate, and that these soulmates were uniformly distributed across the world’s population, and that people routinely found their soulmates. But if those assumptions aren’t made then you can’t say that the fiancee is necessarily believing in something improbable.

Lincoln Peirce’s Big Nate: First Class sees Nate looking for help with his mathematics homework. The strip originally ran the 16th of December, 1994.

And that covers the comic strips of last week! I figure on Sunday to have a fresh Reading the Comics post at this link. And I’m thinking whether, or what, to have later this week. Thanks for reading.

## Reading the Comics, December 2, 2019: Laconic Week Edition

You know, I had picked these comic strips out as the ones that, last week, had the most substantial mathematics content. And on preparing this essay I realize there’s still not much. Maybe I could have skipped out on the whole week instead.

Bill Amend’s FoxTrot for the 1st is mostly some wordplay. Jason’s finding ways to represent the counting numbers with square roots. The joke plays more tightly than one might expect. Root beer was, traditionally, made with sassafras root, hence the name. (Most commercial root beers don’t use actual sassafras anymore as the safrole in it is carcinogenic.) The mathematical term root, meanwhile, derives from the idea that the root of a number is the thing which generates it. That 2 is the fourth root of 16, because four 2’s multiplied together is 16. That idea. This draws on the metaphor of the roots of a plant being the thing which lets the plant grow. This isn’t one of those cases where two words have fused together into one set of letters.

Jef Mallett’s Frazz for the 1st is set up with an exponential growth premise. The kid — I can’t figure out his name — promises to increase the number of push-ups he does each day by ten percent, with exciting forecasts for how many that will be before long. As Frazz observes, it’s not especially realistic. It’s hard to figure someone working themselves up from nothing to 300 push-ups a day in only two months.

Also much else of the kid’s plan doesn’t make sense. On the second day he plans to do 1.1 push-ups? On the third 1.21 push-ups? I suppose we can rationalize that, anyway, by taking about getting a fraction of the way through a push-up. But if we do that, then, I make out by the end of the month that he’d be doing about 15.863 push-ups a day. At the end of two months, at this rate, he’d be at 276.8 push-ups a day. That’s close enough to three hundred that I’d let him round it off. But nobody could be generous enough to round 15.8 up to 90.

An alternate interpretation of his plans would be to say that each day he’s doing ten percent more, and round that up. So that, like, on the second day he’d do 1.1 rounded up to 2 push-ups, and on the third day 2.2 rounded up to 3 push-ups, and so on. Then day thirty looks good: he’d be doing 94. But the end of two months is a mess as by then he’d be doing 1,714 push-ups a day. I don’t see a way to fit all these pieces together. I’m curious what the kid thought his calculation was. Or, possibly, what Jef Mallett thought the calculation was.

Zach Weinersmith’s for the 2nd has a kid rejecting accounting in favor of his art. But, wanting to do that art with optimum efficiency … ends up doing accounting. It’s a common story. A common question after working out that someone can do a thing is how to do it best. Best has many measures, yes. But the logic behind how to find it stays the same. Here I admit my favorite kinds of games tend to have screen after screen of numbers, with the goal being to make some number as great as possible considering. If they ever made Multiple Entry Accounting Simulator none of you would ever hear from me again.

Which may be some time! Between Reading the Comics, A to Z, recap posts, and the occasional bit of filler I’ve just finished slightly over a hundred days in a row posting something. That is, however, at its end. I don’t figure to post anything tomorrow. I may not have anything before Sunday’s Reading the Comics post, at this link. I’ll be letting my typing fingers sleep in instead. Thanks for reading.

## Reading the Comics, November 30, 2019: Big Embarrassing Mistake Edition

See if you can spot where I discover my having made a big embarrassing mistake. It’s fun! For people who aren’t me!

Lincoln Peirce’s Big Nate for the 24th has boy-genius Peter drawing “electromagnetic vortex flow patterns”. Nate, reasonably, sees this sort of thing as completely abstract art. I’m not precisely sure what Peirce means by “electromagnetic vortex flow”. These are all terms that mathematicians, and mathematical physicists, would be interested in. That specific combination, though, I can find only a few references for. It seems to serve as a sensing tool, though.

No matter. Electromagnetic fields are interesting to a mathematical physicist, and so mathematicians. Often a field like this can be represented as a system of vortices, too, points around which something swirls and which combine into the field that we observe. This can be a way to turn a continuous field into a set of discrete particles, which we might have better tools to study. And to draw what electromagnetic fields look like — even in a very rough form — can be a great help to understanding what they will do, and why. They also can be beautiful in ways that communicate even to those who don’t undrestand the thing modelled.

Megan Dong’s Sketchshark Comics for the 25th is a joke based on the reputation of the Golden Ratio. This is the idea that the ratio, $1:\frac{1}{2}\left(1 + \sqrt{5}\right)$ (roughly 1:1.6), is somehow a uniquely beautiful composition. You may sometimes see memes with some nice-looking animal and various boxes superimposed over it, possibly along with a spiral. The rectangles have the Golden Ratio ratio of width to height. And the ratio is kind of attractive since $\frac{1}{2}\left(1 + \sqrt{5}\right)$ is about 1.618, and $1 \div \frac{1}{2}\left(1 + \sqrt{5}\right)$ is about 0.618. It’s a cute pattern, and there are other similar cute patterns.. There is a school of thought that this is somehow transcendently beautiful, though.

It’s all bunk. People may find stuff that’s about one-and-a-half times as tall as it is wide, or as wide as it is tall, attractive. But experiments show that they aren’t more likely to find something with Golden Ratio proportions more attractive than, say, something with $1:1.5$ proportions, or $1:1.8$, or even to be particularly consistent about what they like. You might be able to find (say) that the ratio of an eagle’s body length to the wing span is something close to $1:1.6$. But any real-world thing has a lot of things you can measure. It would be surprising if you couldn’t find something near enough a ratio you liked. The guy is being ridiculous.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th builds on the idea that everyone could be matched to a suitable partner, given a proper sorting algorithm. I am skeptical of any “simple algorithm” being any good for handling complex human interactions such as marriage. But let’s suppose such an algorithm could exist.

This turns matchmaking into a problem of linear programming. Arguably it always was. But the best possible matches for society might not — likely will not be — the matches everyone figures to be their first choices. Or even top several choices. For one, our desired choices are not necessarily the ones that would fit us best. And as the punch line of the comic implies, what might be the globally best solution, the one that has the greatest number of people matched with their best-fit partners, would require some unlucky souls to be in lousy fits.

Although, while I believe that’s the intention of the comic strip, it’s not quite what’s on panel. The assistant is told he’ll be matched with his 4,291th favorite choice, and I admit having to go that far down the favorites list is demoralizing. But there are about 7.7 billion people in the world. This is someone who’ll be a happier match with him than 6,999,995,709 people would be. That’s a pretty good record, really. You can fairly ask how much worse that is than the person who “merely” makes him happier than 6,999,997,328 people would

And that’s all I have for last week. Sunday I hope to publish another Reading the Comics post, one way or another. And later this week I’ll have closing thoughts on the Fall 2019 A-to-Z sequence. And I do sincerely apologize to Lincoln Peirce for getting his name wrong, and this on a comic strip I’ve been reading since about 1991.

## Reading the Comics, November 21, 2019: Computational Science Edition

There were just a handful of comic strips that mentioned mathematical topics I found substantial. Of those that did, computational science came up a couple times. So that’s how we got to here.

Rick Detorie’s One Big Happy for the 17th has Joe writing an essay on the history of computing. It’s basically right, too, within the confines of space and understandable mistakes like replacing Pennsylvania with an easier-to-spell state. And within the confines of simplification for the sake of getting the idea across briefly. Most notable is Joe explaining ENIAC as “the first electronic digital computer”. Anyone calling anything “the first” of an invention is simplifying history, possibly to the point of misleading. But we must simplify any history to have it be understandable. ENIAC is among the first computers that anyone today would agree is of a kind with the laptop I use. And it’s certainly the one that, among its contemporaries, most captured the public imagination.

Incidentally, Heman Hollerith was born on Leap Day, 1860; this coming year will in that sense see only his 39th birthday.

Ryan North’s Dinosaur Comics for the 18th is based on the question of whether P equals NP. This is, as T-Rex says, the greatest unsolved problem in computer science. These are what appear to be two different kinds of problems. Some of them we can solve in “polynomial time”, with the number of steps to find a solution growing as some polynomial function of the size of the problem. Others seem to be “non-polynomial”, meaning the number of steps to find a solution grows as … something not a polynomial.

You see one problem. Not knowing a way to solve a problem in polynomial time does not necessarily mean there isn’t a solution. It may mean we just haven’t thought of one. If there is a way we haven’t thought of, then we would say P equals NP. And many people assume that very exciting things would then follow. Part of this is because computational complexity researchers know that many NP problems are isomorphic to one another. That is, we can describe any of these problems as a translation of another of these problems. This is the other part which makes this joke: the declaration that ‘whether God likes poutine’ is isomorphic to the question ‘does P equal NP’.

We tend to assume, also, that if P does equal NP then NP problems, such as breaking public-key cryptography, are all suddenly easy. This isn’t necessarily guaranteed. When we describe something as polynomial or non-polynomial time we’re talking about the pattern by which the number of steps needed to find the solution grows. In that case, then, an algorithm that takes one million steps plus one billion times the size-of-the-problem to the one trillionth power is polynomial time. An algorithm that takes two raised to the size-of-the-problem divided by one quintillion (rounded up to the next whole number) is non-polynomial. But for most any problem you’d care to do, this non-polynomial algorithm will be done sooner. If it turns out P does equal NP, we still don’t necessarily know that NP problems are practical to solve.

Bil Keane and Jeff Keane’s The Family Circus for the 20th has Dolly explaining to Jeff about the finiteness of the alphabet and infinity of numbers. I remember in my childhood coming to understand this and feeling something unjust in the difference between the kinds of symbols. That we can represent any of those whole numbers with just ten symbols (thirteen, if we include commas, decimals, and a multiplication symbol for the sake of using scientific notation) is an astounding feat of symbolic economy.

Zach Weinersmth’s Saturday Morning Breakfast cereal for the 21st builds on the statistics of genetics. In studying the correlations between one thing and another we look at something which varies, usually as the result of many factors, including some plain randomness. If there is a correlation between one variable and another we usually can describe how much of the change in one quantity depends on the other. This is what the scientist means on saying the presence of this one gene accounts for 0.1% of the variance in eeeeevil. The way this is presented, the activity of one gene is responsible for about one-thousandth of the level of eeeeevil in the person.

As the father observes, this doesn’t seem like much. This is because there are a lot of genes describing most traits. And that before we consider epigenetics, the factors besides what is in DNA that affect how an organism develops. I am, unfortunately, too ignorant of the language of genetics to be able to say what a typical variation for a single gene would be, and thus to check whether Weinersmith has the scale of numbers right.

This finishes the mathematically-themed comic strips from this past week. If all goes to my plan, Tuesday and Thursday will find the last of this year’s A-to-Z postings for this year. And Wednesday? I’ll try to think of something for Wednesday. It’d be a shame to just leave it hanging loose like it might.

## Reading the Comics, November 13, 2019: I Could Have Posted This Wednesday Edition

Now let me discuss the comic strips from last week with some real meat to their subject matter. There weren’t many: after Wednesday of last week there were only casual mentions of any mathematics topic. But one of the strips got me quite excited. You’ll know which soon enough.

Mac King and Bill King’s Magic in a Minute for the 10th uses everyone’s favorite topological construct to do a magic trick. This one uses a neat quirk of the Möbius strip: that if sliced along the center of its continuous loop you get not two separate shapes but one Möbius strip of greater length. There are more astounding feats possible. If the strip were cut one-third of the way from an edge it would slice the strip into two shapes, one another Möbius strip and one a simple loop.

Or consider not starting with a Möbius strip. Make the strip of paper by taking one end and twisting it twice around, for a full loop, before taping it to the other end. Slice this down the center and what results are two interlinked rings. Or place three twists in the original strip of paper before taping the ends together. Then, the shape, cut down the center, unfolds into a trefoil knot. But this would take some expert hand work to conceal the loops from the audience while cutting. It’d be a neat stunt if you could stage it, though.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th uses mathematics as obfuscation. We value mathematics for being able to make precise and definitely true statements. And for being able to describe the world with precision and clarity. But this has got the danger that people hear mathematical terms and tune out, trusting that the point will be along soon after some complicated talk.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 11th would be a Pi Day joke if it hadn’t run in November. But when this strip first ran, in 2010, Pi Day was not such a big event in the STEM/Internet community. The Boychuks couldn’t have known.

The formulas on the blackboard are nearly all legitimate, and correct, formulas for the value of π. The upper-left and the lower-right formulas are integrals, and ones that correspond to particular trigonometric formulas. The The middle-left and the upper-right formulas are series, the sums of infinitely many terms. The one in the upper right, $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$, was roughly proven by Leonhard Euler. Euler developed a proof that’s convincing, but that assumed that infinitely-long polynomials behave just like finitely-long polynomials. In this context, he was correct, but this can’t be generally trusted to happen. We’ve got proofs that, to our eyes, seem rigorous enough now.

The center-left formula doesn’t look correct to me. To my eye, this looks like a mistaken representation of the formula

$\pi = 2 \sum_{k = 0}^{\infty} \frac{2^k \cdot k!^2}{\left(2k + 1\right)!}$

But it’s obscured by Haskins’s head. It may be that this formula’s written in a format that, in full, would be correct. There are many, many formulas for π (here’s Mathworld’s page of them and here’s Wikipedia’s page of π formulas); it’s impossible to list them all.

The center-right formula is interesting because, in part, it looks weird. It’s written out as

$\pi = \frac{4}{6+}\frac{1^2}{6+}\frac{3^2}{6+}\frac{5^2}{6+}\frac{7^2}{6+} \cdots$

That looks at first glance like something’s gone wrong with one of those infinite-product series for π. Not so; this is a notation used for continued fractions. A continued fraction has a string of denominators that are typically some whole number plus another fraction. Often the denominator of that fraction will itself be a whole number plus another fraction. This gets to be typographically challenging. So we have this notation instead. Its syntax is that

$a + \frac{b}{c + \frac{d}{e + \frac{f}{g}}} = a + \frac{b}{c+} \frac{d}{e+} \frac{f}{g}$

There are many attractive formulas for π. It’s temping to say this is because π is such a lovely number it naturally has beautiful formulas. But more likely humans are so interested in π we go looking for formulas with some appealing sequence to them. There are some awful-looking formulas out there too. I don’t know your tastes, but for me I feel my heart cool when I see that π is equal to four divided by this number:

$\sum_{n = 0}^{\infty} \frac{(-1)^n (4n)! (21460n + 1123)}{(n!)^4 441^{2n + 1} 2^{10n + 1}}$

however much I might admire the ingenuity which found that relationship, and however efficiently it may calculate digits of π.

Glenn McCoy and Gary McCoy’s The Duplex for the 13th uses skill at arithmetic as shorthand for proving someone’s a teacher. There’s clearly some implicit idea that this is a school teacher, probably for elementary schools, and doesn’t have a particular specialty. But it is only three panels; they have to get the joke done, after all.

And that’s all for the comic strips this week. Come Sunday I should have another Reading the Comics post. And the Fall 2019 A-to-Z draws closer to its conclusion with two more essays, trusting that I can indeed write them, for Tuesday and Thursday. I also have something disturbing to write about for Wednesday. Can’t wait.

## Reading the Comics, November 9, 2019: Two Pairs Edition

So finally I get to the mathematically-themed comic strips of last week. There were four strips which group into natural pairings. So let’s use that as the name for this edition.

Vic Lee’s Pardon My Planet for the 3rd puts forth “cookie and cake charts”, as a riff on pie charts. There’s always room for new useful visual representations of data, certainly, although quite a few of the ones we do use are more than two centuries old now. Pie charts, which we trace to William Playfair’s 1801 Statistical Breviary, were brought to the public renown by Florence Nightingale. She wanted her reports on the causes of death in the Crimean War to communicate well, and illustrations helped greatly.

Wayno and Piraro’s Bizarro for the 9th is another pie chart joke. If I weren’t already going on about pie charts this week I probably would have relegated this to the “casual mentions” heap. I love the look of the pie, though.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th jokes about stereotypes of mathematics and English classes. Or exams, anyway. There is some stabbing truth in the presentation of English-as-math-class. Many important pieces of mathematics are definitions or axioms. In an introductory class there’s not much you can usefully say about, oh, why we’d define a limit to be this rather than that. The book surely has its reasons and we’ll avoid confusion by trusting in them.

I dislike the stereotype of English as a subject rewarding longwinded essays that avoid the question. It seems at least unfair to what good academic writing strives for. (If you wish to argue about bad English writing, you have your blog for that, but let’s not pretend mathematics lacks fundamentally bad papers.) And writing an essay about why a thing should be true, or interesting, is certainly worthwhile. I’m reminded of a mathematical logic professor I had, who spoke of a student who somehow could not do a traditional proper-looking proof. But could write a short essay explaining why a thing should be true which convinced the professor that the student deserved an A. The professor was sad that the student was taking the course pass-fail.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th shows off a bit of mathematical modeling. The specific problem is silly, yes. But the approach is dead on: identify the things that affect what you’re interested in, and how they interact. Add to this estimates of the things’ values and you’ll get at least a provisional answer. You can then use that answer to guide the building of a more precise model, if you need one.

This little bugs-on-Superman problem makes note of the units everything’s measured in. Paying attention to the units is often done in dimensional analysis, a great tool for building simple models. I ought to write an essay sequence about that sometime.

Mark Anderson’s Andertoons for the 9th is the Mark Anderson’s Andertoons for the week. This one plays on the use of the same word to measure an angle and a temperature. Degree, etymologically, traces back to “a step”, like you might find in stairs. This, taken to represent a stage of progress, got into English in the 13th century. By the late 14th century “degree” was used to describe this 1/360th slice of a circle. By the 1540s it was a measure of heat. Making the degree the unit of temperature, as on a thermometer, seems to be written down only as far back as the 1720s.

And for a last strip of the week, Gary Wise and Lance Aldrich’s Real Life Adventures for the 7th mentions an advantage of being a cartoonist “instead of an engineer” is how cartooning doesn’t require math. Also I guess this means the regular guy in Real Life Adventures represents one (or both?) of the creators? I guess that makes the name Real Life Adventures make more sense. I just thought he was a generic comic strip male. And, of course, there’s nothing about mathematics that keeps one from being a cartoonist, although I don’t know of any current daily-syndicated cartoonists with strong mathematics backgrounds. Bill Amend, of FoxTrot, and Bud Grade, of The Piranha Club/Ernie, were both physics majors, which is a heavy-mathematics program.

And that covers last week’s comics. Reading the Comics should return Sunday at this link. And tomorrow I hope to get tothe Fall 2019 A to Z’s exploration of the letter ‘U’. Thanks for reading.

## Reading the Comics, November 2, 2019: Eugene the Jeep Edition

I knew by Thursday this would be a brief week. The number of mathematically-themed comic strips has been tiny. I’m not upset, as the days turned surprisingly full on me once again. At some point I would have to stop being surprised that every week is busier than I expect, right?

Anyway, the week gives me plenty of chances to look back to 1936, which is great fun for people who didn’t have to live through 1936.

Elzie Segar’s Thimble Theatre rerun for the 28th of October is part of the story introducing Eugene the Jeep. The Jeep has astounding powers which, here, are finally explained as being due to it being a fourth-dimensional creature. Or at least able to move into the fourth dimension. This is amazing for how it shows off the fourth dimension being something you could hang a comic strip plot on, back in the day. (Also back in the day, humor strips with ongoing plots that might run for months were very common. The only syndicated strips like it today are Gasoline Alley, Alley Oop, the current storyline in Safe Havens where they’ve just gone and terraformed Mars, and Popeye, rerunning old daily stories.) The Jeep has many astounding powers, including that he can’t be kept inside — or outside — anywhere against his will, and he’s able to forecast the future.

Could there be a fourth-dimensional animal? I dunno, I’m not a dimensional biologist. It seems like we need a rich chemistry for life to exist. Lots of compounds, many of them long and complicated ones. Can those exist in four dimensions? I don’t know the quantum mechanics of chemical formation well enough to say. I think there’s obvious problems. Electrical attraction and repulsion would fall off much more rapidly with distance than they do in three-dimensional space. This seems like it argues chemical bonds would be weaker things, which generically makes for weaker chemical compounds. So probably a simpler chemistry. On the other hand, what’s interesting in organic chemistry is shapes of molecules, and four dimensions of space offer plenty of room for neat shapes to form. So maybe that compensates for the chemical bonds. I don’t know.

But if we take the premise as given, that there is a four-dimensional animal? With some minor extra assumptions then yeah, the Jeep’s powers fit well enough. Not being able to be enclosed follows almost naturally. You, a three-dimensional being, can’t be held against your will by someone tracing a line on the floor around you. The Jeep — if the fourth dimension is as easy to move through as the third — has the same ability.

Forecasting the future, though? We have a long history of treating time as “the” fourth dimension. There’s ways that this makes good organizational sense. But we do have to treat time as somehow different from space, even to make, for example, general relativity work out. If the Jeep can see and move through time? Well, yeah, then if he wants he can check on something for you, at least if it’s something whose outcome he can witness. If it’s not, though? Well, maybe the flow of events from the fourth dimension is more obvious than it is from a mere three, in the way that maybe you can spot something coming down the creek easily, from above, in a way that people on the water can’t tell.

Olive Oyl and Popeye use the Jeep to tease one another, asking for definite answers about whether the other is cute or not. This seems outside the realm of things that the fourth dimension could explain. In the 1960s cartoons he even picks up the power to electrically shock offenders; I don’t remember if this was in the comic strips at all.

Elzie Segar’s Thimble Theatre rerun for the 29th of October has Wimpy doing his best to explain the fourth dimension. I think there’s a warning here for mathematician popularizers here. He gets off to a fair start and then it all turns into a muddle. Explaining the fourth dimension in terms of the three dimensions we’re familiar with seems like a good start. Appealing to our intuition to understand something we have to reason about has a long and usually successful history. But then Wimpy goes into a lot of talk about the mystery of things, and it feels like it’s all an appeal to the strangeness of the fourth dimension. I don’t blame Popeye for not feeling it’s cleared anything up. Segar would come back, in this storyline, to several other attempted explanations of the Jeep’s powers, although they do come back around to, y’know, it’s a magical animal. They’re all over the place in the Popeye comic universe.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th of October is a riff on predictability and encryption. Good encryption schemes rely on randomness. Concealing the content of a message means matching it to an alternate message. Each of the alternate messages should be equally likely to be transmitted. This way, someone who hasn’t got the key would not be able to tell what’s being sent. The catch is that computers do not truly do randomness. They mostly rely on quasirandom schemes that could, in principle, be detected and spoiled. There are ways to get randomness, mostly involving putting in something from the real world. Sensors that detect tiny fluctuations in temperature, for example, or radio detectors. I recall one company going for style and using a wall of lava lamps, so that the rise and fall of lumps were in some way encoded into unpredictable numbers.

Robb Armstrong’s JumpStart for the 2nd of November is a riff on the Birthday “Paradox”, the thing where you’re surprised to find someone shares a birthday with you. (I have one small circle of friends featuring two people who share my birthday, neatly enough.) Paradox is in quotes because it defies only intuition, not logic. The logic is clear that you need only a couple dozen people before some pair will probably share a birthday. Marcie goes overboard in trying to guess how many people at her workplace would share their birthday on top of that. Birthdays are nearly uniformly spread across all days of the year. There are slight variations; September birthdays are a little more likely than, say, April ones; the 13th of any month is a less likely birthday than the 12th or the 24th are. But this is a minor correction, aptly ignored when you’re doing a rough calculation. With 615 birthdays spread out over the year you’d expect the average day to be the birthday of about 1.7 people. (To be not silly about this, a ten-day span should see about 17 birthdays.) However, there are going to be “clumps”, days where three or even four people have birthdays. There will be gaps, days nobody has a birthday, or even streaks of days where nobody has a birthday. If there weren’t a fair number of days with a lot of birthdays, and days with none, we’d have to suspect birthdays weren’t random here.

There were also a handful of comic strips just mentioning mathematics, that I can’t make anything in depth about. Here’s two.

T Shepherd’s Snow Sez for the 1st of November nominally talks about how counting can be a good way to meditate. It can also become a compulsion, with hazards, though.

Terri Libenson’s The Pajama Diaries for the 2nd of November uses mathematics as the sort of indisputably safe topic that someone can discuss in place of something awkward.

And that is all I have to say for last week’s comics. Tuesday I should publish the next Fall 2019 A to Z essay. I also figure to open the end of the alphabet up to nominations this week. My next planned Reading the Comic post should be Sunday. Thanks for reading.

## Reading the Comics, October 22, 2019: Bifurcated Week Edition

The past week started strong for mathematically-themed comics. Then it faded out into strips that just mentioned the existence of mathematics. I have no explanation for this phenomenon. It makes dividing up the week’s discussion material easy enough, though.

John Zakour and Scott Roberts’s Maria’s Day rerun for the 19th is a lottery joke. Maria’s come up with a scheme to certainly win the grand prize in a lottery. There’s no disputing that one could, on buying enough tickets, get an appreciable chance of winning. Even, in principle, get a certain win. There’s no guaranteeing a solo win, though. But sometimes lottery jackpots will grow large enough that even if you had to split the prize two or three ways it’d be worth it.

Tom Horacek’s Foolish Mortals for the 21st plays on the common wisdom that mathematicians’ best work is done when they’re in their 20s. Or at least their most significant work. I don’t like to think that’s so, as someone who went through his 20s finding nothing significant. But my suspicion is that really significant work is done when someone with fresh eyes looks at a new problem. Young mathematicians are in a good place to learn, and are looking at most everything with fresh eyes, and every problem is new. Still, experienced mathematicians, bringing the habits of thought that served well one kind of problem, looking at something new will recreate this effect. We just need to find ideas to think about that we haven’t worn down.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st has a petitioner asking god about whether P = NP. This is shorthand for a famous problem in the study of algorithms. It’s about finding solutions to problems, and how much time it takes to find the solution. This time usually depends on the size of whatever it is you’re studying. The question, interesting to mathematicians and computer scientists, is how fast this time grows. There are many classes of these problems. P stands for problems solvable in polynomial time. Here the number of steps it takes grows at, like, the square or the cube or the tenth power of the size of the thing. NP is non-polynomial problems, growing, like, with the exponential of the size of the thing. (Do not try to pass your computer science thesis defense with this description. I’m leaving out important points here.) We know a bunch of P problems, as well as NP problems.

Like, in this comic, God talks about the problem of planning a long delivery route. Finding the shortest path that gets to a bunch of points is an NP problem. What we don’t know about NP problems is whether the problem is we haven’t found a good solution yet. Maybe next year some bright young 68-year-old mathematician will toss of a joke on a Reddit subthread and then realize, oh, this actually works. Which would be really worth knowing. One thing we know about NP problems is there’s a big class of them that are all, secretly, versions of each other. If we had a good solution for one we’d have a solution for all of them. So that’s why a mathematician or computer scientist would like to hear God’s judgement on how the world is made.

Hector D. Cantú and Carlos Castellanos’s Baldo for the 22nd has Baldo asking his sister to do some arithmetic. I fancy he’s teasing her. I like doing some mental arithmetic. If nothing else it’s worth having an expectation of the answer to judge whether you’ve asked the computer to do the calculation you actually wanted.

Mike Thompson’s Grand Avenue for the 22nd has Gabby demanding to know the point of learning Roman numerals. As numerals, not much that I can see; they serve just historical and decorative purposes these days, mostly as a way to make an index look more fancy. As a way to learn that how we represent numbers is arbitrary, though? And that we can use different schemes if that’s more convenient? That’s worth learning, although it doesn’t have to be Roman numerals. They do have the advantage of using familiar symbols, though, which (say) the Babylonian sexagesimal system would not.

And that’s the comic strips with enough mathematics for me to discuss from the first half of last week. I plan tomorrow to at least mention the strips with just mentions of mathematics. And then Tuesday, The A-to-Z reaches the letter Q. I’m interested to see how that turns out too.

## Reading the Comics, September 28, 2019: Modeling Edition

The second half of last week’s mathematically-themed comic strips had an interesting range of topics. Two of them seemed to circle around the making of models. So that’s my name for this installment.

Ryan North’s Dinosaur Comics for the 26th has T-Rex trying to build a model. In this case, it’s to project how often we should expect to see a real-life Batman. T-Rex is building a simple model, which is fine. Simple models, first, are usually easier to calculate with. How they differ from reality can give a guide to how to make a more complex model. Or they can indicate the things that have to be learned in order to make a more complex model. The difference between a model’s representation and the observed reality (or plausibly expected reality) can point out problems in one’s assumptions, too.

For example, T-Rex supposes that a Batman needs to have billionaire parents. This makes for a tiny number of available parents. But surely what’s important is that a Batman be wealthy enough he doesn’t have to show up to any appointments he doesn’t want to make. Having a half-billion dollars, or a “mere” hundred million, would allow that. Even a Batman who had “only” ten million dollars would be about as free to be a superhero. Similarly, consider the restriction to Olympic athletes. Astronaut Ed White, who on Gemini IV became the first American to walk in space, was not an Olympic athlete; but he certainly could have been. He missed by a split-second in the 400 meter hurdles race. Surely someone as physically fit as Ed White would be fit enough for a Batman. Not to say that “Olympic athletes or NASA astronauts” is a much bigger population than “Olympic athletes”. (And White was unusually fit even for NASA astronauts.) But it does suggest that merely counting Olympic athletes is too restrictive.

But that’s quibbling over the exact numbers. The process is a good rough model. List all the factors, suppose that all the factors are independent of one another, and multiply how likely it is each step happens by the population it could happen to. It’s hard to imagine a simpler model, but it’s a place to start.

Greg Wallace’s Nothing Is Not Something for the 26th is a bit of a geometry joke. It’s built on the idiom of the love triangle, expanding it into more-sided shapes. Relationships between groups of people like this can be well-represented in graph theory, with each person a vertex, and each pair of involved people an edge. There are even “directed graphs”, where each edge contains a direction. This lets one represent the difference between requited and unrequited interests.

Brian Anderson’s Dog Eat Doug for the 27th has Sophie the dog encounter some squirrels trying to disprove a flat Earth. They’re not proposing a round Earth either; they’ve gone in for a rhomboid. Sophie’s right to point out that drilling is a really hard way to get through the Earth. That’s a practical matter, though.

Is it possible to tell something about the shape of a whole thing from a small spot? In the terminology, what kind of global knowledge can we get from local information? We can do some things. For example, we can draw a triangle on the surface of the Earth and measure the interior angles to see what they sum to. If this could be done perfectly, finding that the interior angles add up to more than 180 degrees would show the triangle’s on a spherical surface. But that also has practical limitations. Like, if we find that locally the planet is curved then we can rule out it being entirely flat. But it’s imaginable that we’d be on the one dome of an otherwise flat planet. At some point you have to either assume you’re in a typical spot, or work out ways to find what’s atypical. In the Conspiracy Squirrels’ case, that would be the edge between two faces of the rhomboid Earth. Then it becomes something susceptible to reason.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th has the mathematician making another model. And this is one of the other uses of a model: to show a thing can’t happen, show that it would have results contrary to reason. But then you have to validate the model, showing that its premises do represent reality so well that its conclusion should be believed. This can be hard. There’s some nice symbol-writing on the chalkboard here, although I don’t see that they parse. Particularly, the bit on the right edge of the panel, where the writing has a rotated-by-180-degrees ‘E’ followed by an ‘x’, a rotated-by-180-degrees ‘A’, and then a ‘z’, is hard to fit inside an equation like this. The string of symbols mean “there exists some x for which, for all z, (something) is true”. This fits at the start of a proof, or before an equation starts. It doesn’t make grammatical sense in the middle of an equation. But, in the heat of writing out an idea, mathematicians will write out ungrammatical things. As with plain-text writing, it’s valuable to get an idea down, and edit it into good form later.

Tom Batiuk’s Funky Winkerbean Vintage for the 28th sees the school’s Computer explaining the nature of its existence, and how it works. Here the Computer claims to just be filled with thousands of toes to count on. It’s silly, but it is the case that there’s no operation a computer does that isn’t something a human can do, manually. If you had the paper and the time you could do all the steps of a Facebook group chat, a game of SimCity, or a rocket guidance computer’s calculations. The results might just be impractically slow.

And that’s finished the comic strips of last week! Sunday I should have a new Reading the Comics post. And then tomorrow I hope to resume the Fall 2019 A to Z series with ‘J’. Thanks for reading.