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.

Presenting as a magic trick: cutting a loop of paper in half, along the loop's center, with the result being a single yet larger loop. The trick is to make the paper into a Moebius strip, and conceal the 'twist' so that your audience does not know what they see.
Mac King and Bill King’s Magic in a Minute for the 10th of November, 2019. This and other mathematics-based tricks featured in Magic In A Minute are at this link.

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.

Confession Tip: Use Statistics. Kid: 'Mom! Dad! Did you know that, in your immediate area, teen pregnancy may be as high as 100 percent?!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th of November, 2019. This and many other essays discussing Saturday Morning Breakfast Cereal are at this link.

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.

On the blackboard several calculus formulas for the value of pi. At the table a scientist type says to another, while eating a slice of pie, 'I don't know why, Haskins, but I've had a craving for this all day.'
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 11th of November, 2019. It originally ran the 29th of November, 2010. This and other essays mentioning The Chuckle Brothers, which has gone into perpetual reruns, are at this link.

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.

Eno, on a coffee date: 'So you claim you're a teacher, huh?' Teacher: 'What do you mean, 'claim'?' Eno; 'What's 30 divided by 5?' Teacher: 'Six!' Eno: 'OK, you check out.'
Glenn McCoy and Gary McCoy’s The Duplex for the 13th of November, 2019. This surprises me by not being a new comic tag. Essays mentioning The Duplex are at this link.

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, August 29, 2018: The Week I Missed One Edition


Have you ever wondered how I gather comic strips for these Reading the Comics posts? Sure, why not go along with me. Well, I do it by this: I read a lot of comic strips. When I run across one that’s got some mathematical theme, I copy the URL for it over to a page of notes. Then I go back to those notes and write up a paragraph or so on each. That is, I do it exactly the way you might imagine if you weren’t very imaginative or trying hard. I explain all this to say that I made a note that I then didn’t notice. So I missed a comic strip. And opened myself up to wondering if there’s an etymological link between “note” and “notice”. Anyway, it’s here. I’m just explaining why it’s late.

Jim Toomey’s Sherman’s Lagoon for the 19th of August is the belated inclusion. It’s a probability strip. It’s built partly on how badly people estimate probability, especially of rare events. And of how badly people estimate the consequences of rare events. For anything that isn’t common, our feelings about the likelihood of events are garbage. And even for common events we’re not that good.

Sherman: 'See that guy up on the beach?' Fillmore: 'Yeah.' Sherman: 'He's not swimming. Wanna know why? He's afraid of sharks. With the help of my friend Ernest, I will argue that his fear is irrational. Ernest, what are this guy's chances of getting attacked by a shark?' Ernest: 'One in 3.7 million.' Sherman: 'And what are his chances of getting struck by lightning?' Ernest: 'One in 600,000.' Sherman: 'I rest my case.' (Crack of lightning strikes Sherman on this nearly cloudless day.) Sherman: 'What are the chances of a shark getting struck by lightning?' Ernest: 'Oddly enough, pretty good.'
Jim Toomey’s Sherman’s Lagoon for the 19th of August, 2018. This strip caused me to learn that the comic strip is indeed set in a specific if fictional place, the Kapupu Lagoon, set near the Palau archipelago. I had kind of just figured, you know, ‘The Pacific Ocean somewhere’, but the strip is too ocean-aware to just leave it at that.

But then it’s hard to quantify a low-probability event too. Take the claim that a human has one chance in 3.7 million of being attacked by a shark. We’ll pretend that’s the real number; I don’t know what is. (I’m suspicious of the ‘3-7’. People picking a random two-digit number are surprisingly likely to pick 37 because, I guess, it ‘feels’ random.) Is that over their lifetime? Over a summer? In a single swimming event? In any case it’s such a tiny chance it’s not worth serious worry. But even then, a person who lives in Wisconsin and only ever swims in Lake Michigan has a considerably smaller chance of shark attack than a person from New Jersey who swims at the Shore. At least some of these things are probabilities we can affect.

So the fellow may be irrational, denying himself something he’d enjoy based on a fantastically unlikely event. But he is acting to avoid something he’s decided he doesn’t want to risk. And, you know, we all act irrationally at times, or else I couldn’t justify buying a lottery ticket every eight months or so. Also is Fillmore (the turtle) the person who needs to hear this argument?

Eno: 'I finally got our checkbook to balance, but I had to invent my own kind of math, where zero equals $235.37.'
Gary McCoy and Glenn McCoy’s The Duplex for the 26th of August, 2018. At the risk of taking out my protractor — an old rec.arts.comics.strips quip about trying to demand unreasonable precision in comic strip art, based on a For Better Or For Worse panel where some folks thought a telescope pointed at the wrong part of the sky — why is nothing on Eno’s table there a checkbook?

Gary McCoy and Glenn McCoy’s The Duplex for the 26th is an accounting joke. And a cry about poverty, with the idea that one could make the adding up of one’s assets and debts work only by making mathematics logically inconsistent. Or maybe inconsistent. Arithmetic modulo a particular number could be said to make zero equal to some other number, after all, and that’s all valid. Useful, too, especially in enciphering messages and in generating random numbers. It’s less useful for accounting, though. At least it would draw attention if used unilaterally.

Hayden, thinking about his assignment: 'Ugh. Again. Find x. It's always x. I don't understand this obsession with unmasking x.' (Aloud.) 'Honestly, Miss Hansen, what's wrong with a little mystery in life?'
Steve Kelley and Jeff Parker’s Dustin for the 28th of August, 2018. I’m not sure that Hayden isn’t young enough that the unknown quantity couldn’t be represented with a box or a blank line.

Steve Kelley and Jeff Parker’s Dustin for the 28th is roughly a student-resisting-the-homework problem. From the first panel I thought Hayden might be complaining that ‘x’ was used, once again, as the variable to be solved for. It is the default choice, made because we all grew up learning of ‘x’ as the first choice for a number with a not-yet-known identity. ‘y’ and ‘z’ come in as second and third choices, most likely because they’re quite close to ‘x’. Sometimes another letter stands out, usually because the problem compels it. If the framing of the problem is about when events happen then ‘t’ becomes the default choice. If the problem suggests circular motion then ‘r’ or ‘θ’ — radius and angle — become compelling. But if we know no context, and have only the one variable, then ‘x’ it is. It seems weird to do otherwise.

[ Fi's Math Talk For Schools. ] Fi: 'Numbers are actually about relationships. Take this 'equal' sign. Both sides of the equation must be in balance. One side can't be toxic or abusive.' (Quiet.) Audience question: 'Are we still talking about math?' Fi: 'You'd think all humans could meet a bar that low.'
Bill Holbrook’s On The Fastrack for the 28th of August, 2018. But yeah, this does seem like a curious leap. Misanthropy is one of Fi’s defining characteristics but I can usually follow why she’s gotten from the mathematical to the social.

Bill Holbrook’s On The Fastrack for the 28th is part of a week of Fi talking about mathematics to kids. She occasionally delivers seminars meant to encourage enthusiasm about mathematics. I love the principle although I don’t know how long the effect lasts. (Although it is kind of what I’m doing here. Except I think maybe Fi gets paid.) Holbrook’s strips of this mode often include nice literal depictions of metaphors. This week didn’t offer much chance for that particular artistic play.


I have at least one, and often several, Reading the Comics posts, each week. They should all appear at this link. Other essays with Sherman’s Lagoon will appear at this link when they’re written. I’m surprised to learn that’s a new tag. Essays that mention The Duplex are at this link. Other appearances by Dustin, a character who does not appear in this particular essay’s strips, are at this link. And On The Fastrack mentions should appear at this link. Thank you.

Reading the Comics, July 21, 2018: Infinite Hotels Edition


Ryan North’s Dinosaur Comics for the 18th is based on Hilbert’s Hotel. This is a construct very familiar to eager young mathematicians. It’s an almost unavoidable pop-mathematics introduction to infinitely large sets. It’s a great introduction because the model is so mundane as to be easily imagined. But you can imagine experiments with intuition-challenging results. T-Rex describes one of the classic examples in the third through fifth panels.

The strip made me wonder about the origins of Hilbert’s Hotel. Everyone doing pop mathematics uses the example, but who created it? And the startling result is, David Hilbert, kind of. My reference here is Helge Kragh’s paper The True (?) Story of Hilbert’s Infinite Hotel. Apparently in a 1924-25 lecture series in Göttingen, Hilbert encouraged people to think of a hotel with infinitely many rooms. He apparently did not use it for so many examples as pop mathematicians would. He just used the question of how to accommodate a single new guest after the infinitely many rooms were first filled. And then went to imagine an infinite dance party. I don’t remember ever seeing the dance party in the wild; perhaps it’s a casualty of modern rave culture.

T-Rex: 'David Hilbert was a mathematician and hotelier who was born in 1892. He built an infinite hotel, you guys! THE INFINITE HOTEL: A TRUE STORY. So Hilbert built this infinite hotel that was infinitely big and had infinitely many rooms; I believe this was a matter of some investment. But build it he did, and soon after a bus with infinity people in it showed up, with each of them wanting a room! Lucky for Hilbert he had his infinite hotel, so each guest got a room, and the hotel was filled up to capacity. Nice! But just then another friggin' bus showed up, and it ALSO had infinity people in it!' Utahraptor: 'Nobody builds for TWO infinite buses showing up right after the other!' T-Rex: 'Turns out they do! He just told every guest already there to move into the room that was double their current room number. So the guest in room 3 moved into room 6, and so on! Thus, only the even-numbered rooms were occupied, and everyone on the new bus could have an odd-numbered room!' Utahraptor: 'Amazing!' T-Rex: 'Yep! Anyway! It's my understanding he died an infinitely rich man infinity years later.'
Ryan North’s Dinosaur Comics for the 18th of July, 2018. The strip likely ran sometime before on North’s own web site; I don’t know when.

Hilbert’s Hotel seems to have next seen print in George Gamow’s One, Two Three … Infinity. Gamow summoned the hotel back from the realms of forgotten pop mathematics with a casual, jokey tone that fooled Kragh into thinking he’d invented the model and whimsically credited Hilbert with it. (Gamow was prone to this sort of lighthearted touch.) He came back to it in The Creation Of The Universe, less to make readers consider the modern understanding of infinitely large sets than to argue for a universe having infinitely many things in it.

And then it disappeared again, except for cameo appearances trying to argue that the steady-state universe would be more bizarre than what we actually see. The philosopher Pamela Huby seems to have made Hilbert’s Hotel a thing to talk about again, as part of a debate about whether a universe could be infinite in extent. William Lane Craig furthered using the hotel, as part of the theological debate about whether there could be an infinite temporal regress of events. Rudy Rucker and Eli Maor wrote descriptions of the idea in the 1980s, with vague ideas about whether Hilbert actually had anything to do with the place. And since then it’s stayed, a famous fictional hotel.

David Hilbert was born in 1862; T-Rex misspoke.

Teacher: 'Sluggo --- describe an octagon.' Sluggo: 'A figure with eight sides and eight angles.' Teacher: 'Correct. Now, Nancy --- describe a sphere'. (She blows a bubble-gum bubble.)
Ernie Bushmiller’s Nancy Classics for the 20th of July, 2018. Originally run, it looks to me, like the 18th of October, 1953.

Ernie Bushmiller’s Nancy Classics for the 20th gets me out of my Olivia Jaimes rut. We could probably get a good discussion going about whether giving an example of a sphere is an adequate description of a sphere. Granted that a bubble-gum bubble won’t be perfectly spherical; neither will any example that exists in reality. We always trust that we can generalize to an ideal example of this thing.

I did get to wondering, in Sluggo’s description of the octagon, why the specification of eight sides and eight angles. I suspect it’s meant to avoid calling an octagon something that, say, crosses over itself, thus having more angles than sides. Not sure, though. It might be a phrasing intended to make sure one remembers that there are sides and there are angles and the polygon can be interesting for both sets of component parts.

Literal Figures: a Venn diagram of two circles, their disjoint segments labelled 'Different' and their common area labelled 'Same'. A graph, 'Height of Rectangles', a bar chart with several rectangles. A graph, 'Line Usage': a dashed line labelled Dashed; a jagged line labelled Jagged; a curvy line labelled Curvy. A map: 'Global Dot Concentration', with dots put on a map of the world.
John Atkinson’s Wrong Hands for the 20th of July, 2018. So this spoils a couple good ideas for my humor blog’s Statistics Saturdays now that you know I’ve seen this somewhere.

John Atkinson’s Wrong Hands for the 20th is the Venn Diagram joke for the week. The half-week anyway. Also a bunch of other graph jokes for the week. Nice compilation of things. I love the paradoxical labelling of the sections of the Venn Diagram.

Ziggy: 'I wish I'd paid more attention in math class! I can't even count the number of times I've had trouble with math!'
Tom II Wilson’s Ziggy for the 20th of July, 2018. Tom Wilson’s still credited with the comic strip, though he died in 2011. I don’t know whether this indicates the comic is in reruns or what.

Tom II Wilson’s Ziggy for the 20th is a plaintive cry for help from a despairing soul. Who’s adding up four- and five-digit numbers by hand for some reason. Ziggy’s got his projects, I guess is what’s going on here.

Cop: 'You were travelling at 70 miles per hour. How much later would you have arrived if you were only going 60?' Eno: 'No fair --- I hate word problems!'
Glenn McCoy and Gary McCoy’s The Duplex for the 21st of July, 2018. So the strip is named The Duplex because it’s supposed to be about two families in the same, uh, duplex: this guy with his dog, and a woman with her cat. I was reading the strip for years before I understood that. (The woman doesn’t show up nearly so often, or at least it feels like that.)

Glenn McCoy and Gary McCoy’s The Duplex for the 21st is set up as an I-hate-word-problems joke. The cop does ask something people would generally like to know, though: how much longer would it take, going 60 miles per hour rather than 70? It turns out it’s easy to estimate what a small change in speed does to arrival time. Roughly speaking, reducing the speed one percent increases the travel time one percent. Similarly, increasing speed one percent decreases travel time one percent. Going about five percent slower should make the travel time a little more than five percent longer. Going from 70 to 60 miles per hour reduces the speed about fifteen percent. So travel time is going to be a bit more than 15 percent longer. If it was going to be an hour to get there, now it’ll be an hour and ten minutes. Roughly. The quality of this approximation gets worse the bigger the change is. Cutting the speed 50 percent increases the travel time rather more than 50 percent. But for small changes, we have it easier.

There are a couple ways to look at this. One is as an infinite series. Suppose you’re travelling a distance ‘d’, and had been doing it at the speed ‘v’, but now you have to decelerate by a small amount, ‘s’. Then this is something true about your travel time ‘t’, and I ask you to take my word for it because it has been a very long week and I haven’t the strength to argue the proposition:

t = \frac{d}{v - s} = \frac{d}{v}\left(1 + \left(\frac{s}{v}\right) + \left(\frac{s}{v}\right)^2 + \left(\frac{s}{v}\right)^3 + \left(\frac{s}{v}\right)^4 + \left(\frac{s}{v}\right)^5 + \cdots \right)

‘d’ divided by ‘v’ is how long your travel took at the original speed. And, now, \left(\frac{s}{v}\right) — the fraction of how much you’ve changed your speed — is, by assumption, small. The speed only changed a little bit. So \left(\frac{s}{v}\right)^2 is tiny. And \left(\frac{s}{v}\right)^3 is impossibly tiny. And \left(\frac{s}{v}\right)^4 is ridiculously tiny. You make an error in dropping these \left(\frac{s}{v}\right) squared and cubed and forth-power and higher terms. But you don’t make much of one, not if s is small enough compared to v. And that means your estimate of the new travel time is:

\frac{d}{v} \left(1 + \frac{s}{v}\right)

Or, that is, if you reduce the speed by (say) five percent of what you started with, you increase the travel time by five percent. Varying one important quantity by a small amount we know as “perturbations”. Working out the approximate change in one quantity based on a perturbation is a key part of a lot of calculus, and a lot of mathematical modeling. It can feel illicit; after a lifetime of learning how mathematics is precise and exact, it’s hard to deliberately throw away stuff you know is not zero. It gets you to good places, though, and fast.

Wellington: 'First our teacher says 25 plus 25 equals 50. Then she says 30 and 20 equals 50. Then she says 10 and 40 equals 50. Finally she says 15 and 35 equals 50. Shouldn't we have a teacher who can make up her mind?'
Morrie Turner’s Wee Pals rerun for the 21st of July, 2018. Originally ran the 22nd of July, 2013.

Morrie Turner’s Wee Pals for the 21st shows Wellington having trouble with partitions. We can divide any counting number up into the sum of other counting numbers in, usually, many ways. I can kind of see his point; there is something strange that we can express a single idea in so many different-looking ways. I’m not sure how to get Wellington where he needs to be. I suspect that some examples with dimes, quarters, and nickels would help.

And this is marginal but the “Soul Circle” personal profile for the 20th of July — rerun from the 20th of July, 2013 — was about Dr Cecil T Draper, a mathematics professor.


You can get to this and more Reading the Comics posts at this link. Other essays mentioning Dinosaur Comics are at this link. Essays that describe Nancy, vintage and modern, are at this link. Wrong Hands gets discussed in essays on this link. Other Ziggy-based essays are at this link. The Duplex will get mentioned in essays at this link if any other examples of the strip get tagged here. And other Wee Pals strips get reviewed at this link.