Let’s see if I can’t close out the first week of June’s comics. I’d rather have published this either Tuesday or Thursday, but I didn’t have the time to write my statistics post for May, not yet. I’ll get there.
One of Gary Larson’s The Far Side reprints for the 4th is one I don’t remember seeing before. The thing to notice is the patient has a huge right brain and a tiny left one. The joke is about the supposed division between left-brained and right-brained people. There are areas of specialization in the brain, so that the damage or destruction of part can take away specific abilities. The popular imagination has latched onto the idea that people can be dominated by specialties of the either side of the brain. I’m not well-versed in neurology. I will hazard the guess that neurologists see “left-brain” and “right-brain” as amusing stuff not to be taken seriously. (My understanding is the division of people into “type A” and “type B” personalities is also entirely bunk unsupported by any psychological research.)
Bud Blake’s Tiger rerun for the 6th has Tiger complaining about his arithmetic homework. And does it in pretty nice form, really, doing some arithmetic along the way. It does imply that he’s starting his homework at 1 pm, though, so I guess it’s a weekend afternoon. It seems like rather a lot of homework for that age. Maybe he’s been slacking off on daily work and trying to make up for it.
John McPherson’s Close To Home for the 6th has a cheat sheet skywritten. It’s for a geometry exam. Any subject would do, but geometry lets cues be written out in very little space. The formulas are disappointingly off, though. We typically use ‘r’ to mean the radius of a circle or sphere, but then would use C for its circumference. That would be . The area of a circle, represented with A, would be . I’m not sure what ‘Vol.C’ would mean, although ‘Volume of a cylinder’ would make sense … if the next line didn’t start “Vol.Cyl”. The volume of a circular cylinder is , where r is the radius and h the height. For a non-circular cylinder, it’s the area of a cross-section times the height. So that last line may be right, if it extends out of frame.
Granted, though, a cheat sheet does not necessarily make literal sense. It needs to prompt one to remember what one needs. Notes that are incomplete, or even misleading, may be all that one needs.
Last week saw a modest number of mathematically-themed comic strips. Then it threw in a bunch of them all on Thursday. I’m splitting the week partway through that, since it gives me some theme to this collection.
Tim Rickard’s Brewster Rockit for the 3rd of May is a dictionary joke, with Brewster naming each kind of chart and making a quick joke about it. The comic may help people who’ve had trouble remembering the names of different kinds of graphs. I doubt people are likely to confuse a pie chart with a bar chart, admittedly. But I could imagine thinking a ‘line graph’ is what we call a bar chart, especially if the bars are laid out horizontally as in the second panel here.
The point of all these graphs is to understand data geometrically. We have fair intuitions about relatives lengths and areas. Bar charts represent relative magnitudes in lengths. Pie charts and bubble charts represent magnitudes in area. We have okay skills in noticing structures in complex shapes. Line graphs and scatter plots use that skill. So these pictures can help us understand some abstraction or something we can’t sense using a sense we do have. It’s not necessarily great; note that I said our intuitions were ‘fair’ and ‘okay’. But we hope to use reason helped by intuition to better understand what we are doing.
And, yes, in the greater scheme of things, any homework or classwork problem is trivial. It’s meant to teach how to calculate things we would like to know. The framing of the story is meant to give us a reason to want to know a thing. But they are practice, and meant to be practice. One practices on something of no consequence, where errors in one’s technique can be corrected without breaking anything.
It happens a round of story problems broke out among my family. My sister’s house has some very large trees. There turns out to be a poorly-organized process for estimating the age of these trees from their circumference. This past week saw a lot of chatter and disagreement about what the ages of these trees might be.
Michael Fry’s Committed rerun for the 7th finally gets us to golf. The Lazy Parent tries to pass off watching golf as educational, with working out the distance to the pin as a story problem. Structurally this is just fine, though: a golfer would be interested to know how far the ball has yet to go. All the information needed is given. It’s the question of whether anyone but syndicated cartoonists cares about golf that’s a mystery.
I think few will oppose me if I say the best part of March 2020 was that it ended. Let me close out nearly all my March business by getting through the last couple comic strips which mentioned some mathematics topic that month. I’ll still have my readership review, probably to post Friday, and then that finishes my participation in the month at last.
Connie Sun’s Connie to the for the 30th features the title character trying to explain what “exponential growth” is. She struggles. Appropriately, as it’s something we see very rarely in ordinary life.
They turn up in mathematics all the time. And mathematical physics, and such. Any process with a rate of change that’s proportional to the current amount of the thing tends to be exponential. This whether growing or decaying. Even circular motion, periodic motion, can be understood as exponential growth with imaginary numbers. So anyone doing mathematics gets trained to see, and expect, exponentials. They have great analytic properties, too. You can use them to solve differential equations. And differential equations are so much of science that it’s easy to forget they’re not.
In ordinary life, though? Well, yes, a lot of quantities will change at rates which depend on their current quantity. But in anything that’s been around a while, the quantity will usually be at, or near enough, an equilibrium. Some kind of balance. It may move away from that balance, but usually, it’ll move back towards it. (I am skipping some complicating factors. Don’t worry about them.) A mathematician will see the hidden exponentials in this. But to anyone else? The thing may start growing, but then it peters out and slows to a stop. Or it might collapse, but that change also peters out. Maybe it’ll hit a new equilibrium; maybe it’ll go back to the old. We rarely see something changing without the sorts of limits that tamp the change back down.
Even the growth of infection rates for Covid-19 will not stay exponential forever, even if there were no public health measures responding to it. There can’t be more people infected than there are people in the world. At some point, the curve representing number of infected people versus time would stop growing more and more, and would level out, from a pattern called the logistic equation. But the early stages of this are almost indistinguishable from exponential growth.
Todd Clark’s Lola for the 30th has a student asking what the end of mathematics is. And learning how after algebra comes geometry, trigonometry, calculus, topology, and more. All fair enough, though I’m surprised to see it put for that that of course someone who does enough mathematics will do topology. (I only have a casual brush with it myself, mostly in service to other topics.) But it’s nice to have it acknowledged that, if you want, you can go on learning new mathematics fields, practically without limit.
Ashleigh Brilliant’s Pot-Shots for the 30th just declares infinity to be a favorite number. Is it a number? … We have to be careful what exactly we mean by number. Allow that we are careful, though. It’s certainly at least number-adjacent.
Finally we get to last week’s comics. This past one wasn’t nearly so busy a week for mathematically-themed comic strips. But there’s still just enough that I can split them across two days. This fits my schedule well, too.
Rick Detorie’s One Big Happy for the 9th is trying to be the anthropomorphized numerals joke of the week. It’s not quite there, but it also uses some wordplay. … And I’ll admit being impressed any of the kids could do much with turning any of the numerals into funny pictures. I remember once having a similar assignment, except that we were supposed to use the shape of our state, New Jersey, as the basis for the picture. I grant I am a dreary and literal-minded person. But there’s not much that the shape of New Jersey resembles besides itself, “the shape of Middlesex County, New Jersey”, and maybe a discarded sock. I’m not still upset about this.
The choice of ‘z’ to mean a snore is an arbitrary choice, no more inherent to the symbol than that ‘2’ should mean two. Christopher Miller’s American Cornball, which tracks a lot of (American) comedic conventions of the 20th century, notes a 1911 comic postcard representing snoring as “Z-Z-Z-Z-R-R-R-R-Z-Z-Z-Z-R-R-R-R”, which captures how the snore is more than a single prolonged sound.
Dave Blazek’s Loose Parts for the 11th has the traditional blackboard full of symbols. And two mathematics-types agreeing that they could make up some more symbols. Well, mathematics is full of symbols. Each was created by someone. Each had a point, which was to express some concept better. Usually the goal is to be more economical: it’s fewer strokes of the pen to write = instead of “equals”, and = is quicker even than “eq”. Or we want to talk a lot about a complicated concept, which is how we get, say, for “a representative of the set of angles with sine equal to x”.
I suspect every mathematician has made up a couple symbols in their notes. In the excitement of working out a problem there’ll be something they want to refer to a lot. That gets reduced to an acronym or a repeated scribble soon enough. Sometimes it’s done by accident: for a while when I needed a dummy variable I would call on “ksee”, a Greek letter so obscure that it does not even exist. It looks like a cross between zeta and xi. The catch is, always, getting anyone else to use the symbol. Most of these private symbols stay private, because they don’t do work that can’t be better done by a string of symbols we already have (letters included). Or at least they don’t to well enough to be worth the typesetting trouble. I’d be surprised if any of the students I used “ksee” in front of reused the letter, even if they did find a need for a dummy variable. Founding a field, or writing a definitive text in a field, helps your chances.
I am curious how the modern era of digital typesetting will affect symbol creation. It’s relatively easy to put in a new symbol — or to summon one in the Unicode universe not currently used for mathematics — in a document and have it copied. Certainly it’s easy compared to what it was like in typewriter and Linotype days, when you might need to rely on a friend who knows a guy at the type foundry. On the other hand, it’s hard enough to get the raw file in LaTeX — a long-established standard mathematics typesetting computer language — from another person and have it actually work, even without adding in new symbols. I don’t see that changing just because several people have found that a bubble tea emoji quite helps their paper on sedimentation rates.
Pedro Martin’s Mexikid Stories for the 11th recounts childhood memories and anxieties of being matched, boys versus girls, in various activities. This includes mathematics quizzes. Here, the mathematics is done as a class game, which is a neat coincidence as I’d been thinking of similar public mathematics quiz-games that I’d done. I liked them, but then, I was almost always at top or second in the class rankings, and — after the initial couple rounds — never fell below third. My recent thoughts were for how much less fun this must have been for the kids in 26th place, especially if they’re ones who can do the work just fine, given time and space. We do value speed, in working, and that comes from practicing a task so often that we do it in the slightest time possible. And we value ability to perform under pressure, so we put people into anxiety-producing states until they can do a particular task anyway.
And here’s the other half of last week’s comic strips that name-dropped mathematics in such a way that I couldn’t expand it to a full paragraph. We’ll likely be back to something more normal next week.
David Malki’s Wondermark for the 20th is built on the common idiom of giving more than 100%. I’m firmly on the side of allowing “more than 100%” in both literal and figurative uses of percent, so there’s not much more to say.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses Big Numbers as the sort of thing that need a down-to-earth explanation. The strip is about explanations that don’t add clarity. It shows my sense of humor that I love explanations that are true but explain nothing. The more relevant and true without helping the better. Right up until it’s about something I could be explaining instead.
Tom Batiuk’s vintage Funky Winkerbean for the 21st is part of a week of strips from the perspective of a school desk. It includes a joke about football players working mathematics problems. The strip originally ran the 8th of February, 1974, looks like.
I’m finding it surprisingly good for my workflow to use Sundays for the comic strips which mention mathematics only casually. Tomorrow or so I’ll get to the ones with substantial material, in an essay available at this link.
Jim Meddick’s Monty for the 19th is a sudoku joke, with Monty filling in things that aren’t numerals. Many of them are commonly used mathematical symbols. The ones that I don’t recognize I suspect come from physics applications, especially particle physics. These rely heavily on differential equations and group theory and are likely where Meddick got things like the and the from.
Chris Browne’s Raising Duncan rerun for the 23rd has a man admitting bad mathematics skills for why he can’t count the ways he loves his wife. The strip originally ran the 27th of September, 2003. (The strip was short-lived, and is in perpetual reruns. It may be worth reading at least one time through, though, since the pairs of main characters in it are eagerly in love, without being sappy about it, and it’s pleasant seeing people enthusiastic about each other. This is the strip that had the exchange “Marry me!” “I did!” “Marry me more!” “Okay!” that keeps bringing me cheer and relationship goals.)
There were a couple more comic strips than made a good fit in yesterday’s recap. Here’s the two that I had much to write about.
Jason Poland’s Robbie and Bobby for the 18th is another rerun. I mentioned it back in December of 2016. Zeno’s Paradoxical Pasta plays on the most famous of Zeno’s Paradoxes, about how to get to a place one has to get halfway there, but to get halfway there requires getting halfway to halfway. This goes on in infinite regression. The paradox is not a failure to understand that we can get to a place, or finish swallowing a noodle.
A couple years back we needed to patch a bunch of weak spots in the roof. We found all the spots that needed shoring up and measured how long they were, and went to buy some wood and get it cut to fit. I turned over the list of sizes and the guy told us we’d have to buy more than one of the standard-size sheets of plywood to do it. I thought, wait, no, that can’t be, and sketched out possible ways to cut the wood and fit pieces together. Finally I concluded that, oh, yes, the guy whose job it was to figure out how much wood was needed for particular tasks knew what he was talking about. His secret? I don’t know. What finally convinced me was adding up the total area of the wood we’d need, and finding that it was more than what one sheet would be.
Dave Blazek’s Loose Parts for the 19th uses a whiteboard full of mathematics as visual shorthand for “some really complicated subject”. It’s a good set of mathematics symbols on the whiteboard. They don’t mean anything in the combination shown, though. It’s just meant to bewilder.
The key is the Mover’s claim that he can look at any amount of stuff and tell you whether it fits in the moving bins. Working out something like this is a version of the knapsack problem. The knapsack problem is … well, the problem you imagine it might be, if someone told you “some mathematicians study a thing called the knapsack problem”? That’s about right. Formally, it’s about selecting from a set of things of different value. How hard is it to pick a subset of things with exactly that value? Or find that there is no such subset?
Well, in a sense, not hard at all. You can just keep trying combinations. Eventually you’ll either find a set that works, or you’ll try every possibility and find none of them work. This is known as “exhaustion”, and correctly. If there are ten things, there are 3,628,800 possibilities. Then it gets really bad. If there are twenty things, there are 2,432,902,008,176,640,000 possibilities. Finding the one that works? That could take a while.
So being able to tell whether a collection of things can fit within a particular space? That’s a form of the knapsack problem. Being able to always solve that any faster than just “try out every combination until you find one that works”? That would be incredible. The problem is hard. That’s a technical term. It means what you imagine it means, but more precisely.
So why the mathematician’s response? It’s because the problem of hacking the common Internet security algorithms is also hard. (I am discussing here how difficult hacking would be if the algorithm were implemented perfectly. There are many hacking techniques available because of bugs. Programs are not written perfectly. Compilers do not translate them to computer code perfectly. Computers are not built perfectly. These and more flaws make hacking more possible than it should be.) It’s the same kind of hard as this knapsack problem. I mean “the same” more technically than you might imagine. If you had a method to quickly solve this knapsack problem, then, you could use this to break computer encryption quickly. And, it turns out, vice-versa, so at least there’s some fairness to things. So if the the Mover can, truly, always instantly tell whether a set of things fit in the moving bins, then hacking e-mails should be possible to. The Mover would have to team up with a mathematician who studies computational problems like this. I don’t know how to do it, myself. I think about the how to do this and feel lost, myself.
So is the Mover full of it? Let’s put this more nicely. Is he at least unduly optimistic about his claims?
Nah. What makes the knapsack problem hard is that you have to find a solution that quickly finds answers for every possible set of things. But the Mover doesn’t have to deal with that. Most of the stuff is in boxes. It’s in mostly simple polygonal shapes. There’s not, like, 400 million items, each the size of a Cheerio. The Mover may plausibly have never encountered a set of things to move where he couldn’t tell whether it fits.
And, yes, there’s selection bias. Suppose he declared that no, this load had to fit into two vans. But that actually a sufficiently clever arrangement would have let it fit in one. Who would ever know he was wrong? He’d only ever know his intuition was wrong if he declared something would fit in one van and, in fact, it couldn’t.
Percy Crosby’s Skippy for the 21st is a student-at-the-board problem. It’s using the punch line that “I don’t know” might be a true answer to any problem. There are many real mathematics problems for which nobody really knows an answer.
But Miss Larkin has good advice here. Maybe you don’t know the final answer. But do you know anything? Write it down. It’s good for partial credit, at least. Working out a part of the problem might also be useful, too. Often you can work out how to do a hard problem by looking at a similar but simpler problem. If Skippy is lost at 8 + 4 + 7 + 5, could he do at least 8 + 4 + 7? Could he do 8 + 4? Maybe this wouldn’t help him get to the ultimate answer. Often a difficult problem turns out to be solved by solving a circle of simple problems, that starve out the hard.
Our roof patches held up for their need, which was just to last a couple months while we contracted for a replacement roof. And, happily, the roof replacement got done speedily and during a week that did not rain. (Back in grad school the apartment I was in had its roof replaced on a day that, it turns out, would get a spontaneous downpour halfway through. My apartment was on the top floor. This made for an exciting afternoon.)
We continue to be in the summer vacation doldrums for mathematically-themed comic strips. But there’ve been a couple coming out. I could break this week’s crop into two essays, for example. All of today’s strips are comics that turn up in my essays a lot. It’s like hanging out with a couple of old friends.
Samson’s Dark Side of the Horse for the 17th uses the motif of arithmetic expressions as “difficult” things. The expressions Samson quotes seem difficult for being syntactically weird: What does the colon under the radical sign mean in ? Or they’re difficult for being indirect, using a phrase like “50%” for “half”. But with some charity we can read this as Horace talking about 3:33 am to about 6:30 am. I agree that those are difficult hours.
It also puts me in mind of a gift from a few years back. An aunt sent me an Irrational Watch, with a dial that didn’t have the usual counting numbers on it. Instead there were various irrational numbers, like the Golden Ratio or the square root of 50 or the like. Also the Euler-Mascheroni Constant, a number that may or may not be irrational. Nobody knows. It’s likely that it is irrational, but it’s not proven. It’s a good bit of fun, although it does make it a bit harder to use the watch for problems like “how long is it until 4:15?” This isn’t quite what’s going on here — the square root of nine is a noticeably rational number — but it seems in that same spirit.
I may need to rewrite that old essay. An “improper” form satisfies all the required conditions for the term. But it misses some of the connotation of the term. It’s true that, say, the new process takes “a fraction of the time” of the old, if the old process took one hour and the new process takes fourteen years. But if you tried telling someone that they would assume you misunderstood something. The ordinary English usage of “fraction” carries the connotation of “a fraction between zero and one”, and that’s what makes a “proper fraction”.
In practical terms, improper fractions are fine. I don’t know of any mathematicians who seriously object to them, or avoid using them. The hedging word “seriously” is in there because of a special need. That need is: how big is, say, ? Is it bigger than five? Is it smaller than six? An improper fraction depends on you knowing, in this case, your fourteen-times tables to tell. Switching that to a mixed fraction, , helps figure out what the number means. That’s as far as we have to worry about the propriety of fractions.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses the form of a Fermi problem for its joke. Fermi problems have a place in mathematical modeling. The idea is to find an estimate for some quantity. We often want to do this. The trick is to build a simple model, and to calculate using a tiny bit of data. The Fermi problem that has someone reached public consciousness is called the Fermi paradox. The question that paradox addresses is, how many technologically advanced species are there in the galaxy? There’s no way to guess. But we can make models and those give us topics to investigate to better understand the problem. (The paradox is that reasonable guesses about the model suggest there should be so many aliens that they’d be a menace to air traffic. Or that the universe should be empty except for us. Both alternatives seem unrealistic.) Such estimates can be quite wrong, of course. I remember a Robert Heinlein essay in which he explained the Soviets were lying about the size of Moscow, his evidence being he didn’t see the ship traffic he expected when he toured the city. I do not remember that he analyzed what he might have reasoned wrong when he republished this in a collection of essays he didn’t seem to realize were funny.
So the interview question presented is such a Fermi problem. The job applicant, presumably, has not committed to memory the number of employees at the company. But there would be clues. Does the company own the whole building it’s in, or just a floor? Just an office? How large is the building? How large is the parking lot? Are there people walking the hallways? How many desks are in the offices? The question could be answerable. The applicant has a pretty good chain of reasoning too.
Bill Amend’s FoxTrot Classics for the 20th has several mathematical jokes in it. One is the use of excessively many decimal points to indicate intelligence. Grant that someone cares about the hyperbolic cosines of 15.2. There is no need to cite its wrong value to nine digits past the decimal. Decimal points are hypnotic, though, and listing many of them has connotations of relentless, robotic intelligence. That is what Amend went for in the characters here. That and showing how terrible nerds are when they find some petty issue to rage over.
Eugene is correct about the hyperbolic cosine being wrong, there, though. He’s not wrong to check that. It’s good form to have some idea what a plausible answer should be. It lets one spot errors, for one. No mathematician is too good to avoid making dumb little mistakes. And computing tools will make mistakes too. Fortunately they don’t often, but this strip originally ran a couple years after the discovery of the Pentium FDIV bug. This was a glitch in the way certain Pentium chips handled floating-point division. It was discovered by Dr Thomas Nicely, at Lynchberg College, who found inconsistencies in some calculations when he added Pentium systems to the computers he was using. This Pentium bug may have been on Amend’s mind.
Eugene would have spotted right away that the hyperbolic cosine was wrong, though, and didn’t need nine digits for it. The hyperbolic cosine is a function. Its domain is the real numbers. It range is entirely numbers greater than or equal to one, or less than or equal to minus one. A 0.9 something just can’t happen, not as the hyperbolic cosine for a real number.
And what is the hyperbolic cosine? It’s one of the hyperbolic trigonometric functions. The other trig functions — sine, tangent, arc-sine, and all that — have their shadows too. You’ll see the hyperbolic sine and hyperbolic tangent some. You will never see the hyperbolic arc-cosecant and anyone trying to tell you that you need it is putting you on. They turn up in introductory calculus classes because you can differentiate them, and integrate them, the way you can ordinary trig functions. They look just different enough from regular trig functions to seem interesting for half a class. By the time you’re doing this, your instructor needs that.
The ordinary trig functions come from the unit circle. You can relate the Cartesian coordinates of a point on the circle described by to the angle made between that point and the center of the circle and the positive x-axis. Hyperbolic trig functions we can relate the Cartesian coordinates of a point on the hyperbola described by to angles instead. The functions … don’t have a lot of use at the intro-to-calculus level. Again, other than that they let you do some quite testable differentiation and integration problems that don’t look exactly like regular trig functions do. They turn up again if you get far enough into mathematical physics. The hyperbolic cosine does well in describing catenaries, that is, the shape of flexible wires under gravity. And the family of functions turn up in statistical mechanics, often, in the mathematics of heat and of magnetism. But overall, these functions aren’t needed a lot. A good scientific calculator will offer them, certainly. But it’ll be harder to get them.
There is another oddity at work here. The cosine of 15.2 degrees is about 0.965, yes. But mathematicians will usually think of trigonometric functions — regular or hyperbolic — in terms of radians. This is just a different measure of angles. A right angle, 90 degrees, is measured as radians. The use of radians makes a good bit of other work easier. Mathematicians get to accustomed to using radians that to use degrees seems slightly alien. The cosine of 15.2 radians, then, would be about -0.874. Eugene has apparently left his calculator in degree mode, rather than radian mode. If he weren’t so worked up about the hyperbolic cosine being wrong he might have noticed. Perhaps that will be another exciting error to discover down the line.
This strip was part of a several-months-long story Bill Amend did, in which Jason has adventures at Math Camp. I don’t remember the whole story. But I do expect the strip to have several more appearances here this summer.
When I collected last week’s mathematically-themed comic strips I thought this set an uninspiring one. That changed sometime while I wrote. That’s the sort of week I like to have.
Richard Thompson’s Richard’s Poor Almanac for the 28th is a repeat; all these strips are. And I’ve featured it here before too. But never before in color, so I’ll take this chance to show it one last time. One of the depicted plants is the “Non-Euclidean Creeper”, which “ignores the geometry of the space-time continuum”. Non-Euclidean is one of those few geometry-related words that people recognize — maybe even only learn — in their adulthood. It has connotations of the bizarre and the weird and the wrong.
And it is a bit weird. While we live in a non-Euclidean space, we never really notice. Euclidean space is the geometry we’re used to from drawing shapes on paper and putting boxes in the corners of basements. And from this we’ve given “non-Euclidean” this sinister reputation. We credit it with defying common sense and even logic itself, although it’s geometry. It can’t defy logic. It can defy intuition. Non-Euclidean geometries have the idea that there are no such things as parallel lines. Or the idea that there are too many parallel lines. And it can get to weird results, particularly if we look at more than three dimensions of space. Those also tax the imagination. It will get a weed a bad reputation.
Chen Weng’s Messycow Comics for the 30th is about a child’s delight in learning how to count. I don’t remember ever being so fascinated by counting that it would distract me permanently. I do remember thinking it was amazing that once a pattern was established it kept on, with no reason to ever stop, or even change. My recollection is I thought this somehow unfair to the alphabet, which had a very sudden sharp end.
The counting numbers — counting in general — seem to be things we’ve evolved to understand. Other animals know how to count. Here I recommend again Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics, which describes some of the things we know about how animals do mathematics. It also describes how children come to understand it.
Samson’s Dark Side of the Horse for the 31st is a bit of play with arithmetic. Horace simplifies his problem by catching all the numerals with loops in them — the zeroes and the eights — and working with what’s left. Evidently he’s already cast out all the nines. (This is me making a joke. Casting out nines is a simple checksum that you can do which can guard against some common arithmetic mistakes. It doesn’t catch everything. But it is simple enough to do that it can be worth using.)
The part that disappoints me is that to load the problem up with digits with loops, we get a problem that’s not actually hard: 100 times anything is easy. If the problem were, say, 189 times 80008005 then you’d have a problem someone might sensibly refuse to do. But without those zeroes at the start it’d be harder to understand what Horace was doing. Maybe if it were 10089 times 800805 instead.
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st is the anthropomorphic numerals joke for the week. Also the anthropomorphic letters joke. The capital B sees occasional use in mathematics. It can represent the ball, that is, the set of all points that represent the interior of a sphere of a set radius. Usually a radius of 1. It also sometimes appears in equations as a parameter, a number whose value is fixed for the length of the problem but whose value we don’t care about. I had thought there were a few other roles for B alone, such as a label to represent the Bessel functions. These are a family of complicated-looking polynomials with some nice properties it’s too great a diversion for me to discuss just now. But they seem to more often be labelled with a capital J for reasons that probably seemed compelling at the time. It’ll also get used in logic, where B might stand for the second statement of some argument. 4, meanwhile, is that old familiar thing.
This clears out last week’s comic strips. This present week’s strips should be at this link on Sunday. I haven’t yet read Friday or Saturday’s comics, so perhaps there’s been a flood, but this has been a slow week so far.
John Graziano’s Ripley’s Believe It or Not for the 26th mentions several fairly believable things. The relevant part is about naming the kind of surface that a Pringles chip represents. That is, the surface a Pringles chip would be if it weren’t all choppy and irregular, and if it continued indefinitely.
The shape is, as Graziano’s Ripley’s claims, a hypberbolic paraboloid. It’s a shape you get to know real well if you’re a mathematics major. They turn up in multivariable calculus and, if you do mathematical physics, in dynamical systems. It’s also a shape mathematics majors get to calling a “saddle shape”, because it looks enough like a saddle if you’re not really into horses.
The shape is one of the “quadratic surfaces”. These are shapes which can be described as the sets of Cartesian coordinates that make a quadratic equation true. Equations in Cartesian coordinates will have independent variables x, y, and z, unless there’s a really good reason. A quadratic equation will be the sum of some constant times x, and some constant times x2, and some constant times y, and some constant times y2, and some constant times z, and some constant times z2. Also some constant times xy, and some constant times yz, and some constant times xz. No xyz, though. And it might have some constant added to the mix at the end of all this.
There are seventeen different kinds of quadratic surfaces. Some of them are familiar, like ellipsoids or cones. Some hardly seem like they could be called “quadratic”, like intersecting planes. Or parallel planes. Some look like mid-century modern office lobby decor, like elliptic cylinders. And some have nice, faintly science-fictional shapes, like hyperboloids or, as in here, hyperbolic paraboloids. I’m not a judge of which ones would be good snack shapes.
Bud Blake’s Tiger for the 31st is a rerun, of course. Blake died in 2005 and no one else drew his comic strip. It’s a funny-answer-to-a-story-problem joke. And, more, it’s a repeat of a Tiger strip I’ve already run here. I admit a weird pride when I notice a comic strip doing a repeat. It gives me some hope that I might still be able to remember things. But this is also a special Tiger repeat. It’s the strip which made me notice Bud Blake redrawing comics he had already used. This one is not a third iteration of the strip which reran in April 2015 and June 2016. It’s a straight repeat of the June 2016 strip.
The mystery to me now is why King Features apparently has less than three years’ worth of reruns in the bank for Tiger. The comic ran from 1965 to 2003, and it’s not as though the strip made pop culture references or jokes ripped from the headlines. Even if the strip changed its dimensions over the decades, to accommodate shrinking newspapers, there should be a decade at least of usable strips to rerun.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 31st uses a chart to tease mathematicians, both in the comic and in the readership. The joke is in the format of the graph. The graph is supposed to argue that the Mathematician’s pedantry is increasing with time, and it does do that. But it is customary in this sort of graph for the independent variable to be the horizontal axis and the dependent variable the vertical. So, if the claim is that the pedantry level rises as time goes on, yes, this is a … well, I want to say wrong way to arrange the axes. This is because the chart, as drawn, breaks a convention. But convention is a tool to help people’s comprehension. We are right to ignore convention if doing so makes the chart better serve its purpose. Which, the punch line is, this does.
It’s happened again: another slow week around here. My supposition is that Comic Strip Master Command was snowed in about a month ago, and I’m seeing the effects only now. There’s obviously no other reason that more comic strips didn’t address my particular narrow interest in one seven-day span.
Samson’s Dark Side of the Horse for the 18th is a numerals joke. The mathematics content is slight, I admit, but I’ve always had a fondness for Dark Side of the Horse. (I know it sounds like I have a fondness for every comic strip out there. I don’t quite, but I grant it’s close.) Conflating numerals and letters, and finding words represented by numerals, is an old tradition. It was more compelling in ancient days when letters were used as numerals so that it was impossible not to find neat coincidences. I suppose these days it’s largely confined to typefaces that make it easy to conflate a letter and a numeral. I mean moreso than the usual trouble telling apart 1 and l, 0 and O, or 5 and S. Or to special cases like hexadecimal numbers where, for ease of representation, we use the letters A through F as numerals.
Jef Mallett’s Frazz for the 18th is built on an ancient problem. I remember being frustrated with it. How is “questions 15 to 25” eleven questions when the difference between 15 and 25 is ten? The problem creeps into many fields. Most of the passion has gone out of the argument but around 1999 you could get a good fight going about whether the new millennium was to begin with January 2000 or 2001. The kind of problem is called a ‘fencepost error’. The name implies how often this has complicated someone’s work. Divide a line into ten segments. There are nine cuts on the interior of the line and the two original edges. I’m not sure I could explain to an elementary school student how the cuts and edges of a ten-unit-long strip match up to the questions in this assignment. I might ask how many birthdays someone’s had when they’re nine years old, though. And then flee the encounter.
Mark Parisi’s Off The Mark for the 19th is another numerals joke. This one’s also the major joke to make about an ice skater doing a figure eight: write the eight some other way. (I’d have sworn there was an M-G-M Droopy cartoon in which Spike demonstrates his ability to skate a figure 8, and then Droopy upstages him by skating ‘4 + 4’. I seem to be imagining it; the only cartoon where this seems to possibly fit is 1950’s The Chump Champ, and the joke isn’t in that one. If someone knows the cartoon I am thinking of, please let me know.) Here, the robot is supposed to be skating some binary numeral. It’s nothing close to an ‘8’, but perhaps the robot figures it needs to demonstrate some impressive number to stand out.
Bud Blake’s Tiger for the 21st has Tiger trying to teach his brother arithmetic. Working it out with fingers seems like a decent path to try, given Punkinhead’s age and background. And Punkinhead has a good point: why is the demonstration the easy problem and the homework the hard problem? I haven’t taught in a while, but do know I would do that sort of thing. My rationalization, I think, would be that a hard problem is usually hard because it involves several things. If I want to teach a thing, then I want to highlight just that thing. So I would focus on a problem in which that thing is the only tricky part, and everything else is something the students are so familiar with they don’t notice it. The result is usually an easy problem. There isn’t room for toughness. I’m not sure if that’s a thing I should change, though. Demonstrations of how to work harder problems are worth doing. But I usually think of those as teaching “how to use these several things we already know”. Using a tough problem to show one new thing, plus several already-existing tricky things, seems dangerous. It might be worth it, though.
That there were twelve comic strips making my cut as mention-worthy this week should have let me do three essays of four comics each. But the desire to include all the comics from the same day in one essay leaves me one short here. So be it. Three of the four cartoonists featured here have a name of Sansom or Samson, so, that’s an edition title for you. No, Sam and Silo do not appear here.
Art Sansom and Chip Sansom’s Born Loser for the 6th uses arithmetic as a test of deference. Will someone deny a true thing in order to demonstrate loyalty? Arithmetic is full of things that are inarguably true. If we take the ordinary meanings of one, plus, equals, and three, it can’t be that one plus one equals three. Most fields of human endeavor are vulnerable to personal taste, or can get lost in definitions and technicalities. Or the advance of knowledge: my love and I were talking last night how we remembered hearing, as kids, the trivia that panda bears were not really bears, but a kind of raccoon. (Genetic evidence has us now put giant pandas with the bears, and red pandas as part of the same superfamily as raccoons, but barely.) Or even be subject to sarcasm. Arithmetic has a harder time of that. Mathematical ideas do evolve in time, certainly. But basic arithmetic is pretty stable. Logic is also a reliable source of things we can be confident are true. But arithmetic is more familiar than most logical propositions.
Samson’s Dark Side of the Horse for the 8th is the Roman Numerals joke for the week. It’s also a bit of a wordplay joke, although the music wordplay rather tha mathematics. Me, I still haven’t heard a clear reason why ‘MIC’ wouldn’t be a legitimate Roman numeral representation of 1099. I’m not sure whether ‘MIC’ would step on or augment the final joke, though.
Pab Sungenis’s New Adventures of Queen Victoria for the 8th has a comedia dell’arte-based structure for its joke. (The strip does that, now and then.) The comic uses a story problem, with the calculated answer rejected for the nonsense it would be. I suppose it must be possible for someone to eat eighty apples over a long enough time that it’s not distressing, and yet another twenty apples wouldn’t spoil. I wouldn’t try it, though.
Brian Fies’s The Last Mechanical Monster for the 24th is a repeat. I included it last October, when I first saw it on GoComics. Still, the equations in it are right, for ballistic flight. Ballistic means that something gets an initial velocity in a particular direction and then travels without any further impulse. Just gravity. It’s a pretty good description for any system where acceleration’s done for very brief times. So, things fired from guns. Rockets, which typically have thrust for a tiny slice of their whole journey and coast the rest of the time. Anything that gets dropped. Or, as in here, a mad scientist training his robot to smash his way through a bank, and getting flung so.
The symbols in the equations are not officially standardized. But they might as well be. ‘v’ here means the speed that something’s tossed into the air. It really wants to be ‘velocity’, but velocity, in the trades, carries with it directional information. And here that’s buried in ‘θ’, the angle with respect to vertical that the thing starts flight in. ‘g’ is the acceleration of gravity, near enough constant if you don’t travel any great distance over the surface of the Earth. ‘y0‘ is the height from which the thing started to fly. And so then ‘d’ becomes the distance travelled, while ‘t’ is the time it takes to travel. I’m impressed the mad scientist (the one from the original Superman cartoon, in 1941; Fies wrote a graphic novel about that man after his release from jail in the present day.)
Greg Cravens’s Hubris! for the 24th jokes about the dangers of tangled earbuds. For once, mathematics can help! There’s even a whole field of mathematics about this. Not earbuds specifically, but about knots. It’s called knot theory. I trust field was named by someone caught by surprise by the question. A knot, in this context, is made of a loop of thread that’s assumed to be infinitely elastic, so you can always stretch it out or twist it around some. And it’s frictionless, so you can slide the surface against itself without resistance. And you can push it along an end. These are properties that real-world materials rarely have.
But. They can be close enough. And knot theory tells us some great, useful stuff. Among them: your earbuds are never truly knotted. To be a knot at all, the string has to loop back and join itself. That is, it has to be like a rubber band, or maybe an infinity scarf. If it’s got loose ends, it’s no knot. It’s topologically just a straight thread with some twists made in the surface. They can come loose.
All that holds these earbuds together is the friction of the wires against each other. (That the earbud wire splits into a left and a right bud doesn’t matter, here.) They can be loosened. Let me share how.
My love owns, among other things, a marionette dragon. And once, despite it being stored properly, the threads for it got tangled, and those things are impossible to untangle on purpose. I, having had one (1) whole semester of knot theory in grad school, knew an answer. I held the marionette upside-down, by the dragon. The tangled wires and the crossed sticks that control it hung loose underneath. And then shook the puppet around. This made the wires, and the sticks, shake around. They untangled, quickly.
What held the marionette strings, and what holds earbuds, together, is just friction. It’s hard to make the wire slide loosely against itself. Shaking it around, though? That gives it some energy. That gives the wire some play. And here we have one of the handful of cases where entropy does something useful for us. There’s a limit to how tightly a wire can loop around itself. There’s no limit to how loosely it can go. Little, regular, random shakes will tend to loosen the wire. When it’s loose enough, it untangles naturally.
You can help this along. We all know how. Use a pen-point or a toothpick a needle to pry some of the wires apart. That makes the “knot” easier to remove. This works by the same principle. If you reduce how much the wire contacts itself, you reduce the friction on the wire. The wire can slide more easily into the un-knot that it truly is. The comic’s tech support guy gave up too easily.
Samson’s Dark Side of the Horse for the 25th is the Roman numerals joke for this essay. And a cute bit about coincidences between what you can spell out with Roman numerals and sounds people might make. Writing out calculations evokes peculiar, magical prowess. When they include, however obliquely, words? Or parts of words? Can’t blame people for seeing the supernatural in it.
For the second part of last week’s comics, there’s several strips whose authors prefer to use a single name. I’m relieved. Somehow my writing seems easier when I don’t have a long authorial credit to give. I can take writing “Zach Weinersmith” fourteen times a week. It’s all those appearances of, like, “Corey Pandolph and Phil Frank and Joe Troise” (The Elderberries) that slow me way up.
Darrin Bell’s Candorville for the 4th shows off one of the things statistics can do. Tracking some measurable thing lets one notice patterns. These patterns might signify something important. At the least they can suggest things that deserve more scrutiny. There’s dangers, of course. If you’re measuring something that’s rare, or that naturally fluctuates a lot, you might misinterpret changes. You could suppose the changes represent some big, complicated, and invariably scary pattern that isn’t actually there. You can take steps to avoid how much weight you give to little changes. For example, you could look at running averages. Instead of worrying about how often Lemont has asked for his clippers this year versus last, look at how often he’s asked for it, on average, each of the last three years, compared to the average of the three years before that. Changes in that are more likely to be meaningful. But doing this does mean that a sudden change, or a slight but persistent change, is harder to notice. There are always mistakes to be made, when analyzing data. You have to think about what kinds of mistakes you would rather make, and how likely you want to make them.
C-Dog talks about fitting Lemont’s hair growth to a curve. This means looking at the data one has as points in space. What kinds of curves will come as close as possible to including all those points? It turns out infinitely many curves will, and you can fit a curve to all the data points you have. (Unless you have some inconsistent data, like, in 2017 Lemont asked both 14 times and 18 times.) So to do an interpolation you need to make some suppositions. Suppose that the data is really a straight line, with some noise in it. Or is really a parabola. Really a sine wave. Or, drawing from a set of plausible curves, which of those best fits the data?
The Bézier Curve mentioned here is a family of shapes. They’re named for Pierre Bézier, an engineer with Renault who in the 1950s pioneered the using of these curves. There are infinitely many of them. But they’re nice to work with. You can make great-looking curves as sharply curved or as smoothly curved as you like, using them. Most modern fonts use Bézier Curves to compute the shapes of letters. If you have a drawing program, it’s got some kind of Bézier curve in there. It’s the weird tool with a bunch of little dots, most of which are nowhere near the curve they draw. But moving the dots changes the way the curve looks.
A Bézier curve can be linear; indeed, it can just be a line. C-Dog’s showing off by talking about a linear Bézier curve. Or he means something that looks a lot like a line, to the casual eye. Negative-sloped means what it would in high school algebra when you talk about lines: it’s a thing with a value that decreases as the independent variable increases. Something getting rarer in time, for example.
Thaves’s Frank and Ernest for the 5th uses arithmetic, particularly simple addition, as emblematic of the basics of life. Hard to argue that this isn’t some of the first things anyone would learn, and that mathematics as it’s taught builds from that. A mathematician might see other fields — particularly set theory and category theory — as more fundamental than arithmetic. That is, that you can explain arithmetic in terms of set theory, and set theory in terms of category theory. So one could argue that those are the more basic. But if we mean basic as in the first things anyone learns, yeah, it’s arithmetic. Definitely.
Kliban’s Kliban Cartoons for the 5th speaks of proofs. A good bit of mathematics is existence proofs, which is to say, showing that a thing with desired properties does exist. Sometimes they actually show you the thing. Such a “constructive proof” — showing how you make an example of the thing — pretty well proves the thing exists. But sometimes the best you can do is show that there is an answer. In any case, an example of a fish would convince all but the most hardcore skeptics that fish do exist.
For a while I thought this essay would include only the mathematically-themed strips which Comic Strip Master Command sent out through to June 26th, which is picking up the nickname Stitch Day (for 6-26, the movie character’s experiment number). And then I decided some from last Sunday weren’t on-point enough (somehow), and there were enough that came later in the week that I couldn’t do a June 26th Only edition. Which is my longwinded way of saying this one doesn’t have a nonsense name. It just has a name that’s only partially on point.
Mike Baldwin’s Cornered for the 26th is the Rubik’s Cube/strange geometry joke for the week. It seems to me I ought to be able to make some link between the number of various ways to arrange a Rubik’s Cube — which pieces can and which ones cannot be neighbors to a red piece, say, no matter how one scrambles the cube — and the networking between people that you can get from an office where people have to see each other. But I’m not sure that I can make that metaphor work. I’m blaming the temperature, both mine (I have a cold) and the weather’s (it’s a heat wave).
Mark Leiknes’s Cow and Boy for the 26th makes literal the trouble some people have with the phrase “110 percent”. Read uncharitably, yes, “110 of a hundred” doesn’t make sense, if 100 percent is all that could conceivably be of the thing. But if we can imagine, say, the number of cars passing a point on the highway being 90 percent of the typical number, surely we can imagine the number of cars also being 110 percent. To give an example of why I can’t side with pedants in objecting to the phrase.
Jef Mallett’s Frazz for the 26th is just itching for a fight. From me and from the Creative Writing department. Yes, mathematics rewards discipline. All activities do. At the risk of making a prescription: if you want to do something well, spend time practicing the boring parts. For arithmetic, that’s times tables and regrouping calculations and factoring and long division. For writing, that’s word choice and sentence structure and figuring how to bring life to describing dull stuff. Do the fun stuff too, yes, but because it is fun. Getting good at the boring stuff makes you an expert. When you discover that the boring stuff is also kinda fun, you will do the fun stuff masterfully.
But to speak of mathematics as pursuing a single right answer — well, perhaps. In an elementary-school problem there is typically just the one right answer, and the hope is that students learn how to get there efficiently. But if the subject is something well-worn, then there are many ways to do any problem. All are legitimate and the worst one can say of a method is maybe it’s not that efficient, or maybe it’s good here but doesn’t generally work. If the subject is on the edge of what mathematics we know, there may be only one way to get there. But there are many things to find, including original ways to understand what we have already found. To not see that mathematics is creative is to not see mathematics. Or, really, any field of human activity.
Samson’s Dark Side of the Horse for the 27th edges up to being the anthropomorphic numerals joke for the week. I need a good name for this sort of joke about mathematical constructs made tangible, even if they aren’t necessarily characters.
So. A “graph” is a thing that’s turned up in my A To Z serieses. In this context a graph is a collection of points, called “vertices”, and a collection of “edges” that connect vertices. Often the vertices represent something of interest and the edges ways to turn one thing into another. Sometimes the edges are the thing of interest and the vertices are just there to be manipulated in some way by edges. It’s a way to make visual the studying of how stuff is connected, and how things can pass from one to another.
A “stochastic process” is about random variables. Random variables are some property about a system. And you know some things about that variable’s value. You know maybe the range of possible values it could have. You know whether some values are more likely than others. But you do not know what the value is at any particular moment. Consider, say, the temperature outside where you live at a particular time of day. You may have no idea what that is. But you can say, for example, whether today it is more likely to be 90 degrees Fahrenheit or 60 degrees Fahrenheit or 20 degrees Fahrenheit. For a stochastic process we have some kind of index. We can say, for example, which values of temperature are more likely today, the 1st of July, and which ones will be more likely the 1st of August, and which ones will be less likely the 1st of December. Calling it a “process”, to my intuition, makes it sound like we expect something to happen that causes the likelihood of some temperatures to change. And many processes are time-indexed. They study problems where something interesting changes in time, predictable in aggregate but not in detail.
So a graph like this, representing a stochastic process, is a shorthand. Each vertex is a state that something might be in. Each edge is a way to get from one state to another when — something — happens. Doesn’t matter what thing.
A “drunk walk”, or as it’s known to tenderer writers a “random walk”, is a term of art. Not a deep one. It’s meant to evoke the idea of a severely drunk person who yes, can move, but has no control over which way. Thus he wanders around, reaching any point only by luck. Many things look like random walks, in which there is no overall direction, just an unpredictable shuffling around. A drunk walk on this graph would be, well, start at any of the vertices. Then follow edges, chosen randomly. If you start at the uppermost point of the triangle on top, for example, there’s two places to go on the second step: the lower-left or the center-right vertex on the upper triangle. Suppose you go to the center-right vertex. On the next step, you might go right back where you started. You might go to the lower-left vertex on the triangle. You might drop down that bridge to the top of that quadrilateral. And so on, for another step.
Do that some presumably big number of times. Where are you? … Anywhere, of course. But are there vertices you’re more likely to be on? Ones you’re less likely to be on? How does the shape of the graph affect that likelihood? How does how long you spend walking affect that? These tell us things about the process, and are why someone would draw this graph and talk about a random walk on it.
Once again the name of this Reading the Comics edition has nothing to do with any of the strips. I’m just aware that Weezer’s cover of Africa is quite popular right now and who am I to deny people things they want? (I like the cover, but it’s not different enough for me to feel satisfied by it. I tend to like covers that highlight something minor in the original, or that go in a strange direction. Shifting a peppy song into a minor key doesn’t count anymore. But bear in mind, I’m barely competent at listening to music. Please now enjoy my eight hours of early electronica in which various beeps and whistles are passed off as music.)
Samson’s Dark Side of the Horse for the 3rd is the Roman numerals joke for the week. And a welcome return for Dark Side of the Horse. It feels like it’s been gone a while. I wouldn’t try counting by Roman numerals to lull myself to sleep; it seems like too much fussy detail work. But I suppose if you’ve gotten good at it, it’s easy.
Jef Mallett’s Frazz for the 3rd builds on removing statistics from their context. It’s a common problem. It’s possible to measure so very many things. Without a clear idea of what we should expect as normal the measurement doesn’t tell us much. And it can be hard to know what the right context for something even is. Let me deconstruct Caulfield’s example. We’re supposed to reflect on and consider that 40% of all weekdays are Monday and Friday too. But it’s not only weekdays that people work. Even someone working a Sunday might take a sick day. Monday and Friday are a bit over 28% of the whole week. But more people do work Monday-to-Friday than do Saturdays and Sundays, so the Sunday sick day is surely rarer than the Monday. So even if we grant Caulfield’s premise, what does it tell us?
Jason Chatfield’s Ginger Meggs for the 3rd is a bit of why-learn-mathematics propaganda. Megg’s father has a good answer. But it does shift the question back one step. Also I see in the top row that Meggs has one of those comic-strip special editions where the name of the book is printed on the back cover instead. (I’m also skeptical of the photo and text layout on the newspaper Megg’s father is reading. But I don’t know the graphic design style of Australian, as opposed to United States, newspapers.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd may belong on some philosopher’s Reading the Comics blog instead. No matter. There’s some mathematical-enough talk going on here. There’s often many ways to approach the same problem. For example, approaching a system as a handful of items. Or as a huge number of them. Or as infinitely many things. Or as a continuum of things. There are advantages each way. A handful of things, for example, we can often model as interactions between pairs of things. We can model a continuum as a fluid. A vast number of things can let one’s computer numerically approximate a fluid. Or infinitely many particles if that’s more convenient.
To describe all these different models as sharing an “ontology-space” is good mathematical jargon too. In this context the “-space” would mean the collection of all these things that are built by the same plan but with different values of whichever parameter matters.
Bud Blake’s Tiger for the 6th of August, 1965 features Einstein’s famous equation. I suppose it’s showing how well-informed Julian is, that he knows and can present such a big result. There is beauty in mathematics (and physics). Mathematicians (and physicists) find the subject beautiful to start with, and try to find attractive results. I’m curious what the lay reader makes of mathematical symbols, though, just as pieces of art. I remember as a child finding this beauty in a table of integrals in the front of one of my mother’s old college textbooks. All those parallel rows of integral symbols drew me in though nothing I’d seen in mathematics had prepared me to even read it. I still find that beautiful, but I can’t swear that I would even if I hadn’t formed that impression early in life. Are lay and professional readers’ views of mathematical-expression beauty similar? How are they different?
February’s been a flooding month. Literally (we’re about two blocks away from the Voluntary Evacuation Zone after the rains earlier this week) and figuratively, in Comic Strip Master Command’s suggestions about what I might write. I have started thinking about making a little list of the comics that just say mathematics in some capacity but don’t give me much to talk about. (For example, Bob the Squirrel having a sequence, as it does this week, with a geometry tutor.) But I also know, this is unusually busy this month. The problem will recede without my having to fix anything. One of life’s secrets is learning how to tell when a problem’s that kind.
Ham’s Life on Earth for the 12th has a science-y type giving a formula as “something you should know”. The formula’s gibberish, so don’t worry about it. I got a vibe of it intending to be some formula from statistics, but there’s no good reason for that. I’ve had some statistical distribution problems on my mind lately.
Eric Teitelbaum and Bill Teitelbaum’s Bottomliners for the 12th maybe influenced my thinking. It has a person claiming to be a former statistician, and his estimate of how changing his job’s affected his happiness. Could really be any job that encourages people to measure and quantify things. But “statistician” is a job with strong connotations of being able to quantify happiness. To have that quantity feature a decimal point, too, makes him sound more mathematical and thus, more surely correct. I’d be surprised if “two and a half times” weren’t a more justifiable estimate, given the margin for error on happiness-measurement I have to imagine would be there. (This seems to be the first time I’ve featured Bottomliners at least since I started tagging the comic strips named. Neat.)
Ruben Bolling’s Super-Fun-Pak Comix for the 12th reprinted a panel called The Uncertainty Principal that baffled commenters there. It’s a pun on “Uncertainty Principle”, the surprising quantum mechanics result that there are some kinds of measurements that can’t be taken together with perfect precision. To know precisely where something is destroys one’s ability to measure its momentum. To know the angular momentum along one axis destroys one’s ability to measure it along another. This is a physics result (note that the panel’s signed “Heisenberg”, for the name famously attached to the Uncertainty Principle). But the effect has a mathematical side. The operations that describe finding these incompatible pairs of things are noncommutative; it depends what order you do them in.
We’re familiar enough with noncommutative operations in the real world: to cut a piece of paper and then fold it usually gives something different to folding a piece of paper and then cutting it. To pour batter in a bowl and then put it in the oven has a different outcome than putting batter in the oven and then trying to pour it into the bowl. Nice ordinary familiar mathematics that people learn, like addition and multiplication, do commute. These come with partners that don’t commute, subtraction and division. But I get the sense we don’t think of subtraction and division like that. It’s plain enough that ‘a’ divided by ‘b’ and ‘b’ divided by ‘a’ are such different things that we don’t consider what’s neat about that.
In the ordinary world the Uncertainty Principle’s almost impossible to detect; I’m not sure there’s any macroscopic phenomena that show it off. I mean, that atoms don’t collapse into electrically neutral points within nanoseconds, sure, but that isn’t as compelling as, like, something with a sodium lamp and a diffraction grating and an interference pattern on the wall. The limits of describing certain pairs of properties is about how precisely both quantities can be known, together. For everyday purposes there’s enough uncertainty about, say, the principal’s weight (and thus momentum) that uncertainty in his position won’t be noticeable. There’s reasons it took so long for anyone to suspect this thing existed.
Dana Simpson’s Ozy and Millie rerun for the 14th has the title characters playing “logical fallacy tag”. Ozy is, as Millie says, making an induction argument. In a proper induction argument, you characterize something with some measure of size. Often this is literally a number. You then show that if it’s true that the thing is true for smaller problems than you’re interested in, then it has to also be true for the problem you are interested in. Add to that a proof that it’s true for some small enough problem and you’re done. In this case, Ozy’s specific fallacy is an appeal to probability: all but one of the people playing tag are not it, and therefore, any particular person playing the game isn’t it. That it’s fallacious really stands out when there’s only two people playing.
Alex Hallatt’s Arctic Circle for the 16th riffs on the mathematics abilities of birds. Pigeons, in this case. The strip starts from their abilities understanding space and time (which are amazing) and proposes pigeons have some insight into the Grand Unified Theory. Animals have got astounding mathematical abilities, should point out. Don’t underestimate them. (This also seems to be the first time I’ve tagged Arctic Circle which doesn’t seem like it could be right. But I didn’t remember naming the penguins before so maybe I haven’t? Huh. Mind, I only started tagging the comic strip titles a couple months ago.)
Tony Cochrane’s Agnes for the 17th has the title character try bluffing her way out of mathematics homework. Could there be a fundamental flaw in mathematics as we know it? Possibly. It’s hard to prove that any field complicated enough to be interesting is also self-consistent. And there’s a lot of mathematics out there. And mathematics subjects often develop with an explosion of new ideas and then a later generation that cleans them up and fills in logical gaps. Symplectic geometry is, if I’m following the news right, going into one of those cleaning-up phases now. Is it likely to be uncovered by a girl in elementary school? I’m skeptical, and also skeptical that she’d have a replacement system that would be any better. I admire Agnes’s ambition, though.
Mike Baldwin’s Cornered for the 17th plays on the reputation for quantum mechanics as a bunch of mathematically weird, counter-intuitive results. In fairness to the TV program, I’ve had series run longer than I originally planned too.
The last couple days of last week saw a rush of comics, although most of them were simpler things to describe. Bits of play on words, if you like.
Samson’s Dark Side of the Horse for the 4th of January, 2018, is one that plays on various meanings of “average”. The mean, alluded to in the first panel, is the average most people think of first. Where you have a bunch of values representing instances of something, add up the values, and divide by the number of instances. (Properly that’s the arithmetic mean. There’s some others, such as the geometric mean, but if someone’s going to use one of those they give you clear warning.) The median, in the second, is the midpoint, the number that half of all instances are less than. So you see the joke. If the distribution of intelligence is normal — which is a technical term, although it does mean “not freakish” — then the median and the mean should be equal. If you had infinitely many instances, and they were normally distributed, the two would be equal. With finitely many instances, the mean and the median won’t be exactly in line, for the same reason if you fairly toss a coin two million times it won’t turn up heads exactly one million times.
Dark Side of the Horse for the 5th delivers the Roman numerals joke of the year. And I did have to think about whether ‘D’ is a legitimate Roman numeral. This would be easier to remember before 1900.
Johnny Hart’s Back to BC for the 5th goes to the desire to quantify and count things. And to double-check what other people tell you about this counting. It’s easy, today, to think of the desire to quantify things as natural to humans. I’m not confident that it is. The history of statistics shows this gradual increase in the number and variety of things getting tracked. This strip originally ran the 11th of July, 1960.
Bill Watterson’s Calvin and Hobbes for the 5th talks about averages again. And what a population average means for individuals. It doesn’t mean much. The glory of statistics is that groups are predictable in a way that individuals are not.
Mark O’Hare’s Citizen Dog rerun for the 6th is part of a sequence in which Fergus takes a (human) child’s place in school. Mathematics gets used as a subject that’s just a big pile of unfamiliar terms if you just jump right in. Most subjects are like this if you take them seriously, of course. But mathematics has got an economy of technical terms to stuff into people’s heads, and that have to be understood to make any progress. In grad school my functional analysis professor took great mercy on us, and started each class with re-writing the definitions of all the technical terms introduced the previous class. Also of terms that might be a bit older, but that are important to get right, which is why I got through it confident I knew what a Sobolev Space was. (It’s a collection of functions that have enough derivatives to do your differential equations problem.) Numerator and denominator, we’re experts on by now.
November closed out with another of those weeks not quite busy enough to justify splitting into two. I blame Friday and Saturday. Nothing mathematically-themed was happening them. Suppose some days are just like that.
Johnny Hart’s Back To BC for the 26th is an example of using mathematical truths as profound statements. I’m not sure that I’d agree with just stating the Pythagorean Theorem as profound, though. It seems like a profound statement has to have some additional surprising, revelatory elements to it. Like, knowing the Pythagorean theorem is true means we can prove there’s exactly one line parallel to a given line and passing through some point. Who’d see that coming? I don’t blame Hart for not trying to fit all that into one panel, though. Too slow a joke. The strip originally ran the 4th of September, 1960.
Zach Weinersmith’s Saturday Morning Breakfast Cereal makes its first appearance for the week on the 26th. It’s an anthropomorphic-numerals joke and some wordplay. Interesting trivia about the whole numbers that never actually impresses people: a whole number is either a perfect square, like 1 or 4 or 9 or 16 are, or else its square root is irrational. There’s no whole number with a square root that’s, like, 7.745 or something. Maybe I just discuss it with people who’re too old. It seems like the sort of thing to reveal to a budding mathematician when she’s eight.
Saturday Morning Breakfast Cereal makes another appearance the 29th. The joke’s about using the Greek ε, which has a long heritage of use for “a small, positive number”. We use this all the time in analysis. A lot of proofs in analysis are done by using ε in a sort of trick. We want to show something is this value, but it’s too hard to do. Fine. Pick any ε, a positive number of unknown size. So then we’ll find something we can calculate, and show that the difference between the thing we want and the thing we can do is smaller than ε. And that the value of the thing we can calculate is that. Therefore, the difference between what we want and what we can do is smaller than any positive number. And so the difference between them must be zero, and voila! We’ve proved what we wanted to prove. I have always assumed that we use ε for this for the association with “error”, ideally “a tiny error”. If we need another tiny quantity we usually go to δ, probably because it’s close to ε and ‘d’ is still a letter close to ‘e’. (The next letter after ε is ζ, which carries other connotations with it and is harder to write than δ is.) Anyway, Weinersmith is just doing a ha-ha, your penis is small joke.
There seems to be no Mark Anderson’s Andertoons for this week. There’ve been some great ones (like on the 26th or the 28th and the 29th) but they’re not at all mathematical. I apologize for the inconvenience and am launching an investigation into this problem.
It was a normal-paced week at Comic Strip Master Command. It was also one of those weeks that didn’t have anything from Comics Kingdom or Creators.Com. So I’m afraid you’ll all just have to click the links for strips you want to actually see. Sorry.
Bill Amend’s FoxTrot for the 12th has Jason and Marcus creating “mathic novels”. They, being a couple of mathematically-gifted smart people, credit mathematics knowledge with smartness. A “chiliagon” is a thousand-sided regular polygon that’s mostly of philosophical interest. A regular polygon with a thousand equal sides and a thousand equal angles looks like a circle. There’s really no way to draw one so that the human eye could see the whole figure and tell it apart from a circle. But if you can understand the idea of a regular polygon it seems like you can imagine a chilagon and see how that’s not a circle. So there’s some really easy geometry things that can’t be visualized, or at least not truly visualized, and just have to be reasoned with.
Rick Detorie’s One Big Happy for the 12th is a story-problem-subversion joke. The joke’s good enough as it is, but the supposition of the problem is that the driving does cover fifty miles in an hour. This may not be the speed the car travels at the whole time of the problem. Mister Green is maybe speeding to make up for all the time spent travelling slower.
Shannon Wheeler’s Too Much Coffee Man for the 15thalso uses a blackboard full of equations to represent the deep thinking being done on a less silly subject. It’s a really good-looking blackboard full of equations, by the way. Beyond the appearance of our old friend E = mc2 there’s a lot of stuff that looks like legitimate quantum mechanics symbols there. They’re at least not obvious nonsense, as best I can tell without the ability to zoom the image in. I wonder if Wheeler didn’t find a textbook and use some problems from it for the feeling of authenticity.
Jef Mallett’s Frazz for the 18th talks about making a bet on the World Series, which wrapped up a couple weeks ago. It raises the question: can you bet on an already known outcome? Well, sure, you can bet on anything you like, given a willing partner. But there does seem to be something fundamentally different between betting on something whose outcome isn’t in principle knowable, such as the winner of the next World Series, and betting on something that could be known but happens not to be, such as the winner of the last. We see this expressed in questions like “is it true the 13th of a month is more likely to be Friday than any other day of the week?” If you know which month and year is under discussion the chance the 13th is Friday is either 1 or 0. But we mean something more like, if we don’t know what month and year it is, what’s the chance this is a month with a Friday the 13th? Something like this is at work in this World Series bet. (The Astros won the recently completed World Series.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is also featured on some underemployed philosopher’s “Reading the Comics” WordPress blog and fair enough. Utilitarianism exists in an odd triple point, somewhere on the borders of ethics, economics, and mathematics. The idea that one could quantize the good or the utility or the happiness of society, and study how actions affect it, is a strong one. It fits very well the modern mindset that holds everything can be quantified even if we don’t know how to do it well just yet. And it appeals strongly to a mathematically-minded person since it sounds like pure reason. It’s not, of course, any more than any ethical scheme can be. But it sounds like the ethics a Vulcan would come up with and that appeals to a certain kind of person. (The comic is built on one of the implications of utilitarianism that makes it seem like the idea’s gone off the rails.)
There’s some mathematics symbols on The Utilitarian’s costume. The capital U on his face is probably too obvious to need explanation. The on his chest relies on some mathematical convention. For maybe a half-millennium now mathematicians have been using the capital sigma to mean “take a sum of things”. The things are whatever the expression after that symbol is. Usually, the Sigma will have something below and above which carries meaning. It says what the index is for the thing after the symbol, and what the bounds of the index are. Here, it’s not set. This is common enough, though, if this is understood from context. Or if it’s obvious. The small ‘u’ to the right suggests the utility of whatever’s thought about. (“Utility” being the name for the thing measured and maximized; it might be happiness, it might be general well-being, it might be the number of people alive.) So the symbols would suggest “take the sum of all the relevant utilities”. Which is the calculation that would be done in this case.
It’s another week where everything I have to talk about comes from GoComics.com. So, no pictures. The Comics Kingdom and the Creators.com strips are harder for non-subscribers to read so I feel better including those pictures. There’s not an overarching theme that I can fit to this week’s strips either, so I’m going to name it for the one that was most visually interesting to me.
Charlie Pondrebarac’s CowTown for the 22nd I just knew was a rerun. It turned up the 26th of August, 2015. Back then I described it as also “every graduate students’ thesis defense anxiety dream”. Now I wonder if I have the possessive apostrophe in the right place there. On reflection, if I have “every” there, then “graduate student” has to be singular. If I dropped the “every” then I could talk about “graduate students” in the plural and be sensible. I guess that’s all for a different blog to answer.
Mike Thompson’s Grand Avenue for the 22nd threatened to get me all cranky again, as Grandmom decided the kids needed to do arithmetic worksheets over the summer. The strip earned bad attention from me a few years ago when a week, maybe more, of the strip was focused on making sure the kids drudged their way through times tables. I grant it’s a true attitude that some people figure what kids need is to do a lot of arithmetic problems so they get better at arithmetic problems. But it’s hard enough to convince someone that arithmetic problems are worth doing, and to make them chores isn’t helping.
John Zakour and Scott Roberts’s Maria’s Day for the 22nd name-drops fractions as a worse challenge than dragon-slaying. I’m including it here for the cool partial picture of the fire-breathing dragon. Also I take a skeptical view of the value of slaying the dragons anyway. Have they given enough time for sanctions to work?
Eric the Circle for the 24th, this one by Dennill, gets in here by throwing some casual talk about arcs around. That and π. The given formula looks like nonsense to me. has parts that make sense. The first part will tell you what radian measure corresponds to 94 degrees, and that’s fine. Mathematicians will tend to look for radian measures rather than degrees for serious work. The sine of 94 degrees they might want to know. Subtracting the two? I don’t see the point. I dare to say this might be a bunch of silliness.
Cathy Law’s Claw for the 25th writes off another Powerball lottery loss as being bad at math and how it’s like algebra. Seeing algebra in lottery tickets is a kind of badness at mathematics, yes. It’s probability, after all. Merely playing can be defended mathematically, though, at least for the extremely large jackpots such as the Powerball had last week. If the payout is around 750 million dollars (as it was) and the chance of winning is about one in 250 million (close enough to true), then the expectation value of playing a ticket is about three dollars. If the ticket costs less than three dollars (and it does; I forget if it’s one or two dollars, but it’s certainly not three), then, on average you could expect to come out slightly ahead. Therefore it makes sense to play.
Except that, of course, it doesn’t make sense to play. On average you’ll lose the cost of the ticket. The on-average long-run you need to expect to come out ahead is millions of tickets deep. The chance of any ticket winning is about one in 250 million. You need to play a couple hundred million times to get a good enough chance of the jackpot for it to really be worth it. Therefore it makes no sense to play.
Mathematical logic therefore fails us: we can justify both playing and not playing. We must study lottery tickets as a different thing. They are (for the purposes of this) entertainment, something for a bit of disposable income. Are they worth the dollar or two per ticket? Did you have other plans for the money that would be more enjoyable? That’s not my ruling to make.
Grouping together three decimal digits as a block is as old, in the Western tradition, as decimal digits are. Leonardo of Pisa, in Liber Abbaci, groups the thousands and millions and thousands of millions and such together. By 1228 he had the idea to note this grouping with an arc above the set of digits, like a tie between notes on a sheet of music. This got cut down, part of the struggle in notation to write as little as possible. Johannes de Sacrobosco in 1256 proposed just putting a dot every third digit. In 1636 Thomas Blundeville put a | mark after every third digit. (I take all this, as ever, from Florian Cajori’s A History Of Mathematical Notations, because it’s got like everything in it.) We eventually settled on separating these stanzas of digits with a , or . mark. But that it should be three digits goes as far back as it could.
It was a busy week at Comic Strip Master Command last week, since they wanted to be sure I was overloaded ahead of the start of the Summer 2017 A To Z project. So here’s the couple of comics I didn’t have time to review on Sunday.
Mort (“Addison”) Walker’s Boner’s Ark for the 7th of September, 1971 was rerun the 27th of July. It mentions mathematics but just as a class someone might need more work on. Could be anything, but mathematics has the connotations of something everybody struggles with, and in an American comic strip needs only four letters to write. Most economical use of word balloon space.
Neil Kohney’s The Other End for the 28th also mentions mathematics without having any real mathematics content. Barry tries to make the argument that mathematics has a timeless and universal quality that makes for good aesthetic value. I support this principle. Art has many roles. One is to make us see things which are true which are not about ourselves. This mathematics does. Whether it’s something as instantly accessible as, say, RobertLovesPi‘s illustrations of geometrical structures, or something as involved as the five-color map theorem mathematics gives us something. This isn’t any excuse to slum, though.
Rob Harrell’s Big Top rerun for the 29th features a word problem. It’s cast in terms of what a lion might find interesting. Cultural expectations are inseparable from the mathematics we do, however much we might find universal truths about them. Word problems make the cultural biases more explicit, though. Also, note that Harrell shows an important lesson for artists in the final panel: whenever possible, draw animals wearing glasses.
Samson’s Dark Side Of The Horse for the 29th is another sheep-counting joke. As Samson will often do this includes different representations of numbers before it all turns to chaos in the end. This is why some of us can’t sleep.
Comic Strip Master Command just barely missed being busy enough for me to split the week’s edition. Fine for them, I suppose, although it means I’m going to have to scramble together something for the Tuesday or the Thursday posting slot. Ah well. As befits the comics, there’s a fair bit of mathematics as an icon in the past week’s selections. So let’s discuss.
Mark Anderson’s Andertoons for the 11th is our Mark Anderson’s Andertoons for this essay. Kind of a relief to have that in right away. And while the cartoon shows a real disaster of a student at the chalkboard, there is some truth to the caption. Ruling out plausible-looking wrong answers is progress, usually. So is coming up with plausible-looking answers to work out whether they’re right or wrong. The troubling part here, I’d say, is that the kid came up with pretty poor guesses about what the answer might be. He ought to be able to guess that it’s got to be an odd number, and has to be less than 10, and really ought to be less than 7. If you spot that then you can’t make more than two wrong guesses.
Patrick J Marrin’s Francis for the 12th starts with what sounds like a logical paradox, about whether the Pope could make an infallibly true statement that he was not infallible. Really it sounds like a bit of nonsense. But the limits of what we can know about a logical system will often involve questions of this form. We ask whether something can prove whether it is provable, for example, and come up with a rigorous answer. So that’s the mathematical content which justifies my including this strip here.
Niklas Eriksson’s Carpe Diem for the 13th is a traditional use of the blackboard full of mathematics as symbolic of intelligence. Of course ‘E = mc2‘ gets in there. I’m surprised that both π and 3.14 do, too, for as little as we see on the board.
Mark Anderson’s Andertoons for the 14th is a nice bit of reassurance. Maybe the cartoonist was worried this would be a split-week edition. The kid seems to be the same one as the 11th, but the teacher looks different. Anyway there’s a lot you can tell about shapes from their perimeter alone. The one which most startles me comes up in calculus: by doing the right calculation about the lengths and directions of the edge of a shape you can tell how much area is inside the shape. There’s a lot of stuff in this field — multivariable calculus — that’s about swapping between “stuff you know about the boundary of a shape” and “stuff you know about the interior of the shape”. And finding area from tracing the boundary is one of them. It’s still glorious.
Samson’s Dark Side Of The Horse for the 14th is a counting-sheep joke and a Pi Day joke. I suspect the digits of π would be horrible for lulling one to sleep, though. They lack the just-enough-order that something needs for a semiconscious mind to drift off. Horace would probably be better off working out Collatz sequences.
Dan Barry’s Flash Gordon for the 31st of July, 1962 and rerun the 16th I’m including just because I love the old-fashioned image of a mathematician in Professor Quita here. At this point in the comic strip’s run it was set in the far-distant future year of 1972, and the action here is on one of the busy multinational giant space stations. Flash himself is just back from Venus where he’d set up some dolphins as assistants to a fish-farming operation helping to feed that world and ours. And for all that early-60s futurism look at that gorgeous old adding machine he’s still got. (Professor Quinta’s discovery is a way to peer into alternate universes, according to the next day’s strip. I’m kind of hoping this means they’re going to spend a week reading Buck Rogers.)
Can’t say this was too fast or too slow a week for mathematically-themed comic strips. A bunch of the strips were panel comics, so that’ll do for my theme.
Norm Feuti’s Retail for the 21st mentions every (not that) algebra teacher’s favorite vague introduction to group theory, the Rubik’s Cube. Well, the ways you can rotate the various sides of the cube do form a group, which is something that acts like arithmetic without necessarily being numbers. And it gets into value judgements. There exist algorithms to solve Rubik’s cubes. Is it a show of intelligence that someone can learn an algorithm and solve any cube? — But then, how is solving a Rubik’s cube, with or without the help of an algorithm, a show of intelligence? At least of any intelligence more than the bit of spatial recognition that’s good for rotating cubes around?
I don’t see that learning an algorithm for a problem is a lack of intelligence. No more than using a photo reference shows a lack of drawing skill. It’s still something you need to learn, and to apply, and to adapt to the cube as you have it to deal with. Anyway, I never learned any techniques for solving it either. Would just play for the joy of it. Here’s a page with one approach to solving the cube, if you’d like to give it a try yourself. Good luck.
Nate Fakes’s Break of Day for the 24th features the traditional whiteboard full of mathematics scrawls as a sign of intelligence. The scrawl on the whiteboard looks almost meaningful. The integral, particularly, looks like it might have been copied from a legitimate problem in polar or cylindrical coordinates. I say “almost” because while I think that some of the r symbols there are r’ I’m not positive those aren’t just stray marks. If they are r’ symbols, it’s the sort of integral that comes up when you look at surfaces of spheres. It would be the electric field of a conductive metal ball given some charge, or the gravitational field of a shell. These are tedious integrals to solve, but fortunately after you do them in a couple of introductory physics-for-majors classes you can just look up the answers instead.
Slow week around here for mathematically-themed comic strips. These happen. I suspect Comic Strip Master Command is warning me to stop doing two-a-week essays on reacting to comic strips and get back to more original content. Message received. If I can get ahead of some projects Monday and Tuesday we’ll get more going.
Patrick Roberts’s Todd the Dinosaur for the 20th is a typical example of mathematics being something one gets in over one’s head about. Of course it’s fractions. Is there anything in elementary school that’s a clearer example of something with strange-looking rules and processes for some purpose students don’t even know what they are? In middle school and high school we get algebra. In high school there’s trigonometry. In high school and college there’s calculus. In grad school there’s grad school. There’s always something.
Jeff Stahler’s Moderately Confused for the 21st is the usual bad-mathematics-of-politicians joke. It may be a little more on point considering the Future Disgraced Former President it names, but the joke is surely as old as politicians and hits all politicians with the same flimsiness.
John Graziano’s Ripley’s Believe It Or Not for the 22nd names Greek mathematician Pythagoras. That’s close enough to on-point to include here, especially considering what a slow week it’s been. It may not be fair to call Pythagoras a mathematician. My understanding is we don’t know that actually did anything in mathematics, significant or otherwise. His cult attributed any of its individuals’ discoveries to him, and may have busied themselves finding other, unrelated work to credit to their founder. But there’s so much rumor and gossip about Pythagoras that it’s probably not fair to automatically dismiss any claim about him. The beans thing I don’t know about. I would be skeptical of anyone who said they were completely sure.
Vic Lee’s Pardon My Planet for the 23rd is the usual sort of not-understanding-mathematics joke. In this case it’s about percentages, which are good for baffling people who otherwise have a fair grasp on fractions. I wonder if people would be better at percentages if they learned to say “percent” as “out of a hundred” instead. I’m sure everyone who teaches percentages teaches that meaning, but that doesn’t mean the warning communicates.
Samson’s Dark Side Of The Horse for the 25th mentions sudokus, and that’s enough for a slow week like this. I thought Horace was reaching for a calculator in the last panel myself, and was going to say that wouldn’t help any. But then I checked the numbers in the boxes and that made it all better.
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.
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.
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.
And now I can close out last week’s mathematically-themed comic strips. Two of them are even about counting, which is enough for me to make that the name of this set.
John Allen’s Nest Heads for the 2nd mentions a probability and statistics class and something it’s supposed to be good for. I would agree that probability and statistics are probably (I can’t find a better way to write this) the most practically useful mathematics one can learn. At least once you’re past arithmetic. They’re practical by birth; humans began studying them because they offer guidance in uncertain situations. And one can use many of their tools without needing more than arithmetic.
I’m not so staunchly anti-lottery as many mathematics people are. I’ll admit I play it myself, when the jackpot is large enough. When the expectation value of the prize gets to be positive, it’s harder to rationalize not playing. This happens only once or twice a year, but it’s fun to watch and see when it happens. I grant it’s a foolish way to use two dollars (two tickets are my limit), but you know? My budget is not so tight I can’t spend four dollars foolishly a year. Besides, I don’t insist on winning one of those half-billion-dollar prizes. I imagine I’d be satisfied if I brought in a mere $10,000.
Rick Detorie’s One Big Happy for the 3rd continues my previous essay’s bit of incompetence at basic mathematics, here, counting. But working out that her age is between 22 an a gazillion may be worth doing. It’s a common mathematical challenge to find a correct number starting from little information about it. Usually we find it by locating bounds: the number must be larger than this and smaller than that. And then get the bounds closer together. Stop when they’re close enough for our needs, if we’re numerical mathematicians. Stop when the bounds are equal to each other, if we’re analytic mathematicians. That can take a lot of work. Many problems in number theory amount to “improve our estimate of the lowest (or highest) number for which this is true”. We have to start somewhere.
Samson’s Dark Side of the Horse for the 3rd is a counting-sheep joke and I was amused that the counting went so awry here. On looking over the strip again for this essay, though, I realize I read it wrong. It’s the fences that are getting counted, not the sheep. Well, it’s a cute little sheep having the same problems counting that Horace has. We don’t tend to do well counting more than around seven things at a glance. We can get a bit farther if we can group things together and spot that, say, we have four groups of four fences each. That works and it’s legitimate; we’re counting and we get the right count out of it. But it does feel like we’re doing something different from how we count, say, three things at a glance.
Mick Mastroianni and Mason MastroianniDogs of C Kennel for the 3rd is about the world’s favorite piece of statistical mechanics, entropy. There’s room for quibbling about what exactly we mean by thermodynamics saying all matter is slowly breaking down. But the gist is fair enough. It’s still mysterious, though. To say that the disorder of things is always increasing forces us to think about what we mean by disorder. It’s easy to think we have an idea what we mean by it. It’s hard to make that a completely satisfying definition. In this way it’s much like randomness, which is another idea often treated as the same as disorder.
Bill Amend’s FoxTrot Classics for the 3rd reprinted the comic from the 10th of February, 2006. Mathematics teachers always want to see how you get your answers. Why? … Well, there are different categories of mistakes someone can make. One can set out trying to solve the wrong problem. One can set out trying to solve the right problem in a wrong way. One can set out solving the right problem in the right way and get lost somewhere in the process. Or one can be doing just fine and somewhere along the line change an addition to a subtraction and get what looks like the wrong answer. Each of these is a different kind of mistake. Knowing what kinds of mistakes people make is key to helping them not make these mistakes. They can get on to making more exciting mistakes.
So last week, for schedule reasons, I skipped the Christmas Eve strips and promised to get to them this week. There weren’t any Christmas Eve mathematically-themed comic strips. Figures. This week, I need to skip New Year’s Eve comic strips for similar schedule reasons. If there are any, I’ll talk about them next week.
John Graziano’s Ripley’s Believe It or Not for the 28th presents the quite believable claim that Professor Dwight Barkley created a formula to estimate how long it takes a child to ask “are we there yet?” I am skeptical the equation given means all that much. But it’s normal mathematician-type behavior to try modelling stuff. That will usually start with thinking of what one wants to represent, and what things about it could be measured, and how one expects these things might affect one another. There’s usually several plausible-sounding models and one has to select the one or ones that seem likely to be interesting. They have to be simple enough to calculate, but still interesting. They need to have consequences that aren’t obvious. And then there’s the challenge of validating the model. Does its description match the thing we’re interested in well enough to be useful? Or at least instructive?
Len Borozinski’s Speechless for the 28th name-drops Albert Einstein and the theory of relativity. Marginal mathematical content, but it’s a slow week.
John Allison’s Bad Machinery for the 29th mentions higher dimensions. More dimensions. In particular it names ‘ana’ and ‘kata’ as “the weird extra dimensions”. Ana and kata are a pair of directions coined by the mathematician Charles Howard Hinton to give us a way of talking about directions in hyperspace. They echo the up/down, left/right, in/out pairs. I don’t know that any mathematicians besides Rudy Rucker actually use these words, though, and that in his science fiction. I may not read enough four-dimensional geometry to know the working lingo. Hinton also coined the “tesseract”, which has escaped from being a mathematician’s specialist term into something normal people might recognize. Mostly because of Madeline L’Engle, I suppose, but that counts.
Samson’s Dark Side of the Horse for the 29th is Dark Side of the Horse‘s entry this essay. It’s a fun bit of play on counting, especially as a way to get to sleep.