One is that, as we’ve thought of counting numbers, there is always “one more”. This doesn’t have to be. We could work with perfectly good number systems that have a largest number. We do, in fact. Every computer programming language has some largest integer that it will deal with. If you need a larger number, you have to do something clever. Your clever idea will let you address some range of bigger numbers, but it too will have a maximum. We’ve set those limits large enough that, usually, they’re not an inconvenience. They’re still there.
But those limits are forced on us by the many failings of matter. What when we get just past Plato’s line’s division, into the reasoning of pure mathematics? There we can set up counting numbers. The standard way to do this is to suppose there is a number “1”. And to suppose that, for any counting number we have, there is a successor, a number one-plus-that. If Joey were to ask why there has to be, all Dennis could do is shrug. This makes an axiom out of there always being one more. If you don’t like it, make some other arithmetic. Anyway we only understand any of this using fallible matter, so good luck.
This progression can be heady, though. The counting numbers are probably the most understandable infinitely large set there is. Thinking about them seriously can induce the sort of dizzy awe that pondering Deep Time or the vastness of space can do. That seems a bit above Dennis’s age level, but some people are stricken with the infinite sooner than others are.
Charles Schulz’s Peanuts Begins rerun for the 2nd has Charlie Brown dismiss arithmetic as impractical. It fits the motif of mathematics as an unworldly subject. There’s the common joke that pure mathematics even dreams of being of no use to anyone. Arithmetic, though, has always been a practical subject. It introduces us to many abstract ideas, particularly group theory. This subject looks at what we can do with systems that work like arithmetic without necessarily having numbers, or anything that works with numbers.
And a couple of comic strips mentioned mathematics, although in too slight a way to discuss. Dana Simpson’s Phoebe and her Unicorn on the 30th of April started a sequence in which doodles on Phoebe’s homework came to life. That it’s mathematics homework was mostly incidental. I’m open to the argument that mathematics encourages doodling in a way that, say, spelling does not. I’d also be open to the argument you aren’t doing geometry if you don’t doodle. Anyway. Dan Thompson’s Brevity for the 2nd of May features Sesame Street’s Count von Count. It’s a bit of wordplay on the use of “numbers” for songs. And, of course, the folkloric tradition of vampires as compulsive counters.
People reading my Reading the Comics post Sunday maybe noticed something. I mean besides my correct, reasonable complaining about the Comics Kingdom redesign. That is that all the comics were from before the 30th of March. That is, none were from the week before the 7th of April. The last full week of March had a lot of comic strips. The first week of April didn’t. So things got bumped a little. Here’s the results. It wasn’t a busy week, not when I filter out the strips that don’t offer much to write about. So now I’m stuck for what to post Thursday.
The strip explains things well enough. The Library holds every book that will ever be written. In the original story there are some constraints. Particularly, all the books are 410 pages. If you wanted, say, a 600-page book, though, you could find one book with the first 410 pages and another book with the remaining 190 pages and then some filler. The catch, as explained in the story and in the comic strip, is finding them. And there is the problem of finding a ‘correct’ text. Every possible text of the correct length should be in there. So every possible book that might be titled Mark Twain vs Frankenstein, including ones that include neither Mark Twain nor Frankenstein, is there. Which is the one you want to read?
Henry Scarpelli and Craig Boldman’s Archie for the 4th features an equal-divisions problem. In principle, it’s easy to divide a pizza (or anything else) equally; that’s what we have fractions for. Making them practical is a bit harder. I do like Jughead’s quick work, though. It’s got the slight-of-hand you expect from stage magic.
Scott Hilburn’s The Argyle Sweater for the 4th takes place in an algebra class. I’m not sure what algebraic principle demonstrates, but it probably came from somewhere. It’s 4,829,210. The exponentials on the blackboard do cue the reader to the real joke, of the sign reading “kick10 me”. I question whether this is really an exponential kicking situation. It seems more like a simple multiplication to me. But it would be harder to make that joke read clearly.
Tony Cochran’s Agnes for the 5th is part of a sequence investigating how magnets work. Agnes and Trout find just … magnet parts inside. This is fair. It’s even mathematics.
Thermodynamics classes teach one of the great mathematical physics models. This is about what makes magnets. Magnets are made of … smaller magnets. This seems like question-begging. Ultimately you get down to individual molecules, each of which is very slightly magnetic. When small magnets are lined up in the right way, they can become a strong magnet. When they’re lined up in another way, they can be a weak magnet. Or no magnet at all.
How do they line up? It depends on things, including how the big magnet is made, and how it’s treated. A bit of energy can free molecules to line up, making a stronger magnet out of a weak one. Or it can break up the alignments, turning a strong magnet into a weak one. I’ve had physics instructors explain that you could, in principle, take an iron rod and magnetize it just by hitting it hard enough on the desk. And then demagnetize it by hitting it again. I have never seen one do this, though.
This is more than just a physics model. The mathematics of it is … well, it can be easy enough. A one-dimensional, nearest-neighbor model, lets us describe how materials might turn into magnets or break apart, depending on their temperature. Two- or three-dimensional models, or models that have each small magnet affected by distant neighbors, are harder.
These days I’ve been preparing these comics posts by making a note of every comic that seems like it might have a mathematical topic. Then at the end of the week I go back and re-read them all and think what I could write something about. This past week’s had two that seemed like nice juicy topics. And then I was busy all day Saturday so didn’t have time to put the thought into them that they needed. So instead I offer some comic strips with at least mentions of mathematical subjects. If they’re not tightly on point, well, I need to post something, don’t I?
Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 24th is the anthropomorphic numerals joke for the week. It did get me thinking about the numbers which (in English) are homophones to other words. There don’t seem to be many, though: one, two, four, six, and eight seem to be about all I could really justify. There’s probably dialects where “ten” and “tin” blend together. There’s probably a good Internet Argument to be had about whether “couple” should be considered the name of a number. That there aren’t more is probably that there, in a sense, only a couple of names for numbers, with a scheme to compound names for a particular number of interest.
Scott Hilburn’s The Argyle Sweater for the 25th mentions algebra, but is mostly aimed at the Reading the Comics for some historian blogger. I kind of admire Hilburn’s willingness to go for the 70-year-old scandal for a day’s strip. But a daily strip demands a lot of content, especially when it doesn’t have recurring characters. The quiz answers as given are correct, and that’s easy to check. But it is typically easy to check whether a putative answer is correct. Finding an answer is the hard part.
Daniel Shelton’s Ben for the 25th has a four-year-old offering his fingers as a way to help his older brother with mathematics work. Counting on fingers can be a fine way to get the hang of arithmetic and at least I won’t fault someone for starting there. Eventually, do enough arithmetic, and you stop matching numbers with fingers because that adds an extra layer of work that doesn’t do anything but slow you down.
Catching my interest though is that Nicholas (the eight-year-old, and I had to look that up on the Ben comic strip web site; GoComics doesn’t have a cast list) had worked out 8 + 6, but was struggling with 7 + 8. He might at some point get experienced enough to realize that 7 + 8 has to be the same thing as 8 + 7, which has to be the same thing as 8 + 6 + 1. And if he’s already got 8 + 6 nailed down, then 7 + 8 is easy. But that takes using a couple of mathematical principles — that addition commutes, that you can substitute one quantity with something equal to it, that you addition associates — and he might not see where those principles get him any advantage over some other process.
Ed Allison’s Unstrange Phenomena for the 25th builds its Dadaist nonsense for the week around repeating numbers. I learn from trying to pin down just what Allison means by “repeating numbers” that there are people who ascribe mystical significance to, say, “444”. Well, if that helps you take care of the things you need to do, all right. Repeating decimals are a common enough thing. They appear in the decimal expressions for rational numbers. These expressions either terminate — they have finitely many digits and then go to an infinitely long sequence of 0’s — or they repeat. (We rule out “repeating nothing but zeroes” because … I don’t know. I would guess it makes the proofs in some corner of number theory less bothersome.)
You could also find interesting properties about numbers made up of repeating strings of numerals. For example, write down any number of 9’s you like, followed by a 6. The number that creates is divisible by 6. I grant this might not be the most important theorem you’ll ever encounter, but it’s a neat one. Like, a strong of 4’s followed by a 9 is not necessarily divisible by 4 or 9. There are bunches of cute little theorem like this, mostly good for making one admit that huh, there’s some neat coincidences(?) about numbers.
Although … Allison’s strip does seem to get at seeing particular numbers over and over. This does happen; it’s probably a cultural thing. One of the uses we put numbers to is indexing things. So, for example, a TV channel gets a number and while the station may have a name, it makes for an easier control to set the TV to channel numbered 5 or whatnot. We also use numbers to measure things. When we do, we get to pick the size of our units. We typically pick them so our measurements don’t have to be numbers too big or too tiny. There’s no reason we couldn’t measure the distance between cities in millimeters, or the length of toes in light-years. But to try is to look like you’re telling a joke. So we get see some ranges — 1 to 5, 1 to 10 — used a lot when we don’t need fine precision. We see, like, 1 to 100 for cases where we need more precision than that but don’t have to pin a thing down to, like, a quarter of a percent. Numbers will spill past these bounds, naturally. But we are more likely to encounter a 20 than a 15,642. We set up how we think about numbers so we are. So maybe it would look like some numbers just follow you.
As I said Sunday, last week was a slow one for mathematically-themed comic strips. Here’s the second half of them. They’re not tightly on point. But that’s all right. They all have titles starting with ‘A’. I mean if you ignore the article ‘the’, the way we usually do when alphabetizing titles.
Tony Cochran’s Agnes for the 11th is basically a name-drop of mathematics. The joke would be unchanged if the teacher asked Agnes to circle all the adjectives in a sentence, or something like that. But there are historically links between religious thinking and mathematics. The Pythagoreans, for example, always a great and incredible starting point for any mathematical topic or just some preposterous jokes that might have nothing to do with their reality, were at least as much a religious and philosophical cult. For a long while in the Western tradition, the people with the time and training to do advanced mathematics work were often working for the church. Even as people were more able to specialize, a mystic streak remained. It’s easy to understand why. Mathematics promises to speak about things that are universally true. It encourages thinking about the infinite. It encourages thinking about the infinitely tiny. It courts paradoxes as difficult as any religious Mystery. It’s easy to snark at someone who takes numerology seriously. But I’m not sure the impulse that sees magic in arithmetic is different to the one that sees something supernatural in a “transfinite” item.
Scott Hilburn’s The Argyle Sweater for the 11th is another mistimed Pi Day joke. π is, famously, an irrational number. But so is every number, except for a handful of strange ones that we’ve happened to find interesting. That π should go on and on follows from what an irrational number means. It’s a bit surprising the 4 didn’t know all this before they married.
I appreciate the secondary joke that the marriage counselor is a “Hugh Jripov”, and the counselor’s being a ripoff is signaled by being a ÷ sign. It suggests that maybe successful reconciliation isn’t an option. I’m curious why the letters ‘POV’ are doubled, in the diploma there. In a strip with tighter drafting I’d think it was suggesting the way a glass frame will distort an image. But Hilburn draws much more loosely than that. I don’t know if it means anything.
Mark Anderson’s Andertoons for the 12th is the Mark Anderson’s Andertoons for the essay. I’m so relieved to have a regular stream of these again. The teacher thinks Wavehead doesn’t need to annotate his work. And maybe so. But writing down thoughts about a problem is often good practice. If you don’t know what to do, or you aren’t sure how to do what you want? Absolutely write down notes. List the things you’d want to do. Or things you’d want to know. Ways you could check your answer. Ways that you might work similar problems. Easier problems that resemble the one you want to do. You find answers by thinking about what you know, and the implications of what you know. Writing these thoughts out encourages you to find interesting true things.
And this was too marginal a mention of mathematics even for me, even on a slow week. But Georgia Dunn’s Breaking Cat News for the 12th has a cat having a nightmare about mathematics class. And it’s a fun comic strip that I’d like people to notice more.
With me wrapping up the mathematically-themed comic strips that ran the first of the year, you can see how far behind I’m falling keeping everything current. In my defense, Monday was busier than I hoped it would be, so everything ran late. Next week is looking quite slow for comics, so maybe I can catch up then. I will never catch up on anything the rest of my life, ever.
Scott Hilburn’s The Argyle Sweater for the 2nd is a bit of wordplay about regular and irregular polygons. Many mathematical constructs, in geometry and elsewhere, come in “regular” and “irregular” forms. The regular form usually has symmetries that make it stand out. For polygons, this is each side having the same length, and each interior angle being congruent. Irregular is everything else. The symmetries which constrain the regular version of anything often mean we can prove things we otherwise can’t. But most of anything is the irregular. We might know fewer interesting things about them, or have a harder time proving them.
I’m not sure what the teacher would be asking for in how to “make an irregular polygon regular”. I mean if we pretend that it’s not setting up the laxative joke. I can think of two alternatives that would make sense. One is to draw a polygon with the same number of sides and the same perimeter as the original. The other is to draw a polygon with the same number of sides and the same area as the original. I’m not sure of the point of either. I suppose polygons of the same area have some connection to quadrature, that is, integration. But that seems like it’s higher-level stuff than this class should be doing. I hate to question the reality of a comic strip but that’s what I’m forced to do.
Bud Fisher’s Mutt and Jeff rerun for the 4th is a gambler’s fallacy joke. Superficially the gambler’s fallacy seems to make perfect sense: the chance of twelve bad things in a row has to be less than the chance of eleven bad things in a row. So after eleven bad things, the twelfth has to come up good, right? But there’s two ways this can go wrong.
Suppose each attempted thing is independent. In this case, what if each patient is equally likely to live or die, regardless of what’s come before? And in that case, the eleven deaths don’t make it more likely that the next will live.
Suppose each attempted thing is not independent, though. This is easy to imagine. Each surgery, for example, is a chance for the surgeon to learn what to do, or not do. He could be getting better, that is, more likely to succeed, each operation. Or the failures could reflect the surgeon’s skills declining, perhaps from overwork or age or a loss of confidence. Impossible to say without more data. Eleven deaths on what context suggests are low-risk operations suggest a poor chances of surviving any given surgery, though. I’m on Jeff’s side here.
Mark Anderson’s Andertoons for the 5th is a welcome return of Wavehead. It’s about ratios. My impression is that ratios don’t get much attention in themselves anymore, except to dunk on stupid Twitter comments. It’s too easy to jump right into fractions, and division. Ratios underlie this, at least historically. It’s even in the name, ‘rational numbers’.
Wavehead’s got a point in literally comparing apples and oranges. It’s at least weird to compare directly different kinds of things. This is one of those conceptual gaps between ancient mathematics and modern mathematics. We’re comfortable stripping the units off of numbers, and working with them as abstract entities. But that does mean we can calculate things that don’t make sense. This produces the occasional bit of fun on social media where we see something like Google trying to estimate a movie’s box office per square inch of land in Australia. Just because numbers can be combined doesn’t mean they should be.
Larry Wright’s Motley rerun for the 5th has the form of a story problem. And one timely to the strip’s original appearance in 1987, during the National Football League players strike. The setup, talking about the difference in weekly pay between the real players and the scabs, seems like it’s about the payroll difference. The punchline jumps to another bit of mathematics, the point spread. Which is an estimate of the expected difference in scoring between teams. I don’t know for a fact, but would imagine the scab teams had nearly meaningless point spreads. The teams were thrown together extremely quickly, without much training time. The tools to forecast what a team might do wouldn’t have the data to rely on.
And then I noticed there were a bunch of comic strips with some kind of mathematical theme on the same day. Always fun when that happens.
Bill Holbrook’s On The Fastrack uses one of Holbrook’s common motifs. That’s the depicting as literal some common metaphor. in this case it’s “massaging the numbers”, which might seem not strictly mathematics. But while numbers are interesting, they’re also useful. To be useful they must connect to something we want to know. They need context. That context is always something of human judgement. If the context seems inappropriate to the listener, she thinks the presenter is massaging the numbers. If the context seems fine, we trust the numbers as showing something truth.
Scott Hilburn’s The Argyle Sweater is a seasonal pun that couldn’t wait for a day closer to Christmas. I’m a little curious why not. It would be the same joke with any subject, certainly. The strip did make me wonder if Ebeneezer Scrooge, in-universe, might have taken calculus. This lead me to see that it’s a bit vague what, precisely, Scrooge, or Scrooge-and-Marley, did. The movies are glad to position him as having a warehouse, and importing and exporting things, and making and collecting on loans and whatnot. These are all trades that mathematicians would like to think benefit from knowing advanced mathematics. The logic of making loans implies attention be paid to compounding interest, risks, and expectation values, as well as projecting cash-flow a fair bit into the future. But in the original text he doesn’t make any stated loans, and the only warehouse anyone enters is Fezziwig’s. Well, the Scrooge and Marley sign stands “above the warehouse door”, but we only ever go in to the counting-house. And yes, what Scrooge does besides gather money and misery is irrelevant to the setting of the story.
Teresa Burritt’s Dadaist strip Frog Applause uses knowledge of mathematics as an emblem of intelligence. “Multivariate analysis” is a term of art from statistics. It’s about measuring how one variable changes depending on two or more other variables. The goal is obvious: we know there are many things that influence anything of interest. Can we find what things have the strongest effects? The weakest effects? There are several ways we might mean “strongest” effect, too. It might mean that a small change in the independent variable produces a big change in the dependent one. Or it might mean that there’s very little noise, that a change in the independent variable produces a reliable change in the dependent one. Or we might have several variables that are difficult to measure precisely on their own, but with a combination that’s noticeable. The basic calculations for this look a lot like those for single-variable analysis. But there’s much more calculation. It’s more tedious, at least. My reading suggests that multivariate analysis didn’t develop much until there were computers cheap enough to do the calculations. Might be coincidence, though. Many machine-learning techniques can be described as multivariate analysis problems.
Greg Evans’s Luann Againn is a Pi Day joke from before the time when Pi Day was a thing. Brad’s magazine flipping like that is an unusual bit of throwaway background humor for the comic strip.
The first half of last week’s comics offered another bunch of chances to think about what mathematics is for. Before I do get into all that, though, may I mention the most recent update of Gregory Taylor’s serial:
Whether you're an American celebrating Thanksgiving, or simply enjoying the end of a week… there's still a chance to vote on the latest serial entry! https://t.co/ZEMWN4ticK
It does conclude with a vote about the next direction to take. So it’s a good chance for people who like to see authors twisting to their audience’s demands.
Mort Walker and Dik Browne’s Hi and Lois for the 23rd of May, 1961 builds off a major use of arithmetic. Budgeting doesn’t get much attention from mathematicians. I suppose it seems to us like all the basic problems are solved: adding? Subtracting? Multiplication? All familiar things. Especially now with decimal currency. There are great unsolved problems in mathematics, but they get into specialized areas of financial mathematics and just don’t matter for ordinary household budgeting.
Hi comes across a bit harsh here. I’m going to suppose he was taken so by surprise by Lois’s problem that he spoke without thinking.
Scott Hilburn’s The Argyle Sweater for the 19th is the anthropomorphic numerals strip for the week. With the title of “improper fractions” it’s wordplay on the common meaning for a mathematical term. Two times over, come to it. That negative refers to a class of numbers as well as disapproval of something is ordinary enough. I’ve mentioned it, I estimate, 840 times this month alone.
Jokes about the technical and common meanings of “improper” are rarer. In a proper fraction, the numerator is a smaller number than the denominator. In an improper fraction, we don’t count on that. I remember a modest bit of time in elementary and middle school working on converting improper fractions into mixed fractions — a whole number plus a proper fraction. And also don’t remember anyone caring about that after calculus. In most arithmetic work, there’s not much that’s easier about “1 + 1/2” than about “3/2”. The one major convenience “1 + 1/2” has is that it’s easy to tell at a glance how big the number is. It’s not mysterious how big a number 3/2 is, but that’s because of long familiarity. If I asked you whether 54/17 or 46/13 was the larger number, you’d be fairly stumped and maybe cranky. So there’s not much reason to worry about improper fractions while you’re doing work. For the final presentation of an answer, proper or mixed fractions may well be better.
Whoever colored that minus symbol before the 5 screwed up and confused the joke. Syndicated cartoonists give precise coloring instructions for Sunday strips. But many of them don’t, or aren’t able to, give coloring instructions for weekday strips like this. And mistakes like that are the unfortunate result.
Pascal Wyse and Joe Berger’s Berger and Wyse for the 19th features a sudoku appearance. It’s labelled a diversion, and so it is, as many mathematics and logic puzzles will be. The lone commenter at GoComics claims to have solved the puzzle, so I will suppose they’re being honest about this.
Brian Fies’s Mom’s Cancer for the 19th I have mentioned before, although not since I started including images for all mentioned comics. It’s set a moment when treatment for Mom’s cancer has been declared a great success.
The trouble is, as Feis lays out, volume is three-dimensional. We are pretty good at measuring the length, or at least the greatest width of something. You might call that the “characteristic length”. A linear dimension. But volume scales as the cube of this characteristic length. And the sad thing is that 0.8 times 0.8 times 0.8 is, roughly, 0.5. This means that the characteristic length dropping by 20% drops the volume by 50%. Or, as Feis is disappointed to see in this strip and its successor, the great news of a 50% reduction in the turmor’s mass is that it’s just 20% less big in every direction. It doesn’t look like enough.
Bill Holbrook’s On The Fastrack for the 20th presents one of Fi’s seminars about why mathematics is a good thing. The offscreen student’s question about why one should learn mathematics goes unanswered. As often happens the question is presented as though it’s too absurd to deserve answering. The questioner is conflating “mathematics” with “calculating arithmetic”, yes. And a computer will be better at these calculations. A related question, sometimes asked (and rarely on-topic for my essays here), is why one needs to learn any specific facts when a computer is so much better at finding them.
Knowing facts is not understanding them, no. But it is hard to understand a thing without knowing facts. More, without loving the knowing of facts. If we don’t need to be good at calculating, we do still need to know what to have calculated. And why to calculate that instead of something else. In calculating we can learn things of great beauty. And some of us do go on to mathematics which cannot be calculated. There is software that will do very well at computing, say, the indefinite integral of functions. I don’t know of any that will even start on a problem like “find the kernel of this ring”. But these are problems we see, and think interesting, because our experience in arithmetic trains us to notice them. Perhaps there is new interesting mathematics that we would notice if we didn’t have preconceptions set by times tables and long division. But it is hard to believe that we can’t find it because we’re not ignorant enough. I wouldn’t risk it.
For today’s entry, Iva Sallay, of Find The Factors, gave me an irresistible topic. I did not resist.
What’s purple and commutes?
An Abelian grape.
Whatever else you say about mathematics we are human. We tell jokes. I will tell some here. You may not understand the words in them. That’s all right. From the Abelian grape there, you gather this is some manner of wordplay. A pun, particularly. It’s built on a technical term. “Abelian groups” come from (not high school) Algebra. In an Abelian group, the group multiplication commutes. That is, if ‘a’ and ‘b’ are any things in the group, then their product “ab” is the same as “ba’. That is, the group works like ordinary addition on numbers does. We say “Abelian” in honor of Niels Henrik Abel, who taught us some fascinating stuff about polynomials. Puns are a common kind of humor. So common, they’re almost base. Even a good pun earns less laughter than groans.
But mathematicians make many puns. A typical page of mathematics jokes has a whole section of puns. “What’s yellow and equivalent to the Axiom of Choice? Zorn’s Lemon.” “What’s nonorientable and lives in the sea?” “Möbius Dick.” “One day Jesus said to his disciples, `The Kingdom of Heaven is like 3x2 + 8x – 9′. Thomas looked very confused and asked peter, `What does the teacher mean?’ Peter replied, `Don’t worry. It’s just another one of his parabolas’.” And there are many jokes built on how it is impossible to tell the difference between the sounds of “π” and “pie”.
It shouldn’t surprise that mathematicians make so many puns. Mathematics trains people to know definitions. To think about precisely what we mean. Puns ignore definitions. They build nonsense out of the ways that sounds interact. Mathematicians practice how to make things interact, even if they don’t know or care what the underlying things are. If you’ve gotten used to proving things about , without knowing what ‘a’ or ‘b’ are, it’s difficult to avoid turning “poles on the half-plane” (which matters in some mathematical physics) to a story about Polish people on an aircraft.
If there’s a flaw to this kind of humor it’s that these jokes may sound juvenile. One of the first things that strikes kids as funny is that a thing might have several meanings. Or might sound like another thing. “Why do mathematicians like parks? Because of all the natural logs!”
Jokes can be built tightly around definitions. “What do you get if you cross a mosquito with a mountain climber? Nothing; you can’t cross a vector with a scalar.” “There are 10 kinds of people in the world, those who understand binary mathematics and those who don’t.” “Life is complex; it has real and imaginary parts.”
There are more sophisticated jokes. Many of them are self-deprecating. “A mathematician is a device for turning coffee into theorems.” “An introvert mathematician looks at her shoes while talking to you. An extrovert mathematician looks at your shoes.” “A mathematics professor is someone who talks in someone else’s sleep”. “Two people are adrift in a hot air balloon. Finally they see someone and shout down, `Where are we?’ The person looks up, and studies them, watching the balloon drift away. Finally, when they are barely in shouting range, the person on the ground shouts back, `You are in a balloon!’ The first passenger curses their luck at running across a mathematician. `How do you know that was a mathematician?’ `Because her answer took a long time, was perfectly correct, and absolutely useless!”’ These have the form of being about mathematicians. But they’re not really. It would be the same joke to say “a poet is a device for turning coffee into couplets”, the sleep-talker anyone who teachers, or have the hot-air balloonists discover a lawyer or a consultant.
Some of these jokes get more specific, with mathematics harder to extract from the story. The tale of the nervous flyer who, before going to the conference, sends a postcard that she has a proof of the Riemann hypothesis. She arrives and admits she has no such thing, of course. But she sends that word ahead of every conference. She knows if she died in a plane crash after that, she’d be famous forever, and God would never give her that. (I wonder if Ian Randal Strock’s little joke of a story about Pierre de Fermat was an adaptation of this joke.) You could recast the joke for physicists uniting gravity and quantum mechanics. But I can’t imagine a way to make this joke about an ISO 9000 consultant.
A dairy farmer knew he could be milking his cows better. He could surely get more milk, and faster, if only the operations of his farm were arranged better. So he hired a mathematician to find the optimal way to configure everything. The mathematician toured every part of the pastures, the milking barn, the cows, everything relevant. And then the mathematician set to work devising a plan for the most efficient possible cow-milking operation. The mathematician declared, “First, assume a spherical cow.”
This joke is very mathematical. I know of no important results actually based on spherical cows. But the attitude that tries to make spheres of cows comes from observing mathematicians. To describe any real-world process is to make a model of that thing. A model is a simplification of the real thing. You suppose that things behave more predictably than the real thing. You trust the error made by this supposition is small enough for your needs. A cow is complicated, all those pointy ends and weird contours. A sphere is easy. And, besides, cows are funny. “Spherical cow” is a funny string of sounds, at least in English.
The spherical cows approach parodying the work mathematicians do. Many mathematical jokes are burlesques of deductive logic. Or not even burlesques. Charles Dodgson, known to humans as Lewis Carroll, wrote this in Symbolic Logic:
“No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.
∴ This party of tourists need not run.”
[ Here is another opportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.
He will reply “Why, it’s perfectly correct, of course! And if your precious Logic-book tells you it isn’t, don’t believe it! You don’t mean to tell me those tourists need to run? If I were one of them, and knew the Premises to be true, I should be quite clear that I needn’t run — and I should walk!”
And you will reply “But suppose there was a mad bull behind you?”
And then your innocent friend will say “Hum! Ha! I must think that over a bit!” ]
The punch line is diffused by the text being so educational. And by being written in the 19th century, when it was bad form to excise any word from any writing. But you can recognize the joke, and why it should be a joke.
Not every mathematical-reasoning joke features some manner of cattle. Some are legitimate:
Claim. There are no uninteresting whole numbers.
Proof. Suppose there is a smalled uninteresting whole number. Call it N. That N is uninteresting is an interesting fact. Therefore N is not an uninteresting whole number.
Three mathematicians step up to the bar. The bartender asks, “you all want a beer?” The first mathematician says, “I don’t know.” The second mathematician says, “I don’t know.” The third says, “Yes”.
Some mock reasoning uses nonsense methods to get a true conclusion. It’s the fun of watching Mister Magoo walk unharmed through a construction site to find the department store exchange counter:
Venn Diagrams are not by themselves jokes (most of the time). But they are a great structure for jokes. And easy to draw, which is great for us who want to be funny but don’t feel sure about their drafting abilities.
And then there are personality jokes. Mathematics encourages people to think obsessively. Obsessive people are often funny people. Alexander Grothendieck was one of the candidates for “greatest 20th century mathematician”. His reputation is that he worked so well on abstract problems that he was incompetent at practical ones. The story goes that he was demonstrating something about prime numbers and his audience begged him to speak about a specific number, that they could follow an example. And that he grumbled a bit and, finally, said, “57”. It’s not a prime number. But if you speak of “Grothendieck’s prime”, many will recognize what you mean, and grin.
There are more outstanding, preposterous personalities. Paul Erdös was prolific, and a restless traveller. The stories go that he would show up at some poor mathematician’s door and stay with them several months. And then co-author a paper with the elevator operator. (Erdös is also credited as the originator of the “coffee into theorems” quip above.) John von Neumann was supposedly presented with this problem:
Two trains are on the same track, 60 miles apart, heading toward each other, each travelling 30 miles per hour. A fly travels 60 miles per hour, leaving one engine flying toward the other. When it reaches the other engine it turns around immediately and flies back to the other engine. This is repeated until the two trains crash. How far does the fly travel before the crash?
The first, hard way to do this is to realize how far the fly travels is a series. The fly starts at, let’s say, the left engine and flies to the right. Add to that the distance from the right to the left train now. Then left to the right again. Right to left. This is a bunch of calculations. Most people give up on that and realize the problem is easier. The trains will crash in one hour. The fly travels 60 miles per hour for an hour. It’ll fly 60 miles total. John von Neumann, say witnesses, had the answer instantly. He recognized the trick? “I summed the series.”
The personalities can be known more remotely, from a handful of facts about who they were or what they did. “Cantor did it diagonally.” Georg Cantor is famous for great thinking about infinitely large sets. His “diagonal proof” shows the set of real numbers must be larger than the set of rational numbers. “Fermat tried to do it in the margin but couldn’t fit it in.” “Galois did it on the night before.” (Évariste Galois wrote out important pieces of group theory the night before a duel. It went badly for him. French politics of the 1830s.) Every field has its celebrities. Mathematicians learn just enough about theirs to know a couple of jokes.
The jokes can attach to a generic mathematician personality. “How can you possibly visualize something that happens in a 12-dimensional space?” “Easy, first visualize it in an N-dimensional space, and then let N go to 12.” Three statisticians go hunting. They spot a deer. One shoots, missing it on the left. The second shoots, missing it on the right. The third leaps up, shouting, “We’ve hit it!” An engineer and a mathematician are sleeping in a hotel room when the fire alarm goes off. The engineer ties the bedsheets into a rope and shimmies out of the room. The mathematician looks at this, unties the bedsheets, sets them back on the bed, declares, “this is a problem already solved” and goes back to sleep. (Engineers and mathematicians pair up a lot in mathematics jokes. I assume in engineering jokes too, but that the engineers make wrong assumptions about who the joke is on. If there’s a third person in the party, she’s a physicist.)
Do I have a favorite mathematics joke? I suppose I must. There are jokes I like better than others, and there are — I assume — finitely many different mathematics jokes. So I must have a favorite. What is it? I don’t know. It must vary with the day and my mood and the last thing I thought about. I know a bit of doggerel keeps popping into my head, unbidden. Let me close by giving it to you.
Integral z-squared dz
From 1 to the cube root of 3
Times the cosine
Of three π over nine
Equals log of the cube root of e.
This may not strike you as very funny. I’m not sure it strikes me as very funny. But it keeps showing up, all the time. That has to add up.
Now I’ve finally had the time to deal with the rest of last week’s comics. I’ve rarely been so glad that Comic Strip Master Command has taken it easy on me for this week.
Tom Toles’s Randolph Itch, 2am for the 20th is about a common daydream, that of soap bubbles of weird shapes. There’s fun mathematics to do with soap bubbles. Most of these fall into the “calculus of variations”, which is good at finding minimums and maximums. The minimum here is a surface with zero mean curvature that satisfies particular boundaries. In soap bubble problems the boundaries have a convenient physical interpretation. They’re the wire frames you dunk into soap film, and pull out again, to see what happens. There’s less that’s proven about soap bubbles than you might think. For example: we know that two bubbles of the same size will join on a flat common surface. Do three bubbles? They seem to, when you try blowing bubbles and fitting them together. But this falls short of mathematical rigor.
Parker and Hart’s Wizard of Id Classics for the 21st is a joke about the ignorance of students. Of course they don’t know basic arithmetic. Curious thing about the strip is that you can read it as an indictment of the school system, failing to help students learn basic stuff. Or you can read it as an indictment of students, refusing the hard work of learning while demanding a place in politics. Given the 1968 publication date I have a suspicion which was more likely intended. But it’s hard to tell; 1968 was a long time ago. And sometimes it’s just so easy to crack an insult there’s no guessing what it’s supposed to mean.
Gene Mora’s Graffiti for the 22nd mentions what’s probably the most famous equation after that thing with two times two in it. It does cry out something which seems true, that was there before Albert Einstein noticed it. It does get at one of those questions that, I say without knowledge, is probably less core to philosophers of mathematics than the non-expert would think. But are mathematical truths discovered or invented? There seems to be a good argument that mathematical truths are discovered. If something follows by deductive logic from the axioms of the field, and the assumptions that go into a question, then … what’s there to invent? Anyone following the same deductive rules, and using the same axioms and assumptions, would agree on the thing discovered. Invention seems like something that reflects an inventor.
But it’s hard to shake the feeling that there is invention going on. Anyone developing new mathematics decides what things seem like useful axioms. She decides that some bundle of properties is interesting enough to have a name. She decides that some consequences of these properties are so interesting as to be named theorems. Maybe even the Fundamental Theorem of the field. And there was the decision that this is a field with a question interesting enough to study. I’m not convinced that isn’t invention.
Mark Anderson’s Andertoons for the 23rd sees Wavehead — waaait a minute. That’s not Wavehead! This throws everything off. Well, it’s using mathematics as the subject that Not-Wavehead is trying to avoid. And it’s not using arithmetic as the subject easiest to draw on the board. It needs some kind of ascending progression to make waiting for some threshold make sense. Numbers rising that way makes sense.
That birds will fly in V-formation has long captured people’s imaginations. We’re pretty confident we know why they do it. The wake of one bird’s flight can make it easier for another bird to stay aloft. This is especially good for migrating birds. The fluid-dynamic calculations of this are hard to do, but any fluid-dynamic calculations are hard to do. Verifying the work was also hard, but could be done. I found and promptly lost an article about how heartbeat monitors were attached to a particular flock of birds whose migration path was well-known, so the sensors could be checked and data from them gathered several times over. (Birds take turns as the lead bird, the one that gets no lift from anyone else’s efforts.)
So far as I’m aware there’s still some mystery as to how they do it. That is, how they know to form this V-formation. A particularly promising line of study in the 80s and 90s was to look at these as self-organizing structures. This would have each bird just trying to pay attention to what made sense for itself, where to fly relative to its nearest-neighbor birds. And these simple rules created, when applied to the whole flock, that V pattern. I do not know whether this reflects current thinking about bird formations. I do know that the search for simple rules that produce rich, complicated patterns goes on. Centuries of mathematics, physics, and to an extent chemistry have primed us to expect that everything is the well-developed result of simple components.
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 has a mirror image near ). 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.
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.
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.
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, . 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.
The title of this installment has nothing to do with anything. My love and I just got to talking about Reader’s Digest Condensed Books and I learned moments ago that they’re still being made. I mean, the title of the series changed from “Condensed Books” to “Select Editions” in 1997, but they’re still going on, as far as anyone can tell. This got us wondering things like how they actually do the abridging. And got me wondering whether any abridged book ended up being better than the original. So I have reasons for only getting partway through last week’s mathematically-themed comics. I don’t say they’re good reasons.
Scott Hilburn’s The Argyle Sweater for the 13th is the Roman Numerals joke for the week, the first one of those in like five days. Also didn’t know that there were still sidewalk theaters that still showed porn movies. I thought they had all been renovated into either respectable neighborhood-revitalization projects that still sometimes show Star Wars films or else become incubator space for startup investment groups.
Corey Pandolph’s The Elderberries for the 13th is a joke about learning fractions. They don’t see to be having much fun thinking about them. Fair enough, I suppose. Once you’ve got the hang of basic arithmetic here come fractions to follow rules for addition and subtraction that are suddenly way more complicated. Multiplication isn’t harder, at least, although it is longer. Same with division. Without a clear idea why this is anything you want to do, yeah, it seems to be unmotivated complicating of stuff.
Dave Whamond’s Reality Check for the 13th is trying to pick a fight with me. I’m not taking the bait. Although by saying ‘likelihood’ the question seems to be setting up a probability question. Those tend to use ‘p’ and ‘q’ as a generic variable name, rather than ‘x’. I bet you imagine that ‘p’ gets used to represent a possibly-unknown ‘probability’ because, oh yeah, first letter. Well … so far as I know that’s why. I’m away from my references right now so I can’t look them over and find no quite satisfactory answer. But that sure seems like it. ‘q’ gets called in if you need a second probability, and don’t want to deal with subscripts, then it’s a nice convenient letter close to ‘p’ in the alphabet. Again, so far as I know.
The other half of last week’s mathematically-themed comics were on familiar old themes. I’ll see what I can do with them anyway.
Scott Hilburn’s The Argyle Sweater for the 9th is the anthropomorphic numerals joke for the week. I’m curious why the Middletons would need multiple division symbols, but I suppose that’s their business. It does play on the idea that “division” and “splitting up” are the same thing. And that fits the normal use of these words. We’re used to thinking, say, of dividing a desired thing between several parties. While that’s probably all right in introducing the idea, I do understand why someone would get very confused when they first divide by one-half or one-third or any number between zero and one. And then negative numbers make things even more confusing.
Thaves’s Frank and Ernest for the 9th is the anthropomorphic geometric figures joke for the week. I think I can wrangle a way by which Circle’s question has deeper mathematical context. Mathematicians use the idea of “space” a lot. The use is inspired by how, you know, the geometry of a room works. Euclidean space, in the trade. A Euclidean space is a collection of points that obey a couple simple rules. You can take two points and add them, and get something in the space. You can take any scalar and multiply it by any point and get a point in the space. A scalar is something that acts like a real number. For example, real numbers. Maybe complex numbers, if you’re feeling wild.
A Euclidean space can be two-dimensional. This is the geometry of stuff you draw on paper. It can be three-dimensional. This is the geometry of stuff in the real world, or stuff you draw on paper with shading. It can be four-dimensional. This is the geometry of stuff you draw on paper with big blobby lines around it. Each of these is an equally good space, though, as legitimate and as real as any other. Context usually puts an implicit “three dimensional” before most uses of the word “space”. But it’s not required to be there. There’s many kinds of spaces out there.
And “space” describes stuff that doesn’t look anything like rooms or table tops or sheets of paper. These are spaces built of things like functions, or of sets of things, or of ways to manipulate things. Spaces built of the ways you can subdivide the integers. The details vary. But there’s something in common in all these ideas that communicates.
Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. I think we’ve all seen this joke go across our social media feed and it’s reassuring to know Mark Anderson has social media too. We do talk about solving for x, using the language of describing how we help someone get past a problem. I wonder if people might like this kind of algebra more if we talked more about finding out what values ‘x’ could have that make the equation true. Well, it won’t stop people feeling they don’t like the mathematics they learned in school. But it might help people feel like they know why they’re doing it.
I have another Reading the Comics post with a title that’s got nothing to do with the post. It has got something to do with how I spent my weekend. Not sure if I’ll ever get around to explaining that since there’s not much mathematical content to that weekend. I’m not sure whether the nonsense titles are any better than trying to find a theme in what Comic Strip Master Command has sent the past week. It takes time to pick something when anything would do, after all.
Scott Hilburn’s The Argyle Sweater for the 10th is the anthropomorphic numerals strip for the week. Also arithmetic symbols. The ÷ sign is known as “the division symbol”, although now and then people try to promote it as the “obelus”. They’re not wrong to call it that, although they are being the kind of person who tries to call the # sign the “octothorp”. Sometimes social media pass around the false discovery that the ÷ sign is a representation of a fraction, , with the numbers replaced by dots. It’s a good mnemonic for linking fractions and division. But it’s wrong to say that’s what the symbol means. ÷ started being used for division in Western Europe in the mid-17th century, in competition with many symbols, including / (still in common use), : (used in talking about ratios or odds), – (not used in this context anymore, and just confusing if you do try to use it so). And ÷ was used in northern Europe to mean “subtraction” for several centuries after this.
Tom Toles’s Randolph Itch, 2am for the 11th is a repeat; the too-short-lived strip has run through several cycles since I started doing these summaries. But it is also one of the great pie chart jokes ever and I have no intention of not telling people to enjoy it.
Pie charts, and the also-mentioned bar charts, come to us originally from the economist William Playfair, who in the late 1700s and early 1800s devised nearly all the good ways to visualize data. But we know them thanks to Florence Nightingale. Among her other works, she recognized in these charts good ways to represent her studies about Crimean War medicine and about sanitation in India. Nightingale was in 1859 named the first woman in the Royal Statistical Society, and was named an honorary member of the American Statistical Association in 1874.
Olivia Jaimes’s Nancy for the 12th uses arithmetic as iconic for classwork nobody wants to do. Algebra, too; I understand the reluctance to start. Simultaneous solutions; the challenge is to find sets of values ‘x’ and ‘y’ that make both equations true together. That second equation is a good break, though. makes it easy to write what ‘y’ has to be in terms of ‘x’. Then you can replace the ‘y’ in the first equation with its expression in terms of ‘x’. In a slightly tedious moment, it’s going to turn out there’s multiple sets of answers. Four sets, if I haven’t missed something. But they’ll be clearly related to each other. Even attractively arranged.
is an equation that’s true if the numbers ‘x’ and ‘y’ are coordinates of the points on a circle. This is if the coordinates are using the Cartesian coordinate system for the plane, which is such a common thing to do that mathematicians can forget they’re doing that. The circle has radius . So you can look at the first equation and draw a circle and write down a note that its radius is and you’ve got it. looks like an equation that’s true if the numbers ‘x’ and ‘y’ are coordinates of the points on a hyperbola. Again in the Cartesian coordinate system. But I have to feel a little uncomfortable saying this. If the equation were (say) then it’d certainly be a hyperbola, which mostly looks like a mirror-symmetric pair of arcs. But equalling zero? That’s called a “degenerate hyperbola”, which makes it sound like the hyperbola is doing something wrong. Unfortunate word, but one we’re stuck with.
The description just reflects that the hyperbola is boring in some way. In this case, it’s boring because the ‘x’ and ‘y’ that make the equation true are just the points on a pair of straight lines that go through the origin, the point with coordinates (0, 0). And they’re going to be mirror-images of each other around the x- and the y-axis. So it seems like a waste to use the form of a hyperbola when we could do just as well using the forms of straight lines to describe the same points. This hyperbola will look like an X, although it might be a pretty squat ‘x’ or a pretty narrow one or something. Depends on the exact equation.
So. The solutions for ‘x’ and ‘y’ are going to be on the points that are on both a circle centered around the origin and on an X centered around the origin. This is a way to see why I would expect four solutions. Also they they would look about the same. There’d be an answer with positive ‘x’ and positive ‘y’, and then three more answers. One answer has ‘x’ with the same size but a minus sign. One answer has ‘y’ with the same size but a minus sign. One has both ‘x’ and ‘y’ with the same values but minus signs.
Sorry I wasn’t there to partner with.
Dave Coverly’s Speed Bump for the 12th is a Rubik’s Cube joke. Here it merges the idea with the struggles of scheduling anything anymore. I’m not sure that the group-theory operations of lining up a Rubik’s cube can be reinterpreted as the optimization problems of scheduling stuff. But there are all sorts of astounding and surprising links between mathematical problems. So I wouldn’t rule it out.
John Allen’s Nest Heads for the 13th is a lotteries joke. I’m less dogmatic than are many mathematicians about the logic of participating in a lottery. At least in the ones as run by states and regional authorities the chance of a major payout are, yes, millions to one against. There can be jackpots large enough that the expectation value of playing becomes positive. In this case the reward for that unlikely outcome is so vast that it covers the hundreds of millions of times you play and lose. But even then, you have the question of whether doing something that just won’t pay out is worth it. My taste is to say that I shall do much more foolish things with my disposable income than buying a couple tickets each year. And while I would like to win the half-billion-dollar jackpot that would resolve all my financial woes and allow me to crush those who had me imprisoned in the Château d’If, I’d also be coming out ahead if I won, like, one of the petty $10,000 prizes.
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.
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.
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.
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.
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 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.
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 . 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. is far more important. Suppose that there is some number, which we give that name, with the property that 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 , 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 would look if people weren’t primed to think it was weird.
And now, closer to deadline than I like, let me wrap up last week’s mathematically-themed comic strips. I had a lot happening, that’s all I can say.
Glenn McCoy and Gary McCoy’s The Flying McCoys for the 10th is another tragic moment in the mathematics department. I’m amused that white lab coats are taken to read as “mathematician”. There are mathematicians who work in laboratories, naturally. Many interesting problems are about real-world things that can be modelled and tested and played with. It’s hardly the mathematics-department uniform, but then, I’m not sure mathematicians have a uniform. We just look like academics is all.
It also shows off that motif of mathematicians as doing anything with numbers in a more complicated way than necessary. I can’t imagine anyone in an emergency trying to evoke 9-1-1 by solving any kind of puzzle. But comic strip characters are expected to do things at least a bit ridiculously. I suppose.
Mark Litzler’s Joe Vanilla for the 11th is about random numbers. We need random numbers; they do so much good. Getting them is hard. People are pretty lousy at picking random numbers in their head. We can say what “lousy” random numbers look like. They look wrong. There’s digits that don’t get used as much as the others do. There’s strings of digits that don’t get used as much as other strings of the same length do. There are patterns, and they can be subtle ones, that just don’t look right.
And yet we have a terrible time trying to say what good random numbers look like. Suppose we want to have a string of random zeroes and ones: is 101010 better or worse than 110101? Or 000111? Well, for a string of digits that short there’s no telling. It’s in big batches that we should expect to see no big patterns. … Except that occasionally randomness should produce patterns. How often should we expect patterns, and of what size? This seems to depend on what patterns we’ve found interesting enough to look for. But how can the cultural quirks that make something seem interesting be a substantial mathematical property?
Olivia Jaimes’s Nancy for the 11th uses mathematics-assessment tests for its joke. It’s of marginal relevance, yes, but it does give me a decent pretext to include the new artist’s work here. I don’t know how long the Internet is going to be interested in Nancy. I have to get what attention I can while it lasts.
Well, that’s got me thinking. Obviously all the sides of a triangle can be rational, and so its perimeter can be too. But … the area of an equilateral triangle is times the square of the length of any side. It can have a rational side and an irrational area, or vice-versa. Just as the circle has. If it’s not an equilateral triangle?
Can you have a triangle that has three rational sides and a rational area? And yes, you can. Take the right triangle that has sides of length 5, 12, and 13. Or any scaling of that, larger or smaller. There is indeed a whole family of triangles, the Heronian Triangles. All their sides are integers, and their areas are integers too. (Sides and areas rational are just as good as sides and areas integers. If you don’t see why, now you see why.) So there’s that at least. The name derives from Heron/Hero, the ancient Greek mathematician whom we credit with that snappy formula that tells us, based on the lengths of the three sides, what the area of the triangle is. Not the Pythagorean formula, although you can get the Pythagorean formula from it.
Still, I’m going to bet that there’s some key measure of even a Heronian Triangle that ends up being irrational. Interior angles, most likely. And there are many ways to measure triangles; they can’t all end up being rational at once. There are over two thousand ways to define a “center” of a triangle, for example. The odds of hitting a rational number on all of them at once? (Granted, most of these triangle centers are unknown except to the center’s discoverer/definer and that discoverer’s proud but baffled parents.)
Carla Ventresca and Henry Beckett’s On A Claire Day for the 12th mentions taking classes in probability and statistics. They’re the classes nobody doubts are useful in the real world. It’s easy to figure probability is more likely to be needed than functional analysis on some ordinary day outside the university. I can’t even compose that last sentence without the language of probability.
I’d kind of agree with calling the courses intense, though. Well, “intense” might not be the right word. But challenging. Not that you’re asked to prove anything deep. The opposite, really. An introductory course in either provides a lot of tools. Many of them require no harder arithmetic work than multiplication, division, and the occasional square root. But you do need to learn which tool to use in which scenario. And there’s often not the sorts of proofs that make it easy to understand which tool does what. Doing the proofs would require too much fussing around. Many of them demand settling finicky little technical points that take you far from the original questions. But that leaves the course as this archipelago of small subjects, each easy in themselves. But the connections between them are obscured. Is that better or worse? It must depend on the person hoping to learn.
Last week had enough mathematically-themed comics for me to split the content. Usually I split the comics temporally, and this time I will too. What’s unusual is that somewhere along the week the URLs that GoComics pages provide switched from http to https. https is the less-openly-insecure version of the messaging protocol that sends web pages around. It’s good practice; we should be using https wherever possible. I don’t know why they switched that on, and why switch it on midweek. I suppose someone there knew what they were doing.
Tom Wilson’s Ziggy for the 6th of May uses mathematical breakthroughs as shorthand for inspiration. In two ways, too, one with a basically geometric figure and one with a bunch of equations. The geometric figure doesn’t seem to have any significance to me. The equations … that’s a bit harder. They’re probably nonsense. But it’s hard to look at ‘a’ and not see acceleration; the letter is often used for that. And it’s hard to look at ‘v’ and not see velocity. ‘x’ is often a position and ‘t’ is often a time. ‘xf – xi‘ looks meaningful too. It almost begs to be read as “position, final, minus position, initial”. “tf – ti” almost begs to be read as “time, final, minus time, initial”. And the difference in position divided by a difference in time suggests a velocity.
So here’s something peculiar inspired by looking at the units that have to follow. If ‘v’ is velocity, then it’s got units of distance over time. and would have units of distance-squared over time-squared. At least unless ‘a ‘or ‘V’ or ‘I’ are themselves measurements. But the square root of their sum then gets us back to distance over time. And then a distance-over-time divided by … well, distance-over-time suggests a pure number. Or something of whatever units ‘R’ carries with it.
So this equation seems arbitrary, and of course the expression doesn’t need to make sense for the joke. But it’s odd that the most-obvious choice of meanings for v and x and t means that the symbols work out so well. At least almost: an acceleration should have units of distance-over-time-squared, and this has units of (nothing). But I may have guessed wrong in thinking ‘a’ meant acceleration here. It might be a description of how something in one direction corresponds to something in another. And that would make sense as a pure number. I wonder whether Wilson got this expression from from anything, or if any readers recognize something that I should have seen right away.
Todd Clark’s Lola for the 7th jokes about being bad at mathematics. The number of days left to the end of school isn’t something that a kid should have trouble working out. However, do remember the first rule of calculating the span between two dates on the calendar: never calculate the span between two dates on the calendar. There is so much that goes wrong trying. All right, there’s a method. That method is let someone else do it.
Bud Fisher’s Mutt and Jeff for the 7th uses the form of those mathematics-magic games. You know, the ones where you ask someone to pick a number, then do some operations, and then tell you the result. From that you reverse-engineer the original number. They’re amusing enough tricks even if they are all basically the same. It’s instructive to figure out how they work. Replace your original number with symbols and follow the steps then. If you just need the number itself you can replace that with ‘x’. If you need the digits of the number then you’d replace it with something like “10*a + b”, to represent the numerals “ab”. Here, yeah, Mutt’s just being arbitrarily mean.
Paul Gilligan and Kory Merritt’s Poptropica for the 7th depicts calculating stuff as the way to act like a robot. Can’t deny; calculation is pretty much what we expect computers to do. It may hide. It may be done so abstractly it looks like we’re playing Mini Metro instead. This is a new comics tag. I’m sad to say this might be the last use of that tag. Poptropica is fun, but it doesn’t touch on mathematics much at all.
Gene Mora’s Graffiti for the 8th mentions arithmetic, albeit obliquely. It’s meant to be pasted on the doors of kindergarten teachers and who am I to spoil the fun?
Scott Hilburn’s The Argyle Sweater for the 9th is the anthropomorphic-numerals joke for this week. Converting between decimals and fractions has been done since decimals got worked out in the late 16th century. There’s advantages to either representation. To my eyes the biggest advantage of fractions is they avoid hypnotizing people with the illusion of precision. 0.25 reads as more exact than 1/4. We can imagine it being 0.2500000000000000 and think we know the quantity to any desired precision. 1/4 reads (to me, anyway) as being open to the possibility we’re rounding off from 0.998 out of 4.00023.
Another advantage fractions do have is flexibility. There are infinitely many ways to express the same number as a fraction. In decimals, there are at most two. If you’re trying to calculate something that would be more easily done with a denominator of 30 than of 5, you’re free to do that. Decimals can have advantages in computing, certainly, especially if you’re already set up to manipulate digits. And you can tell at a glance whether, say, 14/29th is greater or less than 154/317th. In case you ever find reason to wonder, I mean. I’m not saying either is always the right way to go.
So today I am trying out including images for all the mathematically-themed comic strips here. This is because of my discovery that some links even on GoComics.com vanish without warning. I’m curious how long I can keep doing this. Not for legal reasons. Including comics for the purpose of an educational essay about topics raised by the strips is almost the most fair use imaginable. Just because it’s a hassle copying the images and putting them up on WordPress.com and that’s even before I think about how much image space I have there. We’ll see. I might try to figure out a better scheme.
Also in this batch of comics are the various Pi Day strips. There was a healthy number of mathematically-themed comics on the 14th of March. Many of those were just coincidence, though, with no Pi content. I’ll group the Pi Day strips together.
Tom Batiuk’s Funky Winkerbean for the 2nd of April, 1972 is, I think, the first appearance of Funky Winkerbean around here. Comics Kingdom just started running the strip, as well as Bud Blake’s Tiger and Bill Hoest’s Lockhorns, from the beginning as part of its Vintage Comics roster. And this strip really belonged in Sunday’s essay, but I noticed the vintage comics only after that installment went to press. Anyway, this strip — possibly the first Sunday Funky Winkerbean — plays off a then-contemporary fear of people being reduced to numbers in the face of a computerized society. If you can imagine people ever worrying about something like that. The early 1970s were a time in American society when people first paid attention to the existence of, like, credit reporting agencies. Just what they did and how they did it drew a lot of critical examination. Josh Lauer’s recently published Creditworthy: a History of Consumer Surveillance and Financial Identity in America gets into this.
Bob Scott’s Bear With Me for the 14th sees Molly struggling with failure on a mathematics test. Could be any subject and the story would go as well, but I suppose mathematics gets a connotation of the subject everybody has to study for, even the geniuses. (The strip used to be called Molly and the Bear. In either name this seems to be the first time I’ve tagged it, although I only started tagging strips by name recently.)
Bud Fisher’s Mutt and Jeff rerun for the 14th is a rerun from sometime in 1952. I’m tickled by the problem of figuring out how many times Fisher and his uncredited assistants drew Mutt and Jeff. Mutt saying that the boss “drew us 14,436 times” is the number of days in 45 years, so that makes sense if he’s counting the number of strips drawn. The number of times that Mutt and Jeff were drawn is … probably impossible to calculate. There’s so many panels each strip, especially going back to earlier and earlier times. And how many panels don’t have Mutt or don’t have Jeff or don’t have either in them? Jeff didn’t appear in the strip until March of 1908, for example, four months after the comic began. (With a different title, so the comic wasn’t just dangling loose all that while.)
Scott Hilburn’s The Argyle Sweater for the 14th starts the Pi Day off, of course, with a pun and some extension of what makes 3/14 get its attention. And until Hilburn brought it up I’d never thought about the zodiac sign for someone born the 14th of March, so that’s something.
Mark Parisi’s Off The Mark for the 14th riffs on one of the interesting features of π, that it’s an irrational number. Well, that its decimal representation goes on forever. Rational numbers do that too, yes, but they all end in the infinite repetition of finitely many digits. And for a lot of them, that digit is ‘0’. Irrational numbers keep going on with more complicated patterns. π sure seems like it’s a normal number. So we could expect that any finite string of digits appears somewhere in its decimal expansion. This would include a string of digits that encodes any story you like, The Neverending Story included. This does not mean we might ever find where that string is.
Michael Cavna’s Warped for the 14th combines the two major joke threads for Pi Day. Specifically naming Archimedes is a good choice. One of the many things Archimedes is famous for is finding an approximation for π. He’d worked out that π has to be larger than 310/71 but smaller than 3 1/7. Archimedes used an ingenious approach: we might not know the precise area of a circle given only its radius. But we can know the area of a triangle if we know the lengths of its legs. And we can draw a series of triangles that are enclosed by a circle. The area of the circle has to be larger than the sum of the areas of those triangles. We can draw a series of triangles that enclose a circle. The area of the circle has to be less than the sum of the areas of those triangles. If we use a few triangles these bounds are going to be very loose. If we use a lot of triangles these bounds can be tight. In principle, we could make the bounds as close together as we could possibly need. We can see this, now, as a forerunner to calculus. They didn’t see it as such at the time, though. And it’s a demonstration of what amazing results can be found, even without calculus, but with clever specific reasoning. Here’s a run-through of the process.
John Zakour and Scott Roberts’s Working Daze for the 15th is a response to Dr Stephen Hawking’s death. The coincidence that he did die on the 14th of March made for an irresistibly interesting bit of trivia. Zakour and Roberts could get there first, thanks to working on a web comic and being quick on the draw. (I’m curious whether they replaced a strip that was ready to go for the 15th, or whether they normally work one day ahead of publication. It’s an exciting but dangerous way to go.)
The other half of last week’s comic strips didn’t have any prominent pets in them. The six of them appeared on two days, though, so that’s as good as a particular theme. There’s also some π talk, but there’s enough of that I don’t want to overuse Pi Day as an edition name.
Mark Anderson’s Andertoons for the 10th is a classroom joke. It’s built on a common problem in teaching by examples. The student can make the wrong generalization. I like the joke. There’s probably no particular reason seven was used as the example number to have zero interact with. Maybe it just sounded funnier than the other numbers under ten that might be used.
Mike Baldwin’s Cornered for the 10th uses a chalkboard of symbols to imply deep thinking. The symbols on the board look to me like they’re drawn from some real mathematics or physics source. There’s force equations appropriate for gravity or electric interactions. I can’t explain the whole board, but that’s not essential to work out anyway.
Marty Links’s Emmy Lou for the 17th of March, 1976 was rerun the 10th of August. It name-drops the mathematics teacher as the scariest of the set. Fortunately, Emmy Lou went to her classes in a day before Rate My Professor was a thing, so her teacher doesn’t have to hear about this.
Scott Hilburn’s The Argyle Sweater for the 12th is a timely remidner that Scott Hilburn has way more Pi Day jokes than we have Pi Days to have. Also he has octopus jokes. It’s up to you to figure out whether the etymology of the caption makes sense.
John Zakour and Scott Roberts’s Working Daze for the 12th presents the “accountant can’t do arithmetic” joke. People who ought to be good at arithmetic being lousy at figuring tips is an ancient joke. I’m a touch surprised that Christopher Miller’s American Cornball: A Laffopedic Guide to the Formerly Funny doesn’t have an entry for tips (or mathematics). But that might reflect Miller’s mission to catalogue jokes that have fallen out of the popular lexicon, not merely that are old.
It wasn’t like the week wasn’t busy. Comic Strip Master Command sent out as many mathematically-themed comics as I might be able to use. But they were again ones that don’t leave me much to talk about. I’ll try anyway. It was looking like an anthropomorphic-symboles sort of week, too.
Dan Thompson’s Brevity for the 31st is another entry in the anthropomorphic-symbols joke contest. This one sticks to mathematical symbols, so if the Frank and Ernest makes the cut this week so must this one.
Eric the Circle for the 31st, this installment by “T daug”, gives the slightly anthropomorphic geometric figure a joke that at least mentions a radius, and isn’t that enough? What catches my imagination about this panel particularly is that the “fractured radius” is not just a legitimate pun but also resembles a legitimate geometry drawing. Drawing a diameter line is sensible enough. Drawing some other point on the circle and connecting that to the ends of the diameter is also something we might do.
Scott Hilburn’s The Argyle Sweater for the 1st of August is one of the logical mathematics jokes you could make about snakes. The more canonical one runs like this: God in the Garden of Eden makes all the animals and bids them to be fruitful. And God inspects them all and finds rabbits and doves and oxen and fish and fowl all growing in number. All but a pair of snakes. God asks why they haven’t bred and they say they can’t, not without help. What help? They need some thick tree branches chopped down. The bemused God grants them this. God checks back in some time later and finds an abundance of baby snakes in the Garden. But why the delay? “We’re adders,” explain the snakes, “so we need logs to multiply”. This joke absolutely killed them in the mathematics library up to about 1978. I’m told.
John Deering’s Strange Brew for the 1st is a monkeys-at-typewriters joke. It faintly reminds me that I might have pledged to retire mentions of the monkeys-at-typewriters joke. But I don’t remember so I’ll just have to depend on saying I don’t think I retired the monkeys-at-typewriters jokes and trust that someone will tell me if I’m wrong.
Mark Anderson’s Andertoons for the 3rd is the reassuringly normal appearance of Andertoons for this week. It is a geometry class joke about rays, line segments with one point where there’s an end and … a direction where it just doesn’t. And it riffs on the notion of the existence of mathematical things. At least I can see it that way.
No surprise what the recurring theme for this set of mathematics-mentioning comic strips is. Look at the date range. But here goes.
Henry Scarpelli and Craig Boldman’s Archie rerun for the 13th uses algebra as the thing that will stun a class into silence. I know the silence. As a grad student you get whole minutes of instructions on how to teach a course before being sent out as recitation section leader for some professor. And what you do get told is the importance of asking students their thoughts and their ideas. This maybe works in courses that are obviously friendly to opinions or partially formed ideas. But in Freshman Calculus? It’s just deadly. Even if you can draw someone into offering an idea how we might start calculating a limit (say), they’re either going to be exactly right or they’re going to need a lot of help coaxing the idea into something usable. I’d like to have more chatty classes, but some subjects are just hard to chat about.
Steve Skelton’s 2 Cows And A Chicken for the 13th includes some casual talk about probability. As normally happens, they figure the chances are about 50-50. I think that’s a default estimate of the probability of something. If you have no evidence to suppose one outcome is more likely than the other, then that is a reason to suppose the chance of something is 50 percent. This is the Bayesian approach to probability, in which we rate things as more or less likely based on what information we have about how often they turn out. It’s a practical way of saying what we mean by the probability of something. It’s terrible if we don’t have much reliable information, though. We need to fall back on reasoning about what is likely and what is not to save us in that case.
Scott Hilburn’s The Argyle Sweater lead off the Pi Day jokes with an anthropomorphic numerals panel. This is because I read most of the daily comics in alphabetical order by title. It is also because The Argyle Sweater is The Argyle Sweater. Among π’s famous traits is that it goes on forever, in decimal representations, yes. That’s not by itself extraordinary; dull numbers like one-third do that too. (Arguably, even a number like ‘2’ does, if you write all the zeroes in past the decimal point.) π gets to be interesting because it goes on forever without repeating, and without having a pattern easily describable. Also because it’s probably a normal number but we don’t actually know that for sure yet.
Mark Parisi’s Off The Mark panel for the 14th is another anthropomorphic numerals joke and nearly the same joke as above. The answer, dear numeral, is “chained tweets”. I do not know that there’s a Twitter bot posting the digits of π in an enormous chained Twitter feed. But there’s a Twitter bot posting the digits of π in an enormous chained Twitter feed. If there isn’t, there is now.
John Zakour and Scott Roberts’s Working Daze for the 14th is your basic Pi Day Wordplay panel. I think there were a few more along these lines but I didn’t record all of them. This strip will serve for them all, since it’s drawn from an appealing camera angle to give the joke life.
Dave Blazek’s Loose Parts for the 14th is a mathematics wordplay panel but it hasn’t got anything to do with π. I suspect he lost track of what days he was working on, back six or so weeks when his deadline arrived.
Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 15th is some sort of joke about the probability of the world being like what it seems to be. I’m not sure precisely what anyone is hoping to express here or how it ties in to world peace. But the world does seem to be extremely well described by techniques that suppose it to be random and unpredictable in detail. It is extremely well predictable in the main, which shows something weird about the workings of the world. It seems to be doing all right for itself.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th is built on the staggering idea that the Earth might be the only place with life in the universe. The cosmos is a good stand-in for infinitely large things. It might be better as a way to understand the infinitely large than actual infinity would be. Somehow thinking of the number of stars (or whatnot) in the universe and writing out a representable number inspires an understanding for bigness that the word “infinity” or the symbols we have for it somehow don’t seem to, at least to me.
Mikael Wulff and Anders Morgenthaler’s TruthFacts for the 17th gives us valuable information about how long ahead of time the comic strips are working. Arithmetic is probably the easiest thing to use if one needs an example of a fact. But even “2 + 2 = 4” is a fact only if we accept certain ideas about what we mean by “2” and “+” and “=” and “4”. That we use those definitions instead of others is a reflection of what we find interesting or useful or attractive. There is cultural artifice behind the labelling of this equation as a fact.
Jimmy Johnson’s Arlo and Janis for the 18th capped off a week of trying to explain some point about the compression and dilution of time in comic strips. Comic strips use space and time to suggest more complete stories than they actually tell. They’re much like every other medium in this way. So, to symbolize deep thinking on a subject we get once again a panel full of mathematics. Yes, I noticed the misquoting of “E = mc2” there. I am not sure what Arlo means by “Remember the boat?” although thinking on it I think he did have a running daydream about living on a boat. Arlo and Janis isn’t a strongly story-driven comic strip, but Johnson is comfortable letting the setting evolve. Perhaps all this is forewarning that we’re going to jump ahead to a time in Arlo’s life when he has, or has had, a boat. I don’t know.