Reading the Comics, January 26, 2019: The Week Ended Early Edition


Last week started out at a good clip: two comics with enough of a mathematical theme I could imagine writing a paragraph about them each day. Then things puttered out. The rest of the week had almost nothing. At least nothing that seemed significant enough. I’ll list those, since that’s become my habit, at the end of the essay.

Jonathan Lemon and Joey Alison Sayers’s Alley Oop for the 20th is my first chance to show off the new artist and writer team. They’ve decided to make Sunday strips a side continuity about a young Alley Oop and his friends. I’m interested. The strip is built on the bit of pop anthropology that tells us “primitive” tribes will have very few counting words. That you can express concepts like one, two, and three, but then have to give up and count “many”.

Little Alley Oop: 'You think scientists will ever invent a number bigger than three?' Garg: 'I guess it's possible. There are three scientists working around the clock trying to come up with a new number.' Oop: 'Three scientists? Wow! That's a lot.' Garg: 'Maybe someday *we'll* be scientists. Then there'll be *three* scientists.' Oop: 'Nah, I think I want to be a fire-fighter. There are only three of those in the whole world.'
Jonathan Lemon and Joey Alison Sayers’s Alley Oop for the 20th of January, 2019. I don’t seem to have had much cause to mention Alley Oop before; the only previous reference I can find is in this cameo in a different comic. Well, this and any other essays that write about Alley Oop at any length should be at this link. I don’t figure to differentiate between the weekday Alley Oop strip and the Sunday Little Oop line in using this tag.

Perhaps it’s so. Some societies have been found to have, what seem to us, rather few numerals. This doesn’t reflect on anyone’s abilities or intelligence or the like. And it doesn’t mean people who lack a word for, say, “forty-nine” would be unable to compute. It might take longer, but probably just from inexperience. If someone practiced much calculation on “forty-nine” they’d probably have a name for it. And folks raised in the western mathematics use, even enjoy, some vagueness about big numbers too. We might say there are “dozens” of a thing even if there are not precisely 24, 36, or 48 of the thing; “52” is close enough and we probably didn’t even count it up. “Hundred” similarly has gotten the connotation of being a precise number, but it’s used to mean “really quite a lot of a thing”. The words “thousands”, “millions”, and mock-numbers like “zillions” have a similar role. They suggest different ranges of what might be “many”.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a SABRmetrics joke! At least, it’s an optimization joke, built on the idea that you can find an optimum strategy for anything, whether winning baseball games or The War. The principle is hard to argue with. Nobody would doubt that different approaches to a battle affect how likely winning is. We can imagine gathering data on how different tactics affect the outcome. (We can easily imagine combat simulators running these experiments, particularly.)

War party stats nerd: 'We are taking a statistical approach combat. First, don't go for kills. Go for *stabs*. Successful stabs are *much* more valuable to victory than lethal strikes. Look at Birk over here. He averages 22.1 stabs per battle. He alone accounts for an additional 3.8 wins per campaign season. Siegwurst brings in 1.6.' Skeptic: 'But remember when Siegwurst slew two Cossacks with his Dance of the Whirling Blades?' Stats Nerd: 'NO MORE READING THE SAGAS, OK? No spin moves! They're impressive but the expected addition wins per spin is negative. NEGATIVE.' Skeptic :'This is gonna have serious negative effects on morale.' Stats nerd: 'Which correlates with EXACTLY NOTHING. Now GET STABBY!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of January, 2019. Pretty much every Reading the Comics brings up this strip. But the essays that specifically mention Saturday Morning Breakfast Cereal should be at this link.

The catch — well, one catch — is that this tempts one to reward a process. Once it’s taken for granted the process works, then whether it’s actually doing what you want gets forgotten. And once everyone knows what’s being measured it becomes possible to game the system. Famously, in the mid-1960s the United States tried to judge its progress in the Vietnam War by counting the number of enemy soldiers killed. There was then little reason to care about who was killed, or why. And reason to not care whether actual enemy soldiers were being killed. There’s good to be said about testing whether the things you try to do work. There’s great danger in thinking that the thing you can measure guarantees success.

Mark Anderson’s Andertoons for the 21st is a bit of fun with definitions. Mathematicians rely on definitions. It’s hard to imagine a proof about something undefined. But definitions are hard to compose. We usually construct a definition because we want a common term to describe a collection of things, and to exclude another collection of things. And we need people like Wavehead who can find edge cases, things that seem to satisfy a definition while breaking its spirit. This can let us find unstated assumptions that we should pay attention to. Or force us to accept that the definition is so generally useful that we’ll tolerate it having some counter-intuitive implications.

On the blackboard: 'NOT A POLYGON: Fewer than three sides; not connected at ends; lines that cross.' Teacher, to student: 'True, a chicken nugget is also not a polygon, but we're going to focus more on lines and vertices.'
Mark Anderson’s Andertoons for the 21st of January, 2019. Pretty much every Reading the Comics brings up this strip, at least since last fall’s weird gap has ended. But the essays that specifically mention Andertoons should be at this link.

My favorite counter-intuitive implication is in analysis. The field has a definition for what it means that a function is continuous. It’s meant to capture the idea that you could draw a curve representing the function without having to lift the pen that does it. The best definition mathematicians have settled on allows you to count a function that’s continuous at a single point in all of space. Continuity seems like something that should need an interval to happen. But we haven’t found a better way to define “continuous” that excludes this pathological case. So we embrace the weirdness in exchange for general usefulness.

Princess Kat, awestruck at soap bubbles: 'How did you do that, Jackson? Are you a wizard?' Jackson: 'They're just bubbles. You dunk this wand into a cup of soapy water and then blow through it.' Kat: 'Gimme gimme! I wanna blow bubbles too!' Jackson: 'Just take it easy or they'll pop! It's pretty simple.' (Kat blows a soap bubble cube, then tetrahedron and cones and some optical illusion shapes.) Kat: 'I think I'm doing this wrong.' Jackson: 'Suddenly I feel like a round peg in a square hole.'
Charles Brubaker’s Ask A Cat for the 21st of January, 2019. Although I’ve mentioned this comic before, it wasn’t when I was putting up tags naming each comic. I’ll fix that now. This and future essays mentioning Ask A Cat (or its Fuzzy Princess sideline, as long as that hasn’t got its own GoComics presence) should appear at this link. And this strip previously ran the 19th of June, 2016.

Charles Brubaker’s Ask A Cat for the 21st is a guest appearance from Brubaker’s other strip, The Fuzzy Princess. It’s a rerun and I did discuss it earlier. Soap bubbles make for great mathematics. They’re easy to play with, for one thing. That’s good for capturing imagination. And the mathematics behind them is deep, and led to important results analytically and computationally. It happens when this strip first ran I’d encountered a triplet of essays about the mathematics of soap bubbles and wireframe surfaces. My introduction to those essays is here.

Benita Epstein’s Six Chix for the 25th I wasn’t sure I’d include. But Roy Kassinger asked about it, so that tipped the scales. The dog tries to blame his bad behavior on “the algorithm”, bringing up one of the better monsters of the last couple years. An algorithm is just the procedure by which you do something. Mathematically, that’s usually to solve a problem. That might be finding some interesting part of the domain or range of a function. That might be putting a collection of things in order. that might be any of a host of things. And then we go make a decision based on the results of the algorithm.

Dog, explaining a messy room to its horrified humans: 'The algorithm made me do it!'
Benita Epstein’s Six Chix for the 25th of January, 2019. The essays wherein I mention Six Chix, from any of the shared strip’s authors, should appear at this link.

What earns The Algorithm its deserved bad name is mindlessness. The idea that once you have an algorithm that a problem is solved. Worse, that once an algorithm is in place it would be irrational to challenge it. I have seen the process termed “mathwashing”, by analogy with whitewashing, and it’s a good one. The notion that because something is done by computer it must be done correctly is absurd. We knew it was absurd before there were computers as we knew them, as see anyone for the past century who has spoken of a “Kafkaesque” interaction with a large organization. It’s impossible to foresee all the outcomes of any reasonably complicated process, much less to verify that all the outcomes are handled correctly. This is before we consider that there will always be mistakes made in the handling of data. Or in the carrying out of the process. And that’s before we consider bad actors. I’m sure there must be research into algorithms designed to handle gaming of the system. I don’t know that there are any good results yet, though. We certainly need them.


There were a couple comics that didn’t seem to be substantial enough for me to write at length about. You might like them anyway. Connie Sun’s Connie to the Wonnie for the 21st shows off a Venn Diagram. Hector D Cantú and Carlos Castellanos’s Baldo for the 23rd is a bit of wordplay about what mathematicians do. Jonathan Lemon’s Rabbits Against Magic for the 23rd similarly is a bit of wordplay built around percentages. (Lemon is the new artist for Alley Oop.) And Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips features Albert Einstein, and a joke based on one of the symmetries which make relativity such a useful explanation of the world’s workings.


I don’t plan to have another Reading the Comics post until next Sunday. But when I do, it’ll be here.

Reading the Comics, January 16, 2019: Young People’s Mathematics Edition


Today’s quartet of mathematically-themed comic strips doesn’t have an overwhelming theme. There’s some bits about the mathematics that young people do, so, that’s enough to separate this from any other given day’s comics essay.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is built on a bit of mathematical folklore. As Weinersmith’s mathematician (I don’t remember that we’ve been given her name) mentions, there is a belief that “revolutionary” mathematics is done by young people. That isn’t to say that older mathematicians don’t do great work. But the stereotype is that an older mathematician will produce masterpieces in already-established fields. It’s the young that establish new fields. Indeed, one of mathematics’s most prestigious awards, the Fields Medal, is only awarded to mathematicians under the age of forty. I was cheated of mine. Long story.

Mathematician: 'Only young people do revolutionary mathematics. 20 is ancient. 15 is old. 10 is middle-aged.' Kid, holding up two fingers: 'Three is THIS MANY.' Mathematician: 'It's counter-intuitive, but we must accept it.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th of January, 2019. I have many essays inspired by something said in Saturday Morning Breakfast Cereal. You can find them at this link.

There’s intuitive appeal in the idea that revolutions in thinking are for the young. We think that people get set in their ways as they develop their careers. We have a couple dramatic examples, most notably Évariste Galois, who developed what we now see as foundations of group theory and died at twenty. While the idea is commonly held, I don’t know that it’s actually true. That is, that it holds up to scrutiny. It seems hard to create a definition for “revolutionary mathematics” that could be agreed upon by two people. So it would be difficult to test at what age people do their most breathtaking work, and whether it is what they do when young or when experienced.

Is there harm to believing an unprovable thing? If it makes you give up on trying, yes. My suspicion is that true revolutionary work happens when a well-informed, deep thinker comes to a field that hasn’t been studied in that way before. And when it turns out to be a field well-suited to study that way. That doesn’t require youth. It requires skill in one field, and an understanding that there’s another field ready to be studied that way.

Spud: 'Can you help me with this math problem?' Wallace: '10 + 12? It helps if you visualize real things. Say you have ten cans of E-Z Cheez and someone gives you twelve more ... how many cans of E-Z Cheez do you have?' Spud: 'I'm sweating.'
Will Henry’s Wallace the Brave for the 14th of January, 2019. I have only had a few chances to talk about Wallace the Brave so far, but the chances I’ve taken are at this link. (It and Breaking Cat News are the two recently-launched comics I’m most excited by.)

Will Henry’s Wallace the Brave for the 14th is a mathematics anxiety joke. Wallace tries to help by turning an abstract problem into a concrete one. This is often a good way to approach a problem. Even in more advanced mathematics, one can often learn the way to solve a general problem by trying a couple of specific examples. It’s almost as though there’s only a certain amount of abstraction people can deal with, and you need to re-cast problems so they stay within your limits.

Yes, the comments turn to complaining about Common Core. I’m not sure what would help Spud work through this problem (or problems in general). But thinking of alternate problems that estimated or approached what he really wanted might help. If he noticed, for example, that 10 + 12 has to be a little more than 10 + 10, and he found 10 + 10 easy, then he’d be close to a right answer. If he noticed that 10 + 12 had to be 10 + 10 + 2, and he found 10 + 10 easy, then he might find 20 + 2 easy as well. Maybe Spud would be better off thinking of ways to rewrite a problem without changing the result.

Widow, to the party gathered at the gravesite: 'Needless to say, calculus wasn't his best subject.' The epitaph: 'It's a calculated risk, but you only live once!'
Wiley Miller’s Non Sequitur for the 15th of January, 2019. Essays mentioning Non Sequitur should appear at this link.

Wiley Miller’s Non Sequitur for the 15th mentions calculus. It’s more of a probability joke. To speak of a calculated risk is to speak of doing something that’s not certain, but that has enough of a payoff to be worth the cost of failure. But one problem with this attitude is that people are very, very bad at estimating probabilities. We have terrible ideas of how likely losses are and how uncertain rewards can be. But even if we allow that the risks and rewards are calculated right, there’s a problem with things you only do once. Or only can do once. You can get into a good debate about whether there’s even a meaningful idea of probability for things that happen only the one time. Life’s among them.

Kid: 'Dad! Let's tackle my homework!' Moose: 'Later, son. I'm busy.' Kid goes to Westpork Savings and Loan. The bank clerk's sitting under a sign, 'Let us help you with your money problems.' Kid reads: 'If Farmer Smith sells wheat at $1.25 a bushel and Farmer Brown sells it at $1.30, how many bushels must each sell ... '
Bob Weber Sr’s Moose and Molly for the 16th of January, 2019. I haven’t had the chance to talk about Moose and Molly before. But now I have the tag, and will be putting essays mentioning it at this link.

Bob Weber Sr’s Moose and Molly for the 16th is a homework joke. It does actually depend on being mathematics homework, though, or there’d be no grounds for Moose’s kid to go to the savings and loan clerk who’ll help with “money problems”.


I think there’s one more batch of comic strips to discuss this week. When I’ve published it, you should find the essay at this link. And then there’ll be Sunday again.

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


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

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th mentions along its way the Liar Paradox and Zeno’s Paradoxes. Both are ancient problems. The paradoxes arise from thinking with care and rigor about things we seem to understand intuitively. For the Liar Paradox it’s about what we mean to declare a statement true or false. For Zeno’s Paradoxes it’s about whether we think space (and time) are continuous or discrete. And, as the strip demonstrates, there is a particular kind of nerd that declares the obvious answer is the only possible answer and that it’s foolish to think deeper. To answer a question’s literal words while avoiding its point is a grand old comic tradition, of course, predating even the antijoke about chickens crossing roads. Which is what gives these answers the air of an old stage comedian.

A man seeks the Wise Man atop a mountain. He asks: 'Wise master, if a man says 'I am lying' is he telling the truth?' Wise Man: 'Yes, if his name is 'lying'.' Seeker: 'But- ' Wise Man: 'NEXT.' Seeker departs, angrily. Other Seeker: 'Wise master, how can you cross infinite points in finite time?' Wise Man: 'By walking. NEXT!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of November, 2018. Other essays mentioning topics raised by Saturday Morning Breakfast Cereal are at this link. Well, all right, this link, if you want to avoid the maybe four times it hasn’t turned up.

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

Lio, on his ham radio, gets an ALIEN TRANSMISSION: 'INVASION UNDERWAY! TARGET ALL ELEMENTARY SCHOOLS!' Lio gets excited. Next ALIEN TRANSMITION: 'JK! JK! JK! ENJOY YOUR MATH CLASS, STUPID KID! LOL!' Lio grimaces; aliens laugh at their prank call.
Mark Tatulli’s Lio for the 28th of November, 2018. Essays discussing topics raised by Lio should be at this link.

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

Randolph, at a restaurant, asking his date as he holds open his wallet: 'Do you have a twenty? All I seem to have is zeroes.' Footer joke: 'And you can never have enough of those.'
Tom Toles’s Randolph Itch, 2am rerun for the 28th of November, 2018. It first appeared the 12th of April, 2000. Other instances of Randolph Itch, 2 am that I thought worth discussing are at this link.

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

Caption: Mobius Comic Strip. Titled 'Down in the Dumpties with Humpty'; on it a pair of eggs argue about whether they're going in circles and whether they've done all this before and getting sick as the comic 'twists' over.
Dan Collins’s Looks Good On Paper for the 29th of November, 2018. Other appearances by Looks Good On Paper should be at this link.

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

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

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


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

Reading the Comics, November 24, 2018: Origins Edition


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

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

Dethany, to her boyfriend: 'Fi concludes her math talks with a demonstration of the tangible benefits of numbers. By having pizza delivered. Square pizza.' Fi, to the kids, as the pizza guy arrives: 'First, calculate how much more area you get than with a round one.'
Bill Holbrook’s On The Fastrack for the 21st of November, 2018. Essays mentioning topics raised by On The Fastrack are at this link.

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

At the lottery ticket booth. Grimm: 'Hey, why do you always but lottery tickets? The odds of you winning are astronomical!' Goose: 'Yeah, but they're astronomically higher if I don't buy a ticket.'
Mike Peters’s Mother Goose and Grimm for the 22nd of November, 2018. Other essays which mention Mother Goose and Grimm should be at this link. I had thought this was a new link, but it turns out there was a strip in early 2017 and another in mid-2015 that got my attention here.

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

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

Peter: 'If you had three clams and gave one away, then I took two, what would you have?' Curls: 'A worthless reason for being in business.'
Johnny Hart’s Back to BC for the 22nd of November, 2018. It originally appeared the 27th of May, 1961. Essays which discuss topics brought up by B.C., both the current-run and the half-century-old reruns, are at this link.

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

Fat Broad, to a dinosaur: 'How much is one and one?' The dinosaur stops a front foot twice. Then gets ready to stomp a third time. Fat Broad whaps the dinosaur senseless. Broad: 'Isn't it amazing how fast animals learn?'
Johnny Hart’s Back to BC for the 24th of November, 2018. It originally appeared the 30th of May, 1961. If this strip has inspired any essays oh wait, I already said where to find them, didn’t I? Well, you know what to look for, then.

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

Kid: 'How come in old paintings the perspective is really badly drawn?' Dad: 'Perspective didn't exist back then. Sometimes there'd be a whole castle right behind you . Other times you'd sit at a table and the tabletop would face away from you. That's also why portraits were badly drawn. Try holding a brush in a world without three consistent dimensions. Italian architects invented perspective to make it easier to draw buildings. What's why things suddenly look a lot nicer around the 16th century.' Kid: 'Are you sure?' Dad: 'How else do you explain that it took 10,000 years of civilization to invent Cartesian coordinates?' Kid: 'I figured people are just kinda stupid.' Dad: 'How facile.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th of November, 2018. The many essays mentioning topics raised by Saturday Morning Breakfast Cereal are at this link.

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

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

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

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


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

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


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

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

Rabbit, reading the paper: 'Artificial intelligence could make animal testing obsolete.' Polar Bear: 'Thank goodness.' Penguin imagines the Polar Bear in school, being asked by the teacher the square root of 121, with a robot beside him whispering '11'.
Alex Hallatt’s Arctic Circle for the 12th of November, 2018. Other essays based on Arctic Circle should be at this link.

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

In school. Binkley: 'Don't say anything, Ms Harlow, but a giant spotted snorkewacker from my closet full of anxieties has followed me to school and since experience has proven that he plans to grab me, I'd like permission to go home and hide.' Ms Harlow: 'Mr Binkley, that's the stinkiest excuse I've ever heard for getting out of a geometry exam. Go sit down.' Binkley's face-down at his desk; the Giant Spotted Snorklewacker asks, 'Pssst! What's the Pythagorean theorem?'
Berkeley Breathed’s Bloom County rerun for the 13th of November, 2018. It originally ran the 17th of February, 1983. Never mind the copyright notice; those would often show the previous year the first couple weeks of the year. Essays based on topics raised by Bloom County — original or modern continuation — should be at this link.

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

Written into two ring stains on a napkin: 'People who drink coffee'. 'People who drink tea'. Pointing to the intersection: 'People who share napkins.'
Terry Border’s Bent Objects for the 15th of November, 2018. Other essays based on Bent Objects will be at this link. It’s a new tag, so for now, there’s just that.

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

Teacher: 'I need to buy some graph paper for my students. Is there a convenience store near here?' Guy: 'Yeah, just two miles way from campus.' Later: Teacher, driving, realizes: 'Wait, he didn't specify a coordinate system. NOOOOOOO!' as her car leaps into the air.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th of November, 2018. In case there’s ever another essay which mentions Saturday Morning Breakfast Cereal it’ll be at this link.

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

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

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

The Odds. Guy checking his phone after his friend's been knocked down: 'There's tons of stuff about being struck by a bolt of lightning --- nothing about bolts of fabric.' [Title panel extra gag: 'Lucky for you it's soft and silky.']
Hilary Price’s Rhymes With Orange for the 16th of November, 2018. And times I’ve discussed something from Rhymes With Orange should be at this link.

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


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

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

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


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

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

Friend 1: 'That algebra test was awful.' Friend 2: 'Toby just gave up and handed his paper in!' Toby: 'No, I finished. My mom said as long as I studied I didn't have to do any chores.' Friend 1: 'That'd eat up all your gaming hours!' Toby: 'Yep. Hey, did you know algebra letters stand for things?'
Greg Cravens’s The Buckets for the 8th of November, 2018. I’m sorry I can’t figure out the names of Toby’s friends here. Character lists, cartoonists, please.

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

Funtime Activity: Ruining students forever. Teacher: 'Physics students must learn the difference between vectors and scalars is that scalars don't exist.' Student: 'What about 'amount of apples'?' Teacher: 'Huh? Oh, you're referring to 'distance in apple-space'.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th of November, 2018. Also, shouldn’t that be “displacement in apple-space”?

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

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

Billy: 'On the 80s station I think I just heard my favorite song ever! It's about carrying a laser down some road. I think it's called 'Carry a laser' and it's all about lasers!' Cow: 'Actually, it's 'Kyrie Eleison'. It means 'Lord, have Mercy'. It has nothing at all to do with lasers.' Billy: 'Right, and 'Hip to b^2' has nothing to do with algebra.' Cow: 'That I don't know.'s
Mark Leiknes’s Cow and Boy rerun for the 9th of November, 2018. It first ran the 19th of March, 2012.

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

Brad: 'I think I've got this worked out. It takes 500 half-inch hairs to make a good moustache. If I grow one hair a week, and each new hair grows 1/8 inch per month, I can grow a perfect moustache in ... ' Luann: '225 years, not bad!'
Greg Evans’s Luann Againn for the 9th of November, 2018. It first ran the 9th of November, 1990.

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

Test Question: 'Mr Gray drove 55 mph to a city 80 miles away. He made two stops: one for 20 minutes, and one for 5. How long did it take Mr Gray to reach the city?' Student's answer: 'This made my head hurt, so I'm just going to say 'the whole trip'. You can't argue that.'
John Zakour and Scott Roberts’s Maria’s Day for the 9th of November, 2018. Again, character lists. I don’t know which of the characters this is except that he’s either very small or has an enormous pencil.

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


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

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

Reading the Comics, November 5, 2018: November 5, 2018 Edition


This past week included one of those odd days that’s so busy I get a column’s worth of topics from a single day’s reading. And there was another strip (the Cow and Boy rerun) which I might have roped in had the rest of the week been dead. The Motley rerun might have made the cut too, for a reference to E = mc^2 .

Jason Chatfield’s Ginger Meggs for the 5th is a joke about resisting the story problem. I’m surprised by the particulars of this question. Turning an arithmetic problem into counts of some number of particular things is common enough and has a respectable history. But slices of broccoli quiche? I’m distracted by the choice, and I like quiche. It’s a weird thing for a kid to have, and a weird amount for anybody to have.

Mr Crackett: 'Alright, Meggs. Here's one for you. If Fitzcloon had 15 slices of broccoli quiche and you took a third, what would you have?' Meggs: 'A bucket ready to catch my vom---' Crackett: 'MEGGS!'
Jason Chatfield’s Ginger Meggs for the 5th of November, 2018. I’m of the age cohort to remember Real Men Don’t Eat Quiche being a book people had for some reason. Also not understanding why “real men” would not eat quiche. If you named the same dish “Cheddar Bacon Pie” you’d have men lined up for a quarter-mile to get it. Anyway, it took me too long to work out but I think the teacher’s name is Mr Crackett? Cast lists, cartoonists. We need cast lists on your comic’s About pages.

JC Duffy’s Lug Nuts for the 5th uses mathematics as a shorthand for intelligence. And it particularly uses π as shorthand for mathematics. There’s a lot of compressed concepts put into this. I shouldn’t be surprised if it’s rerun come mid-March.

The Thinking Man's Team: The Portland Pi. Shows a baseball cap with the symbol pi on it.
JC Duffy’s Lug Nuts for the 5th of November, 2018. OK, some of these strips I don’t need a cast list for.

Tom Toles’s Randolph Itch, 2 am for the 5th I’ve highlighted before. It’s the pie chart joke. It will never stop amusing me, but I suppose I should take Randolph Itch, 2 am out of my rotation of comics I read to include here.

Randolph dreaming about his presentation: pie chart. Pies have hit him and his podium, per the chart: '28% landed on stage, 13% back wall, 22% glancing blow off torso, 12% hit podium, 25% direct hit in face'. Footer joke: 'I turn now to the bar graph.'
Tom Toles’s Randolph Itch, 2 am for the 5th of November, 2018. I never get to presentations like this. It’s always someone explaining the new phone system.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th is a logic puzzle joke. And a set theory joke. Dad is trying to argue he can’t be surprised by his gift because it’ll belong to one of two sets of things. And he receives nothing. This ought to defy his expectations, if we think of “nothing” as being “the empty set”. The empty set is an indispensable part of set theory. It’s a set that has no elements, has nothing in it. Then suppose we talk about what it means for one set to be contained in another. Take what seems like an uncontroversial definition: set A is contained in set B if there’s nothing in A which is not also in B. Then the empty set is contained inside every set. So Dad, having supposed that he can’t be surprised, since he’d receive either something that is “socks” or something that is “not-socks”, does get surprised. He gets the one thing that is both “socks” and “not-socks” simultaneously.

Kids: 'Daddy, we got you a surprise!' Dad: 'Impossible! I assume the surprise is socks. Thus in case 1 where you get me socks, I am not surprised. In case 2, you got me not-socks. Given that I KNOW you will not give me socks because I'm anticipating socks, it's obvious the gift will be not-socks. Therefore in all cases with your gift, I remain UNSURPRISED!' Kids, after a pause: 'The gift is NOTHING!' Dad curses.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th of November, 2018. I may have mentioned. So my partner in Modern Physics Lab one time figured to organize his dorm room by sorting everything in it into two piles, “pair of socks” and “not a pair of socks”. I asked him how he’d classify two socks that, while mismatched, were bundled together. He informed me that he hated me.

I hate to pull this move a third time in one week (see here and here), but the logic of the joke doesn’t work for me. I’ll go along with “nothing” as being “the empty set” for these purposes. And I’ll accept that “nothing” is definitely “not-socks”. But to say that “nothing” is also “socks” is … weird, unless you are putting it in the language of set theory. I think the joke would be saved if it were more clearly established that Dad should be expecting some definite thing, so that no-thing would defy all expectations.

“Nothing” is a difficult subject to treat logically. I have been exposed a bit to the thinking of professional philosophers on the subject. Not enough that I feel I could say something non-stupid about the subject. But enough to say that yeah, they’re right, we have a really hard time describing “nothing”. The null set is better behaved. I suppose that’s because logicians have been able to tame it and give it some clearly defined properties.

Mega Lotto speaker: 'Hmm, what are the odds? First he wins the lottery and then ... ' A torn-up check and empty shoes are all that's left as a crocodile steps out of panel.
Mike Shiell’s The Wandering Melon for the 5th of November, 2018. I am curious whether this is meant to be the same lottery winner who in August got struck by lightning. It would make the torn, singed check make more direct sense. But what are the odds someone wins the lottery, gets hit by lightning, and then eaten by a crocodile? … Ah well, at least nothing worse is going to happen to him.

Mike Shiell’s The Wandering Melon for the 5th felt like a rerun to me. It wasn’t. But Shiell did do a variation on this joke in August. Both are built on the same whimsy of probability. It’s unlikely one will win a lottery. It’s unlikely one will die in a particular and bizarre way. What are the odds someone would have both things happen to them?


This and every Reading the Comics post should be at this link. Essays that include Ginger Meggs are at this link. Essays in which I discuss Lug Nuts are at this link. Essays mentioning Randolph Itch, 2 am, should be at this link. The many essays with a mention of Saturday Morning Breakfast Cereal are at this link. And essays where I’m inspired by something in The Wandering Melon should be at this link. And, what the heck, when I really discuss Cow and Boy it’s at this link. Real discussions of Motley are at this link. And my Fall 2018 Mathematics A-To-Z averages two new posts a week, now and through December. Thanks again for reading.

Reading the Comics, October 27, 2018: Surprise Rerun Edition


While putting together the last comics from a week ago I realized there was a repeat among them. And a pretty recent repeat too. I’m supposing this is a one-off, but who can be sure? We’ll get there. I figure to cover last week’s mathematically-themed comics in posts on Wednesday and Thursday, subject to circumstances.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th is a joking reminder that educational texts, including in mathematics, don’t have to be boring. We can have narrative thrust and energy. It’s a good reminder.

Caption: 'I wish all educational texts were written like Epictetus wrote.' Textbook: 'What, hapless wretch? Do you suppose int(sqrt(x^2 + x)dx = (x^2 + x)^{-1/2}? And when you eat, do you carry the food to your mouth or to your eyes? Slave!
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th of October, 2018.

As fits the joke, the bit of calculus in this textbook paragraph is wrong. \int \sqrt{x^2 + x} dx does not equal \left(x^2 + x\right)^{-\frac12} . This is even ignoring that we should expect, with an indefinite integral like this, a constant of integration. An indefinite integral like this is equal to a family of related functions. But it’s common shorthand to write out one representative function. But the indefinite integral of \sqrt{x^2 + x} is not \left(x^2 + x\right)^{-\frac12} . You can confirm that by differentiating \left(x^2 + x\right)^{-\frac12} . The result is nothing like \sqrt{x^2 + x} . Differentiating an indefinite integral should get the original function back. Here are the rules you need to do that for yourself.

As I make it out, a correct indefinite integral would be:

\int{\sqrt{x^2 + x} dx} = \frac{1}{4}\left( \left(2x + 1\right)\sqrt{x^2 + x} + \log \left|\sqrt{x} + \sqrt{x + 1} \right| \right)

Plus that “constant of integration” the value of which we can’t tell just from the function we want to indefinitely-integrate. I admit I haven’t double-checked that I’m right in my work here. I trust someone will tell me if I’m not. I’m going to feel proud enough if I can get the LaTeX there to display.

Berle: 'It was just a happy stroll through the gloomy graveyard when suddenly ... ' Spivak's Calculus, 3rd Edition appears. Berle: 'Math jumped out of nowhere!' Harvey: 'Drink it off.'
Stephen Beals’s Adult Children rerun for the 27th of October, 2018. It also appeared the 31st of March, 2018. I’m surprised it was that recent. I can’t blame Beals if he needed a break. This is a Halloween-ready example of the comic.

Stephen Beals’s Adult Children for the 27th has run already. It turned up in late March of this year. Michael Spivak’s Calculus is a good choice for representative textbook. Calculus holds its terrors, too. Even someone who’s gotten through trigonometry can find the subject full of weird, apparently arbitrary rules. And formulas like those in the above paragraph.

Manfred: 'How do you want to divide up the bill?' Wink: 'Lemme see it .. add that and that ... carry the two ... let's just split it htree ways.' Manfred: 'Nice try.' Dusty: 'Mr Surf 'n' Turf needs a calculator!'
Rob Harrell’s Big Top rerun for the 27th of October, 2018. It originally appeared the 13th of December, 2008.

Rob Harrell’s Big Top for the 27th is a strip about the difficulties of splitting a restaurant bill. And they’ve not even got to calculating the tip. (Maybe it’s just a strip about trying to push the group to splitting the bill a way that lets you off cheap. I haven’t had to face a group bill like this in several years. My skills with it are rusty.)

Jack-o-lantern standing on a scale: 'Hey! I weigh exactly 3.14 pounds!' Caption: 'Pumpkin Pi'.
Dave Whamond’s Reality Check for the 27th of October, 2018. Does the weight count if the jack-o-lantern is wearing sneakers?

Dave Whamond’s Reality Check for the 27th is a Pi Day joke shifted to the Halloween season.


And I have more Reading the Comics post at this link. Since it’s not true that every one of these includes a Saturday Morning Breakfast Cereal mention, you can find those that have one at this link. Essays discussing Adult Children, including the first time this particular strip appeared, are at this link. Essays with a mention of Big Top are at this link. And essays with a mention of Reality Check are at this link. Furthermore, this month and the rest of this year my Fall 2018 Mathematics A-To-Z should continue. And it is open for requests for more of the alphabet.

Reading the Comics, October 19, 2018: More Short Things Edition


At least, I’d thought the last half of last week’s comics were mostly things I could discuss quickly. Then Frank and Ernest went and sprawled on me. Such will happen.

Before I get to that, I did want to mention that Gregory Taylor’s paneling for votes for the direction his mathematics-inspired serial takes:

You may enjoy; at least, give it a try.

Thaves’s Frank and Ernest for the 18th is a bit of wordplay. There’s something interesting culturally about phrasing “lots of math, but no chemistry”. Algorithms as mathematics makes sense. Much of mathematics is about finding processes to do interesting things. Algorithms, and the mathematics which justifies them, can at least in principle be justified with deductive logic. And we like to think that the universe must make deductive-logical sense. So it is easy to suppose that something mathematical simply must make logical sense.

Frank: 'It didn't go well? That date was selected for you by a sophisticated statistical algorithm.' Ernest: 'Lots of math, but no chemistry.'
Thaves’s Frank and Ernest for the 18th of October, 2018. One might argue this just represents the algorithm not having enough data. That there are aspects to both people which were poorly modelled. Could happen.

Chemistry, though. It’s a metaphor for whatever the difference is between a thing’s roster of components and the effect of the whole. The suggestion is that it is mysterious and unpredictable. It’s an attitude strange to actual chemists, who have a rather good understanding of why most things happen. My suspicion is that this sense of chemistry is old, dating to before we had a good understanding of why chemical bonds work. We have that understanding thanks to quantum mechanics, and its mathematical representations.

But we can still allow for things that happen but aren’t obvious. When we write about “emergent properties” we describe things which are inherent in whatever we talk about. But they only appear when the things are a large enough mass, or interact long enough. Some things become significant only when they have enough chance to be seen.

Zeno: 'Honey, I'd love to, but it's not as if I can traverse infinite regions in finite time!' Caption: 'Fun Fact: Zeno never took out the garbage.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th of October, 2018. I wrote all this text assuming Weinersmith meant Zeno of Elea. It’d be a heck of a thing if he meant Zeno, the Omni-King of the 12 Universes that I’m told is a thing in Dragon Ball. I guess they have different character designs.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is about mathematicians’ favorite Ancient Greek philosopher they haven’t actually read. (In fairness, Zeno is hard to read, even for those who know the language.) Zeno’s famous for four paradoxes, the most familiar of which is alluded to here. To travel across a space requires travelling across half of it first. But this applies recursively. To travel any distance requires accomplishing infinitely many partial-crossings. How can you do infinitely many things, each of which take more than zero time, in less than an infinitely great time? But we know we do this; so, what aren’t we understanding? A callow young mathematics major would answer: well, pick any tiny interval of time you like. All but a handful of the partial-crossings take less than your tiny interval time. This seems like a sufficient answer and reason to chuckle at philosophers. Fine; an instant has zero time elapse during it. Nothing must move during that instant, then. So when does movement happen, if there is no movement during all the moments of time? Reconciling these two points slows the mathematician down.

Teacher: 'Todd, please come up to the chalkboard and do this fraction problem.' Todd: 'Uh-oh! It's late October, almost November! And hibernation is kicking in! I'll have the answer for ya next spring! Well, see ya!' Teacher: 'Todd! Get up here right now!' Student: 'Teacher, it's not safe to wake a hibernating T-Rex! Especially when I'm the first one he'll see!'
Patrick Roberts’s Todd the Dinosaur for the 19th of October, 2018. I’d imagine he would have a sufficient excuse in his arms not being able to reach the chalkboard, actually.

Patrick Roberts’s Todd the Dinosaur for the 19th mentions fractions. It’s only used to list a kind of mathematics problem a student might feign unconsciousness rather than do. And takes quite little space in the word balloon to describe. It’d be the same joke if Todd were asked to come up and give a ten-minute presentation on the Battle of Bunker Hill.

Penny: 'It says here the Rubik's Cube is still a top-selling gift for kids at Christmas.' Earl: 'Why for kids?? *I* can't even do it!' Burl: 'But it's great for kids. It keeps Timmy busy for hours, even though the poor kid doesn't realize yet that it's impossible to do.
Julie Larson’s The Dinette Set for the 19th of October, 2018. It ran originally the 12th of December, 2007. Among the stray, side, filler jokes, I like best that both Earl and Burl have mugs reading “And Dumber”.

Julie Larson’s The Dinette Set for the 19th mentions the Rubik’s Cube. Sometime I should do a proper essay about its mathematics. Any Rubik’s Cube can be solved in at most 20 moves. And it’s apparently known there are some cube configurations that take at least 20 moves, so, that’s nice to have worked out. But there are many approaches to solving a cube, none of which I am competent to do. Some algorithms are, apparently, easier for people to learn, at the cost of taking more steps. And that’s fine. You should understand something before you try to do it efficiently.

Venn diagram of Content Warning. Bubble A: Violence. Bubble B: Mature Subject Matter. Bubble C: Coarse Language. Intersection of A and B: The Bible. Intersection of B and C: Thanksgiving Dinner. Intersection of C and A: Sports. Intersection of A, B, and C: Life.
John Atkinson’s Wrong Hands for the 19th of October, 2018. So … life is the intersection of Thanksgiving, the Bible, and Sports? … I could see the case for that.

John Atkinson’s Wrong Hands for the 19th is the Venn Diagram joke for the week. Good to have one around.


This and my other Reading the Comics posts are available at this link. The essays mentioning Frank and Ernest should be at this link. For just the Reading the Comics posts with Saturday Morning Breakfast Cereal content try this link. Essays which talk about things raised by Todd the Dinosaur are at this link. Posts that write about The Dinette Set are at this link. And the essays based on Wrong Hands should be at this link. And do please stick around for more of my Fall 2018 Mathematics A-To-Z, with another post due tomorrow that I need to write today.

Reading the Comics, October 11, 2018: Under Weather Edition


I ended up not finding more comics on-topic on GoComics yesterday. So this past week’s mathematically-themed strips should fit into two posts well. I apologize for any loss of coherence in this essay, as I’m getting a bit of a cold. I’m looking forward to what this cold does for the A To Z essays coming Tuesday and Friday this week, too.

Stephen Beals’s Adult Children for the 7th uses Albert Einstein’s famous equation as shorthand for knowledge. I’m a little surprised it’s written out in words, rather than symbols. This might reflect that E = mc^2 is often understood just as this important series of sounds, rather than as an equation relating things to one another. Or it might just reflect the needs of the page composition. It could be too small a word balloon otherwise.

(In a darkened bar.) Harvey: 'Are they going to close?' Berle: 'They'll have to if this isn't fixed.' Harvey: 'E equals MC squared!' (The lights come on.) Berle 'Yay! ... Why did you yell that?' Harvey: 'Knowledge is power.'
Stephen Beals’s Adult Children for the 7th of October, 2018. I’m still not sure how I feel about this strip’s use of slightly offset panels like this.

Julie Larson’s The Dinette Set for the 9th continues the thread of tip-calculation jokes around here. I have no explanation for this phenomenon. In this case, Burl is doing the calculation correctly. If the tip is supposed to be 15% of the bill, and the bill is reduced 10%, then the tip would be reduced 10%. If you already have the tip calculated, it might be quicker to figure out a tenth of that rather than work out 15% of the original bill. And, yes, the characters are being rather unpleasantly penny-pinching. That was just the comic strip’s sense of humor.

Burl: 'So a 15% tip four two Monte Cristo platters would be $1.26' Dale: 'So ours would be the same as yours ... waidda minute! Today is 10% OFF for AARP members! So we times our total by 25% to figure the tip?' Burl: 'It's simple, Dale. Take 10% off the $1.26 tip, which is 12.6 cents, round that up to 13 cents, the minus that fro $1.26, and her tip is now $1.13.' Dale: 'Wow! I'm impressed! You did all that in your head! I'll bet I woulda given too much!'
Julie Larson’s The Dinette Set for the 9th of October, 2018. This is a rerun; it originally ran the 2nd of December, 2007.

Todd Clark’s Lola for the 9th take the form of your traditional grumbling about story problems. It also shows off the motif of updating of the words in a story problem to be awkwardly un-hip. The problem seems to be starting in a confounding direction anyway. The first sentence isn’t out and it’s introducing the rate at which Frank is shedding social-media friends over time and the rate at which a train is travelling, some distancer per time. Having one quantity with dimensions friends-per-time and another with dimensions distance-per-time is begging for confusion. Or for some weird gibberish thing, like, determining something to be (say) ninety mile-friends. There’s trouble ahead.

Lola: 'Sammy boy. How's middle school going?' Sammy: 'Stupid story problems.' Lola: 'Whatcha got? Maybe I can help.' Sammy: 'If Frank unfriends people at a rate of six per hour while on a train travelling ... '
Todd Clark’s Lola for the 9th of October, 2018. Don’t let your eye be distracted by the coloring job done on Lola’s eyeglasses in the last panel there.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th proposes naming a particular kind of series. A series is the sum of a sequence of numbers. It doesn’t have to be a sequence with infinitely many numbers in it, but it usually is, if it’s to be an interesting series. Properly, a series gets defined by something like the symbols in the upper caption of the panel:

\sum_{i = 1}^{\infty} a_i

Here the ‘i’ is a “dummy variable”, of no particular interest and not even detectable once the calculation is done. It’s not that thing with the square roots of -1 in thise case. ‘i’ is specifically known as the ‘index’, since it indexes the terms in the sequence. Despite the logic of i-index, I prefer to use ‘j’, ‘k’, or ‘n’. This avoids confusion with that square-root-of-minus-1 meaning for i. The index starts at some value, the one to the right of the equals sign underneath the capital sigma; in this case, 1. The sequence evaluates whatever the formula described by a_i is, for each whole number between that lowest ‘i’, in this case 1, and whatever the value above the sigma is. For the infinite series, that’s infinitely large. That is, work out a_i for every counting number ‘i’. For the first sum in the caption, that highest number is 4, and you only need to evaluate four terms and add them together. There’s no rule given for a_i in the caption; that just means that, in this case, we don’t yet have reason to care what the formula is.

On the blackboard: 'Solve: 24 + 12 + 6 + 3 + ... = ?' He put down 48. Woman: 'I ... wow. You've never studied series and you got it instantly.' Man: 'The 'plus three dots' part means 'plus 3', right?' Caption: 'New sequence type: Lucky Moron sequences. Definition: any convergent series such that 3 + (the first four terms) = (the infinite series)'.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th of October, 2018. And if you think that’s a not-really-needed name for a kind of series, note that MathWorld has a definition for the “FoxTrot Series” based on one problem from the FoxTrot comic strip from 1998.

This is the way to define a series if we’re being careful, and doing mathematics properly. But there are shorthands, and we fall back on them all the time. On the blackboard is one of them: 24 + 12 + 6 + 3 + \cdots . The \cdots at the end of a summation like this means “carry on this pattern for infinitely many terms”. If it appears in the middle of a summation, like 2 + 4 + 6 + 8 + \cdots + 20 it means “carry on this pattern for the appropriate number of terms”. In that case, it would be 10 + 12 + 14 + 16 + 18 .

The flaw with this “carry on this pattern” is that, properly, there’s no such thing as “the” pattern. There are infinitely many ways to continue from whatever the start was, and they’re all equally valid. What lets this scheme work is cultural expectations. We expect the difference between one term and the next to follow some easy patterns. They increase or decrease by the same amount as we’ve seen before (an arithmetic progression, like 2 + 4 + 6 + 8, increasing by two each time). They increase or decrease by the same ratio as we’ve seen before (a geometric progression, like 24 + 12 + 6 + 3, cutting in half each time). Maybe the sign alternates, or changes by some straightforward rule. If it isn’t one of these, then we have to fall back on being explicit. In this case, it would be that a_i = 24 \cdot \left(\frac{1}{2}\right)^{i - 1} .

The capital-sigma as shorthand for “sum” traces to Leonhard Euler, because of course. I’m finding it hard, in my copy of Florian Cajori’s History of Mathematical Notations, to find just where the series notation as we use it got started. Also I’m not finding where ellipses got into mathematical notation either. It might reflect everybody realizing this was a pretty good way to represent “we’re not going to write out the whole thing here”.

With something marked 30% off. First customer: 'This is normally 100 bucks. How much will it be at 30% off?' Val: '$70.' First customer, to second: 'See, I told you percents are the same as dollars.' Second customer: 'When you're right, you're right.' Val, thinking: 'I pity the next retailer who has to convince him otherwise.'
Norm Feuti’s Retail for the 11th of October, 2018. Yes, the comments include people explaining how people doing Common Core mathematics would never find an answer to “30% off $20”. Also a commenter who explains how one would, probably do it: “30% of 10 is 3. 20 is twice 20, so, 30% off 20 would be twice 3, or 6. So, 20 minus 6, or 14.” Followed by someone saying that if you did it by real math, it would be .7 times 20.

Norm Feuti’s Retail for the 11th riffs on how many people, fundamentally, don’t know what percentages are. I think it reflects thinking of a percentage as some kind of unit. We get used to measurements of things, like, pounds or seconds or dollars or degrees or such that are fixed in value. But a percentage is relative. It’s a fraction of some original quantity. A difference of (say) two pounds in weight is the same amount of weight whatever the original was; why wouldn’t two percent of the weight behave similarly? … Gads, yes, I feel for the next retailer who gets these customers.

I think I’ve already used the story from when I worked in the bookstore about the customer concerned whether the ten-percent-off sticker applied before or after sales tax was calculated. So I’ll only share if people ask to hear it. (They won’t ask.)


When I’m not getting a bit ill, I put my Reading the Comics posts at this link. Essays which mention Adult Children are at this link. Essays with The Dinette Set discussions should be at this link. The essays inspired by Lola are at this link. There’s some mention of Saturday Morning Breakfast Cereal in essays at link, or pretty much every Reading the Comics post. And Retail gets discussed at this link.

Reading the Comics, September 22, 2018: Last Chance Edition


I plan tomorrow to have another of my Mathematics A To Z posts. This weekend I’ll publish this month’s Playful Mathematics Blog Carnival. So if you’ve seen any web site, blog, video, podcast, or other reference that had something that delighted and taught you something, this is your last chance to let me know, and let my audience know about it. Please leave a comment if you know about anything I ought to see. Thank you.

Mark Tatulli’s Lio for the 20th is a numerals and a wordplay joke. It is not hard to make numerals tattooed on a person an alarming thing. But when done with (I trust) the person’s consent, and done whimsically like this, it’s more a slightly odd bit of play.

(Lio walks past a man who's got many different renderings of the numeral '2' on his arms. He comes up to a storefront: 'Tattwos'.)
Mark Tatulli’s Lio for the 20th of September, 2018. It … seems a little out of the usual run of Lio‘s sense of humor, but then I don’t know where else Tatulli could have used the joke.

Tony Cochrane’s Agnes for the 21st is ultimately a strip about motivating someone to learn arithmetic. Agnes’s reasoning is sound, though. If the only reason to learn this unpleasant chore is because your job may need it, why not look at another job? We wouldn’t try to convince someone who didn’t want to learn French that they’ll need it for their job as … a tour guide in Quebec? There’s plenty of work that doesn’t need that. I suspect kids don’t buy “this is good for your future job” as a reason. Even if it were, general education should not be job training either.

Teacher: 'Agnes, please finish the third division problem.' Agnes: 'I don't think so. I'm going to be a diesel mechanic.' Teacher: 'Do you even know what that is?' Agnes: 'No, but I found a brochure in Grandma's sock drawer, and the people in it looked rapturous about work.' Teacher: 'Diesel mechanics need to know division.' Agnes: 'Fine, I'll go with one of the others. How about a mortician?'
Tony Cochrane’s Agnes for the 21st of September, 2018. Agnes doesn’t pursue being a mortician, at least this time, and instead tries out keeping a journal.

Juba’s Viivi and Wagner for the 21st gives Wagner a short-lived ambition to be a wandering mathematician. The abacus serves as badge of office. There are times and places that his ambition wouldn’t be completely absurd. Before the advent of electric and electronic computing, people who could calculate were worth hiring for their arithmetic. In 18th Century London there was a culture of “penny universities”, people with academic training making a living by giving lectures and courses to whatever members of the public cared to come to their talk, often in coffee-houses or barns.

Wagner: 'I decided to be a wandering mathematician. I already bought an abacus. I'll earn my living by performing simple additions and subtractions.' Viivi : 'You'll probably starve to death.' Wagner: 'I don't do probability calculus.'
Juba’s Viivi and Wagner for the 21st of September, 2018. This puts me in mind of a Peanuts where Sally Brown wondered about what if their father lost their job. What could she do to help the family earn its living? She suggested: she could make her bed.

Mathematicians learn that there used to be public spectacles, mathematicians challenging one another to do problems, with real cash or jobs on the line. They learn this because one such challenge figures in to the story of Gerolamo Cardano and Niccolò Fontana, known as Tartaglia. It’s about how we learned formulas to solve some kinds of polynomials. You may sense uncertainty in my claim there. It’s because it turns out it’s hard to find clear records of this sort of challenge outside the Cardano-Tartaglia match. That isn’t to say these things weren’t common. It’s just that I’ve been slowly learning to be careful about my claims.

(I’m aided here by a startling pair of episodes of The History of Philosophy Without Any Gaps podcast. This pair — “Trivial Pursuits: Fourteenth Century Logic” and “Sara Uckleman on Obligations” — describe a fascinating logic game that sounds like it would still be a great party game, for which there’s numerous commentaries and rule sets and descriptions of how to play. But no records of people actually ever playing it, or talking about games they had played, or complaining about being cheated out of a win or stuff like that. It’s a strong reminder to look closely at what your evidence does support.)

Father: 'Son, I know you pretended to be sick today so you could skip school.' Kid: 'But I didn't!' Father: 'Shhh. It's OK. Don't tell Mom, but I got tickets to a luchador show this afternoon.' Kid: 'Oh boy!' [ LATER ... ] Luchador: 'Raaa! I'm the masked Karl Weierstrass and I'm WRESTLING with how to rigorously define the limit of a function!' Kid: 'Nooooooooooooo!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 22nd of September, 2018. So do you think the father is really into the luchador show in the last panel or is he just going along with this while his kid comes to hate promises of fun?

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 22nd is the comforting return of Zach Weinersmith to these essays. And yes, it’s horrible parenting to promise something fun and have it turn out to be a mathematics lecture, but that’s part of the joke.

Karl Weierstrass was a real person, and a great mathematician best known for giving us a good, rigorous idea of what a limit is. We need limits because, besides their being nice things to have, calculus depends on them. At least, calculus depends on thinking about calculations on infinitely many things. Or on things infinitesimally small. Trying to do this works pretty well, much of the time. But you can also start calculating like this and get nonsense. How to tell whether your particular calculation works out or is nonsense?

Weierstrass worked out a good, rigorous idea for what we mean by a limit. It mostly tracks with what we’d intuitively expect. And it avoids all the dangerous spots we’ve noticed so far. Particularly, it doesn’t require us to ever look at anything that’s infinitely vast, or infinitesimally small. Anything we calculate on is done with regular arithmetic, that we’re quite confident in. But it lets us draw conclusions about the infinitely numerous or tiny. It’s brilliant work. When it’s presented to someone in the start of calculus, it leaves them completely baffled but they can maybe follow along with the rules. When it’s presented to mathematics majors in real analysis, it leaves them largely baffled but they can maybe follow along with the reasons. Somewhere around grad school I got comfortable with it, even excited. Weierstrass’s sort of definition turns up all over the place in real and in functional analysis. So at the least you get very comfortable with it.

So it is part of Weinersmith’s joke that this is way above that kid’s class level. As a joke, that fails for me. The luchador might as well be talking complete nonsense and the kid would realize that right away. There’s not the threat that this is something he ought to be able to understand. But it will probably always be funny to imagine mathematician wrestlers. Can count on that. I didn’t mean that as a joke, but you’ll notice I’m letting it stand.


And with that, you know what I figure to post on Sunday. It and my other Reading the Comics posts should be at this tag. Other appearances of Lio should be at this link. The mentions of Agnes should be at this link. Essays with some mention of Viivi and Wagner will be at this link, although it’s a new tag, so who knows how long it’ll take for the next to appear? And other essays with Saturday Morning Breakfast Cereal will be at this link when there’s any to mention.

Reading the Comics, 1 September 2018: Retirement Of A Tag Edition


I figure to do something rare, and retire one of my comic strip tags after today. Which strip am I going to do my best to drop from Reading the Comics posts? Given how many of the ones I read are short-lived comics that have been rerun three or four times since I started tracking them? Read on and see!

Bill Holbrook’s On The Fastrack for the 29th of August continues the sequence of Fi talking with kids about mathematics. My understanding was that she tried to give talks about why mathematics could be fun. That there are different ways to express the same number seems like a pretty fine-grain detail to get into. But this might lead into some bigger point. That there are several ways to describe the same thing can be surprising and unsettling to discover. That you have, when calculating, the option to switch between these ways freely can be liberating. But you have to know the option is there, and where to look for it. And how to see it’ll make something simpler.

Wendy: 'I never thought Fi would have a talent for teaching.' Dethany: 'It surprised her, too. But something about her demeanor appeals to kids.' (At the class.) Fi: 'See? Two-sixth of a zombie is the same as one-third ... '
Bill Holbrook’s On The Fastrack for the 29th of August, 2018. What … what graphic does she have on-screen?

Bill Holbrook’s On The Fastrack for the 30th of August gets onto a thread about statistics. The point of statistics is to describe something complicated with something simple. So detail must be lost. That said, there are something like 2,038 different things called “average”. Each of them has a fair claim to the term, too. In Fi’s example here, 73 degrees (Fahrenheit) could be called the average as in the arithmetic mean, or average as in the median. The distribution reflects how far and how often the temperature is from 73. This would also be reflected in a quantity called the variance, or the standard deviation. Variance and standard deviation are different things, but they’re tied together; if you know one you know the other. It’s just sometimes one quantity is more convenient than the other to work with.

Fi: 'Numbers don't lie ... but the unscrupulous can get them to say whatever they want. Like when the boss claims the average temperature in your office is 73 degrees when it's really kept at 63 degrees in the winter and 83 degrees in the summer.' Kid: 'Our principal does that.'
Bill Holbrook’s On The Fastrack for the 30th of August, 2018. Somebody nag me sometime to tell the story about when I used Skylab’s torn-away meteorite shield for a heat-flow problem in a differential equations class. Thank you.

Bill Holbrook’s On The Fastrack for the 1st of September has Fi argue that apparent irrelevance makes mathematics boring. It’s a common diagnosis. I think I’ve advanced the claim myself. I remember a 1980s probability textbook asking the chance that two transistors out of five had broken simultaneously. Surely in the earlier edition of the textbook, it was two vacuum tubes out of five. Five would be a reasonable (indeed, common) number of vacuum tubes to have in a radio. And it would be plausible that two might be broken at the same time.

Kid: 'I think math's boring.' Fi: 'That's because you've been taught with 75-year-old word problems. They just need a little updating. ... Like, if your tweet is retweeted by ten people ... who each share it with ten MORE people ... at what point can it be said to go viral?' (Everyone has hands up.)
Bill Holbrook’s On The Fastrack for the 1st of September, 2018. Not to question Holbrook’s writing, since he somehow maintains three successful daily comic strips and may be presumed to know what he’s doing, but shouldn’t this have come before the strip from the 29th?

It seems obvious that wanting to know an answer makes it easier to do the work needed to find it. I’m curious whether that’s been demonstrated true. Like, it seems obvious that a reference to a thing someone doesn’t know anything about would make it harder to work on. But does it? Does it distract someone trying to work out the height of a ziggurat based on its distance and apparent angle, if all they know about a ziggurat is their surmise that it’s a thing whose height we might wish to know?

Randolph's dream: a pirate at a schooldesk. 'Algebra pretty much put pirates out of business.' Teacher: 'If ax^2 + bx + c = 0, what is x?' (The pirate looks at the treasure map, marked x, sweating.' Footer joke: the teacher asks, '15 men on a dead man's chest, yo ho ho, and a bottle of rum equals what?'
Tom Toles’s Randolph Itch, 2 am rerun for the 30th of August, 2018. It originally ran the 13th of January, 2000.

Tom Toles’s Randolph Itch, 2 am rerun for the 30th of August is an old friend that’s been here a couple times. I suppose I do have to retire the strip from my Reading the Comics posts, at least, although I’m still amused enough by it to keep reading it daily. Simon Garfield’s On The Map, a book about the history of maps, notes that the X-marks-the-spot thing is an invention of the media. Robert Louis Stevenson’s Treasure Island particularly. Stevenson’s treasure map, Garfield notes, had to be redrawn from the manuscript and the author’s notes. The original went missing in the mail to the publishers. I just mention because I think that adds a bit of wonder to the treasure map. And since, I guess, I won’t have the chance to mention this again.

Kid: 'Mom, Dad, why do you have a giant inflatable Klein bottle hidden in the closet?' Mom: 'Compromise. I'll say nothing more. NOW GO WASH YOUR HANDS.' Underneath, a Venn diagram, with one bubble 'Having Sex Inside', the other 'Having Sex Outside', and the intersection 'Having Sex Near A Non-Orientable Surface'.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 30th of August, 2018. Interesting to me is that either the panel comic by itself, or the Venn diagram by itself, would be a sufficient joke. The panel would be a cryptic one, one that probably attracted ‘I don’t get it’ responses, but it’d be decipherable. The Venn Diagram one would be fine, but wouldn’t have the tension and energy of the full strip.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 30th of August satisfies the need for a Venn Diagram joke this time around. It’s also the strange-geometry joke for the week. Klein bottles were originally described by Felix Klein. They exist in four (or more) dimensions, in much the way that M&oum;bius strips exist in three. And like the M&oum;bius strip the surface defies common sense. You can try to claim some spot on the surface is inside and some other spot outside. But you can get from your inside to your outside spot in a continuous path, one you might trace out on the surface without lifting your stylus.

If you were four-dimensional. Or more. If we were to see one in three dimensions we’d see a shape that intersects itself. As beings of only three spatial dimensions we have to pretend that doesn’t happen. It’s the same we we pretend a drawing of a cube shows six squares all of equal size and connected at right angles to one another, even though the drawing is nothing like that. The bottle-like shape Weinersmith draws is, I think, the most common representation of the Klein bottle. It looks like a fancy bottle, and you can buy one as a novelty gift for a mathematician. I don’t need one but do thank you for thinking of me. MathWorld shows another representation, a figure-eight-based one which looks to me like an advanced pasta noodle. But it doesn’t look anything like a bottle.

Abstract's Bar and Grille. Once again, Eric the Circle's pick-up line backfires ... and he is left confused and speechless. Eric: 'You're acutey. What's your sign?' Triangle: 'Opposite over hypotenuse. What's yours?'
Eric the Circle for the 31st of August, 2018. … Shouldn’t this be with a right triangle?

Eric the Circle for the 31st of August, this one by JohnG, is a spot of wordplay. The pun here is the sine of an angle in a (right) triangle. That would be the length of the leg opposite the angle divided by the length of the hypotenuse. This is still stuff relevant to circles, though. One common interpretation of the cosine and sine of an angle is to look at the unit circle. That is, a circle with radius 1 and centered on the origin. Draw a line segment opening up an angle θ from the positive x-axis. Draw it counterclockwise. That is, if your angle is a very small number, you’re drawing a line segment that’s a little bit above the positive x-axis. Draw the line segment long enough that it touches the unit circle. That point where the line segment and the circle intersect? Look at its Cartesian coordinates. The y-coordinate will be the sine of θ. The x-coordinate will be the cosine of θ. The triangle you’re looking at has vertices at the origin; at x-coordinate cosine θ, y-coordinate 0; and at x-coordinate cosine θ, y-coordinate sine θ.

[ When Zippy was three, he said the darnedest things ] Zippy: 'X plus Y divided by Shirley Booth equals Soft Serve.' [ At the age of 11, he continued to amaze and impress his parents. ] 'If I had the powers of Spider-Man and the costume of Mighty Mouse, I could understand algebra!' [ He mellowed a bit at 16 and thought deep thoughts about the universe and stuff. ] 'If you lined up ALL the jars of Bosco ever manufactured, they'd form a ring all the way around the Earth and wind up conking Einstein on the bean in Newark!' [ As a ADULT, Zippy knows that not everyone appreciates his surreal spoutings, so he often muses to himself. ] Zippy, thinking: 'If I drew Mary Worth, she'd drive a 1958 two-tone Metro!'
Bill Griffith’s Zippy the Pinhead for the 1st of September, 2018. Somebody nag me sometime to save that last panel for my next What’s Going On In Mary Worth plot recap.

Bill Griffith’s Zippy the Pinhead for the 1st of September is its usual sort of nonsense, the kind that’s up my alley. It does spend two panels using arithmetic and algebra as signifiers of intelligence, or at least thoughtfulness.


My other Reading the Comics posts should appear at this link. Other essays with On The Fastrack are at this link. The essays that mentioned Randolph Itch, 2 am, are at this link, and I suppose this will be the last of them. We’ll see if I do succeed in retiring the tag. Other appearances by Saturday Morning Breakfast Cereal are at this link. The strip comes up here a lot. Eric the Circle comics should be at this link. And other essays with Zippy the Pinhead mentions should be at this link. Thank you.

Reading the Comics, August 16, 2018: Recursive Edition


This edition of Reading the Comics can be found at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is a fractals joke. Benoit Mandelbrot became the centerpiece of the big fractals boom in pop mathematics in the 80s and 90s. This was thanks to a fascinating property of complex-valued numbers that he discovered and publicized. The Mandelbrot set is a collection of complex-valued numbers. It’s a border, properly, between two kinds of complex-valued numbers. This boundary has this fascinating shape that looks a bit like a couple kidney beans surrounded by lightning. That’s neat enough.

What’s amazing, and makes this joke anything, is what happens if you look closely at this boundary. Anywhere on it. In the bean shapes or in the lightning bolts. You find little replicas of the original shape. Not precisely the original shape. No two of these replicas are precisely identical (except for the “complex conjugate”, that is, something near the number -1 + 1 \imath has a mirror image near -1 - 1 \imath ). None of these look precisely like the original shape. But they look extremely close to one another. They’re smaller, yes, and rotated relative to the original, and to other copies. But go anywhere on this boundary and there it is: the original shape, including miniature imperfect copies, all over again.

Man: 'Oh, dang it. Here comes Mandelbrot.' Woman: 'Why don't you like him?' Man: 'He's always trying to get people to look at his mole.' Mandelbrot: 'Hey guys, wanna see something?' (On his cheek is a tiny replica of his whole face, including a mole that is presumably another tiny head.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th of August, 2018. This by the way is an acceptable sketch of Mandelbrot, although at least in the picture Wikipedia has of him in 2010 the only thing that could be dubbed a mole looks more like just a shadow to me.

The Mandelbrot Set itself — well, there are a bunch of ways to think of it. One is in terms of something called the Julia Set, named for Gaston Julia. In 1918 he published a massive paper about the iteration of rational functions. That is, start with some domain and a function rule; what’s the range? Now if we used that range as the domain again, and used the same rule for the function, what’s the range of that range? If we use the range-of-that-range as the domain for the same function rule, what’s the range-of-the-range-of-the-range? The particular function here has one free parameter, a single complex-valued number. Depending on what it is, the range-of-the-range-of-the-range-etc becomes a set that’s either one big blob or a bunch of disconnected blobs. The Mandelbrot Set is the locus of parameters separating the one-big-blob from the many-disconnected-blob outcomes.

By the way, yes, Julia published this in 1918. The work was amazing. It was also forgotten. You can study this stuff analytically, but it’s hard. To visualize it you need to do incredible loads of computation. So this is why so much work lay fallow until the 1970s, when Mandelbrot could let computers do incredible loads of computation, and even draw some basic pictures.

A thousand monkeys at a thousand typewriters ... will eventually write 'Hamlet'. A thousand cats at a thousand typewriters ... will tell you go to write your own danged 'Hamlet'.
Doug Savage’s Savage Chickens for the 14th of August, 2018. I appreciate the one monkey in the first panel who thinks he’s on to something here.

Doug Savage’s Savage Chickens for the 14th is another instance of the monkeys-at-typewriters joke. I’ve written about this and the history of the monkeys-at-typewriters bit recently enough to feel comfortable pointing people there. It’s interesting that monkeys should have a connotation of reliably random typewriting, while cats would be reliably not doing something. But that’s a cultural image that’s a little too far from being mathematics for me to spend 800 words discussing.

Cavemen sitting at a stone table. 'It's a calendar, Blork. Till we invent numbers, it has only 'today', 'yesterday', and 'we'll see', see?'
Thom Bleumel’s Birdbrains for the 15th of August, 2018. I question the plausibility of none of these people tuning out the meeting to read their tablets instead.

Thom Bleumel’s Birdbrains for the 15th is a calendars joke. Numbers come into play since, well, it seems odd to try tracking large numbers of dates without some sense of arithmetic. Also, likely, without some sense of geometry. Calendars are much used to forecast coming events, such as New and Full Moons or the seasons. That takes basic understanding of how to locate things in the sky to do at all. It takes sophisticated understanding of how to locate things in the sky to do well.

A 5, holding hands in front of a 3's eyes: 'Don't look, sweetie!' A 9: 'Get a room!' A 2: 'Disgusting!' An 8: 'There are children watching!' The scandal: 4 and 7 standing on either side of an x. And *smiling*.
Scott Hilburn’s The Argyle Sweater for the 16th of August, 2018. Oh, these people would be at least as scandalized by a ÷ sign.

Scott Hilburn’s The Argyle Sweater for the 16th is the first anthropomorphic-numerals joke around here in like three days. Certainly, the scandalous thing is supposed to be these numbers multiplying out in public where anyone might see them. I wonder if any part of the scandal should be that multiplication like this has to include three partners: the 4, the 7, and the x. In algebra we get used to a convention in which we do without the ‘x’. Just placing one term next to another carries an implicit multiplication: ‘4a’ for ‘4 times a’. But that convention fails disastrously with numerals; what should we make of ’47’? We might write 4(7), or maybe (4)(7), to be clear. Or we might put a little centered dot between the two, 4 \cdot 7 . The ‘x’ by that point is reserved for “some variable whose value isn’t specified”. And it would be weird to write ‘4 times x times 7’. It wouldn’t be wrong; it’d just look weird. It would suggest you were trying to emphasize a point. I’ve probably done it in one of my long derivation-happy posts.


Other essays about comic strips are at this link. When I’ve talked about Saturday Morning Breakfast Cereal I’ve tried to make sure it turns up at this link. Essays in which I’ve discussed Savage Chickens should be at this link. The times I’ve discussed Birdbrains should be at this link. And other essays describing The Argyle Sweater are at this link.

Reading the Comics, August 4, 2018: August 4, 2018 Edition


And finally, at last, there’s a couple of comics left over from last week and that all ran the same day. If I hadn’t gone on forever about negative Kelvin temperatures I might have included them in the previous essay. That’s all right. These are strips I expect to need relatively short discussions to explore. Watch now as I put out 2,400 words explaining Wavehead misinterpreting the teacher’s question.

Dave Whamond’s Reality Check for the 4th is proof that my time spent writing about which is better, large numbers or small last week wasn’t wasted. There I wrote about four versus five for Beetle Bailey. Here it’s the same joke, but with compound words. Well, that’s easy to take care of.

[ Caption: Most people have a forehead --- Dave has a Five-Head. ] (Dave has an extremely tall head with lots of space between his eyebrows and his hair.) Squirrel in the corner: 'He'll need a 12-gallon hat.'
Dave Whamond’s Reality Check for the 4th of August, 2018. I’m sure it’s a coincidence that the tall-headed person shares a name with the cartoonist.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is driving me slightly crazy. The equation on the board looks like an electrostatics problem to me. The ‘E’ is a common enough symbol for the strength of an electric field. And the funny-looking K’s look to me like the Greek kappa. This often represents the dielectric constant. That measures how well an electric field can move through a material. The upside-down triangles, known in the trade as Delta, describe — well, that’s getting complicated. By themselves, they describe measuring “how much the thing right after this changes in different directions”. When there’s a x symbol between the Delta and the thing, it measures something called the “curl”. This roughly measures how much the field inspires things caught up in it to turn. (Don’t try passing this off to your thesis defense committee.) The Delta x Delta x E describes the curl of the curl of E. Oh, I don’t like visualizing that. I don’t blame you if you don’t want to either.

Professor Ridley: 'Imagine an infinitely thin rod. Visualize it but don't laugh at it. I know it's difficult. Now, the following equations hold for ... ' [ Caption: Professor Ridley's cry for help goes unnoticed. ]
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th of August, 2018. Really not clear what the cry for help would be about. Just treat the rod as a limiting case of an enormous number of small spheres placed end to end and you’re done.

Anyway. So all this looks like it’s some problem about a rod inside an electric field. Fine enough. What I don’t know and can’t work out is what the problem is studying exactly. So I can’t tell you whether the equation, so far as we see it, is legitimately something to see in class. Envisioning a rod that’s infinitely thin is a common enough mathematical trick, though. Three-dimensional objects are hard to deal with. They have edges. These are fussy to deal with. Making sure the interior, the boundary, and the exterior match up in a logically consistent way is tedious. But a wire? A plane? A single point? That’s easy. They don’t have an interior. You don’t have to match up the complicated stuff.

For real world problems, yeah, you have to deal with the interior. Or you have to work out reasons why the interiors aren’t important in your problem. And it can be that your object is so small compared to the space it has to work in that the fact it’s not infinitely thin or flat or smooth just doesn’t matter. Mathematical models, such as give us equations, are a blend of describing what really is there and what we can work with.

Lotto official looking over a burnt, shattered check: 'What are the ODDS?! First he wins the lottery and then he gets struck by lightning!'
Mike Shiell’s The Wandering Melon for the 4th of August, 2018. Still, impressive watchband that it’s stood up to all that trouble.

Mike Shiell’s The Wandering Melon for the 4th is a probability joke, about two events that nobody’s likely to experience. The chance any individual will win a lottery is tiny, but enough people play them that someone wins just about any given week. The chance any individual will get struck by lightning is tiny too. But it happens to people. The combination? Well, that’s obviously impossible.

In July of 2015, Peter McCathie had this happen. He survived a lightning strike first. And then won the Atlantic Lotto 6/49. This was years apart, but the chance of both happening the same day, or same week? … Well, the world is vast and complicated. Unlikely things will happen.


And that’s all that I have for the past week. Come Sunday I should have my next Reading the Comics post, and you can find it and other essays at this link. Other essays that mention Reality Check are at this link. The many other essays which talk about Saturday Morning Breakfast Cereal are at this link. And other essays about The Wandering Melon are at this link. Thanks.

Reading the Comics, July 7, 2018: Mutt and Jeff Relettering Scandal Edition


I apologize for not having a more robust introduction here. My week’s been chopped up by concern with the health of the older of our rabbits. Today’s proved to be less alarming than we had feared, but it’s still a lot to deal with. I appreciate your kind thoughts. Thank you.

Meanwhile the comics from last week have led me to discover something really weird going on with the Mutt and Jeff reruns.

Charles Schulz’s Peanuts Classics for the 6th has the not-quite-fully-formed Lucy trying to count the vast. She’d spend a while trying to count the stars and it never went well. It does inspire the question of how to count things when doing a simple tally is too complicated. There are many mathematical approaches. Most of them are some kind of sampling. Take a small enough part that you can tally it, and estimate the whole based on what your sample is. This can require ingenuity. For example, when estimating our goldfish population, it was impossible to get a good sample at one time. When tallying the number of visible stars in the sky, we have the problem that the Galaxy has a shape, and there are more stars in some directions than in others. This is why we need statisticians.

Lucy, going out in twilight with a pencil and sheet of paper: 'I'm going to count all the stars even if it kills me! People say I'm crazy, but I know I'm not , and that's what counts! I think I'll just sit here until it gets dark. This way I can take my time counting the stars. I'll mark 'em down as they come out. HA! There's the first one ... dum te ta te dum. There's another one. Two, three, four ... this is a cinch. Five, six, oh oh! SevenEightNineTen ... ElevenTwelveThirteen Oh, MY! They're coming out all over! SLOW DOWN! 21, 22, 23, 24, ... 35, 35, 40! Whew! (Gasp, gasp!) 41, 42 ... ' (Defeated Lucy sitting on the curb, exhausted, beneath the night sky.) 'Rats!'
Charles Schulz’s Peanuts Classics for the 6th of July, 2018. It originally ran the 4th of April, 1954. That is an adorable little adding machine and stool that Lucy has in the title panel there.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th looks initially like it’s meant for a philosophy blog’s Reading the Comics post. It’s often fruitful in the study of ethics to ponder doing something that is initially horrible, but would likely have good consequences. Or something initially good, but that has bad effects. These questions challenge our ideas about what it is to do good or bad things, and whether transient or permanent effects are more important, and whether it is better to be responsible for something (or to allow something) by action or inaction.

It comes to mathematics in the caption, though, and with an assist from the economics department. Utilitarianism seems to offer an answer to many ethical problems. It posits that we need to select a primary good of society, and then act so as to maximize that good. This does have an appeal, I suspect even to people who don’t thrill of the idea of finding the formula that describes society. After all, if we know the primary good of society, why should we settle for anything but the greatest value of that good? It might be difficult in practice, say, to discount the joy a musician would bring over her lifetime with her performances fairly against the misery created by making her practice the flute after school when she’d rather be playing. But we can imagine working with a rough approximation, at least. Then the skilled thinkers point out even worse problems and we see why utilitarianism didn’t settle all the big ethical questions, even in principle.

Professor: 'Suppose you want to kill a baker. But, if you kill him, a bunch of starving people will get access to his bread. Should you do it anyway?' Caption: 'All moral dilemmas can be rephrased as evil-maximization problems.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th of July, 2018. Confess I’m not sure the precise good-maximization reversal of this. I suppose it’s implying that the baker is refusing to give bread to starving people who can’t pay, and the hungry could alleviate the problem a while by eating the rich?

The mathematics, though. As Weinersmith’s caption puts it, we can phrase moral dilemmas as problems of maximizing evil. Typically we pose them as ones of maximizing good. Or at least of minimizing evil. But if we have the mechanism in place to find where evil is maximized, don’t we have the tools to find where good is? If we can find the set of social parameters x, y, and z which make E(x, y, z) as big as possible, can’t we find where -E(x, y, z) is as big, too? And isn’t that then where E(x, y, z) has to be smallest?

And, sure. As long as the maximum exists, or the minimum exists. Maybe we can tell whether or not there is one. But this is why when you look at the mathematics of finding maximums you realize you’re also doing minimums, or vice-versa. Pretty soon you either start referring to what you find as extremums. Or you stop worrying about the difference between a maximum and a minimum, at least unless you need to check just what you have found. Or unless someone who isn’t mathematically expert looks at you wondering if you know the difference between positive and negative numbers.

Jeff: 'You're such a fool, I'll bet you can't solve this simple problem!' Mutt: 'Which problem?' Jeff: 'If five men can eat a ham in five minutes, how long it will take ten men to eat that same ham?' Mutt: 'Well, some people eat slower.' Jeff: 'See? You just can't do it!' Mutt: 'Neither can you! It can't be solved!' Jeff: 'You say it can't be solved? Why?' Mutt: 'Because the first five men have ALREADY eaten the ham!'
Bud Fisher’s Mutt and Jeff for the 7th of July, 2018. So I found a previous iteration of this strip, from the 21st of February, 2015. They had relettered things, changing the wording slightly and making it overall somehow clunkier. The thing is, that 2015 strip looks to me like it might be a computer-lettered typeface too; look at the C’s, and the little loops on top of the letters. On the other hand, there’s some variation in the ? marks there. I understand relettering the more impenetrable old strips, especially if they don’t have the original material and have to go from archived newspaper prints. But the 2015 edition seems quite clear enough; why change that?

Bud Fisher’s Mutt and Jeff for the 7th has run here before. Except that was before they redid the lettering; it was a roast beef in earlier iterations. I was thinking to drop Mutt and Jeff from my Reading the Comics routine before all these mysteries in the lettering turned up. Anyway. The strip’s joke starts with a work-rate problems. Given how long some people take to do a thing, how long does it take a different number of people to do a thing? These are problems that demand paying attention to units, to the dimensions of a thing. That seems to be out of fashion these days, which is probably why these questions get to be baffling. But if eating a ham takes 25 person-minutes to do, and you have ten persons eating, you can see almost right away how long to expect it to take. If the ham’s the same size, anyway.

Teacher: 'Can you tell me how many triangles are in this diagram?' (It's an equilateral triangle, divided into thirds horizontally, and with the angle up top trisected, so that there are nine discrete figures inside.) Nancy, with a dozen scraps of used paper strewn around: 'Can you tell me how many pages we have to waste trying to solve this accursed puzzle?'
Olivia Jaimes’s Nancy for the 7th of July, 2018. There’s some real Old People Complaining in the comments, by the way, about how dare Nancy go sassing her elders like that. So, if you want to read those comments, judge wisely.

Olivia Jaimes’s Nancy for the 7th is built on a spot of recreational mathematics. Also on the frustration one can have when a problem looks like it’s harmless innocent fun and turns out to take just forever and you’re never sure you have the answers just right. The commenters on GoComics.com have settled on 18. I’m content with that answer.


Care for more of this? You can catch all my Reading the Comics posts at this link. Essays with Saturday Morning Breakfast Cereal content are at this link. Essays with Peanuts are at this link. Those with Mutt and Jeff are at this link. And those with Nancy are here. Thank you.

Reading the Comics, June 29, 2018: Chuckle and Breakfast Cereal Edition


The last half of last week was not entirely the work of Chuckle Brothers and Saturday Morning Breakfast Cereal. It seemed like it, though. Let’s review.

Patrick Roberts’s Todd the Dinosaur for the 28th is a common sort of fear-of-mathematics joke. In this case the fear of doing arithmetic even when it is about something one would really like to know. I think the question got away from Todd, though. If they just wanted to know whether they had enough money, well, they need twelve dollars and have seven. Subtracting seven from twelve is only needed if they want to know how much more they need. Which they should want to know, but wasn’t part of the setup.

Kid: 'Do we have enough money to go to the movie?' Todd: 'Let's see! You ahve four dollars and I have three dollars. That's seven. The movie is twelve dollars for both of us. So twelve take away seven is ... *GASP* Oh no! I accidentally did math!' Kid: 'So?' Todd: 'This is SUMMER!' Kid: 'I don't even know you!'
Patrick Roberts’s Todd the Dinosaur for the 28th of June, 2018. I’m sorry, I don’t know the kid’s name.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 28th uses mathematics as the sine qua non of rocket science. As in, well, the stuff that’s hard and takes some real genius to understand. It’s not clear to me that the equations are actually rocket science. There seem to be a shortage of things in exponentials to look quite right to me. But I can’t zoom in on the art, so, who knows just what might be in there.

Professor-type in front of a class labelled Rocket Science 101: 'Doesn't ANYBODY understand this stuff?'
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 28th of June, 2018. It originally ran the 16th of July, 2009. Relatable.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a set theory joke. Or a logic joke, anyway. It refers to some of the mathematics/logic work of Bertrand Russell. Among his work was treating seriously the problems of how to describe things defined in reference to themselves. These have long been a source of paradoxes, sometimes for fun, sometimes for fairy-tale logic, and sometimes to challenge our idea of what we mean by definitions of things. Russell made a strong attempt at describing what we mean when we describe a thing by reference to itself. The iconic example here was the “set of all sets not members of themselves”.

Caption: 'Nobody liked Bertrand Russell's scavenger hunts.' Items to find: 'The list of all lists that do not list themselves. (List here).'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th of June, 2018. Well, among other things, wouldn’t there be infinitely many such lists? Unless this description were enough to describe them all, by being a description of what to do to get you all of them?

Russell started out by trying to find some way to prove Georg Cantor’s theorems about different-sized infinities wrong. He worked out a theory of types, and what kinds of rules you can set about types of things. Most mathematicians these days prefer to solve the paradox with a particular organization of set theory. But Russell’s type theory still has value, particularly as part of the logic behind lambda calculus. This is an approach to organizing relationships between things that can do wonderful things, including in computer programming. It lets one write code that works extremely efficiently and can never be explained to another person, modified, or debugged ever. I may lack the proper training for the uses I’ve made of it.

News anchor: 'In a cruel, bizarre twist of fate, this week's $1 million winning lotto number 579281703 was shared by exactly one million people. In other news ... ' (The person watching the news has a lottery ticket number 579281703.)
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 29th of June, 2018. It originally ran the 17th of July, 2009. You can tell it’s from so long ago because the TV set is pre-HD.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 29th is a lottery joke. It does happen that more than one person wins a drawing; sometimes three or even four people do, for the larger prizes. The chance that there’s a million winners? Frightfully unlikely unless something significant went wrong with the lottery mechanism.

So what are the chances of a million lottery winners? If I’m not mistaken the only way to do this is to work out a binomial distribution. The binomial distribution is good for cases where you have many attempts at doing a thing, where each thing can either succeed or fail, and the likelihood of success or failure is independent of all the other attempts. In this case each lottery ticket is an attempt; it winning is success and it losing is failure. Each ticket has the same chance of winning or losing, and that chance doesn’t depend on how many wins or losses there are. What is that chance? … Well, if each ticket has one chance in a million of winning, and there are a million tickets out there, the chance of every one of them winning is about one-millionth raised to the millionth power. Which is so close to zero it might as well be nothing. … And yet, for all that it’s impossible, there’s not any particular reason it couldn’t happen. It just won’t.

What I Learned This Year. Kid: 'Um ... you can divide a number by 3 if the sum of its digits can be divided by 3.' [ Later ] Frazz: 'So, what'd you learn this year?' Kid: 'Don't go last on what-I-learned-this-year day.'
Jef Mallet’s Frazz for the 29th of June, 2018. Sorry, again, not sure of this kid’s name. The comic is often so good about casually dropping in character names.

Jef Mallet’s Frazz for the 29th is a less dire take on what-you-learned-this-year. In this case it’s trivia, but it’s a neat sort of trivia. Once you understand how it works you can understand how to make all sorts of silly little divisibility rules. The threes rule — and the nines rule — work by the same principle. Suppose you have a three-digit number. Let me call ‘a’ the digit in the hundreds column, ‘b’ the digit in the tens column, and ‘c’ the digit in the ones column. Then the number is equal to 100\cdot a + 10\cdot b + 1\cdot c . And, well, that’s equal to 99\cdot a + 1\cdot a + 9 \cdot b + 1 \cdot b + 1 \cdot c . Which is 99\cdot a + 9 \cdot b + a + b + c . 99 times any whole number is a multiple of 9, and also of 3. 9 times any whole number is a multiple of 9, and also of 3. So whether the original number is divisible by 9, or by 3, depends on whether a + b + c is. And that’s why adding the digits up tells you whether a number is a whole multiple of three.

This has only proven anything for three-digit numbers. But with that proof in mind, you probably can imagine what the proof looks like for two- or four-digit numbers, and would believe there’s one for five- and for 500-digit numbers. Or, for that matter, the proof for an arbitrarily long number. So I’ll skip actually doing that. You can fiddle with it if you want a bit of fun yourself.

Also maybe it’s me, or the kind of person who gets into mathematics. But I find silly little rules like this endearing. It’s a process easy to understand that anyone can do and it tells you something not obvious from when you start. It feels like getting let in on a magic trick. That seems like the sort of thing that endears people to mathematics.

Michael: 'Grandma broke out the math workbooks!' Gabby: 'She does this every summer!' (They hide behind a tree.) Gabby: 'Says she doesn't want us to forget what we learned during the school year.' Michael: 'She has a point. We do need to keep our homework-avoidance skills sharp.'
Mike Thompson’s Grand Avenue for the 29th of June, 2018. At the risk of taking the art too literally: isn’t that tree kind of short to be that fat? Shouldn’t the leaves start higher up?

Mike Thompson’s Grand Avenue for the 29th is trying to pick its fight with me again. I can appreciate someone wanting to avoid kids losing their mathematical skills over summer. It’s just striking how Thompson has consistently portrayed their grandmother as doing this in a horrible, joy-crushing manner.

Greek: 'Why are you the wisest man, Socrates?' Socrates: 'Because I know one thing: that I know nothing.' Greek: 'That's all you know?' Socrates: 'I mean strictly speaking ... ' Greek: 'What about the infinite universe of analytic statements, like if A = A then A = A?' Socrates: 'Okay yeah That stuff. Just that.' Greek: 'Just ALL of math.' (Pause.) Greek: 'Sorry, did I make you sad?' Socrates: 'I can't be certain, but probably.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th of June, 2018. I am curious if anyone in the philosophy department would offer an idea which Ancient Greek might be chatting with Socrates here. If Weinersmith had anyone in mind I would guess whichever one has Socrates getting a slave to do a geometry proof. But there’s also … I want to say Parmenides, where the elder scholar whips the young Socrates in straight syllogisms. Again, if anyone specific was in mind and it wasn’t just “another Ancient Greek type”.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th gets into a philosophy-of-mathematics problem. Also a pure philosophy problem. It’s a problem of what things you can know independently of experience. There are things it seems as though are true, and that seem independent of the person who is aware of them, and what culture that person comes from. All right. Then how can these things be relevant to the specifics of the universe that we happen to be in just now? If ‘2’ is an abstraction that means something independent of our universe, how can there be two books on the table? There’s something we don’t quite understand yet, and it’s taking our philosophers and mathematicians a long while to work out what that is.


And as ever, if you’d like to see more Reading the Comics posts, please look to this page. For essays with Todd the Dinosaur in them, look here. For essays with the Chuckle Brothers, here you go. For some of the many, many essays with Saturday Morning Breakfast Cereal, follow this link. For more talk about Frazz, look here. And for the Grand Avenue comics, try this link please.

Reading the Comics, June 27, 2018: Stitch Day Edition


For a while I thought this essay would include only the mathematically-themed strips which Comic Strip Master Command sent out through to June 26th, which is picking up the nickname Stitch Day (for 6-26, the movie character’s experiment number). And then I decided some from last Sunday weren’t on-point enough (somehow), and there were enough that came later in the week that I couldn’t do a June 26th Only edition. Which is my longwinded way of saying this one doesn’t have a nonsense name. It just has a name that’s only partially on point.

Mike Baldwin’s Cornered for the 26th is the Rubik’s Cube/strange geometry joke for the week. It seems to me I ought to be able to make some link between the number of various ways to arrange a Rubik’s Cube — which pieces can and which ones cannot be neighbors to a red piece, say, no matter how one scrambles the cube — and the networking between people that you can get from an office where people have to see each other. But I’m not sure that I can make that metaphor work. I’m blaming the temperature, both mine (I have a cold) and the weather’s (it’s a heat wave).

Man sitting behind an upside-down desk, to a person standing on a horizontal wall-with-window: 'Hang on --- I've almost got it.' Caption: Rubik's Cubicle.
Mike Baldwin’s Cornered for the 26th of June, 2018. Say what you will; at least it’s not an open-office plan.

Mark Leiknes’s Cow and Boy for the 26th makes literal the trouble some people have with the phrase “110 percent”. Read uncharitably, yes, “110 of a hundred” doesn’t make sense, if 100 percent is all that could conceivably be of the thing. But if we can imagine, say, the number of cars passing a point on the highway being 90 percent of the typical number, surely we can imagine the number of cars also being 110 percent. To give an example of why I can’t side with pedants in objecting to the phrase.

Boy (Billy), playing chess with Cow: 'I hate it when people say they're giving a hundred and ten percent. I mean, how is that even possible? Wouldn't you be trying so hard that your body couldn't contain the extra ten percent of effort and your head would explode?' Cow: 'Check mate!' [ Cow's head explodes. ] Boy: 'OK, but I was only giving it like 35 percent.' Headless Cow: 'Darn.'
Mark Leiknes’s Cow and Boy for the 26th of June, 2018. This strip originally ran the 12th of October, 2011 and it’s not usually so gruesome.

Jef Mallett’s Frazz for the 26th is just itching for a fight. From me and from the Creative Writing department. Yes, mathematics rewards discipline. All activities do. At the risk of making a prescription: if you want to do something well, spend time practicing the boring parts. For arithmetic, that’s times tables and regrouping calculations and factoring and long division. For writing, that’s word choice and sentence structure and figuring how to bring life to describing dull stuff. Do the fun stuff too, yes, but because it is fun. Getting good at the boring stuff makes you an expert. When you discover that the boring stuff is also kinda fun, you will do the fun stuff masterfully.

Student presenting 'What I Learned This Year': 'Writing rewards creativity while math rewards a disciplined pursuit of a single right answer.' Later, Frazz: 'So, what'd you learn this year?' Student: 'Apparently we don't learn how to fudge the numbers until business school.'
Jef Mallett’s Frazz for the 26th of June, 2018. Again I apologize; I don’t know who the student is. Cast lists, cartoonists. Get your cast on your web page.

But to speak of mathematics as pursuing a single right answer — well, perhaps. In an elementary-school problem there is typically just the one right answer, and the hope is that students learn how to get there efficiently. But if the subject is something well-worn, then there are many ways to do any problem. All are legitimate and the worst one can say of a method is maybe it’s not that efficient, or maybe it’s good here but doesn’t generally work. If the subject is on the edge of what mathematics we know, there may be only one way to get there. But there are many things to find, including original ways to understand what we have already found. To not see that mathematics is creative is to not see mathematics. Or, really, any field of human activity.

Horace, reading the newspaper: 'Your horoscope: you will be positively surprised.' A giant + sign drops from the sky, barely missing Horace.
Samson’s Dark Side of the Horse for the 27th of June, 2018. So, how would you rewrite the horoscope to make this work for multiplication? ‘You’re encountering some surprising times’?

Samson’s Dark Side of the Horse for the 27th edges up to being the anthropomorphic numerals joke for the week. I need a good name for this sort of joke about mathematical constructs made tangible, even if they aren’t necessarily characters.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th I hope makes sense if you just know the words “graph” and “drunk”, and maybe “McNugget”. That’s all you truly need to understand why this contains a joke. But there is some good serious mathematical terminology at work here.

Mathematics instructor: 'Here we have a graph which embodies a stochastic process. Now, we perform a random walk on the graph for n steps and --- HEY! [ Curses ] The graph went out for McNuggets!' (The graph looks faintly more like a person, has a basket of McNuggets, and is saying, 'Nuggs nuggs nuggie nuggie nugg WOOH! God you're so hot.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of June, 2018. Can’t be intended, but that graph looks to me like plots of what the constellation Orion is expected to look like after several ten thousand years of stellar movement.

So. A “graph” is a thing that’s turned up in my A To Z serieses. In this context a graph is a collection of points, called “vertices”, and a collection of “edges” that connect vertices. Often the vertices represent something of interest and the edges ways to turn one thing into another. Sometimes the edges are the thing of interest and the vertices are just there to be manipulated in some way by edges. It’s a way to make visual the studying of how stuff is connected, and how things can pass from one to another.

A “stochastic process” is about random variables. Random variables are some property about a system. And you know some things about that variable’s value. You know maybe the range of possible values it could have. You know whether some values are more likely than others. But you do not know what the value is at any particular moment. Consider, say, the temperature outside where you live at a particular time of day. You may have no idea what that is. But you can say, for example, whether today it is more likely to be 90 degrees Fahrenheit or 60 degrees Fahrenheit or 20 degrees Fahrenheit. For a stochastic process we have some kind of index. We can say, for example, which values of temperature are more likely today, the 1st of July, and which ones will be more likely the 1st of August, and which ones will be less likely the 1st of December. Calling it a “process”, to my intuition, makes it sound like we expect something to happen that causes the likelihood of some temperatures to change. And many processes are time-indexed. They study problems where something interesting changes in time, predictable in aggregate but not in detail.

So a graph like this, representing a stochastic process, is a shorthand. Each vertex is a state that something might be in. Each edge is a way to get from one state to another when — something — happens. Doesn’t matter what thing.

A “drunk walk”, or as it’s known to tenderer writers a “random walk”, is a term of art. Not a deep one. It’s meant to evoke the idea of a severely drunk person who yes, can move, but has no control over which way. Thus he wanders around, reaching any point only by luck. Many things look like random walks, in which there is no overall direction, just an unpredictable shuffling around. A drunk walk on this graph would be, well, start at any of the vertices. Then follow edges, chosen randomly. If you start at the uppermost point of the triangle on top, for example, there’s two places to go on the second step: the lower-left or the center-right vertex on the upper triangle. Suppose you go to the center-right vertex. On the next step, you might go right back where you started. You might go to the lower-left vertex on the triangle. You might drop down that bridge to the top of that quadrilateral. And so on, for another step.

Do that some presumably big number of times. Where are you? … Anywhere, of course. But are there vertices you’re more likely to be on? Ones you’re less likely to be on? How does the shape of the graph affect that likelihood? How does how long you spend walking affect that? These tell us things about the process, and are why someone would draw this graph and talk about a random walk on it.


If you’d like to read more of my comic-strip review posts please do! They all should be available at this link, listed in reverse chronological order.

To read more of the individual comics? Here are essays with Cornered in them. These are Cow and Boy comics at this link. Frazz strips are here. Essays including Dark Side of the Horse are here. And Saturday Morning Breakfast Cereal, which is threatening to take over “being the majority of my blog” from Andertoons, I have at that link.

Reading the Comics, March 9, 2018: Some Old Lines Edition


To close out last week’s comics I got a bunch of strips that were repeats, or that touch on topics I’ve discussed quite a bit around these parts already. I’m pretty sure all the words I have here are new in their specific organization. The words themselves are pretty old.

Maria Scrivan’s Half Full for the 4th is the Rubik’s Cube joke for the week. I ought to write up a proper description of the algebra of Rubik’s Cubes. The real stuff is several books’ worth of material, yes. But a couple hundred words about what’s interesting should be doable. … Or I could just ask folks if they’ve read good descriptions of the group theory that cubes show off. I’m always open to learning other people have said stuff better than me. This is part of why I’ve never published an essay about Cantor’s Diagonal Proof; many people have written such essays and I couldn’t add anything useful to that heap of words.

Partly scrambled Rubik's Cube to a solved one: 'Rough week.'
Maria Scrivan’s Half Full for the 4th of June, 2018. Yeah, uh, it me.

Ryan North’s Dinosaur Comics for the 5th is about the heap paradox. Or the sorites paradox, depending on what book you’ve been reading from. The problem is straightforward enough. As God, in the strip says, a big pile of sand is clearly a heap. One or two grains of sand is clearly not. If you remove grains from the heap, eventually, you lose the heap-ness. T-Rex suggests solving the question of when that happens by statistical survey, finding what people on average find to be the range where things shift over.

God: 'T-Rex let's say you have a giant heap of sand and I remove one grain of it at a time.' T-Rex: 'Ooh, let's!' God: 'Clearly when there's only one grain of sand left it's not a heap anymore!' T-Rex: 'Clearly!' God: 'Aha my friend but when precisely did it switch from heap to non-heap?' T-Rex: 'I dunno! At some fuzzy point if would switch for most observers from 'heap' to, say, 'small pine', and there we can draw the line. Language isn't that precise.' God: 'Listen this is a classic paradox of Eubulides of Miletus came up with over 2000 years ago. You need to have your mind blown now okay.' T-Rex: 'Sounds kinda dumb to me!' Utahraptor: 'What does?' T-Rex: 'The point at which a shrinking heap of sand becomes a non-heap. Clearly I'm supposed to struggle with an arbitrary threshold, because piles on either side of it look much the same. But it's just language! Look at statistical usage of the word 'heap', decide using that average, end of story. Oh, snap, philosophers! Did T-Rex just totally school you with his statistically-based descriptivist approach to semantics? IT APPEARS THAT HE TOTALLY DID! It also appears he's speaking in the third person because he's so impressed with his awesome self!'
Ryan North’s Dinosaur Comics for the 5th of June, 2018. I get that part of the setup of these comics is that T-Rex is nerdy-smart, but I can also imagine the philosophers rolling their eyes at how he’s missed the point. Maybe if he were asked about the density of a single molecule of water he’d understand better why the question can’t be obvious. (And T-Rex does sometimes revisit issues with deeper understanding of the issues. This might have happened between when this strip first appeared on qwantz.com and when it appeared on GoComics.com.

As with many attempts to apply statistical, or experimental, methods to philosophical questions it misses the point. There are properties that things seem to have only as aggregations. Where do they come from? How can there be something true about a collection of things that isn’t true about any part of the thing? This is not just about messy real-world properties either; we can say stuff about groups of mathematical objects that aren’t true about individual objects within the set. For example, suppose we want to draw a real number at random, uniformly, from the continuous interval 0 to 10. There’s a 50% chance we’ll draw a number greater than 5. The chance of drawing any specific number greater than 5, though, is zero. But we can always draw one. Something weird is happening here, as often happens with questions we’ve been trying to answer for thousands of years.

Customer: 'How much will this be at 80% off?' Clerk: 'Ten bucks.' Customer: 'How did you do that in your head so fast?' Clerk: '20% of fifty is ten.' Customer: 'Wow! So you're some kind of super math genius?' Customer: 'Sure.'
Norm Feuti’s Retail for the 6th of June, 2018. This joke, though not this strip, was also run the 26th of June, 2017. There I share my one great retail-mathematics anecdote.

Norm Feuti’s Retail for the 6th is a new strip, although the joke’s appeared before. There’s some arithmetic calculations that are easy to do, or that become easy because you do them a lot. Or because you see them done a lot and learn what the patterns are. A handful of basic tricks — like that 80 percent off is 20 percent of something, or that 20 percent of a thing is one-fifth the original thing — can be stunning. Stage magicians find the same effect.

Rita: 'Tell your group I expect them to give me 110%! Keep in mind, reviews are coming!' Jay: 'Rita --- you should realize that it's impossible to give more than 100%!' Rita: 'No --- not with that kind of attitude!'
John Zakour and Scott Roberts’s Working Daze for the 6th of June, 2018. It ran the 22nd of October, 2014, although that was as part of a “Best Of” week. No idea when it originally ran.

John Zakour and Scott Roberts’s Working Daze for the 6th is another chance for me to talk about the supposed folly of giving 110 percent. Or point you to where I did already. I’m forgiving of the use of the phrase.

Abacus at the bar: 'If you ever find yourself working for Weinstein as a bookkeeper, let me offer you sum advice ... never use the phrase, 'Harvey, you can count on me'.' Hostess: 'Thanks for the tip.'
Bob Shannon’s Tough Town for the 7th of June, 2018. The strip is one about all sorts of odd creatures hanging out in the bar, so, you’re not misunderstanding this.

Bob Shannon’s Tough Town for the 7th is the anthropomorphized abacus joke of the week. Been a while since we had one of those. I suppose an adding machine would be at least as good a representative of the abstract concept of doing arithmetic, but it’s likely harder to draw too. This is just tiring to draw.

Cave-person Father: 'Me have method for knowing how many rocks you have. Called 'counting'. Put up fingers, then say --- ' Cave-person Kid: 'We ever use this in REAL LIFE?' Caption: The First Math Class.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th of June, 2018. Admit I do wonder how often cave people needed to track the number of rocks they had. I mean, how often do we need to count our rocks? Aren’t the rocks themselves an adequate representation of the number of rocks around?

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th presents the old complaint about mathematics’s utility, here in an ancient setting. I’m intereste that the caveman presents counting in terms of matching up other things to his fingers. We use this matching of one set of things to another even today. It gets us to ordinal and cardinal numbers, and the to what we feel pretty sure about with infinitely large sets. An idea can be ancient and basic and still be vital.

Karen: 'Uuuhhhhggghh!!! I hate math!!!' Dad: 'First of all, don't say 'hate'. It's a very strong word. Secondly, you will always need math. Even if you're in sales like me. In fact, I'm using math right now. I'm figuring out where I stand against my quota for this quarter. Observe ... I take this number, add it to that one. Take a percentage of this value and subtract it here. See, that's my number ... ... ... I hate math.'
Steve Sicula’s Home and Away rerun for the 9th of June, 2018. The strip originally ran the 6th of March, 2011. … How does Karen there say “Uuuhhhggghh”?

Steve Sicula’s Home and Away for the 9th is about the hatred people profess for mathematics. Some of that is more hatred of how it’s taught, which is too often as a complicated and apparently pointless activity. Some of that is hatred of how it’s used, since it turns up in a lot of jobs. And for some reason we’ve designed society so that we do jobs we don’t like. I don’t know why we think that’s a good idea. We should work on that.

Reading the Comics, June 4, 2018: Weezer’s Africa Edition


Once again the name of this Reading the Comics edition has nothing to do with any of the strips. I’m just aware that Weezer’s cover of Africa is quite popular right now and who am I to deny people things they want? (I like the cover, but it’s not different enough for me to feel satisfied by it. I tend to like covers that highlight something minor in the original, or that go in a strange direction. Shifting a peppy song into a minor key doesn’t count anymore. But bear in mind, I’m barely competent at listening to music. Please now enjoy my eight hours of early electronica in which various beeps and whistles are passed off as music.)

Samson’s Dark Side of the Horse for the 3rd is the Roman numerals joke for the week. And a welcome return for Dark Side of the Horse. It feels like it’s been gone a while. I wouldn’t try counting by Roman numerals to lull myself to sleep; it seems like too much fussy detail work. But I suppose if you’ve gotten good at it, it’s easy.

Horace, counting sheep jumping over the fence: MCDXCVII; MCDXCIX and the sheep falls over the fence; MD and a sheep with a medical bag runs up to tend the fallen sheep.
Samson’s Dark Side of the Horse for the 3rd of June, 2018. Have to say that’s an adorable medical sheep in the third panel.

Jef Mallett’s Frazz for the 3rd builds on removing statistics from their context. It’s a common problem. It’s possible to measure so very many things. Without a clear idea of what we should expect as normal the measurement doesn’t tell us much. And it can be hard to know what the right context for something even is. Let me deconstruct Caulfield’s example. We’re supposed to reflect on and consider that 40% of all weekdays are Monday and Friday too. But it’s not only weekdays that people work. Even someone working a Sunday might take a sick day. Monday and Friday are a bit over 28% of the whole week. But more people do work Monday-to-Friday than do Saturdays and Sundays, so the Sunday sick day is surely rarer than the Monday. So even if we grant Caulfield’s premise, what does it tell us?

Caulfield: 'Did you know 40% of all sick days are taken on Mondays and Fridays?' Three panels of silence. Caulfield: 'Think about it. ... Did you know 60% of some comic strips is filler?' Frazz: 'If the cartoonist can still make it funny and get outside on the first nice day of spring, I'm cool.'
Jef Mallett’s Frazz for the 3rd of June, 2018. So Jef Mallett lives in the same metro area I do, which means I could in principle use this to figure out how far ahead of deadline he wrote this strip. Except that’s a fraud since we never had a first nice day of spring this year. We just had a duplicate of March for all of April and the first three weeks of May, and then had a week of late July before settling into early summer. Just so you know.

Jason Chatfield’s Ginger Meggs for the 3rd is a bit of why-learn-mathematics propaganda. Megg’s father has a good answer. But it does shift the question back one step. Also I see in the top row that Meggs has one of those comic-strip special editions where the name of the book is printed on the back cover instead. (I’m also skeptical of the photo and text layout on the newspaper Megg’s father is reading. But I don’t know the graphic design style of Australian, as opposed to United States, newspapers.)

Ginger Meggs: 'Dad, do I really need to know how to do maths?' Dad: 'Well, of course you need to know how to do mathematics, Ginger! Think about it! Without maths, you could never become an accountant!' (Ginger and his dog stand there stunned for a panel. Next panel, they're gone. Next panel after that ... ) Mom: 'I suppose you know you just blew it.'
Jason Chatfield’s Ginger Meggs for the 3rd of June, 2018. So … I guess Ginger Megg’s father is an accountant? I’m assuming because it makes the joke land better?

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd may belong on some philosopher’s Reading the Comics blog instead. No matter. There’s some mathematical-enough talk going on here. There’s often many ways to approach the same problem. For example, approaching a system as a handful of items. Or as a huge number of them. Or as infinitely many things. Or as a continuum of things. There are advantages each way. A handful of things, for example, we can often model as interactions between pairs of things. We can model a continuum as a fluid. A vast number of things can let one’s computer numerically approximate a fluid. Or infinitely many particles if that’s more convenient.

Professor: 'Monists believe there is no distinction between mind and body.' (Writes 1/1.) 'Dualists believe mind and body are, in some sense, separate aspects of being.' (Writes 1/2.) 'There's a lively debate here, but the important thing to notice is that both are talking about the same human beings. This proves that you can add 1 to the quantity of aspects of being without altering the being itself.' (Writes 1/3, 1/4, 1/5, 1/6, ... ) 'By induction, you can be a monist, dualist, triplist, quadruplist, and so on. There are literally infinite permitted philosophies in ontology-space! Personally, I am a 10-to-the-27th-powerist, in that I believe every one of the atoms in my body is meaningfully distinct.' Student: 'You've taken a difficult philosophy problem and reduced it to a tractable but pointless math problem.' Professor: 'You may also be interested in my work on free will!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd of June, 2018. Also I’m not sure where the professor figures he’s going with this but my understanding is it’s rather key to our understanding of quantum mechanics that, say, every atom of Carbon-12 in our bodies is the same as every other atom. At least apart from accidental properties like which compound it might happen to be in at the moment and where it is in that compound. That is, if you swapped two of the same isotope there’d be no way to tell you had.

To describe all these different models as sharing an “ontology-space” is good mathematical jargon too. In this context the “-space” would mean the collection of all these things that are built by the same plan but with different values of whichever parameter matters.

Julian writes E = mc^2 on a blackboard. He tells Suzy, 'That's Einstein's theory.' Suzy: 'It's real cute, Julian!'
Bud Blake’s Tiger for the 6th of August, 1965. It was rerun the 4th of June, 2018. I confess I’m not sure exactly what the joke is. If it’s not that Suzy has no idea what’s being written but wants to say something nice about Julian’s work … all right, and I guess that’s an unremarkable attitude for a cartoonist to express in 1965, but it’s a weak joke.

Bud Blake’s Tiger for the 6th of August, 1965 features Einstein’s famous equation. I suppose it’s showing how well-informed Julian is, that he knows and can present such a big result. There is beauty in mathematics (and physics). Mathematicians (and physicists) find the subject beautiful to start with, and try to find attractive results. I’m curious what the lay reader makes of mathematical symbols, though, just as pieces of art. I remember as a child finding this beauty in a table of integrals in the front of one of my mother’s old college textbooks. All those parallel rows of integral symbols drew me in though nothing I’d seen in mathematics had prepared me to even read it. I still find that beautiful, but I can’t swear that I would even if I hadn’t formed that impression early in life. Are lay and professional readers’ views of mathematical-expression beauty similar? How are they different?

Reading the Comics, May 30, 2018: Spherical Photos Edition


Last week’s offerings from Comic Strip Master Command got away from me. Here’s some more of the strips that had some stuff worth talking about. I should have another installment this week. I’m back to nonsense edition names; sorry.

Lincoln Pierce’s Big Nate for the 29th of May is about the gambler’s fallacy. Everyone who learns probability learns about it. The fallacy builds on indisputable logic: your chance of losing at something eighteen times in a row is less than the chance of your losing at that thing seventeen times in a row. So it makes sense that if you’ve lost seventeen times in a row then you must be due.

And that’s one of those lies our intuition tells us about probability. What’s important to Nate here is not the chance he’s in an 18-at-bat losing streak. What’s important is the chance that he’s in an 18-at-bat losing streak, given that he’s already failed 17 times in a row. These are different questions. The chance of an 18th at-bat in a row being a failure (for him) is much larger than the chance of an 18-at-bat losing streak starting from scratch.

Nate: 'Time for me to break this 0-and-17 stretch.' Teddy: 'Exactly! You're due, Nate! You're due!' Francis: 'Not necessarily. The chances of Nate getting a hit aren't enhanced by the fact that he's gone five games without one.' Teddy: 'I lied. You're not due.' Francis: 'But miracles happen, so go for it.'
Lincoln Pierce’s Big Nate rerun for the 29th of May, 2018. The strip first ran the 18th of May, 2010. I’ve not heard anything about why Pierce has been away from the strip since the start of the year.

That said I can’t go along with Francis’s claim that the chance of Nate getting a hit isn’t enhanced by his long dry spell. We can, and often do, model stuff like at-bats as though they’re independent. That is, that the chance of getting a hit doesn’t depend on what came before. Doing it this way gives results that look like real sports matches do. But it’s very hard to quantify things like losing streaks or their opposite, hot hands. It’s hard to dismiss the evidence of people who compete, though. Everyone who does has known the phenomenon of being “in the zone”, where things seem easier. I was in it for two games out of five just last night at pinball league. (I was dramatically out of it for the other three. I nearly doubled my best-ever game of Spider-Man and still came in second place. And by so little a margin my opponent thought the bonus might make the difference. Such heartbreak.)

But there is a huge psychological component to how one plays at a game. Nate thinks differently about what he’s doing going up to bat after seventeen failures in a row than he would after, say, three home runs in a row. It’s hard to believe that this has no effect on how he plays, even if it’s hard to track down a consistent signal through the noise. Maybe it does wash out. Maybe sometimes striking out the first three at-bats in a game makes the batter give up on the fourth. Meanwhile other times it makes the batter focus better on the fourth, and there’s no pinning down which effect will happen. But I can’t go along with saying there’s no effect.

Melvin: 'Hold on now --- replacement? Who could you find to do all the tasks only Melvin can perform?' Rita: 'A macaque, in fact. Listen, if an infinite number of monkeys can write all the great works, I'm confident that one will more than cover for you.'
John Zakour and Scott Roberts’s Working Daze for the 29th of May, 2018. Earlier in the sequence they had the Zootopia sloth replacing Ed, but there’s no making that on topic for my blog here.

John Zakour and Scott Roberts’s Working Daze for the 29th is an infinite-monkeys joke. Well, given some reasonable assumptions we can suppose that sufficiently many monkeys on typewriters will compose whatever’s needed, given long enough. Figuring someone’s work will take fewer monkeys and less time is a decent probability-based insult.

Hazel, with mathematics book, asking a bored kid: 'Okay, now what's nine times eight?' Next panel: the kid's coming out and saying 'Next'; a sign reads, 'Need help with your homework? See Hazel 1 to 5 pm Saturdays'.
Ted Key’s Hazel rerun for the 30th of May, 2018. I can’t say when this first ran. I’m not sure what the kid’s name is, sorry.

Ted Key’s Hazel for the 30th has the maid doing a bit of tutoring work. That’s about all I can make of this either. Doesn’t seem like a lot of fun, but there is only so much to do with arithmetic computation like this. It’s convenient to know a times table by memory.

Accessories of Famous Teachers: Einstein's Chalkboard; Galileo's Compass; Confucius's Fortune Cookie; Socrates's Hemlock; Miss Othmar's Trombone.
Scott Hilburn’s The Argyle Sweater for the 30th of May, 2018. Are … Einstein, Galileo, and Confucius really famous teachers? Calling Socrates a teacher is a lesser stretch.

Scott Hilburn’s The Argyle Sweater for the 30th has a chalkboard full of mathematical symbols as iconic for deep thinking. And it’s even Einstein’s chalkboard. And it’s even stuff that could plausibly be on Einstein’s chalkboard at some point. Besides E = mc2 the other formulas are familiar ones from relativity. They’re about the ways our ideas of how much momentum or mass a thing has has to change if we see the thing in motion. (I’m a little less sure about that \Delta t expression, but I think I can work something out.) And as a bonus it includes the circle-drawing compass as Galileo might have used. Well, he surely used a compass; I’m just not sure that the model shown wouldn’t be anachronistic. As though that matters; fortune cookies, after all, are a 20th century American invention and we’re letting that pass.

Mathematical Fun Fact: For each of the possible espresso-to-milk ratios, there exists at least one Italian-sounding name: Just Milk; 1:3 'latte', 1:2 'Cappuccino', 1:1 'Antoccino', 2:1 'Macchiato', 3:1 'Antilatte', Just Espresso. Also: 1/c^2 'Relativisto'; (espresso + milk)/espresso = espresso/milk 'Phicetto'; i:1 'Imaginarati', pi:1 'Irratiognito'; 6.022*10^23 : 1, 'Avogadro'; lim_{milk->0} espresso/milk: 'Infiniccino'.
Zach Weinersmiths’s Saturday Morning Breakfast Cereal for the 30th of May, 2018. Kind of curious what sorts of drinks you get from putting in infinitesimals. (You get milk or espresso with a homeopathic bit of the other.)

Zach Weinersmiths’s Saturday Morning Breakfast Cereal for the 30th builds on a fun premise. Underneath the main line it gets into some whimsical ratios built on important numbers you’d never use for this sort of thing, such as π, and the imaginary unit \imath . The Golden Ratio makes an appearance too, sneaking a definition for φ in in terms of espresso and milk. Here’s a free question: is there a difference between the “infiniccino” and “just espresso” except for the way it’s presented? … Well, presentation can be an important part of a good coffee.

π is well-known. Not sure I have anything interesting to add to its legend. φ is an irrational number a bit larger than 1.6. I’m not sure if I’ve ever called it the Boba Fett of numbers, but I should have. It’s a cute enough number, far more popular than its importance would suggest. \imath is far more important. Suppose that there is some number, which we give that name, with the property that \imath^2 equals -1. Then we get complex-valued numbers, which let us solve problems we’d like to know but couldn’t do before. It’s a great advance.

The name tells you how dubiously people approached this number, when it was first noticed. I wonder if people would be less uneasy with “imaginary numbers” if it weren’t for being told how there’s no such thing as the square root of minus one for years before algebra comes along and says, well, yes there is. It’s hard to think of a way that, say, “negative four” is more real than \imath , after all, and people are mostly all right with -4. And I understand why people are more skeptical of -4 than they are of, say, 6. Still, I wonder how weird \imath would look if people weren’t primed to think it was weird.

Reading the Comics, April 28, 2018: Friday Is Pretty Late Edition


I should have got to this yesterday; I don’t know. Something happened. Should be back to normal Sunday.

Bill Rechin’s Crock rerun for the 26th of April does a joke about picking-the-number-in-my-head. There’s more clearly psychological than mathematical content in the strip. It shows off something about what people understand numbers to be, though. It’s easy to imagine someone asked to pick a number choosing “9”. It’s hard to imagine them picking “4,796,034,621,322”, even though that’s just as legitimate a number. It’s possible someone might pick π, or e, but only if that person’s a particular streak of nerd. They’re not going to pick the square root of eleven, or negative eight, or so. There’s thing that are numbers that a person just, offhand, doesn’t think of as numbers.

Crock to the two prisoners in lockboxes: 'Guess the number I'm thinking and I'll set you free.' First prisoner: '4,796,034,621,322.' Crock: 'Sorry, it's nine.' Second prisoner: 'What made you guess THAT number?' First prisoner: 'It was the first one to pop into my head.'
Bill Rechin’s Crock rerun for the 26th of April, 2018. Going ahead and guessing there’s another Crock with the same setup, except the prisoner guesses nine, and Crock says it was 4,796,034,621,322, and then in the final panel we see that Crock really had thought nine and lied.

Mark Anderson’s Andertoons for the 26th sees Wavehead ask about “borrowing” in subtraction. It’s a riff on some of the terminology. Wavehead’s reading too much into the term, naturally. But there are things someone can reasonably be confused about. To say that we are “borrowing” ten does suggest we plan to return it, for example, and we never do that. I’m not sure there is a better term for this turning a digit in one column to adding ten to the column next to it, though. But I admit I’m far out of touch with current thinking in teaching subtraction.

On the board: 51 - 26, with the 51 rewritten as 4 with a borrowed 11. Wavehead: 'So we're just borrowing 10 no questions asked? What about a credit check? What's the interest rate?'
Mark Anderson’s Andertoons for the 26th of April, 2018. This is Mark Anderson’s Andertoons for the week.

Greg Cravens’s The Buckets for the 26th is kind of a practical probability question. And psychology also, since most of the time we don’t put shirts on wrong. Granted there might be four ways to put a shirt on. You can put it on forwards or backwards, you can put it on right-side-out or inside-out. But there are shirts that are harder to mistake. Collars or a cut around the neck that aren’t symmetric front-to-back make it harder to mistake. Care tags make the inside-out mistake harder to make. We still manage it, but the chance of putting a shirt on wrong is a lot lower than the 75% chance we might naively expect. (New comic tag, by the way.)

Larry: 'Your shirt is on all wrong.' Toby: 'It was bound to happen.' Larry: 'What? Why?' Toby: 'There's FOUR different ways a shirt can go on! That gives me only, like, a 20% chance any time I put it on.'
Greg Cravens’s The Buckets for the 26th of April, 2018. I’m not sure Larry (the father)’s disbelief at his kid figuring putting the shirt on all wrong was bound to happen. It’s a mistake we all make; accepting the inevitability of that doesn’t seem that wrong.

Charles Schulz’s Peanuts rerun for the 27th is surely set in mathematics class. The publication date interests me. I’m curious if this is the first time a Peanuts kid has flailed around and guessed “the answer is twelve!” Guessing the answer is twelve would be a Peppermint Patty specialty. But it has to start somewhere.

Sally, at her schooldesk: 'The answer is twelve! It isn't? How about six? Four? Nine? Two? Ten? ... Do you have the feeling that I'm guessing?'
Charles Schulz’s Peanuts rerun for the 27th of April, 2018. This strip first ran the 30th of April, 1971. It also was rerun the 25th of April, 2003, with a different colorization scheme for some reason.

Knowing nothing about the problem, if I did get the information that my first guess of 12 was wrong, yeah, I’d go looking for 6 or 4 as next guesses, and 12 or 48 after that. When I make an arithmetic mistake, it’s often multiplying or dividing by the wrong number. And 12 has so many factors that they’re good places to look. Subtracting a number instead of adding, or vice-versa, is also common. But there’s nothing in 12 by itself to suggest another place to look, if the addition or subtraction went wrong. It would be in the question which, of course, doesn’t exist.

Venn Diagram. One circle's labelled 'Venn Diagrams'; the second 'Jokes'. The intersection is 'Lazy Cartoonists'.
Maria Scrivan’s Half-Full for the 28th of April, 2018. Hey, cartoonists deserve easy days at work too. And there’s not always a convenient holiday they can have the cast just gather around and wish everyone a happy instance of.

Maria Scrivan’s Half-Full for the 28th is the Venn Diagram joke for this week. It could include an extra circle for bloggers looking for content they don’t need to feel inspired to write. This one isn’t a new comics tag, which surprises me.

Guy: 'Relax. Half the time, job interviewers don't even read your resume. They just see how long it is.' Mathematician: 'Really?' Guy: 'Yeah. Where are you going?' Mathematician: 'To make a Mobius strip.' Interviewer: 'Wow! I've never met someone with *infinite* skills and work experience.' Mathematician: 'I don't like to brag.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th of April, 2018. If I had seen this strip in 2007 maybe I would’ve got that tenure-track posting instead of going into the world of technically being an extant mathematics blog.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th uses the M&oum;bius Strip. It’s an example of a surface that you could just go along forever. There’s nothing topologically special about the M&oum;bius Strip in this regard, though. The mathematician would have as infinitely “long” a résumé if she tied it into a simple cylindrical loop. But the M&oum;bius Strip sounds more exotic, not to mention funnier. Can’t blame anyone going for that instead.

Reading the Comics, April 25, 2018: Coronet Blue Edition


You know what? Sometimes there just isn’t any kind of theme for the week’s strips. I can use an arbitrary name.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st of April, 2018 would have gone in last week if I weren’t preoccupied on Saturday. The joke is aimed at freshman calculus students and then intro Real Analysis students. The talk about things being “arbitrarily small” turns up a lot in these courses. Why? Well, in them we usually want to show that one thing equals another. But it’s hard to do that. What we can show is some estimate of how different the first thing can be from the second. And if you can show that that difference can be made small enough by calculating it correctly, great. You’ve shown the two things are equal.

Delta and epsilon turn up in these a lot. In the generic proof of this you say you want to show the difference between the thing you can calculate and the thing you want is smaller than epsilon. So you have the thing you can calculate parameterized by delta. Then your problem becomes showing that if delta is small enough, the difference between what you can do and what you want is smaller than epsilon. This is why it’s an appropriately-formed joke to show someone squeezed by a delta and an epsilon. These are the lower-case delta and epsilon, which is why it’s not a triangle on the left there.

Mad scientist cackling at a man being crushed between giant delta and epsilon figure: 'And now, good doctor, we will see how you fit between this delta and this epsilon!' Caption: Soon, soon the calculus teacher would become arbitrarily small.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st of April, 2018. I feel vaguely personally called out by the calculus teacher wearing cargo shorts, tall white socks, and sandals.

For example, suppose you want to know how long the perimeter of an ellipse is. But all you can calculate is the perimeter of a polygon. I would expect to make a proof of it look like this. Give me an epsilon that’s how much error you’ll tolerate between the polygon’s perimeter and the ellipse’s perimeter. I would then try to find, for epsilon, a corresponding delta. And that if the edges of a polygon are never farther than delta from a point on the ellipse, then the perimeter of the polygon and that of the ellipse are less than epsilon away from each other. And that’s Calculus and Real Analysis.

John Zakour and Scott Roberts’s Maria’s Day for the 22nd is the anthropomorphic numerals joke for this week. I’m curious whether the 1 had a serif that could be wrestled or whether the whole number had to be flopped over, as though it were a ruler or a fat noodle.

Maria at her desk challenges a giant number 4 to arm wrestling; she slams its 'arm' down easily. Other numerals flee as she yells out: 'Okay, anyone else wanna take me on? Huh? --- Yeah, didn't think so!' Reality: she's at her desk with a book and some paper and says, "Whew! This math homework was tough --- but I think I got it down.'
John Zakour and Scott Roberts’s Maria’s Day for the 22nd of April, 2018. I’m curious whether Zakour and Roberts deliberately put 2 and 3 to the left, with pain stars indicating they’ve been beaten already, while the bigger numbers are off to the side. Or was it just an arbitrary choice? The numbers are almost in order, left to right, except that the 7’s out of place. So maybe the ordering is just coincidence?

Anthony Blades’s Bewley for the 23rd offers advice for what to do if you’ve not got your homework. This strip’s already been run, and mentioned here. I might drop this from my reading if it turns out the strip is done and I’ve exhausted all the topics it inspires.

Bea: 'Aaaah! I forgot to do my maths homework!' Tonus: 'I did mine.' Bea: 'Can I copy yours?' Tonus: 'Of course you can. I didn't know the answers so I drew a picture of a scary dinosaur.' [ Silent penultimate panel. ] Bea: 'Better than nothing.' Tonus: 'Remember the big teeth. Big teeth make it scary.'
Anthony Blades’s Bewley for the 23rd of April, 2018. Whenever a comic strip with this setup begins I think of the time in geometry class when I realized I hadn’t done any homework and wondered if I could get something done in the time it took papers to be passed up. This in a class of twelve students. No, there was not, even though the teacher started from the other side of the classroom.

Dave Whamond’s Reality Check for the 23rd is designed for the doors of mathematics teachers everywhere. It does incidentally express one of those truths you barely notice: that statisticians and mathematicians don’t seem to be quite in the same field. They’ve got a lot of common interest, certainly. But they’re often separate departments in a college or university. When they do share a department it’s named the Department of Mathematics and Statistics, itself an acknowledgement that they’re not quite the same thing. (Also it seems to me it’s always Mathematics-and-Statistics. If there’s a Department of Statistics-and-Mathematics somewhere I don’t know of it and would be curious.) This has to reflect historical influence. Statistics, for all that it uses the language of mathematics and that logical rigor and ideas about proofs and all, comes from a very practical, applied, even bureaucratic source. It grew out of asking questions about the populations of nations and the reliable manufacture of products. Mathematics, even the mathematics that is about real-world problems, is different. A mathematician might specialize in the equations that describe fluid flows, for example. But it could plausibly be because they have interesting and strange analytical properties. It’d be only incidental that they might also say something enlightening about why the plumbing is stopped up.

[ Clown Statistician vs Clown mathematician. ] The Clown Statistician holds up a pie chart, ready to throw it. The mathematician holds up a pi symbol, ready to throw it. Corner squirrel's comment: 'There's always room for more pi.'
Dave Whamond’s Reality Check for the 23rd of April, 2018. I’m not sure I’ve laughed more at a dumb joke than I have at this in a long while.

Neal Rubin and Rod Whigham’s Gil Thorp for the 24th seems to be setting out the premise for the summer storyline. It’s sabermetrics. Or at least the idea that sports performance can be quantized, measured, and improved. The principle behind that is sound enough. The trick is figuring out what are the right things to measure, and what can be done to improve them. Also another trick is don’t be a high school student trying to lecture classmates about geometry. Seriously. They are not going to thank you. Even if you turn out to be right. I’m not sure how you would have much control of the angle your ball comes off the bat, but that’s probably my inexperience. I’ve learned a lot about how to control a pinball hitting the flipper. I’m not sure I could quantize any of it, but I admit I haven’t made a serious attempt to try either. Also, when you start doing baseball statistics you run a roughly 45% chance of falling into a deep well of calculation and acronyms of up to twelve letters from which you never emerge. Be careful. (This is a new comic strip tag.)

[ With rain delaying (baseball) practice, Kevin Pelwecki expounds on his new favorite subject --- ] Kevin: 'Launch angle! You want the ball coming off the bat at 25 degrees.' Teammate: 'Anyone else notice we're taking math lessons --- from a guy who barely passed geometry?'
Neal Rubin and Rod Whigham’s Gil Thorp for the 24th of April, 2018. Are … both word balloons coming from the same guy? In the last panel there. I understand one guy starting and another closing a thought but that’s usually something you do with an established in-joke that anyone can feed and anyone else can finish. A spontaneous insult like this seems like it only needs the one person, but the word balloon tails are weird if they’re both from the same guy.

Randy Glasbergen’s Glasbergen Cartoons rerun for the 25th feels a little like a slight against me. Well, no matter. Use the things that get you in the mood you need to do well. (Not a new comic strip tag because I’m filing it under ‘Randy Glasbergen’ which I guess I used before?)

Kid with guitar: 'I start every song by counting 1-2-3-4 because it reminds me of math. Math depresses me and that helps me sing the blues.'
Randy Glasbergen’s Glasbergen Cartoons rerun for the 25th of April, 2018. OK, but what’s his guitar plugged in to?

Reading the Comics, March 21, 2018: Old Mathematics Jokes Edition


For this, the second of my Reading the Comics postings with all the comics images included, I’ve found reason to share some old and traditional mathematicians’ jokes. I’m not sure how this happened, but sometimes it just does.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th brings to mind a traditional mathematics joke. A dairy hires a mathematician to improve operations. She tours the place, inspecting the cows and their feeding and the milking machines. She speaks with the workers. She interviews veterinarians. She talks with the truckers who haul out milk. She interviews the clients. Finally she starts to work on a model of better milk production. The first line: “Assume a spherical cow.”

[Pro Tip: this is the answer to any thermodynamics question that requires you to determine an object's temperature: ] T = 2.73 K (assume well-mixed Cosmos)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th of March, 2018. Temperature’s a great subject though, and I’ve been thinking for ages about doing a series on it just because I want to explain negative temperatures Kelvin.

One big field of mathematics is model-building. When doing that you have to think about the thing you model. It’s hard. You have to throw away all the complicating stuff that makes your questions too hard to answer. But you can’t throw away all the complicating stuff or you have a boring question to answer. Depending on what kinds of things you want to know, you’ll need different models. For example, for some atmosphere problems you’ll do fine if you assume the air has no viscosity. For others that’s a stupid assumption. For some you can ignore that the planet rotates and is heated on one side by the sun. For some you don’t dare do that. And so on. The simplifications you can make aren’t always obvious. Sometimes you can ignore big stuff; a satellite’s orbit, for example, can be treated well by pretending that the whole universe except for the Earth doesn’t exist. Depends what you’re looking for. If the universe were homogenous enough, it would all be at the same temperature. Is that useful to your question? That’s the trick.

On the board: 1/2 - 1/8 = ?. Student: 'Apropos of nothing, I have two cats.'
Mark Anderson’s Andertoons for the 20th of March, 2018. Okay, but why the poster with the octopus on it?

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for this essay. It’s just a student trying to distract the issue from fractions. I suppose mathematics was chosen for the blackboard problem because if it were, say, a history or an English or a science question someone would think that was part of the joke and be misled. Fractions, though, those have the signifier of “the thing we’d rather not talk about”.

Woman: 'And if you haven't figured it out yet, this is the math department lavatory'. The door reads 1 +/- 2
Daniel Beyer’s Long Story Short for the 21st of March, 2018. Not to nitpick but shouldn’t it be 1½ ± ½?

Daniel Beyer’s Long Story Short for the 21st is a mathematicians-mindset sort of joke. Let me offer another. I went to my love’s college reunion. On the mathematics floor of the new sciences building the dry riser was labelled as “N Bourbaki”. Let me explain why is a correctly-formed and therefore very funny mathematics joke. “Nicolas Bourbaki” was the pseudonym used by the mathematical equivalent of an artist’s commune, in France, through several decades of the mid-20th century. Their goal was setting mathematics on a rigorous and intuition-free basis, the way mathematicians sometimes like to pretend it is. Bourbaki’s influential nonexistence lead to various amusing-for-academia problems and you can see why a fake office is appropriately named so, then. (This is the first time I’ve tagged this strip, looks like.)

Employee: 'Cool 'power tie' boss'. The tie reads E = mc^2.
Harley Schwadron’s 9 to 5 for the 21st of March, 2018. I understand the tie has to face the audience to make the joke work, but isn’t it more fun to imagine that it’s actually a pyramidal tie, like, a solid triangular projection of tie material, and we see one side of it and maybe there’s another equation written on the other side? Please vote in the comments.

Harley Schwadron’s 9 to 5 for the 21st is a name-drop of Einstein’s famous equation as a power tie. I must agree this meets the literal specification of a power tie since, you know, c2 is in it. Probably something more explicitly about powers wouldn’t communicate as well. Possibly Fermat’s Last Theorem, although I’m not sure that would fit and be legible on the tie as drawn.

Clare: 'How many cylinders with length 3 and diameter 1.5 equal the volume of a sphere with diameter 3?' Neil: 'Um ... 2.6. no, 2.7!' Clare: 'Neil, how on earth did you know that?' Neil: 'It's simple, Clare! I converted the cylinder to 'Ho Hos' and the sphere to Hostess 'Sno Balls', then I imagined eating them!' Clare: 'Um ... wow.' Neil: 'My brain's only average, but my tummy's a genius!'
Mark Pett’s Lucky Cow for the 21st of March, 2018. I preferred Ding Dongs eater myself. But my heart was with the Suzy Q’s, if we’re not letting Tastykake into the discussion.

Mark Pett’s Lucky Cow rerun for the 21st has the generally inept Neil work out a geometry problem in his head. The challenge is having a good intuitive model for what the relationship between the shapes should be. I’m relieved to say that Neil is correct, to the number of decimal places given. I’m relieved because I’ve spent embarrassingly long at this. My trouble was missing, twice over, that the question gave diameters instead of radiuses. Pfaugh. Saving me was just getting answers that were clearly crazy, including at one point 21 1/3.

Professor in girl's daydream: 'But don't take my word for it. It's Euler's theorem.' (Points to e^{i pi} + 1 = 0 on the board.) Girl: 'Greg! Greg! I've changed my mind! Let's be colleagues again! ... Greg?' (Sees a closet jammed shut by a door.) Person inside: 'Help! I'm stuck!' (She unjams the door.) Person inside: 'Did she leave? Where's ray? Someone has to stop her!' Girl: 'That's like trying to stop a yeti!' Person inside: 'By my calculations it's far worse.' (Looks over sheet labelled 'Monster Unit Conversions', with Wray worked out to be 8 orcs or 3 trolls or 6 werewolves or werebears or 2.788 Yetis.)
Zach Weinersmith, Chris Jones and James Ashby’s Snowflakes for the 21st of March, 2018. I would like to give you more context for this but I confess I haven’t been able to follow the storyline. I don’t know why but this is one of the strips I don’t get the flow of.

Zach Weinersmith, Chris Jones and James Ashby’s Snowflakes for the 21st mentions Euler’s Theorem in the first panel. Trouble with saying “Euler’s Theorem” is that Euler had something like 82 trillion theorems. If you ever have to bluff your way through a conversation with a mathematician mention “Euler’s Theorem”. You’ll probably have said something on point, if closer to the basics of the problem than people figured. But the given equation — e^{\imath \pi} + 1 = 0 — is a good bet for “the” Euler’s Theorem. It’s a true equation, and it ties together a lot of interesting stuff about complex-valued numbers. It’s the way mathematicians tie together exponentials and simple harmonic motion. It makes so much stuff easier to work with. It would not be one of the things presented in a Distinctly Useless Mathematics text. But it would be mentioned along the way to something fascinating and useless. It turns up everywhere. (This is another strip I’m tagging for the first time.)

[ Cybil used to teach at MIT ] Cybil, teaching: 'If you've got pi/2 x 4 apples, and you eat Sigma x square root of cos(68) apples, how many apples do you have?' The class looks baffled.
Wulff and Morgenthaler’s WuMo for the 21st of March, 2018. Fun fact: since 68 is a rational number, the cosine of 68 has to be transcendental. All right, but it’s fun to me and whose blog is this? Thank you. But the cosine of any rational number other than zero is transcendental. Ditto the sine and the tangent.

Wulff and Morgenthaler’s WuMo for the 21st uses excessively complicated mathematics stuff as a way to signify intelligence. Also to name-drop Massachusetts Institute of Technology as a signifier of intelligence. (My grad school was Rensselaer Polytechnic Institute, which would totally be MIT’s rival school if we had enough self-esteem to stand up to MIT. Well, on a good day we can say snarky stuff about the Rochester Institute of Technology if we don’t think they’re listening.) Putting the “Sigma” in makes the problem literally nonsense, since “Sigma” doesn’t signify any particular number. The rest are particular numbers, though. π/2 times 4 is just 2π, a bit more than 6.28. That’s a weird number of apples to have but it’s perfectly legitimate a number. The square root of the cosine of 68 … ugh. Well, assuming this is 68 as in radians I don’t have any real idea what that would be either. If this is 68 degrees, then I do know, actually; the cosine of 68 degrees is a little smaller than ½. But mathematicians are trained to suspect degrees in trig functions, going instead for radians.

Well, hm. 68 would be between 11 times 2π and 12 times 2π. I think that’s just a little more than 11 times 2π. Oh, maybe it is something like ½. Let me check with an actual calculator. Huh. It is a little more than 0.440. Well, that’s a once-in-a-lifetime shot. Anyway the square root of that is a little more than 0.663. So you’d be left with about five and a half apples. Never mind this Sigma stuff. (A little over 5.619, to be exact.)

Reading the Comics, March 13, 2018: One Of My Assumptions Is Shaken Edition


I learn, from reading not-yet-dead Usenet group rec.arts.comics.strips, that Rick Stromoski is apparently ending the comic Soup To Nutz. This is sad enough. But worse, GoComics.com has removed all but the current day’s strip from its archives. I had trusted that GoComics.com links were reliable in a way that Comics Kingdom and Creators.com weren’t. Now I learn that maybe I need to include images of the comics I review and discuss here lest my essays become unintelligible in the future? That’s not a good sign. I can do it, mind you. I just haven’t got started. You’ll know when I swing into action.

Norm Feuti, of Retail, still draws Sunday strips for Gil. They’re to start appearing on GoComics.com soon, and I can talk about them from my regular sources after that. But for now I follow the strip on Twitter. And last Sunday he posted this one.

It’s sort of a protesting-the-problem question. It’s also a reaction a lot of people have to “explain how you found the answer” questions. In a sense, yeah, the division shows how the answer was found. But what’s wanted — and what’s actually worth learning — is to explain why you did this calculation. Why, in this case, 216 divided by 8? Why not 216 times 8? Why not 8 divided by 216? Why not 216 minus 8? “How you found your answer” is probably a hard question to make interesting on arithmetic, unfortunately. If you’re doing a long sheet of problems practicing division, it’s not hard to guess that dividing is the answer. And that it’s the big number divided by the small. It can be good training to do blocks of problems that use the same approach, for the same reason it can be good training to focus on any exercise a while. But this does cheat someone of the chance to think about why one does this rather than that.

Patrick Roberts’s Todd the Dinosaur for the 11th has mathematics as the thing Todd’s trying to get out of doing. (I suppose someone could try to argue the Y2K bug was an offshoot of mathematics, on the grounds that computer science has so much to do with mathematics. I wouldn’t want to try defending that, though.) I grant that most fraction-to-decimal conversion problems hit that sweet spot of being dull, tedious, and seemingly pointless. There’s some fun decimal expansions of fractions. The sevenths and the elevenths and 1/243 have charm to them. There’s some kid who’ll become a mathematician because at the right age she was told about \frac{1}{8991} . 3/16th? Eh.

Teacher: 'Who would like to come up here and work this converting-fractions-to-decimals problem on the board? Let's see ... how about you, Todd?' Todd: 'Look out! Y2K! AAAGH! This is terrible! Just terrible! It finally caught up with us! Goodbye, electricity! Goodbye, civilized society!' Todd: 'Nice try, Todd. Y2K never happened!' Todd: 'Uh, yeah, I knew that. I was just saying' that Y2K is the answer to that problem on the board!' Teacher: 'Also a nice try. Now get up here!'
Patrick Roberts’s Todd the Dinosaur for the 11th of March, 2018. I’m not sure that the loss of electricity would actually keep someone from doing chalkboard work, especially if there’s as many windows as we see here to let light in. I mean, yes, there’d be problems after school, but just during school? The end of civilization is not the cure-all people present it as being.

Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. I don’t remember seeing a spinny wheel like this used to introduce probability. It’s a good prop, though. I would believe in a class having it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 11th is built on the Travelling Salesman Problem. It’s one of the famous unsolved and hard problems of mathematics. Weinersmith’s joke is a nice gag about one way to “solve” the problem, that of making it irrelevant. But even if we didn’t need to get to a collection of places efficiently mathematicians would still like to know good ways to do it. It turns out that finding the shortest (quickest, cheapest, easiest, whatever) route connecting a bunch of places is great problem. You can phrase enormously many problems about doing something as well as possible as a Travelling Salesman Problem. It’s easy conceptually to find the answer: try out all the possibilities and pick the best one. But if there’s more than a handful of cities, there are so many possible routes there’s no checking them all, not before you die of old age. We can do very well finding approximate answers, including by my specialization of Monte Carlo methods. In those you take a guess at an answer. Then make, randomly, a change. You’ll either have made things better or worse. If you’ve made it better, keep the change. If you’ve made it worse, usually you reject the change but sometimes you keep it. And repeat. In surprisingly little time you’ll get a really good answer. Maybe not the best possible, but a great answer for how straightforward setting it up was.

Dan Thompson’s Brevity for the 12th is a Rubik’s Cube joke. There’s not a lot of mathematics to that. But I do admire how Thompson was careful enough to draw a Rubik’s Cube that actually looks like the real article; it’s not just an isometric cube with thick lines partitioning it. Look at the corners of each colored sub-cube. I may be the only reader to notice this but I’m glad Thompson did the work.

Mason Mastroianni’s The Wizard of Id for the 12th gets Sir Rodney in trouble with the King for doing arithmetic. I haven’t read the comments on GoComics.com. I’d like to enter “three” as my guess for how many comments one would have to read before finding the “weapons of math instruction” joke in there.

Jef Mallett’s Frazz for the 13th has mathematics homework given as the thing lost by the time change. It’s just a cameo mention.

Steve Moore’s In The Bleachers for the 13th features a story problem as a test of mental acuity. When the boxer can’t work out what the heck the trains-leaving-Penn-Station problem even means he’s ruled unfit to keep boxing. The question is baffling, though. As put, the second train won’t ever overtake the first. The question: did Moore just slip up? If the first train were going 30 miles per hour and the second 40 there would be a perfectly good, solvable question in this. Or was Moore slipping in an extra joke, making the referee’s question one that sounds like it was given wrong? Don’t know, so I’ll suppose the second.

Reading the Comics, February 24, 2018: My One Boring Linear Algebra Anecdote Edition


Wait for it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st mentions mathematics — geometry, primarily — as something a substitute teacher has tried teaching with the use of a cucumber and condom. These aren’t terrible examples to use to make concrete the difference between volumes and surface areas. There are limitations, though. It’s possible to construct a shape that has a finite volume but an infinitely large surface area, albeit not using cucumbers.

There’s also a mention of the spring constant, and physics. This isn’t explicitly mathematical. But the description of movement on a spring are about the first interesting differential equation of mathematical physics. The solution is that of simple harmonic motion. I don’t think anyone taking the subject for the first time would guess at the answer. But it’s easy enough to verify it’s right. And this motion — sine waves — just turns up everywhere in mathematical physics.

Bud Blake’s Tiger rerun for the 23rd just mentions mathematics as a topic Hugo finds challenging, and what’s challenging about it. So a personal story: when I took Intro to Linear Algebra my freshman year one day I spaced on the fact we had an exam. So, I put the textbook on the shelf under my desk, and then forgot to take it when I left. The book disappeared, of course, and the professor never heard of it being turned in to lost-and-found or anything. Fortunately the homework was handwritten questions passed out on photocopies (ask your parents), so I could still do the assignments, but for all those, you know, definitions and examples I had to rely on my own notes. I don’t know why I couldn’t ask a classmate. Shyness, probably. Came through all right, though.

Hugo: 'These math problems we got for homework are gonna be hard to do.' Tiger: 'Because you don't understand them?' Hugo: 'Because I brought home my history book by mistake.'
Bud Blake’s Tiger rerun for the 23rd of February, 2018. Whose house are they at? I mean, did Tiger bring his dog to Hugo’s, or did Hugo bring his homework to Tiger’s house? I guess either’s not that odd, especially if they just got out of school, but then Hugo’s fussing with his homework when he’s right out of school and with Tiger?

Cathy Law’s Claw for the 23rd technically qualifies as an anthropomorphic-numerals joke, in this panel about the smothering of education by the infection of guns into American culture.

Jim Meddick’s Monty for the 23rd has wealthy child Wedgwick unsatisfied with a mere ball of snow. He instead has a snow Truncated Icosahedron (the hyphens in Jarvis’s word balloon may baffle the innocent reader). This is a real shape, one that’s been known for a very long time. It’s one of the Archimedean Solids, a set of 13 solids that have convex shapes (no holes or indents or anything) and have all vertices the same, the identical number of edges coming in to each point in the same relative directions. The truncated icosahedron you maybe also know as the soccer ball shape, at least for those old-style soccer balls made of patches that were hexagons and pentagons. An actual truncated icosahedron needs twelve pentagons, so the figure drawn in the third panel isn’t quite right. At least one pentagonal face would be visible. But that’s also tricky to draw. The aerodynamics of a truncated icosahedron are surely different from those of a sphere. But in snowball-fight conditions, probably not different enough to even notice.

Mark Litzler’s Joe Vanilla for the 24th uses a blackboard full of formulas to represent an overcomplicated answer. The formulas look, offhand, like gibberish to me. But I’ll admit uncertainty since the odd capitalization of “iG(p)” at the start makes me think of some deeper group theory or knot theory symbols. And to see an “m + p” and an “m – p” makes me think of quantum mechanics of atomic orbitals. (But then an “m – p2” is weird.) So if this were anything I’d say it was some quantum chemistry formula. But my gut says if Litzler did take the blackboard symbols from anything, it was without going back to references. (Which he has no need to do, I should point out; the joke wouldn’t be any stronger — or weaker — if the blackboard meant anything.)

Reading the Comics, February 10, 2018: I Meant To Post This Thursday Edition


Ah, yes, so, in the midst of feeling all proud that I’d gotten my Reading the Comics workflow improved, I went out to do my afternoon chores without posting the essay. I’m embarrassed. But it really only affects me looking at the WordPress Insights page. It publishes this neat little calendar-style grid that highlights the days when someone’s posted and this breaks up the columns. This can only unnerve me. I deserve it.

Tom Thaves’s Frank and Ernest for the 8th of February is about the struggle to understand zero. As often happens, the joke has a lot of truth to it. Zero bundles together several ideas, overlapping but not precisely equal. And part of that is the idea of “nothing”. Which is a subtly elusive concept: to talk about the properties of a thing that does not exist is hard. As adults it’s easy to not notice this anymore. Part’s likely because mastering a concept makes one forget what it took to understand. Part is likely because if you don’t have to ponder whether the “zero” that’s “one less than one” is the same as the “zero” that denotes “what separates the count of thousands from the count of tens in the numeral 2,038” you might not, and just assume you could explain the difference or similarity to someone who has no idea.

John Zakour and Scott Roberts’s Maria’s Day for the 8th has maria and another girl bonding over their hatred of mathematics. Well, at least they’re getting something out of it. The date in the strip leads me to realize this is probably a rerun. I’m not sure just when it’s from.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th proposes a prank based on mathematical use of the word “arbitrarily”. This is a word that appears a lot in analysis, and the strip makes me realize I’m not sure I can give a precise definition. An “arbitrarily large number”, for example, would be any number that’s large enough. But this also makes me realize I’m not sure precisely what joke Weinersmith is going for. I suppose that if someone were to select an arbitrarily large number they might pick 53, or a hundred, or million billion trillion. I suppose Weinersmith’s point is that in ordinary speech an arbitrarily made choice is one selection from all the possible alternatives. In mathematical speech an arbitrarily made choice reflects every possible choice. To speak of an arbitrarily large number is to say that whatever selection is made, we can go on to show this interesting stuff is true. We’d typically like to prove the most generically true thing possible. But picking a single example can be easier to prove. It can certainly be easier to visualize. 53 is probably easier to imagine than “every number 52 or larger”, for example.

Quincy: 'Someday I'm gonna write a book, Gran.' Grandmom: 'Wonderful. Will you dedicate it to me?' Quincy: 'Sure. In fact, if you want, I'll dedicate this math homework to you.'
Ted Shearer’s Quincy for the 16th of December, 1978 and reprinted the 9th of February, 2018. I’m not sure just what mathematics homework Quincy could be doing to inspire him to write a book, but then, it’s not like my mind doesn’t drift while doing mathematics either. And book-writing’s a common enough daydream that most people are too sensible to act on.

Ted Shearer’s Quincy for the 16th of December, 1978 was rerun the 9th of February. It just shows Quincy at work on his mathematics homework, and considering dedicating it to his grandmother. Mathematics books have dedications, just as any other book does. I’m not aware of dedications of proofs or other shorter mathematics works, but there’s likely some. There’s often a note of thanks, usually given to people who’ve made the paper’s writers think harder about the subjects. But I don’t think there’s any reason a paper wouldn’t thank someone who provided “mere” emotional support. I just don’t have examples offhand.

Jef Mallet’s Frazz for the 9th looks like one of those creative-teaching exercises I sometimes see in Mathematics Education Twitter: the teacher gives answers and the students come up with story problems to match. That’s not a bad project. I’m not sure how to grade it, but I haven’t done anything that creative when I’ve taught. I’m sorry I haven’t got more to say about it since the idea seems fun.

Redeye: 'C'mon, Pokey. Time for your lessons. Okay, what do you get when you divide 5,967,342 by 973 ... ?' Pokey: 'A headache!'
Gordon Bess’s Redeye for the 30th of September, 1971 and reprinted the 10th of February, 2018. I realized I didn’t know the father’s name and looked it up, and Wikipedia revealed to me that he’s named Redeye. You know, like the comic strip implies right there in the title. Look, I just read the comics, I can’t be expected to think about the comics too.

Gordon Bess’s Redeye for the 30th of September, 1971 was rerun the 10th. It’s a bit of extremely long division and I don’t blame Pokey for giving up on that problem. Starting from 5,967,342 divided by 973 I’d say, well, that’s about six million divided by a thousand, so the answer should be near six thousand. I don’t think the last digits of 2 and 3 suggest anything about what the final digit should be, if this divides evenly. So the only guidance I have is that my answer ought to be around six thousand and then we have to go into actually working. It turns out that 973 doesn’t go into 5,967,342 a whole number of times, so I sympathize more with Pokey. The answer is a little more than 6,132.9311.

Reading the Comics, February 3, 2018: Overworked Edition


And this should clear out last week’s mathematically-themed comic strips. I didn’t realize just how busy last week had been until I looked at what I thought was a backlog of just two days’ worth of strips and it turned out to be about two thousand comics. I exaggerate, but as ever, not by much. This current week seems to be a more relaxed pace. So I’ll have to think of something to write for the Tuesday and Thursday slots. Hm. (I’ll be all right. I’ve got one thing I need to stop bluffing about and write, and there’s usually a fair roundup of interesting tweets or articles I’ve seen that I can write. Those are often the most popular articles around here.)

Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st of February, 2018 gives us an anthropomorphic geometric figures joke for the week. Also a side of these figures that I don’t think I’ve seen in the newspaper comics before. It kind of raises further questions.

The Geometry. A pair of parallel lines, one with a rectangular lump. 'Not true --- parallel lines *do* meet. In fact, Peter and I are expected.' ('We met at a crossroads in both our lives.')
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st of February, 2018. All right, but they’re line segments, but I suppose you can’t reasonably draw infinitely vast things in a daily newspaper strip’s space. The lean of that triangle makes it look way more skeptical, even afraid, than I think Price and Piccolo intended, but I’m not sure there’s a better way to get these two in frame without making the composition weird.

Jason Chatfield’s Ginger Meggs for the 1st just mentions that it’s a mathematics test. Ginger isn’t ready for it.

Mark Tatulli’s Heart of the City rerun for the 1st finally has some specific mathematics mentioned in Heart’s efforts to avoid a mathematics tutor. The bit about the sum of adjacent angles forming a right line being 180 degrees is an important one. A great number of proofs rely on it. I can’t deny the bare fact seems dull, though. I know offhand, for example, that this bit about adjacent angles comes in handy in proving that the interior angles of a triangle add up to 180 degrees. At least for Euclidean geometry. And there are non-Euclidean geometries that are interesting and important and for which that’s not true. Which inspires the question: on a non-Euclidean surface, like say the surface of the Earth, is it that adjacent angles don’t add up to 180 degrees? Or does something else in the proof of a triangle’s interior angles adding up to 180 degrees go wrong?

The Eric the Circle rerun for the 2nd, by JohnG, is one of the occasional Erics that talk about π and so get to be considered on-topic here.

Bill Whitehead’s Free Range for the 2nd features the classic page full of equations to demonstrate some hard mathematical work. And it is the sort of subject that is done mathematically. The equations don’t look to me anything like what you’d use for asteroid orbit projections. I’d expect forecasting just where an asteroid might hit the Earth to be done partly by analytic formulas that could be done on a blackboard. And then made precise by a numerical estimate. The advantage of the numerical estimate is that stuff like how air resistance affects the path of something in flight is hard to deal with analytically. Numerically, it’s tedious, but we can let the computer deal with the tedium. So there’d be just a boring old computer screen to show on-panel.

Bud Fisher’s Mutt and Jeff reprint for the 2nd is a little baffling. And not really mathematical. It’s just got a bizarre arithmetic error in it. Mutt’s fiancee Encee wants earrings that cost ten dollars (each?) and Mutt takes this to be fifty dollars in earring costs and I have no idea what happened there. Thomas K Dye, the web cartoonist who’s done artwork for various article series, has pointed out that the lettering on these strips have been redone with a computer font. (Look at the letters ‘S’; once you see it, you’ll also notice it in the slightly lumpy ‘O’ and the curly-arrow ‘G’ shapes.) So maybe in the transcription the earring cost got garbled? And then not a single person reading the finished product read it over and thought about what they were doing? I don’t know.

Zach Weinersmith’s Saturday Morning Breakfast Cereal reprint for the 2nd is based, as his efforts to get my attention often are, on a real mathematical physics postulate. As the woman postulates: given a deterministic universe, with known positions and momentums of every particle, and known forces for how all these interact, it seems like it should be possible to predict the future perfectly. It would also be possible to “retrodict” the past. All the laws of physics that we know are symmetric in time; there’s no reason you can’t predict the motion of something one second into the past just as well as you an one second into the future. This fascinating observation took a lot of battery in the 19th century. Many physical phenomena are better described by statistical laws, particularly in thermodynamics, the flow of heat. In these it’s often possible to predict the future well but retrodict the past not at all.

But that looks as though it’s a matter of computing power. We resort to a statistical understanding of, say, the rings of Saturn because it’s too hard to track the billions of positions and momentums we’d need to otherwise. A sufficiently powerful mathematician, for example God, would be able to do that. Fair enough. Then came the 1890s. Henri Poincaré discovered something terrifying about deterministic systems. It’s possible to have chaos. A mathematical representation of a system is a bit different from the original system. There’s some unavoidable error. That’s bound to make some, larger, error in any prediction of its future. For simple enough systems, this is okay. We can make a projection with an error as small as we need, at the cost of knowing the current state of affairs with enough detail. Poincaré found that some systems can be chaotic, though, ones in which any error between the current system and its representation will grow to make the projection useless. (At least for some starting conditions.) And so many interesting systems are chaotic. Incredibly simplified models of the weather are chaotic; surely the actual thing is. This implies that God’s projection of the universe would be an amusing but almost instantly meaningless toy. At least unless it were a duplicate of the universe. In which case we have to start asking our philosopher friends about the nature of identity and what a universe is, exactly.

Ruben Bolling’s Super-Fun-Pak Comix for the 2nd is an installment of Guy Walks Into A Bar featuring what looks like an arithmetic problem to start. It takes a turn into base-ten jokes. There are times I suspect Ruben Bolling to be a bit of a nerd.

Nate Fakes’s Break of Day for the 3rd looks like it’s trying to be an anthropomorphic-numerals joke. At least it’s an anthropomorphic something joke.

Percy Crosby’s Skippy for the 3rd originally ran the 8th of December, 1930. It alludes to one of those classic probability questions: what’s the chance that in your lungs is one of the molecules exhaled by Julius Caesar in his dying gasp? Or whatever other event you want: the first breath you ever took, or something exhaled by Jesus during the Sermon on the Mount, or exhaled by Sue the T-Rex as she died. Whatever. The chance is always surprisingly high, which reflects the fact there’s a lot of molecules out there. This also reflects a confidence that we can say one molecule of air is “the same” as some molecule if air in a much earlier time. We have to make that supposition to have a problem we can treat mathematically. My understanding is chemists laugh at us if we try to suggest this seriously. Fair enough. But whether the air pumped out of a bicycle tire is ever the same as what’s pumped back in? That’s the same kind of problem. At least some of the molecules of air will be the same ones. Pretend “the same ones” makes sense. Please.

Reading the Comics, January 27, 2018: Working Through The Week Edition


And today I bring the last couple mathematically-themed comic strips sent my way last week. GoComics has had my comics page working intermittently this week. And I was able to get a response from them, by e-mailing their international sales office, the only non-form contact I could find. Anyway, this flood of comics does take up the publishing spot I’d figured for figuring how I messed up Wronski’s formula. But that’s all right, as I wanted to spend more time thinking about that. Here’s hoping spending more time thinking works out for me.

Nate Fakes’s Break of Day for the 24th was the big anthropomorphic numerals joke for the week. And it’s even dubbed the numbers game.

Mark Tatulli’s Heart of the City from the 24th got into a storyline about Heart needing a mathematics tutor. It’s a rerun sequence, although if you remember a particular comic storyline from 2009 you’re doing pretty well. Nothing significantly mathematical has turned up in the story so far, past the mention of fractions as things that exist and torment students. But the stories are usually pretty good for this sort of strip.

Mikael Wulff and Anders Morganthaler’s WuMo for the 24th includes a story problems freak out. I’m not sure what’s particularly implausible about buying nine apples. I’d agree a person is probably more likely to buy an even number of things, since we seem to like numbers like “ten” and “eight” so well, but it’s hardly ridiculous.

Tim Rickard’s Brewster Rockit for the 25th is an arithmetic class on the Snowman Planet. So there’s some finger-counting involved.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a reminder that most of my days are spent seeing how Zach Weinersmith wants my attention. It also includes what I suppose is a legitimate attempt to offer a definition for what all mathematics is. It’s hard to come up with something that does cover all the stuff mathematicians do. Bear in mind, this includes counting, calculating how far the Sun is based on the appearance of a lunar eclipse, removing static from a recording, and telling how many queens it’s possible to place eight queens on a chess board that’s wrapped around a torus without any being able to capture another, among other problems. My instinct is to dismiss the proposed “anything you can think deeply about that has no reference to the real world”. That seems over-broad, and to cover a lot of areas that are really philosophy’s beat. And I think there’s something unseemly in mathematicians gloating about their work having no “practical” use. I grant I come from an applied school, and I came to there through an interest in physics. But to build up “inapplicability to the real word” as if it were some ideal, as opposed to just how something has turned out to be right now, strikes me as silly. Applicability is so dependent on context, on culture, and accidents of fate that there’s no way it can be important to characterizing mathematics. And it would imply that once we found a use for something it would stop being mathematically interesting. I don’t see evidence of that in mathematical history.

Mikael Wulff and Anders Morganthaler’s WuMo pops back in on the 27th with an appearance of sudoku, presenting the logic puzzle as one of the many things beyond the future Disgraced Former President’s abilities.

Reading the Comics, January 23, 2018: Adult Content Edition


I was all set to say how complaining about GoComics.com’s pages not loading had gotten them fixed. But they only worked for Monday alone; today they’re broken again. Right. I haven’t tried sending an error report again; we’ll see if that works. Meanwhile, I’m still not through last week’s comic strips and I had just enough for one day to nearly enough justify an installment for the one day. Should finish off the rest of the week next essay, probably in time for next week.

Mark Leiknes’s Cow and Boy rerun for the 23rd circles around some of Zeno’s Paradoxes. At the heart of some of them is the question of whether a thing can be divided infinitely many times, or whether there must be some smallest amount of a thing. Zeno wonders about space and time, but you can do as well with substance, with matter. Mathematics majors like to say the problem is easy; Zeno just didn’t realize that a sum of infinitely many things could be a finite and nonzero number. This misses the good question of how the sum of infinitely many things, none of which are zero, can be anything but infinitely large? Or, put another way, what’s different in adding \frac11 + \frac12 + \frac13 + \frac14 + \cdots and adding \frac11 + \frac14 + \frac19 + \frac{1}{16} + \cdots that the one is infinitely large and the other not?

Or how about this. Pick your favorite string of digits. 23. 314. 271828. Whatever. Add together the series \frac11 + \frac12 + \frac13 + \frac14 + \cdots except that you omit any terms that have your favorite string there. So, if you picked 23, don’t add \frac{1}{23} , or \frac{1}{123} , or \frac{1}{802301} or such. That depleted series does converge. The heck is happening there? (Here’s why it’s true for a single digit being thrown out. Showing it’s true for longer strings of digits takes more work but not really different work.)

J C Duffy’s Lug Nuts for the 23rd is, I think, the first time I have to give a content warning for one of these. It’s a porn-movie advertisement spoof. But it mentions Einstein and Pi and has the tagline “she didn’t go for eggheads … until he showed her a new equation!”. So, you know, it’s using mathematics skill as a signifier of intelligence and riffing on the idea that nerds like sex too.

John Graziano’s Ripley’s Believe It or Not for the 23rd has a trivia that made me initially think “not”. It notes Vince Parker, Senior and Junior, of Alabama were both born on Leap Day, the 29th of February. I’ll accept this without further proof because of the very slight harm that would befall me were I to accept this wrongly. But it also asserted this was a 1-in-2.1-million chance. That sounded wrong. Whether it is depends on what you think the chance is of.

Because what’s the remarkable thing here? That a father and son have the same birthday? Surely the chance of that is 1 in 365. The father could be born any day of the year; the son, also any day. Trusting there’s no influence of the father’s birthday on the son’s, then, 1 in 365 it is. Or, well, 1 in about 365.25, since there are leap days. There’s approximately one leap day every four years, so, surely that, right?

And not quite. In four years there’ll be 1,461 days. Four of them will be the 29th of January and four the 29th of September and four the 29th of August and so on. So if the father was born any day but leap day (a “non-bissextile day”, if you want to use a word that starts a good fight in a Scrabble match), the chance the son’s birth is the same is 4 chances in 1,461. 1 in 365.25. If the father was born on Leap Day, then the chance the son was born the same day is only 1 chance in 1,461. Still way short of 1-in-2.1-million. So, Graziano’s Ripley’s is wrong if that’s the chance we’re looking at.

Ah, but what if we’re looking at a different chance? What if we’re looking for the chance that the father is born the 29th of February and the son is also born the 29th of February? There’s a 1-in-1,461 chance the father’s born on Leap Day. And a 1-in-1,461 chance the son’s born on Leap Day. And if those events are independent, the father’s birth date not influencing the son’s, then the chance of both those together is indeed 1 in 2,134,521. So Graziano’s Ripley’s is right if that’s the chance we’re looking at.

Which is a good reminder: if you want to work out the probability of some event, work out precisely what the event is. Ordinary language is ambiguous. This is usually a good thing. But it’s fatal to discussing probability questions sensibly.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 23rd presents his mathematician discovering a new set of numbers. This will happen. Mathematics has had great success, historically, finding new sets of things that look only a bit like numbers were understood. And showing that if they follow rules that are, as much as possible, like the old numbers, we get useful stuff out of them. The mathematician claims to be a formalist, in the punch line. This is a philosophy that considers mathematical results to be the things you get by starting with some symbols and some rules for manipulating them. What this stuff means, and whether it reflects anything of interest in the real world, isn’t of interest. We can know the results are good because they follow the rules.

This sort of approach can be fruitful. It can force you to accept results that are true but intuition-defying. And it can give results impressive confidence. You can even, at least in principle, automate the creating and the checking of logical proofs. The disadvantages are that it takes forever to get anything done. And it’s hard to shake the idea that we ought to have some idea what any of this stuff means.

Reading the Comics, January 16, 2017: Better Workflow Edition


So one little secret of my Reading the Comics posts is I haven’t been writing them in a way that makes sense to me. To me, I should take each day’s sufficiently relevant comics, describe them in a paragraph or two, and then have a nice pile of text all ready for the posting Sunday and, if need be, later. I haven’t been doing that. I’ve let links pile up until Friday or Saturday, and then try to process them all, and if you’ve ever wondered why the first comic of the week gets 400 words about some subtlety while the last gets “this is a comic that exists”, there you go. This time around, let me try doing each day’s strips per day and see how that messes things up.

Jef Mallett’s Frazz for the 14th of January is another iteration of the “when will we ever use mathematics” complaint. The answer of “you’ll use it on the test” is unsatisfactory. But somehow, the answer of “you’ll use it to think deeply about something you had never considered before” also doesn’t satisfy. Anyway I’d like to see the idea that education is job-training abolished; I think it should be about making a person conversant with the history of human thought. That can’t be done perfectly, and we might ask whether factoring 32 is that important a piece, but it should certainly be striven for.

Ham’s Life on Earth for the 14th is a Gary Larsonesque riff on that great moment of calculus and physics history, Newton’s supposition that gravity has to follow a universally true law. I’m not sure this would have made my cut if I reviewed a week’s worth of strips at a time. Hm.

Mason Mastroianni’s B.C. for the 15th is a joke about story problem construction, and how the numbers in a story problem might be obvious nonsense. It’s also a cheap shot at animal hoarders, I suppose, but that falls outside my territory here.

Anthony Blades’s Bewley rerun for the 15th riffs on the natural number sense we all have. And we do have a number sense, remarkably. We might not be able to work out 9 times 6 instantly. But asked to pick from a list of possible values, we’re more likely to think that 58 is credible than that 78 or 38 are. It’s quite imprecise, but isn’t it amazing that it’s there at all?

Bill Amend’s FoxTrot Classics for the 15th is a story problem joke, in this case, creating one with a strong motivation for its solution to be found. The strip originally ran the 22nd of January, 1996.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 16th is maybe marginal to include, too. It’s about the kinds of logic puzzles that mathematicians grow up reading and like to pass around. And the way you can fake out someone by presenting a problem with too obvious a solution. It’s not just professors who’ll be stymied by having the answer look too obvious, by the way. Everyone’s similarly vulnerable. To see anything, including an abstract thing like the answer to a puzzle, you need some idea of what you are looking at. If you don’t think the answer could be something that simple, you won’t see it there.

Paw: 'It's four o'clock ... what time are we going to eat?' Maw :'About five.' Paw: 'Good! That gives me two hours to work with Pokey on his arithmeteic.'
Gordon Bess’s Redeye for the 6th of September, 1971. That’s the sort of punch line that really brings out the comically-anachronistic Old West theme.

Gordon Bess’s Redeye for the 6th of September, 1971, was reprinted the 17th. It’s about the fun of teaching a subject you aren’t all that good on yourself. The mathematics is a name-drop here, but the joke wouldn’t make sense if it were about social studies.

Popeye: 'King, they's one thing I wants to know. How much is a pezozee?' King Blozo: 'Why bring that up?' Popeye: 'Yer men hired me to help lick yer emeny at a thousing pezozees a week - tha's why I'd like to know what is a pezozee.' Blozo: 'A pezozee is two pazookas.' Popeye: 'What's a pazooky?' Blozo: 'A pazooka is two pazinkas.' Popeye: 'What's a pazinky?' Blozo: 'A pazinka is two pazoonies.' Popeye: 'What's a pazeenya?' Blozo: 'Phooey! I wish you would quit following me! A pazooney is two pazeenyas.' Popeye: 'what's a pazeenya?' Blozo: 'Two pazimees.' Popeye: 'Hey! What's a pazimee worth?' Blozo: 'Absolutely nothing!' Popeye: 'Blow me down, I'm glad I ain't gettin' paid in pazimees!'
Elzie Segar’s Thimble Theatre for the 10th of August, 1931. Not listed: the rate of exchange for paczki, which reappeared this week.

Elzie Segar’s Thimble Theatre for the 10th of August, 1931, was also reprinted the 17th. It’s an old gag, even back when it was first run. But I suppose there’s some numerical-conversion mathematics to wring out of it. Given the rate of exchange, a pezozee would seem to be 24 pazimees. I’m not sure we need so many units in-between the pazimee and the pezozee, but perhaps King Blozo’s land set its units in a time when fractions were less familiar to the public. The punch line depends on the pazimee being worth nothing and, taken literally, that has sad implications for the pezozee too. If you take the King as speaking roughly, though, sixteen times a small amount is … at least a less small amount. It wouldn’t take many doublings to go from an infinitesimally tiny sum to a respectable one.

And it turns out there were enough comic strips I need to split this into two segments. So I should schedule that to appear. It’s already written and everything.

Reading the Comics, January 13, 2018: Barney Google Is Messing With My Head For Some Reason Edition


I do not know what’s possessed John Rose, cartoonist for Barney Google and Snuffy Smith — possibly the oldest syndicated comic strip not in perpetual reruns — to decide he needs to mess with my head. So far as I’m aware we haven’t ever even had any interactions. While I’ll own up to snarking about the comic strip here and there, I mean, the guy draws Barney Google and Snuffy Smith. He won’t attract the snark community of, say, Marmaduke, but he knew the job was dangerous when he took it. There’s lots of people who’ve said worse things about the comic than I ever have. He can’t be messing with them all.

There’s no mathematical content to it, but here, continuing the curious thread of Elviney and Miss Prunelly looking the same, and Elviney turning out to have a twin sister, is the revelation that Elviney’s husband also has a twin.

Loweezey: 'I know YOU have always been yore maw's fav'rit, Snuffy. Who is yore paw's?' Snuffy: 'Paw!!' Loweezey: 'Elviney, who's that wif Lukey?' Elviney: 'His brother Lucious!! They ain't seen each other fer years! But look at 'em. Thar able to pick up right whar they left off! It's like they've never been apart!' Lukey: 'Did not! Did not! Did not!' Lucius: 'Did too! Did too! Did too!'
John Rose’s Barney Google and Snuffy Smith for the 14th of January, 2018. The commenters at Comics Kingdom don’t know where this Lucius character came from so I guess now suddenly everybody in Hootin Holler is a twin and we never knew it before I started asking questions?

This means something and I don’t know what.

To mathematics:

Zach Weinersmith’s Saturday Morning Breakfast Cereal gets my attention again for the 10th. There is this famous quotation from Leopold Kronecker, one of the many 19th century German mathematicians who challenged, and set, our ideas of what mathematics is. In debates about what should count as a proof Kronecker said something translated in English to, “God created the integers, all else is the work of man”. He favored proofs that only used finite numbers, and only finitely many operations, and was skeptical of existence proofs. Those are ones that show something with desired properties must exist, without necessarily showing how to find it. Most mathematicians accept existence proofs. If you can show how to find that thing, that’s a constructive proof. Usually mathematicians like those better.

Mark Tatulli’s Heart of the City for the 11th uses a bunch of arithmetic and word problems to represent all of Dean’s homework. All looks like reasonable homework for my best guess about his age.

Jon Rosenberg’s Scenes From A Multiverse for the 11th is a fun, simple joke with some complex stuff behind it. It’s riffing on the kind of atheist who wants moral values to come from something in the STEM fields. So here’s a mathematical basis for some moral principles. There are, yes, ethical theories that have, or at least imply having, mathematics behind them. Utilitarianism at least supposes that ethical behavior can be described as measurable and computable quantities. Nobody actually does that except maybe to make video games more exciting. But it’s left with the idea that one could, and hope that this would lead to guidance that doesn’t go horribly wrong.

Don Asmussen’s Bad Reporter for the 12th uses knowledge of arithmetic as a signifier of intelligence. Common enough joke style.

Thom Bluemel’s Birdbrains for the 13th starts Pi Day observances early, or maybe supposed the joke would be too out of season were it to come in March.

Greg Evans and Karen Evans’s Luann for the 13th uses mathematics to try building up the villainy of one of the strip’s designated villains. Ann Eiffel, there, uses a heap of arithmetic to make her lingerie sale sound better. This isn’t simply a riff on people not wanting to do arithmetic, although I understand people not wanding to work out what five percent of a purchase of over $200 is. There’s a good deal of weird psychology in getting people to buy things. Merely naming a number, for example, gets people to “anchor” their expectations to it. To speak of a free gift worth $75 makes any purchase below $75 seem more economical. To speak of a chance to win $1,000 prepares people to think they’ve got a thousand dollars coming in, and that they can safely spend under that. It’s amazing stuff to learn about, and it isn’t all built on people being too lazy to figure out what five percent off of $220 would be.

T Lewis and Michael Fry’s Over the Hedge for the 13th uses &infty; along the way to making nonsense out of ice-skating judging. It’s a good way to make a hash of a rating system. Most anything done with infinitely large numbers or infinitely large sets challenges one’s intuition at least. This is part of what Leopold Kronecker was talking about.

Reading the Comics, December 16, 2017: Andertoons Drought Ended Edition


And now, finally, we get what we’ve been waiting so long for: my having enough energy and time to finish up last week’s comics. And I make excuses to go all fanboy over Elzie Segar’s great Thimble Theatre. Also more attention to Zach Weinersmith. You’ve been warned.

Mark Anderson’s Andertoons for the 13th is finally a breath of Mark Anderson’s Andertoons around here. Been far too long. Anyway it’s an algebra joke about x’s search for identity. And as often happens I’m sympathetic here. It’s not all that weird to think of ‘x’ as a label for some number. Knowing whether it means “a number whose value we haven’t found yet” or “a number whose value we don’t care about” is one trick, though. It’s not something you get used to from learning about, like, ‘6’. And knowing whether we can expect ‘x’ to have held whatever value it represented before, or whether we can expect it to be something different, is another trick.

Doug Bratton’s Pop Culture Shock Therapy for the 13th I feel almost sure has come up here before. Have I got the energy to find where? Oh, yes. It ran the 5th of September, 2015.

Buckles: Bark! ... Bark bark! ... Bark bark bark! ... (Dazzled.) 'It's difficult to bark sequentially when you don't know how to count.'
David Gilbert’s Buckles for the 14th of December, 2017. I quite like Buckles’s little off-put look in the final panel. It’s very dog considering the situation.

David Gilbert’s Buckles for the 14th is a joke on animals’ number sense. In fairness, after that start I wouldn’t know whether to go for four or five barks myself.

Hugo: 'Adding a long column of numbers is hard. Maybe it'll be easier if I write smaller. Then the column will be shorter.'
Bud Blake’s Tiger for the 15th of December, 2017. One of my love’s favorite recurring motifs in Peanuts is when Sally works out some ridiculous string of not-quite-reasoning and Charlie Brown just sits and watches and kind of stares at the reader through it. Tiger is definitely doing that same “… what?” look as Hugo figures out his strategy.

Bud Blake’s Tiger for the 15th is a bit of kid logic about how to make a long column of numbers easier to add. I endorse the plan of making the column shorter, although I’d do that by trying to pair up numbers that, say, add to 10 or 20 or something else easy to work with. Partial sums can make the overall work so much easier. And probably avoid mistakes.

Bunzo: 'You mean to say I was hit by just one man?' Referee: 'Yes, one man - you must get up, the count will soon be to ten. My gosh, General, you must get up - I'm running out of fractions. 8 19/20 - 9 - 9 1/25 - 9 2/25 - 9 3/25 --- ' Bunzo: 'Use hundredths.' (Getting up.) 'You rat! Everybody's laughing at me! Me, the great chief General!! You're not supposed to do me like this!' Popeye: 'Don't get sore, General. Come on, it's your turn to sock me.' Bunzo: 'Hold still so I can bust your chin.' Popeye: 'Okay, shoot.' Bunzo: 'That'll finish you!' (Smacking Popeye on the chin. It's not very effective.) Popeye: 'You should eat more spinach.' Bunzo: 'Great guns! Are you still standing?!!'
Elzie Segar’s Thimble Theatre for the 8th of July, 1931, and rerun the 15th of December, 2017. If I’m not missing, this week has included Popeye’s first claims about spinach providing him with superior strength. And I know you’re looking at the referee there and thinking J Wellington Wimpy. I’m not sure, since I haven’t checked the complete collection to read ahead in the story, but I think this is merely a proto-Wimpy. (Mind, the Wikipedia entry on this is a complete mess. Bud Sagendorf’s Popeye: The First Fifty Years says Wimpy was derived from a minor character in Segar’s earlier The Five-Fifteen strip, which would itself turn into Sappo. But that proto-Wimpy didn’t have much personality or even a name.)

Elzie Segar’s Thimble Theatre for the 8th of July, 1931, is my most marginal inclusion yet. It was either that strip or the previous day’s worth including. I’m throwing it in here because Segar’s Thimble Theatre keeps being surprisingly good. And, heck, slowing a count by going into fractions is viable way to do it. As the clobbered General Bunzo points out, you can drag this out longer by going into hundredths. Or smaller units. There is no largest real number less than ten; if it weren’t incredibly against the rules, boxers could make use of that.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th is about those mathematics problems with clear and easy-to-understand statements whose answers defy intuition. Weinersmith is completely correct about all of this. I’m surprised he doesn’t mention the one about how you could divide an orange into five pieces, reassemble the pieces, and get back two spheres each the size of a sun.

Reading the Comics, December 9, 2017: Zach Weinersmith Wants My Attention Edition


If anything dominated the week in mathematically-themed comic strips it was Zach Weinersmith’s Saturday Morning Breakfast Cereal. I don’t know how GoComics selects the strips to (re?)print on their site. But there were at least four that seemed on-point enough for me to mention. So, okay. He’s got my attention. What’s he do with it?

On the 3rd of December is a strip I can say is about conditional probability. The mathematician might be right that the chance someone will be murdered by a serial killer are less than one in ten million. But that is the chance of someone drawn from the whole universe of human experiences. There are people who will never be near a serial killer, for example, or who never come to his attention or who evade his interest. But if we know someone is near a serial killer, or does attract his interest? The information changes the probability. And this is where you get all those counter-intuitive and somewhat annoying logic puzzles about, like, the chance someone’s other child is a girl if the one who just walked in was, and how that changes if you’re told whether the girl who just entered was the elder.

On the 5th is a strip about sequences. And built on the famous example of exponential growth from doubling a reward enough times. Well, you know these things never work out for the wise guy. The “Fibonacci Spiral” spoken of in the next-to-last panel is a spiral, like you figure. The dimensions of the spiral are based on those of golden-ratio rectangles. It looks a great deal like a logarithmic spiral to the untrained eye. Also to the trained eye, but you knew that. I think it’s supposed to be humiliating that someone would call such a spiral “random”. But I admit I don’t get that part.

The strip for the 6th has a more implicit mathematical content. It hypothesizes that mathematicians, given the chance, will be more interested in doing recreational puzzles than even in eating and drinking. It’s amusing, but I’ll admit I’ve found very few puzzles all that compelling. This isn’t to say there aren’t problems I keep coming back to because I’m curious about them, just that they don’t overwhelm my common sense. Don’t ask me when I last received actual pay for doing something mathematical.

And then on the 9th is one more strip, about logicians. And logic puzzles, such as you might get in a Martin Gardner collection. The problem is written out on the chalkboard with some shorthand logical symbols. And they’re symbols both philosophers and mathematicians use. The letter that looks like a V with a crossbar means “for all”. (The mnemonic I got was “it’s an A-for-all, upside-down”. This paired with the other common symbol, which looks like a backwards E and means there exists: “E-for-exists, backwards”. Later I noticed upside-down A and backwards E could both be just 180-degree-rotated A and E. But try saying “180-degree-rotated” in a quick way.) The curvy E between the letters ‘x’ and ‘S’ means “belongs to the set”. So that first line says “for all x that belong to the set S this follows”. Writing out “isLiar(x)” instead of, say, “L(x)”, is more a philosopher’s thing than a mathematician’s. But it wouldn’t throw anyway. And the T just means emphasizing that this is true.

And that is as much about Saturday Morning Breakfast Cereal as I have to say this week.

Sam Hurt’s Eyebeam for the 4th tells a cute story about twins trying to explain infinity to one another. I’m not sure I can agree with the older twin’s assertion that infinity means there’s no biggest number. But that’s just because I worry there’s something imprecise going on there. I’m looking forward to the kids learning about negative numbers, though, and getting to wonder what’s the biggest negative real number.

Percy Crosby’s Skippy for the 4th starts with Skippy explaining a story problem. One about buying potatoes, in this case. I’m tickled by how cranky Skippy is about boring old story problems. Motivation is always a challenge. The strip originally ran the 7th of October, 1930.

Dave Whamond’s Reality Check for the 6th uses a panel of (gibberish) mathematics as an example of an algorithm. Algorithms are mathematical, in origin at least. The word comes to us from the 9th century Persian mathematician Al-Khwarizmi’s text about how to calculate. The modern sense of the word comes from trying to describe the methods by which a problem can be solved. So, legitimate use of mathematics to show off the idea. The symbols still don’t mean anything.

Joe: 'Grandpa, what's 5x7?' Grandpa: 'Why do you wanna know?' Joe: 'I'm testing your memory.' Grandpa: 'Oh! The answer's 35.' Joe: 'Thanks! Now what is 8x8?' Grandpa: 'Joe, is that last night's homework?' Joe: 'We're almost done! Only 19 more!'
Rick Detorie’s One Big Happy for the 7th of December, 2017. And some attention, please, for Ruthie there. She’s completely irrelevant to the action, but it makes sense for her to be there if Grandpa is walking them to school, and she adds action — and acting — to the scenes.

Rick Detorie’s One Big Happy for the 7th has Joe trying to get his mathematics homework done at the last minute. … And it’s caused me to reflect on how twenty multiplication problems seems like a reasonable number to do. But there’s only fifty multiplications to even do, at least if you’re doing the times tables up to the 10s. No wonder students get so bored seeing the same problems over and over. It’s a little less dire if you’re learning times tables up to the 12s, but not that much better. Yow.

Olivia Walch’s Imogen Quest for the 8th looks pretty legitimate to me. It’s going to read as gibberish to people who haven’t done parametric functions, though. Start with the plane and the familiar old idea of ‘x’ and ‘y’ representing how far one is along a horizontal and a vertical direction. Here, we’re given a dummy variable ‘t’, and functions to describe a value for ‘x’ and ‘y’ matching each value of ‘t’. The plot then shows all the points that ever match a pair of ‘x’ and ‘y’ coordinates for some ‘t’. The top drawing is a shape known as the cardioid, because it kind of looks like a Valentine-heart. The lower figure is a much more complicated parametric equation. It looks more anatomically accurate,

Still no sign of Mark Anderson’s Andertoons and the drought is worrying me, yes.

But they’re still going on the cartoonist’s web site, so there’s that.

Reading the Comics, December 2, 2017: Showing Intelligence Edition


November closed out with another of those weeks not quite busy enough to justify splitting into two. I blame Friday and Saturday. Nothing mathematically-themed was happening them. Suppose some days are just like that.

Johnny Hart’s Back To BC for the 26th is an example of using mathematical truths as profound statements. I’m not sure that I’d agree with just stating the Pythagorean Theorem as profound, though. It seems like a profound statement has to have some additional surprising, revelatory elements to it. Like, knowing the Pythagorean theorem is true means we can prove there’s exactly one line parallel to a given line and passing through some point. Who’d see that coming? I don’t blame Hart for not trying to fit all that into one panel, though. Too slow a joke. The strip originally ran the 4th of September, 1960.

Tom Toles’s Randolph Itch, 2 am rerun for the 26th is a cute little arithmetic-in-real-life panel. I suppose arithmetic-in-real-life. Well, I’m amused and stick around for the footer joke. The strip originally ran the 24th of February, 2002.

Zach Weinersmith’s Saturday Morning Breakfast Cereal makes its first appearance for the week on the 26th. It’s an anthropomorphic-numerals joke and some wordplay. Interesting trivia about the whole numbers that never actually impresses people: a whole number is either a perfect square, like 1 or 4 or 9 or 16 are, or else its square root is irrational. There’s no whole number with a square root that’s, like, 7.745 or something. Maybe I just discuss it with people who’re too old. It seems like the sort of thing to reveal to a budding mathematician when she’s eight.

Saturday Morning Breakfast Cereal makes another appearance the 29th. The joke’s about using the Greek ε, which has a long heritage of use for “a small, positive number”. We use this all the time in analysis. A lot of proofs in analysis are done by using ε in a sort of trick. We want to show something is this value, but it’s too hard to do. Fine. Pick any ε, a positive number of unknown size. So then we’ll find something we can calculate, and show that the difference between the thing we want and the thing we can do is smaller than ε. And that the value of the thing we can calculate is that. Therefore, the difference between what we want and what we can do is smaller than any positive number. And so the difference between them must be zero, and voila! We’ve proved what we wanted to prove. I have always assumed that we use ε for this for the association with “error”, ideally “a tiny error”. If we need another tiny quantity we usually go to δ, probably because it’s close to ε and ‘d’ is still a letter close to ‘e’. (The next letter after ε is ζ, which carries other connotations with it and is harder to write than δ is.) Anyway, Weinersmith is just doing a ha-ha, your penis is small joke.

Samson’s Dark Side of the Horse for the 28th is a counting-sheep joke. It maybe doesn’t belong here but I really, really like the art of the final panel and I want people to see it.

Arnoldine: 'If you're so SMART, what's the SQUARE ROOT of a million?!' Arnold, after a full panel's thought: 'FIVE!' Arnoldine: 'OK! What's the square root of TWO MILLION?!'
Bud Grace’s Piranha Club for the 29th of November, 2017. So do always remember the old advice for attorneys and people doing investigative commissions: never ask a question you don’t already know the answer to.

Bud Grace’s Piranha Club for the 29th is, as with Back to BC, an attempt at showing intelligence through mathematics. There are some flaws in the system. Fun fact: since one million is a perfect square, Arnold could have answered within a single panel. (Also fun fact: I am completely unqualified to judge whether something is a “fun” fact.)

Jason Chatfield’s Ginger Meggs for the 29th is Ginger subverting the teacher’s questions, like so many teacher-and-student jokes will do.

Dan Thompson’s Brevity for the 30th is the anthropomorphic geometric figures joke for the week.

There seems to be no Mark Anderson’s Andertoons for this week. There’ve been some great ones (like on the 26th or the 28th and the 29th) but they’re not at all mathematical. I apologize for the inconvenience and am launching an investigation into this problem.

Reading the Comics, November 18, 2017: Story Problems and Equation Blackboards Edition


It was a normal-paced week at Comic Strip Master Command. It was also one of those weeks that didn’t have anything from Comics Kingdom or Creators.Com. So I’m afraid you’ll all just have to click the links for strips you want to actually see. Sorry.

Bill Amend’s FoxTrot for the 12th has Jason and Marcus creating “mathic novels”. They, being a couple of mathematically-gifted smart people, credit mathematics knowledge with smartness. A “chiliagon” is a thousand-sided regular polygon that’s mostly of philosophical interest. A regular polygon with a thousand equal sides and a thousand equal angles looks like a circle. There’s really no way to draw one so that the human eye could see the whole figure and tell it apart from a circle. But if you can understand the idea of a regular polygon it seems like you can imagine a chilagon and see how that’s not a circle. So there’s some really easy geometry things that can’t be visualized, or at least not truly visualized, and just have to be reasoned with.

Rick Detorie’s One Big Happy for the 12th is a story-problem-subversion joke. The joke’s good enough as it is, but the supposition of the problem is that the driving does cover fifty miles in an hour. This may not be the speed the car travels at the whole time of the problem. Mister Green is maybe speeding to make up for all the time spent travelling slower.

Brandon Sheffield and Dami Lee’s Hot Comics for Cool People for the 13th uses a blackboard full of equations to represent the deep thinking being done on a silly subject.

Shannon Wheeler’s Too Much Coffee Man for the 15th also uses a blackboard full of equations to represent the deep thinking being done on a less silly subject. It’s a really good-looking blackboard full of equations, by the way. Beyond the appearance of our old friend E = mc2 there’s a lot of stuff that looks like legitimate quantum mechanics symbols there. They’re at least not obvious nonsense, as best I can tell without the ability to zoom the image in. I wonder if Wheeler didn’t find a textbook and use some problems from it for the feeling of authenticity.

Samson’s Dark Side of the Horse for the 16th is a story-problem subversion joke.

Jef Mallett’s Frazz for the 18th talks about making a bet on the World Series, which wrapped up a couple weeks ago. It raises the question: can you bet on an already known outcome? Well, sure, you can bet on anything you like, given a willing partner. But there does seem to be something fundamentally different between betting on something whose outcome isn’t in principle knowable, such as the winner of the next World Series, and betting on something that could be known but happens not to be, such as the winner of the last. We see this expressed in questions like “is it true the 13th of a month is more likely to be Friday than any other day of the week?” If you know which month and year is under discussion the chance the 13th is Friday is either 1 or 0. But we mean something more like, if we don’t know what month and year it is, what’s the chance this is a month with a Friday the 13th? Something like this is at work in this World Series bet. (The Astros won the recently completed World Series.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is also featured on some underemployed philosopher’s “Reading the Comics” WordPress blog and fair enough. Utilitarianism exists in an odd triple point, somewhere on the borders of ethics, economics, and mathematics. The idea that one could quantize the good or the utility or the happiness of society, and study how actions affect it, is a strong one. It fits very well the modern mindset that holds everything can be quantified even if we don’t know how to do it well just yet. And it appeals strongly to a mathematically-minded person since it sounds like pure reason. It’s not, of course, any more than any ethical scheme can be. But it sounds like the ethics a Vulcan would come up with and that appeals to a certain kind of person. (The comic is built on one of the implications of utilitarianism that makes it seem like the idea’s gone off the rails.)

There’s some mathematics symbols on The Utilitarian’s costume. The capital U on his face is probably too obvious to need explanation. The \sum u on his chest relies on some mathematical convention. For maybe a half-millennium now mathematicians have been using the capital sigma to mean “take a sum of things”. The things are whatever the expression after that symbol is. Usually, the Sigma will have something below and above which carries meaning. It says what the index is for the thing after the symbol, and what the bounds of the index are. Here, it’s not set. This is common enough, though, if this is understood from context. Or if it’s obvious. The small ‘u’ to the right suggests the utility of whatever’s thought about. (“Utility” being the name for the thing measured and maximized; it might be happiness, it might be general well-being, it might be the number of people alive.) So the symbols would suggest “take the sum of all the relevant utilities”. Which is the calculation that would be done in this case.