Reading the Comics, November 24, 2018: Origins Edition


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


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

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

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

Lois: 'How much is 7 plus 19, Hi?' Hi: 'Golly! Don't you know how to add?' Lois: 'I guess I've forgotten.' (She's holding up a book marked Home Budget.) 'All I usually get to do is subtract.'
Mort Walker and Dik Browne’s Hi and Lois for the 23rd of May, 1961 and rerun the 19th of November, 2018. Essays that mention topics raised by Hi and Lois, both current-run and vintage, should be at this link.

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.

Caption: Improper Fractions. An anthropomorphic -5, teacher, dragging a 3/2: 'After I wash your mouth out, you're going down to the principal's office!' 3/2: 'Don't be so negative!'
Scott Hilburn’s The Argyle Sweater for the 19th of November, 2018. The many essays with mention of The Argyle Sweater will be at this link.

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.

A sign at the split in the road reads, 'Diversion'. It's a large sudoku, with stopped cars and people gathered around.
Pascal Wyse and Joe Berger’s Berger and Wyse for the 19th of November, 2018. Essays that bring up topics raised by Berger and Wyse will be this link.. It’s a new tag, though.

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.

Mom in Mathematic Land: 'One dimension, line A is 2 times as long as line B. Two dimensions: area varies with the square of length. The area of square A is 4 times that of square B. Three dimensions: Volume varies with the cube of length. Cube A has volume 8 times that of cube B. So when you see that two months of hard-fought chemotherapy and radiation have transformed THIS ... into THIS ... your crushing disappointment only betrays your mathematical ignorance.'
Brian Fies’s Mom’s Cancer for the 19th of November, 2018. The handful of essays inspired by Mom’s Cancer are at this link.

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.

One of Fi's audience: 'Why do I need to learn math? In the computer age, I just have to know ones and zeroes.' (Fi fumes, smoke steaming from her ears.) At the office Dethany reports: 'Fi texts 'every time I consider giving up these math seminars, I'm reminded why I can't'.'
Bill Holbrook’s On The Fastrack for the 20th of November, 2018. This and other essays inspired by On The Fastrack are at this link.

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

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


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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

Reading the Comics, August 29, 2018: The Week I Missed One Edition


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

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

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

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

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

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

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

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

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

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

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


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

Reading the Comics, July 11, 2018: GoComics Hardly Needs Me Edition


The first half of last week’s comics are mostly ones from Comics Kingdom and Creators.com. That’s unusual. GoComics usually far outranks the other sites. Partly for sheer numbers; they have an incredible number of strips, many of them web-only, that Comics Kingdom and Creators.com don’t match. I think the strips on GoComics are more likely to drift into mathematical topics too. But to demonstrate that would take so much effort. Possibly any effort at all. Hm.

Bill Holbrook’s On the Fastrack for the 8th of July is premised on topographic maps. These are some of the tools we’ve made to understand three-dimensional objects with a two-dimensional representation. When topographic maps come to the mathematics department we tend to call them “contour maps” or “contour plots”. These are collections of shapes. They might be straight lines. They might be curved. They often form a closed loop. Each of these curves is called a “contour curve” or a “contour line” (even if it’s not straight). Or it’s called an “equipotential curve”, if someone’s being all fancy, or pointing out the link between potential functions and these curves.

Dethany standing, in perspective, on a white surface with black curves traced on. The camera pulls out, revealing more and more curves, until they finally form an outline of her boss, Rose Trellis. Cut to the actual meeting, where Dethany is listening to Trellis speak. Dethany thinks: 'If only there was a topographic map showing how high a priority this is to her ... '
Bill Holbrook’s On the Fastrack for the 8th of July, 2018. I do like Holbrook’s art here, in evoking a figure standing vertically upon a most horizontal surface. There’s never enough intriguing camera angles in comic strips.

Their purpose is in thinking of three-dimensional surfaces. We can represent a three-dimensional surface by putting up some reasonable coordinate system. For the sake of simplicity let’s suppose the “reasonable coordinate system” is the Cartesian one. So every point in space has coordinates named ‘x’, ‘y’, and ‘z’. Pick a value for ‘x’ and ‘y’. There’s at most one ‘z’ that’ll be on the surface. But there might be many sets of values of ‘x’ and ‘y’ together which have that height ‘z’. So what are all the values of ‘x’ and ‘y’ which match the same height ‘z’? Draw the curve, or curves, which match that particular value of ‘z’.

Topographical maps are a beloved example of this, to mathematicians, because we imagine everyone understands them. A particular spot on the ground at some given latitude and longitude is some particular height above sea level. OK. Imagine the slice of a hill representing all the spots that are exactly 10 feet above sea level, or whatever. That’s a curve. Possibly several curves, but we just say “a curve” for simplicity.

A topographical map will often include more than one curve. Often at regular intervals, say with one set of curves representing 10 feet elevation, another 20 feet, another 30 feet, and so on. Sometimes these curves will be very near one another, where a hill is particularly steep. Sometimes these curves will be far apart, where the ground is nearly level. With experience one can learn to read the lines and their spacing. One can see where extreme values are, and how far away they might be.

Topographical maps date back to 1789. These sorts of maps go back farther. In 1701 Edmond Halley, of comet fame, published maps showing magnetic compass variation. He had hopes that the difference between magnetic north and true north would offer a hint at how to find longitude. (The principle is good. But the lines of constant variation are too close to lines of latitude for the method to be practical. And variation changes over time, too.) And that shows how the topographical map idea can be useful to visualize things that aren’t heights. Weather maps include “isobars”, contour lines showing where the atmospheric pressure is a set vale. More advanced ones will include “isotherms”, each line showing a particular temperature. The isobar and isotherm lines can describe the weather and how it can be expected to change soon.

This idea, rendering three-dimensional information on a two-dimensional surface, is a powerful one. We can use it to try to visualize four-dimensional objects, by looking at the contour surfaces they would make in three dimensions. We can also do this for five and even more dimensions, by using the same stuff but putting a note that “D = 16” or the like in the corner of our image. And, yes, if Cartesian coordinates aren’t sensible for the problem you can use coordinates that are.

If you need a generic name for these contour lines that doesn’t suggest lines or topography or weather or such, try ‘isogonal curves’. Nobody will know what you mean, but you’ll be right.

Hazel, sitting at a table, with a bunch of society women, as she works a calculator: ' ... making a total of $77.60. Fifteen percent for the tip, divided four ways ... '
Ted Key’s Hazel for the 9th of July, 2018. It’s a rerun, as all Hazel strips are. Ted Key, creator of Peabody’s Improbable History, died in 2008, and even then he’d retired in 1993. (I’m not clear whether someone else took up the strip in now-unpublished reruns or whether its original run ended then.)

Ted Key’s Hazel for the 9th is a joke about the difficulties in splitting the bill. It is archetypical of the sort of arithmetic people know they need to do in the real world. Despite that at least people in presented humor don’t get any better at it. I suppose real-world people don’t either, given some restaurants now list 15 and 20 percent tips on the bill. Well, at least everybody has a calculator on their phone so they can divide evenly. And I concede that, yeah, there isn’t really specifically a joke here. It’s just Hazel being competent, like the last time she showed up here.

Wavehead entering class: 'My dad said to tell you that geometry is squaresville. I don't understand what that means but he assured me that was comedy gold.'
Mark Anderson’s Andertoons for the 11th of July, 2018. I think Wavehead’s dad is underestimating triangles here. (There is a lot that we do with triangles, and extend to other polygons by breaking them into triangles.)

Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. And it’s a bit of geometry wordplay, too. Also about how you can carry a joke over well enough even without understanding it, or the audience understanding it, if it’s delivered right.

Dad: 'Joe, I gave you a five-dollar bill. The ice cream sandwich was a dollar fifty. How much change do you owe me?' Joe: 'Dad, you KNOW I don't like math. It's got so many problems!'
Rick DeTorie’s One Big Happy for the 11th of July, 2018. GoComics.com has a different strip for the day, as DeTorie publishes the new strips on Creators.com and uses several-years-old reruns on GoComics.

Rick DeTorie’s One Big Happy for the 11th is another strip about arithmetic done in the real world. I’m also amused by Joe’s attempts to distract from how no kid that age has ever not known precisely how much money they have, and how much of it is fairly won.

[ Toonie Excelsior Cornstarch thought green tea would make him smarter. ] Cornstarch: 'Also greener! And that's th'color of money! And most algae!' [ He downed 20 to 30 bottles of the stuff every day. ] Cornstarch: 'I already understand ALGEBRA! It comes from aliens!' [ Soon he began to think he knew everything about everything ... even quantum physics. ] Cornstarch: 'Dark matter just got much lighter!' [ But, being a TOONIE, he couldn't get a job at MIT, so he took to the streets to protest. That's when he was arrested by the INCORRECT SPELLING POLICE. ] (Cop dressed in a blend of Zippy the Pinhead gown and Keystone Cops uniform has his hand on the naked Cornstarch, who wears the sign 'MY ELEKTRONS CAN BEAT YOUR FOTONS!'
Bill Griffith’s Zippy the Pinhead for the 11th of July, 2018. This is part of a relatively new running sequence, perhaps a spinoff of Griffith’s very long Dingburg obsession, about people who are kind of generically golden-age-of-cartoon characters.

Bill Griffith’s Zippy the Pinhead for the 11th is another example of using understanding algebra as a show of intelligence. And it follows that up with undrestanding quantum physics as a show of even greater intelligence. One can ask what’s meant by “understanding” quantum physics. Someday someone might even answer. But it seems likely that the ability to do calculations based on a model has to be part of fully understanding it.


I have even more Reading the Comics posts, gathered in reverse chronological order at this link. Other essays with On The Fastrack tagged are at this link. Other Reading the Comics posts that mention Hazel are at this link. Some of the many, many essays mentioning Andertoons are at this link. Posts with mention of One Big Happy, both then-current and then-rerun, are at this link. And other mentions of Zippy the Pinhead are at this link.

Reading the Comics, March 11, 2017: Accountants Edition


And now I can wrap up last week’s delivery from Comic Strip Master Command. It’s only five strips. One certainly stars an accountant. one stars a kid that I believe is being coded to read as an accountant. The rest, I don’t know. I pick Edition titles for flimsy reasons anyway. This’ll do.

Ryan North’s Dinosaur Comics for the 6th is about things that could go wrong. And every molecule of air zipping away from you at once is something which might possibly happen but which is indeed astronomically unlikely. This has been the stuff of nightmares since the late 19th century made probability an important part of physics. The chance all the air near you would zip away at once is impossibly unlikely. But such unlikely events challenge our intuitions about probability. An event that has zero chance of happening might still happen, given enough time and enough opportunities. But we’re not using our time well to worry about that. If nothing else, even if all the air around you did rush away at once, it would almost certainly rush back right away.

'The new SAT multiple-choice questions have 4 answers instead of 5, with no penalty for guessing.' 'Let's see ... so if I took it now ... that would be one chance in four, which would be ... 25%?' 'Yes.' 'But back when I took it, my chances were ... let's see ... um ...' 'Remember, there's no penalty for guessing.'
Steve Kelley and Jeff Parker’s Dustin for the 7th of March, 2017. It’s the title character doing the guessing there. Also, Kelley and Parker hate their title character with a thoroughness you rarely see outside Tom Batiuk and Funky Winkerbean. This is a mild case of it but, there we are.

Steve Kelley and Jeff Parker’s Dustin for the 7th of March talks about the SATs and the chance of picking right answers on a multiple-choice test. I haven’t heard about changes to the SAT but I’ll accept what the comic strip says about them for the purpose of discussion here. At least back when I took it the SAT awarded one point to the raw score for a correct answer, and subtracted one-quarter point for a wrong answer. (The raw scores were then converted into a 200-to-800 range.) I liked this. If you had no idea and guessed on answers you should expect to get one in five right and four in five wrong. On average then you would expect no net change to your raw score. If one or two wrong answers can be definitely ruled out then guessing from the remainder brings you a net positive. I suppose the change, if it is being done, is meant to be confident only right answers are rewarded. I’m not sure this is right; it seems to me there’s value in being able to identify certainly wrong answers even if the right one isn’t obvious. But it’s not my test and I don’t expect to need to take it again either. I can expression opinions without penalty.

Mark Anderson’s Andertoons for the 7th is the Mark Anderson’s Andertoons for last week. It’s another kid-at-the-chalkboard panel. What gets me is that if the kid did keep one for himself then shouldn’t he have written 38?

Brian Basset’s Red and Rover for the 8th mentions fractions. It’s just there as the sort of thing a kid doesn’t find all that naturally compelling. That’s all right I like the bug-eyed squirrel in the first panel.

'The happy couple is about to cut the cake!' 'What kind is it?' 'A math cake.' (It has a square root of 4 sign atop it.)
Bill Holbrook’s On The Fastrack for the 9th of March, 2017. I confess I’m surprised Holbrook didn’t think to set the climax a couple of days later and tie it in to Pi Day.

Bill Holbrook’s On The Fastrack for the 9th concludes the wedding of accountant Fi. It uses the square root symbol so as to make the cake topper clearly mathematical as opposed to just an age.

Reading the Comics, March 4, 2017: Frazz, Christmas Trees, and Weddings Edition


It was another of those curious weeks when Comic Strip Master Command didn’t send quite enough comics my way. Among those they did send were a couple of strips in pairs. I can work with that.

Samson’s Dark Side Of The Horse for the 26th is the Roman Numerals joke for this essay. I apologize to Horace for being so late in writing about Roman Numerals but I did have to wait for Cecil Adams to publish first.

In Jef Mallett’s Frazz for the 26th Caulfield ponders what we know about Pythagoras. It’s hard to say much about the historical figure: he built a cult that sounds outright daft around himself. But it’s hard to say how much of their craziness was actually their craziness, how much was just that any ancient society had a lot of what seems nutty to us, and how much was jokes (or deliberate slander) directed against some weirdos. What does seem certain is that Pythagoras’s followers attributed many of their discoveries to him. And what’s certain is that the Pythagorean Theorem was known, at least a thing that could be used to measure things, long before Pythagoras was on the scene. I’m not sure if it was proved as a theorem or whether it was just known that making triangles with the right relative lengths meant you had a right triangle.

Greg Evans’s Luann Againn for the 28th of February — reprinting the strip from the same day in 1989 — uses a bit of arithmetic as generic homework. It’s an interesting change of pace that the mathematics homework is what keeps one from sleep. I don’t blame Luann or Puddles for not being very interested in this, though. Those sorts of complicated-fraction-manipulation problems, at least when I was in middle school, were always slogs of shuffling stuff around. They rarely got to anything we’d like to know.

Jef Mallett’s Frazz for the 1st of March is one of those little revelations that statistics can give one. Myself, I was always haunted by the line in Carl Sagan’s Cosmos about how, in the future, with the Sun ageing and (presumably) swelling in size and heat, the Earth would see one last perfect day. That there would most likely be quite fine days after that didn’t matter, and that different people might disagree on what made a day perfect didn’t matter. Setting out the idea of a “perfect day” and realizing there would someday be a last gave me chills. It still does.

Richard Thompson’s Poor Richard’s Almanac for the 1st and the 2nd of March have appeared here before. But I like the strip so I’ll reuse them too. They’re from the strip’s guide to types of Christmas trees. The Cubist Fur is described as “so asymmetrical it no longer inhabits Euclidean space”. Properly neither do we, but we can’t tell by eye the difference between our space and a Euclidean space. “Non-Euclidean” has picked up connotations of being so bizarre or even horrifying that we can’t hope to understand it. In practice, it means we have to go a little slower and think about, like, what would it look like if we drew a triangle on a ball instead of a sheet of paper. The Platonic Fir, in the 2nd of March strip, looks like a geometry diagram and I doubt that’s coincidental. It’s very hard to avoid thoughts of Platonic Ideals when one does any mathematics with a diagram. We know our drawings aren’t very good triangles or squares or circles especially. And three-dimensional shapes are worse, as see every ellipsoid ever done on a chalkboard. But we know what we mean by them. And then we can get into a good argument about what we mean by saying “this mathematical construct exists”.

Mark Litzler’s Joe Vanilla for the 3rd uses a chalkboard full of mathematics to represent the deep thinking behind a silly little thing. I can’t make any of the symbols out to mean anything specific, but I do like the way it looks. It’s quite well-done in looking like the shorthand that, especially, physicists would use while roughing out a problem. That there are subscripts with forms like “12” and “22” with a bar over them reinforces that. I would, knowing nothing else, expect this to represent some interaction between particles 1 and 2, and 2 with itself, and that the bar means some kind of complement. This doesn’t mean much to me, but with luck, it means enough to the scientist working it out that it could be turned into a coherent paper.

'Has Carl given you any reason not to trust him?' 'No, not yet. But he might.' 'Fi ... you seek 100% certainty in people, but that doesn't exist. In the end,' and Dethany is drawn as her face on a pi symbol, 'we're *all* irrational numbers.'
Bill Holbrook’s On The Fastrack for the 3rd of March, 2017. Fi’s dress isn’t one of those … kinds with the complicated pattern of holes in it. She got it torn while trying to escape the wedding and falling into the basement.

Bill Holbrook’s On The Fastrack is this week about the wedding of the accounting-minded Fi. And she’s having last-minute doubts, which is why the strip of the 3rd brings in irrational and anthropomorphized numerals. π gets called in to serve as emblematic of the irrational numbers. Can’t fault that. I think the only more famously irrational number is the square root of two, and π anthropomorphizes more easily. Well, you can draw an established character’s face onto π. The square root of 2 is, necessarily, at least two disconnected symbols and you don’t want to raise distracting questions about whether the root sign or the 2 gets the face.

That said, it’s a lot easier to prove that the square root of 2 is irrational. Even the Pythagoreans knew it, and a bright child can follow the proof. A really bright child could create a proof of it. To prove that π is irrational is not at all easy; it took mathematicians until the 19th century. And the best proof I know of the fact does it by a roundabout method. We prove that if a number (other than zero) is rational then the tangent of that number must be irrational, and vice-versa. And the tangent of π/4 is 1, so therefore π/4 must be irrational, so therefore π must be irrational. I know you’ll all trust me on that argument, but I wouldn’t want to sell it to a bright child.

'Fi ... humans are complicated. Like the irrational number pi, we can go on forever. You never get to the bottom of us! But right now, upstairs, there are two variables who *want* you in their lives. Assign values to them.' Carl, Fi's fiancee, is drawn as his face with a y; his kid as a face on an x.
Bill Holbrook’s On The Fastrack for the 4th of March, 2017. I feel bad that I completely forgot Carl had a kid and that the face on the x doesn’t help me remember anything.

Holbrook continues the thread on the 4th, extends the anthropomorphic-mathematics-stuff to call people variables. There’s ways that this is fair. We use a variable for a number whose value we don’t know or don’t care about. A “random variable” is one that could take on any of a set of values. We don’t know which one it does, in any particular case. But we do know — or we can find out — how likely each of the possible values is. We can use this to understand the behavior of systems even if we never actually know what any one of it does. You see how I’m going to defend this metaphor, then, especially if we allow that what people are likely or unlikely to do will depend on context and evolve in time.

Reading the Comics, January 21, 2017: Homework Edition


Now to close out what Comic Strip Master Command sent my way through last Saturday. And I’m glad I’ve shifted to a regular schedule for these. They ordered a mass of comics with mathematical themes for Sunday and Monday this current week.

Karen Montague-Reyes’s Clear Blue Water rerun for the 17th describes trick-or-treating as “logarithmic”. The intention is to say that the difficulty in wrangling kids from house to house grows incredibly fast as the number of kids increases. Fair enough, but should it be “logarithmic” or “exponential”? Because the logarithm grows slowly as the number you take the logarithm of grows. It grows all the slower the bigger the number gets. The exponential of a number, though, that grows faster and faster still as the number underlying it grows. So is this mistaken?

I say no. It depends what the logarithm is, and is of. If the number of kids is the logarithm of the difficulty of hauling them around, then the intent and the mathematics are in perfect alignment. Five kids are (let’s say) ten times harder to deal with than four kids. Sensible and, from what I can tell of packs of kids, correct.

'Anne has six nickels. Sue has 41 pennies. Who has more money?' 'That's not going to be easy to figure out. It all depends on how they're dressed!'
Rick Detorie’s One Big Happy for the 17th of January, 2017. The section was about how the appearance and trappings of wealth matter for more than the actual substance of wealth so everyone’s really up to speed in the course.

Rick Detorie’s One Big Happy for the 17th is a resisting-the-word-problem joke. There’s probably some warning that could be drawn about this in how to write story problems. It’s hard to foresee all the reasonable confounding factors that might get a student to the wrong answer, or to see a problem that isn’t meant to be there.

Bill Holbrook’s On The Fastrack for the 19th continues Fi’s story of considering leaving Fastrack Inc, and finding a non-competition clause that’s of appropriate comical absurdity. As an auditor there’s not even a chance Fi could do without numbers. Were she a pure mathematician … yeah, no. There’s fields of mathematics in which numbers aren’t all that important. But we never do without them entirely. Even if we exclude cases where a number is just used as an index, for which Roman numerals would be almost as good as regular numerals. If nothing else numbers would keep sneaking in by way of polynomials.

'Uh, Fi? Have you looked at the non-compete clause in your contract?' 'I wouldn't go to one of Fastrack's competitors.' 'No, but, um ... you'd better read this.' 'I COULDN'T USE NUMBERS FOR TWO YEARS???' 'Roman numerals would be okay.'
Bill Holbrook’s On The Fastrack for the 19th of January, 2017. I feel like someone could write a convoluted story that lets someone do mathematics while avoiding any actual use of any numbers, and that it would probably be Greg Egan who did it.

Dave Whamond’s Reality Check for the 19th breaks our long dry spell without pie chart jokes.

Mort Walker and Dik Browne’s Vintage Hi and Lois for the 27th of July, 1959 uses calculus as stand-in for what college is all about. Lois’s particular example is about a second derivative. Suppose we have a function named ‘y’ and that depends on a variable named ‘x’. Probably it’s a function with domain and range both real numbers. If complex numbers were involved then the variable would more likely be called ‘z’. The first derivative of a function is about how fast its values change with small changes in the variable. The second derivative is about how fast the values of the first derivative change with small changes in the variable.

'I hope our kids are smart enough to win scholarships for college.' 'We can't count on that. We'll just have to save the money!' 'Do you know it costs about $10,000 to send one child through college?!' 'That's $40,000 we'd have to save!' Lois reads to the kids: (d^2/dx^2)y = 6x - 2.
Mort Walker and Dik Browne’s Vintage Hi and Lois for the 27th of July, 1959. Fortunately Lois discovered the other way to avoid college costs: simply freeze the ages of your children where they are now, so they never face student loans. It’s an appealing plan until you imagine being Trixie.

The ‘d’ in this equation is more of an instruction than it is a number, which is why it’s a mistake to just divide those out. Instead of writing it as \frac{d^2 y}{dx^2} it’s permitted, and common, to write it as \frac{d^2}{dx^2} y . This means the same thing. I like that because, to me at least, it more clearly suggests “do this thing (take the second derivative) to the function we call ‘y’.” That’s a matter of style and what the author thinks needs emphasis.

There are infinitely many possible functions y that would make the equation \frac{d^2 y}{dx^2} = 6x - 2 true. They all belong to one family, though. They all look like y(x) = \frac{1}{6} 6 x^3 - \frac{1}{2} 2 x^2 + C x + D , where ‘C’ and ‘D’ are some fixed numbers. There’s no way to know, from what Lois has given, what those numbers should be. It might be that the context of the problem gives information to use to say what those numbers should be. It might be that the problem doesn’t care what those numbers should be. Impossible to say without the context.

Reading the Comics, January 16, 2017: Numerals Edition


Comic Strip Master Command decreed that last week should be busy again. So I’m splitting its strips into two essays. It’s a week that feels like it had more anthropomorphic numerals jokes than usual, but see if I actually count these things.

2 asks 4: 'Six, six, six, can't you think of anything but six?'
Mike Peters’s Mother Goose and Grimm for the 15th of January, 2017. I understand that sometimes you just have to use the idea you have instead of waiting for something that can best use the space available, but really, a whole Sunday strip for a single panel? And a panel that’s almost a barren stage?

Mike Peters’s Mother Goose and Grimm for the 15th I figured would be the anthropomorphic numerals joke for the week. Shows what I know. It is an easy joke, but I do appreciate the touch of craft involved in picking the numerals. The joke is just faintly dirty if the numbers don’t add to six. If they were a pair of 3’s, there’d be the unwanted connotations of a pair of twins talking about all this. A 6 and a 0 would make at least one character weirdly obsessed. So it has to be a 4 and a 2, or a 5 and a 1. I imagine Peters knew this instinctively, at this point in his career. It’s one of the things you learn in becoming an expert.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 15th is mostly physical comedy, with a touch of — I’m not sure what to call this kind of joke. The one where a little arithmetic error results in bodily harm. In this sort of joke it’s almost always something not being carried that’s the error. I suppose that’s a matter of word economy. “Forgot to carry the (number)” is short, and everybody’s done it. And even if they don’t remember making this error, the phrasing clarifies to people that it’s a little arithmetic mistake. I think in practice mistaking a plus for a minus (or vice-versa) is the more common arithmetic error. But it’s harder to describe that clearly and concisely.

Jef Mallett’s Frazz for the 15th puzzled me. I hadn’t heard this thing the kid says about how if you can “spew ten random lines from a classic movie” to convince people you’ve seen it. (I don’t know the kid’s name; it happens.) I suppose that it would be convincing, though. I certainly know a couple lines from movies I haven’t seen, what with living in pop culture and all that. But ten would be taxing for all but the most over-saturated movies, like any of the Indiana Jones films. (There I’m helped by having played the 90s pinball machine a lot.) Anyway, knowing ten random mathematics things isn’t convincing, especially since you can generate new mathematical things at will just by changing a number. But I would probably be convinced that someone who could describe what’s interesting about ten fields of mathematics had a decent understanding of the subject. That requires remembering more stuff, but then, mathematics is a bigger subject than even a long movie is.

In Bill Holbrook’s On The Fastrack for the 16th Fi speaks of tallying the pluses and minuses of her life. Trying to make life into something that can be counted is an old decision-making technique. I think Benjamin Franklin explained how he found it so useful. It’s not a bad approach if a choice is hard. The challenging part is how to weight each consideration. Getting into fractions seems rather fussy to me, but some things are just like that. There is the connotation here that a fraction is a positive number smaller than 1. But the mathematically-trained (such as Fi) would be comfortable with fractions larger than 1. Or also smaller than zero. “Fraction” is no more bounded than “real number”. So, there’s the room for more sweetness here than might appear to the casual reader.

'In a couple of weeks I'm getting married, so I'm taking stock of my life, adding up the pluses and minuses that factor into my goals.' 'Am I a positive or a negative integer?' 'You're a fraction.' 'How presumptuous of me.'
Bill Holbrook’s On The Fastrack for the 16th of January, 2017. Were I in Dethany’s position I would have asked about being a positive or negative number, but then that would leave Holbrook without a third panel. Dethany knows what her author needs most.

Scott Hilburn’s The Argyle Sweater for the 16th is the next anthropomorphic numerals joke for this week. I’m glad Hilburn want to be in my pages more. 5’s concern about figuring out x might be misplaced. We use variables for several purposes. One of them is as a name to give a number whose value we don’t know but wish to work out, and that’s how we first see them in high school algebra. But a variable might also be a number whose value we don’t particularly care about and will never try to work out. This could be because the variable is a parameter, with a value that’s fixed for a problem but not what we’re interested in. We don’t typically use ‘x’ for that, though; usually parameter are something earlier in the alphabet. That’s merely convention, but it is convention that dates back to René Descartes. Alternatively, we might use ‘x’ as a dummy variable. A dummy variable serves the same role that falsework on a building or a reference for an artistic sketch does. We use dummy variables to organize and carry out work, but we don’t care what its values are and we don’t even see the dummy variable in the final result. A dummy variable can be any name, but ‘x’ and ‘t’ are popular choices.

Terry LaBan and Patty LaBan’s Edge City rerun for the 16th plays on the idea that mathematics people talk in algebra. Funny enough, although, “the opposing defense is a variable of 6”? That’s an idiosyncratic use of “variable”. I’m going to suppose that Charles is just messing with Len’s head because, really, it’s fun doing a bit of that.