Three of the strips I have for this installment feature kids around mathematics talk. That’s enough for a theme name.
Gary Delainey and Gerry Rasmussen’s Betty for the 23rd is a strip about luck. It’s easy to form the superstitious view that you have a finite amount of luck, or that you have good and bad lucks which offset each other. It feels like it. If you haven’t felt like it, then consider that time you got an unexpected $200, hours before your car’s alternator died.
If events are independent, though, that’s just not so. Whether you win $600 in the lottery this week has no effect on whether you win any next week. Similarly whether you’re struck by lightning should have no effect on whether you’re struck again.
Except that this assumes independence. Even defines independence. This is obvious when you consider that, having won $600, it’s easier to buy an extra twenty dollars in lottery tickets and that does increase your (tiny) chance of winning again. If you’re struck by lightning, perhaps it’s because you tend to be someplace that’s often struck by lightning. Probability is a subtler topic than everyone acknowledges, even when they remember that it is such a subtle topic.
Darrin Bell’s Candorville for the 23rd jokes about the uselessness of arithmetic in modern society. I’m a bit surprised at Lemont’s glee in not having to work out tips by hand. The character’s usually a bit of a science nerd. But liking science is different from enjoying doing arithmetic. And bad experiences learning mathematics can sour someone on the subject for life. (Which is true of every subject. Compare the number of people who come out of gym class enjoying physical fitness.)
If you need some Internet Old, read the comments at GoComics, which include people offering dire warnings about what you need in case your machine gives the wrong answer. Which is technically true, but for this application? Getting the wrong answer is not an immediately awful affair. Also a lot of cranky complaining about tipping having risen to 20% just because the United States continues its economic punishment of working peoples.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is some wordplay. Mathematicians often need to find minimums of things. Or maximums of things. Being able to do one lets you do the other, as you’d expect. If you didn’t expect, think about it a moment, and then you expect it. So min and max are often grouped together.
Paul Trap’s Thatababy for the 26th is circling around wordplay, turning some common shape names into pictures. This strip might be aimed at mathematics teachers’ doors. I’d certainly accept these as jokes that help someone learn their shapes.
There were a decent number of mathematically-themed comic strips this past week. This figures, because I’ve spent this past week doing a lot of things, and look to be busier this coming week. Nothing to do but jump into it, then.
Jason Chatfield’s Ginger Meggs for the 21st is your usual strip about the student resisting the story problem. Story problems are hard to set. Ideally, they present problems like mathematicians actually do, proposing the finding of something it would be interesting to learn. But it’s hard to find different problems like this. You might be fairly interested in how long it takes a tub filling with water to overflow, but the third problem of this kind is going to look a lot like the first two. And it’s also hard to find problems that allow for no confounding alternate interpretations, like this. Have some sympathy and let us sometimes just give you an equation to solve.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st is a pun built on two technical definitions for “induction”. The one used in mathematics, and logic, is a powerful tool for certain kinds of proof. It’s hard to teach how to set it up correctly, though. It’s a way to prove an infinitely large number of logical propositions, though. Let me call those propositions P1, P2, P3, P4, and so on. Pj for every counting number j. The first step of the proof is showing that some base proposition is true. This is usually some case that’s really easy to do. This is the fun part of a proof by induction, because it feels like you’ve done half the work and it amounts to something like, oh, showing that 1 is a triangular number.
The second part is hard. You have to show that whenever Pj is true, this implies that Pj + 1 is also true. This is usually a step full of letters representing numbers rather than anything you can directly visualize with, like, dots on paper. This is usually the hard part. But put those two halves together? And you’ve proven that all your propositions are true. Making things line up like that is so much fun.
Mark Anderson’s Andertoons for the 22nd is the Mark Anderson’s Andertoons for the week. It’s again your student trying to get out of not really knowing mathematics in class. Longtime readers will know, though, that I’m fond of rough drafts in mathematics. I think most mathematicians are. If you are doing something you don’t quite understand, then you don’t know how to do it well. It’s worth, in that case, doing an approximation of what you truly want to do. This is for the same reason writers are always advised to write something and then edit later. The rough draft will help you find what you truly want. In thinking about the rough draft, you can get closer to the good draft.
Stephen Bentley’s Herb and Jamaal for the 22nd is one lost on me. I grew up when Schoolhouse Rock was a fun and impossible-to-avoid part of watching Saturday Morning cartoons. So there’s a lot of simple mathematics that I learned by having it put to music and played often.
Still, it’s surprising Herb can’t think of why it might be easier to remember something that’s fun, that’s put to a memory-enhancing tool like music, and repeated often, than it is to remember whether 8 times 7 is 54. Arithmetic gets easier to remember when you notice patterns, and find them delightful. Even fun. It’s a lot like everything else humans put any attention to, that way.
I hoped I’d get a Reading the Comics post in for Tuesday, and even managed it. With this I’m all caught up to the syndicated comic strips which, last week, brought up some mathematics topic. I’m open for nominations about what to publish here Thursday. Write in quick.
Hilary Price’s Rhymes With Orange for the 30th is a struggling-student joke. And set in summer school, so the comic can be run the last day of June without standing out to its United States audience. It expresses a common anxiety, about that point when mathematics starts using letters. It superficially seems strange that this change worries students. Students surely had encountered problems where some term in an equation was replaced with a blank space and they were expected to find the missing term. This is the same work as using a letter. Still, there are important differences. First is that a blank line (box, circle, whatever) has connotations of “a thing to be filled in”. A letter seems to carry meaning in to the problem, even if it’s just “x marks the spot”. And a letter, as we use it in English, always stands for the same thing (or at least the same set of things). That ‘x’ may be 7 in one problem and 12 in another seems weird. I mean weird even by the standards of English orthography.
A letter might represent a number whose value we wish to know; it might represent a number whose value we don’t care about. These are different ideas. We usually fall into a convention where numbers we wish to know are more likely x, y, and z, while those we don’t care about are more likely a, b, and c. But even that’s no reliable rule. And there may be several letters in a single equation. It’s one thing to have a single unknown number to deal with. To have two? Three? I don’t blame people fearing they can’t handle that.
Mark Leiknes’s Cow and Boy for the 30th has Billy and Cow pondering the Prisoner’s Dilemma. This is one of the first examples someone encounters in game theory. Game theory sounds like the most fun part of mathematics. It’s the study of situations in which there’s multiple parties following formal rules which allow for gains or losses. This is an abstract description. It means many things fit a mathematician’s idea of a game.
The Prisoner’s Dilemma is described well enough by Billy. It’s built on two parties, each — separately and without the ability to coordinate — having to make a choice. Both would be better off, under interrogation, to keep quiet and trust that the cops can’t get anything significant on them. But both have the temptation that if they rat out the other, they’ll get off free while their former partner gets screwed. And knowing that their partner has the same temptation. So what would be best for the two of them requires them both doing the thing that maximizes their individual risk. The implication is unsettling: everyone acting in their own best interest is supposed to produce the best possible result for society. And here, for the society of these two accused, it breaks down entirely.
Exponents have been written as numbers in superscript following a base for a long while now. The notation developed over the 17th century. I don’t know why mathematicians settled on superscripts, as opposed to the many other ways a base and an exponent might fit together. It’s a good mnemonic to remember, say, “z raised to the 10th” is z with a raised 10. But I don’t know the etymology of “raised” in a mathematical context well enough. It’s plausible that we say “raised” because that’s what the notation suggests.
The proof of the Pythagorean Theorem is one of the very many known to humanity. This one is among the family of proofs that are wordless. At least nearly wordless. You can get from here to with very little prompting. If you do need prompting, it’s this: there are two expressions for how much area of the square with sides a-plus-b. One of these expressions uses only terms of a and b. The other expression uses terms of a, b, and c. If this doesn’t get a bit of a grin out of you, don’t worry. There’s, like, 2,037 other proofs we already know about. We might ask whether we need quite so many proofs of the Pythagorean theorem. It doesn’t seem to be under serious question most of the time.
And then a couple comic strips last week just mentioned mathematics. Morrie Turner’s Wee Pals for the 1st of July has the kids trying to understand their mathematics homework. Could have been anything. Mike Thompson’s Grand Avenue for the 5th started a sequence with the kids at Math Camp. The comic is trying quite hard to get me riled up. So far it’s been the kids agreeing that mathematics is the worst, and has left things at that. Hrmph.
Some months stretch my pop-mathematics writing skills, tasking me with finding new insights into the things I thought I understood and new ways to present them. Some months I’ve written about comic strips a lot. This was one of the latter. Here, let me nearly finish writing about the comic strips of June 2019 that had some mathematical content.
Jonathan Lemon’s Rabbits Against Magic for the 23rd is the Venn Diagram meta-joke for the week. Properly speaking, yes, Eight-Ball hasn’t drawn a Venn Diagram here. Representing two sets in a Venn Diagram, by the proper rules, requires two circles with one overlap. Indicating that both sets have the same elements means noting that there are no elements outside the intersection of these circles. One point of a Venn Diagram is showing all the possible logical relations between sets and maybe then marking off the ones that happen to be relevant to the problem. What Eight-Ball is drawing is an Euler Diagram, which has looser requirements. There’s no sense fighting this terminology battle, though. It makes cleaner pictures to draw a Venn Diagram modified to only show the relations that actually exist. If the goal is to communicate information, clarity counts. A joke counts as information.
Eight-Ball’s propositions are … well, a bit muddled. His first set is “people who like to think they are good at math”. His second set is “which of those people like Venn Diagrams”. This implies the second set can’t be anything but a subset of the first. So this we’d represent as one circle inside another, at least if we allow that there exists at least one person who likes to think they’re good at math, but still doesn’t like Venn Diagrams. It’s fine for the purposes of comic hyperbole to claim there is no such thing, of course, and I don’t quarrel with that.
Why not have the second group be “people who like Venn Diagrams”, without the restriction that they already think they’re good at math? Here I think there is a serious logical constraint. My suspicion is that Venn Diagrams are liked by people who don’t think they’re good at math. Also by people who aren’t good at math. Venn Diagrams are a wonderful tool because they present the relationships of sets in a way that uses our spatial intuitions. They wouldn’t make a good Internet joke format if they were liked only by people who think they’re good at math. Which is why Jonathan Lemon had to write the joke that way. It’s plausible comic hyperbole to say everyone who thinks they’re good at math likes Venn Diagrams. But there are too many people who react to explicit mathematics content with a shudder, but who like Venn Diagram jokes, to make “everyone who likes Venn Diagrams thinks they’re good at math” plausible.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 23rd is a lying-with-statistics joke. The median is an average of a data set. It’s “an” average because, in English, we mean several different things by “average”. Translated into mathematics these different things are, really, completely unrelated. The “median” is the midpoint of the ordered list of the data set. So, as the Man In Black says, half the data in the set is below that value, and half is above. This can be a better measure of “average” than the arithmetic mean is. It tells us a slight something about the distribution, about how the data is arranged. Not much, but then, it’s just one number. What do you want? It has an advantage over the arithmetic mean, which is the thing normal people intend when they say “average”. That advantage is that it’s relatively insensitive to outliers. One or two really large, or tiny, data points can throw the mean way off. The classic example we use these days is to look at the average wealth of twenty people in the room. If Bill Gates enters the room, the mean jumps way up. The median? Doesn’t alter much. (Bill Gates is the figure I see used these days, but it could be anyone impossibly wealthy. I imagine there are versions where it’s Jeff Bezos entering the room. I imagine a century ago, the proposition would be to imagine J P Morgan entering the room, except that a century ago he had been dead six years.)
Steve Skelton’s 2 Cows and a Chicken for the 26th shows off a counting chicken as a wonder. Animals do have some sense of mathematics. We know in some detail how well crows and ravens can count, and do simple arithmetic. This is partly because we know good ways to test crow and raven arithmetic skills. And we’ve come to appreciate their intelligence as deep and surprising. Chickens, to my knowledge, have gotten less study. But I would expect they’ve got skills. If nothing else, I would expect chickens to have a good understanding of the transitive property. This is the rule that if ‘a’ is greater than ‘b’, and ‘b’ is greater than ‘c’, then it follows that ‘a’ is greater than ‘c’. Chickens have a pecking order, and animals with that kind of hierarchy tend to know transitivity. I don’t know that the reasons for that link have been proven, but, c’mon. And animals doing arithmetic, like the cook says, have been good sideshow attractions or performances for a long while. They’ve also been good starts for scientific study, as people try to work out questions like how intelligence formed, and what other ways it might have formed.
Greg Cravens’s The Buckets for the 27th is a joke about the representation of numbers. Cravens has a good observation here about learning the differences between representations, and of not being able to express just what representation you want. I love Eddie’s horrified face as his mother (Sarah) tries to spell out the word. There’s probably a good exercise to be done in thinking of as many ways to represent fifteen as possible.
Etymologically, “fifteen” has exactly the origin you would say if you were dragged out of a sound sleep by someone demanding the history of the word RIGHT NOW, THERE’S NO TIME TO EXPLAIN. In Old English it was “fiftyne”, with “fif” meaning “five” and “tyne” meaning “ten more than”. This construction, pretty much five-and-ten, has fallen out of favor in English. Once we get past nineteen we more commonly write out, like, “twenty-one” and “thirty-five” and such. The alternate construction, which would be, like, one-and-twenty, or nine-and-sixty, or such, seems to have fallen out of use except as a more poetic way to express the idea. I don’t know why, say, five-and-twenty would have shifted to twenty-five while the equivalent five-and-ten didn’t shift to … teenfive(?). I would make an uninformed guess that words used more commonly tend to be more stable, and we tend to need smaller numbers more than bigger ones.
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.
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.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a joke about holographic cosmology, proving that there are such things as jokes about holographic cosmology. Cosmology is about the big picture stuff, like, why there is a universe and why it looks like that. It’s a rather mathematical field, owing to the difficulty of doing controlled experiments. Holograms are that same technology used back in the 80s to put shoddy three-dimensional-ish pictures of eagles on credit cards. (In the United States. I imagine they were other animals in other countries.) Holograms, at least when they’re well-made, encode the information needed to make a three-dimensional image in a two-dimensional surface. (Please pretend that anything made of matter is two-dimensional like that.)
Holographic cosmology is a mathematical model for the universe. It represents the things in a space with a description of information on the boundary of this space. This seems bizarre and it won’t surprise you that key inspiration was in the strange physics of black holes. Properties of everything which falls into a black hole manifest in the event horizon, the boundary between normal space and whatever’s going on inside the black hole. The black hole is this three-dimensional volume, but in some way everything there is to say about it is the two-dimensional edge.
Dr Leonard Susskind did much to give this precise mathematical form. You didn’t think the character name was just a bit of whimsy, did you? Susskind’s work showed how the information of a particle falling into a black hole — information here meaning stuff like its position and momentum — turn into oscillations in the event horizon. The holographic principle argues this can be extended to ordinary space, the whole of the regular universe. Is this so? It’s hard to say. It’s a corner of string theory. It’s difficult to run experiments that prove very much. And we are stuck with an epistemological problem. If all the things in the universe and their interactions are equally well described as a three-dimensional volume or as a two-dimensional surface, which is “real”? It may seem intuitively obvious that we experience a three-dimensional space. But that intuition is a way we organize our understanding of our experiences. That’s not the same thing as truth.
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 22nd is a joke about power, and how it can coerce someone out of truth. Arithmetic serves as an example of indisputable truth. It could be any deductive logic statement, or for that matter a definition. Arithmetic is great for the comic purpose needed here, though. Anyone can understand, at least the simpler statements, and work out their truth or falsity. And need very little word balloon space for it.
Bill Griffith’s Zippy the Pinhead for the 25th also features a quick mention of algebra as the height of rationality. Also as something difficult to understand. Most fields are hard to understand, when you truly try. But algebra works well for this writing purpose. Anyone who’d read Zippy the Pinhead has an idea of what understanding algebra would be like, the way they might not have an idea of holographic cosmology.
Teresa Logan’s Laughing Redhead Comics for the 25th is the Venn diagram joke for the week, this one with a celebrity theme. Your choice whether the logic of the joke makes sense. Ryan Reynolds and John Krasinski are among those celebrities that I keep thinking I don’t know, but that it turns out I do know. Ryan Gosling I’m still not sure about.
And now I’ll cover the handful of comic strips which ran last week and which didn’t fit in my Sunday report. And link to a couple of comics that ultimately weren’t worth discussion in their own right, mostly because they were repeats of ones I’ve already discussed. I have been trimming rerun comics out of my daily reading. But there are ones I like too much to give up, at least not right now.
Bud Blake’s Tiger for the 25th has Tiger quizzing Punkinhead on counting. The younger kid hasn’t reached the point where he can work out numbers without a specific physical representation. It would come, if he were in one of those comics where people age.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is an optimization problem, and an expectation value problem. The wisdom-seeker searches for the most satisfying life. The mathematician-guru offers an answer based in probability and expectation values. List all the possible outcomes, and how probable each are, and how much of the relevant quantity you get (or lose) with each outcome. This is a quite utilitarian view of life-planning. Finding the best possible outcome, given certain constraints, is another big field of mathematics.
John Atkinson’s Wrong Hands for the 26th is a nonsense-equation panel. It’s built on a cute idea. If you do wan to know how many bears you can fit in the kitchen you would need something like this. Not this, though. You can tell by the dimensions. ‘x’, as the area of the kitchen, has units of, well, area. Square feet, or square meters, or square centimeters, or whatever is convenient to measure its area. The average volume of a bear, meanwhile, has units of … volume. Cubic feet, or cubic meters, or cubic centimeters, or the like. The one divided by the other has units of one-over-distance.
And I don’t know what the units of desire to have bears in your kitchen are, but I’m guessing it’s not “bear-feet”, although that would be worth a giggle. The equation would parse more closely if y were the number of bears that can fit in a square foot, or something similar. I say all this just to spoil Atkinson’s fine enough bit of nonsense.
Percy Crosby’s Skippy for the 26th is a joke built on inappropriate extrapolation. 3520 seconds is a touch under an hour. Skippy’s pace, if he could keep it up, would be running a mile every five minutes, 52 seconds. That pace isn’t impossible — I find it listed on charts for marathon runners. But that would be for people who’ve trained to be marathon or other long-distance runners. They probably have different fifty-yard run times.
And now for some of the recent comics that didn’t seem worth their own discussion, and why they didn’t.
Niklas Eriksson’s Carpe Diem for the 20th features reciting the digits of π as a pointless macho stunt. There are people who make a deal of memorizing digits of π. Everyone needs hobbies, and memorizing meaningless stuff is a traditional fanboy’s way of burying oneself in the thing appreciated. Me, I can give you π to … I want to say sixteen digits. I might have gone farther in my youth, but I was heartbroken when I learned one of the digits I had memorized I got wrong, and so after correcting that mess I gave up going farther.
The first, important, thing is that I have not disappeared or done something worse. I just had one of those weeks where enough was happening that something had to give. I could either write up stuff for my mathematics blog, or I could feel guilty about not writing stuff up for my mathematics blog. Since I didn’t have time to do both, I went with feeling guilty about not writing, instead. I’m hoping this week will give me more writing time, but I am fooling only myself.
Second is that Comics Kingdom has, for all my complaining, gotten less bad in the redesign. Mostly in that the whole comics page loads at once, now, instead of needing me to click to “load more comics” every six strips. Good. The strips still appear in weird random orders, especially strips like Prince Valiant that only run on Sundays, but still. I can take seeing a vintage Boner’s Ark Sunday strip six unnecessary times. The strips are still smaller than they used to be, and they’re not using the decent, three-row format that they used to. And the archives don’t let you look at a week’s worth in one page. But it’s less bad, and isn’t that all we can ever hope for out of the Internet anymore?
And finally, Comic Strip Master Command wanted to make this an easy week for me by not having a lot to write about. It got so light I’ve maybe overcompensated. I’m not sure I have enough to write about here, but, I don’t want to completely vanish either.
Dave Whamond’s Reality Check for the 15th is … hm. Well, it’s not an anthropomorphic-numerals joke. It is some kind of wordplay, making concrete a common phrase about, and attitude toward, numbers. I could make the fussy difference between numbers and numerals here but I’m not sure anyone has the patience for that.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 17th touches around mathematics without, I admit, necessarily saying anything specific. The angel(?) welcoming the man to heaven mentions creating new systems of mathematics as some fit job for the heavenly host. The discussion of creating self-consistent physics systems seems mathematical in nature too. I’m not sure whether saying one could “attempt” to create self-consistent physics is meant to imply that our universe’s physics are not self-consistent. To create a “maximally complex reality using the simplest possible constructions” seems like a mathematical challenge as well. There are important fields of mathematics built on optimizing, trying to create the most extreme of one thing subject to some constraints or other.
I think the strip’s premise is the old, partially a joke, concept that God is a mathematician. This would explain why the angel(?) seems to rate doing mathematics or mathematics-related projects as so important. But even then … well, consider. There’s nothing about designing new systems of mathematics that ordinary mortals can’t do. Creating new physics or new realities is beyond us, certainly, but designing the rules for such seems possible. I think I understood this comic better then I had thought about it less. Maybe including it in this column has only made trouble for me.
Doug Savage’s Savage Chickens for the 17th amuses me by making a strip out of a logic paradox. It’s not quite your “this statement is a lie” paradox, but it feels close to that, to me. To have the first chicken call it “Birthday Paradox” also teases a familiar probability problem. It’s not a true paradox. It merely surprises people who haven’t encountered the problem before. This would be the question of how many people you need to have in a group before there’s a 50 percent (75 percent, 99 percent, whatever you like) chance of at least one pair sharing a birthday.
And I notice on Wikipedia a neat variation of this birthday problem. This generalization considers splitting people into two distinct groups, and how many people you need in each group to have a set chance of a pair, one person from each group, sharing a birthday. Apparently both a 32-person group of 16 women and 16 men, or a 49-person group of 43 women and six men, have a 50% chance of some woman-man pair sharing a birthday. Neat.
Mark Parisi’s Off The Mark for the 18th sports a bit of wordplay. It’s built on how multiplication and division also have meanings in biology. … If I’m not mis-reading my dictionary, “multiply” meant any increase in number first, and the arithmetic operation we now call multiplication afterwards. Division, similarly, meant to separate into parts before it meant the mathematical operation as well. So it might be fairer to say that multiplication and division are words that picked up mathematical meaning.
There were a handful of comic strips from last week which I didn’t already discuss. Two of them inspire me to write about how we know how to do things. That makes a good theme.
Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 27th gets into deep territory. How does we could count to a million? Maybe some determined soul has actually done it. But it would take the better part of a month. Things improve some if we allow that anything a computing machine can do, a person could do. This seems reasonable enough. It’s heady to imagine that all the computing done to support, say, a game of Roller Coaster Tycoon could be done by one person working alone with a sheet of paper. Anyway, a computer could show counting up to a million, a billion, a trillion, although then we start asking whether anyone’s checked that it hasn’t skipped some numbers. (Don’t laugh. The New York Times print edition includes an issue number, today at 58,258, at the top of the front page. It’s meant to list the number of published daily editions since the paper started. They mis-counted once, in 1898, and nobody noticed until 1999.)
Anyway, allow that. Nobody doubts that, if we put enough time and effort into it, we could count up to any positive whole number, or as they say in the trade, “counting number”. But … there is some largest number that we could possibly count to, even if we put every possible resource and all the time left in the universe to that counting. So how do we know we “could” count to a number bigger than that? What does it mean to say we “could” if the circumstances of the universe are such that we literally could not?
Counting up to a number seems uncontroversial enough. If I wanted to prove it I’d say something like “if we can count to the whole number with value N, then we can count to the whole number with value N + 1 by … going one higher.” And “We can count to the whole number 1”, proving that by enunciating as clearly as I can. The induction follows. Fine enough. That’s a nice little induction proof.
But … what if we needed to do more work? What if we needed to do a lot of work? There is a corner of logic which considers infinitely long proofs, or infinitely long statements. They’re not part of the usual deductive logic that any mathematician knows and relies on. We’re used to, at least in principle, being able to go through and check every step of a proof. If that becomes impossible is that still a proof? It’s not my field, so I feel comfortable not saying what’s right and what’s wrong. But it is one of those lectures in your Mathematical Logic course that leaves you hanging your jaw open.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th is a joke about algorithms. These are the processes by which we know how to do a thing. Here, Hansel and Gretel are shown using what’s termed a “greedy algorithm” to follow pebbles back home. This kind of thing reflects trying to find an acceptable solution, in this case, finding a path somewhere. What makes it “greedy” is each step. You’re at a pebble. You can see other pebbles nearby. Which one do you go to? Go to some extreme one; in this case, the nearest. It could instead have been the biggest, or the shiniest, the one at the greatest altitude, the one nearest a water source. Doesn’t matter. You choose your summum bonum and, at each step, take the move that maximizes that.
The wicked mother knows something about this sort of algorithm, one that promises merely a solution and not the best solution. And that is that all these solutions can be broken. You can set up a problem that the algorithm can’t solve. Greedy algorithms are particularly vulnerable to this. They’re called “local maximums”. You find the best answer of the ones nearby, but not the best one you possibly could locate.
Why use an algorithm like this, that can be broken so? That’s because we often want to do problems like finding a path through the woods. There are so many possible paths that it’s hard to find one of the acceptable ones. But there are processes that will, typically, find an acceptable answer. Maybe processes that will let us take an acceptable answer and improve it to a good answer. And this is getting into my field.
Actual persons encountering one of these pebble rings would (probably) notice they were caught in a loop. And what they’d do, then, is suspend the greedy rule: instead of going to the nearest pebble they could find, they’d pick something else. Maybe simply the nearest pebble they hadn’t recently visited. Maybe the second-nearest pebble. Maybe they’d give up and strike out in a random direction, trusting they’ll find some more pebbles. This can lead them out of the local maximum they don’t want toward the “global maximum”, the path home, that they do. There’s no reason they can’t get trapped again — this is why the wicked mother made many loops — and no reason they might not get caught in a loop of loops again. Every algorithm like this can get broken by some problem, after all. But sometimes taking the not-the-best steps can lead you to a better solution. That’s the insight at the heart of “Metropolis-Hastings” algorithms, which was my field before I just read comic strips all the time.
I think there are just barely enough comic strips from the past week to make three essays this time around. But one of them has to be a short group, only three comics. That’ll be for the next essay when I can group together all the strips that ran in February. One strip that I considered but decided not to write at length about was Ed Allison’s dadaist Unstrange Phenomena for the 28th. It mentions Roman Numerals and the idea of sneaking message in through them. But that’s not really mathematics. I usually enjoy the particular flavor of nonsense which Unstrange Phenomena uses; you might, too.
John McPherson’s Close to Home for the 29th uses an arithmetic problem as shorthand for an accomplished education. The problem is solvable. Of course, you say. It’s an equation with quadratic polynomial; it can hardly not be solved. Yes, fine. But McPherson could easily have thrown together numbers that implied x was complex-valued, or had radicals or some other strange condition. This is one that someone could do in their heads, at least once they practiced in mental arithmetic.
I feel reasonably confident McPherson was just having a giggle at the idea of putting knowledge tests into inappropriate venues. So I’ll save the full rant. But there is a long history of racist and eugenicist ideology that tried to prove certain peoples to be mentally incompetent. Making an arithmetic quiz prerequisite to something unrelated echoes that. I’d have asked McPherson to rework the joke to avoid that.
(I’d also want to rework the composition, since the booth, the swinging arm, and the skirted attendant with the clipboard don’t look like any tollbooth I know. But I don’t have an idea how to redo the layout so it’s more realistic. And it’s not as if that sort of realism would heighten the joke.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th riffs on the problem of squaring the circle. This is one of three classical problems of geometry. The lecturer describes it just fine: is it possible to make a square that’s got the same area as a given circle, using only straightedge and compass? There are shapes it’s easy to do this for, such as rectangles, parallelograms, triangles, and (why not?) this odd crescent-moon shaped figure called the lune. Circles defied all attempts. In the 19th century mathematicians found ways to represent the operations of classical geometry with algebra, and could use the tools of algebra to show squaring the circle was impossible. The squaring would be equivalent to finding a polynomial, with integer coefficients, that has as a root. And we know from the way algebra works that this can’t be done. So squaring the circle can’t be done.
Which feeds to the secondary joke, of making the philosophers sad. Often philosophy problems test one’s intuition about an idea by setting out a problem, often with unpleasant choices. A common problem with students that I’m going ahead and guessing are engineers is then attacking the setup of the question, trying to show that the problem couldn’t actually happen. You know, as though there were ever a time significant numbers of people were being tied to trolley tracks. (By the way, that thing about silent movie villains tying women to railroad tracks? Only happened in comedies spoofing Victorian melodramas. It’s always been a parody.) Attacking the logic of a problem may make for good movie drama. But it makes for a lousy student and a worse class discussion.
Ted Shearer’s Quincy rerun for the 30th uses a bit of mathematics and logic talk. It circles the difference between the feeling one can have about the rational meaning of a situation and how the situation feels to someone. It seems like a jump that Quincy goes from being asked about logic to talking about arithmetic. Possibly Quincy’s understanding of logic doesn’t start from the sort of very abstract concept that makes arithmetic hard to get to, though.
Last week started out at a good clip: two comics with enough of a mathematical theme I could imagine writing a paragraph about them each day. Then things puttered out. The rest of the week had almost nothing. At least nothing that seemed significant enough. I’ll list those, since that’s become my habit, at the end of the essay.
Perhaps it’s so. Some societies have been found to have, what seem to us, rather few numerals. This doesn’t reflect on anyone’s abilities or intelligence or the like. And it doesn’t mean people who lack a word for, say, “forty-nine” would be unable to compute. It might take longer, but probably just from inexperience. If someone practiced much calculation on “forty-nine” they’d probably have a name for it. And folks raised in the western mathematics use, even enjoy, some vagueness about big numbers too. We might say there are “dozens” of a thing even if there are not precisely 24, 36, or 48 of the thing; “52” is close enough and we probably didn’t even count it up. “Hundred” similarly has gotten the connotation of being a precise number, but it’s used to mean “really quite a lot of a thing”. The words “thousands”, “millions”, and mock-numbers like “zillions” have a similar role. They suggest different ranges of what might be “many”.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a SABRmetrics joke! At least, it’s an optimization joke, built on the idea that you can find an optimum strategy for anything, whether winning baseball games or The War. The principle is hard to argue with. Nobody would doubt that different approaches to a battle affect how likely winning is. We can imagine gathering data on how different tactics affect the outcome. (We can easily imagine combat simulators running these experiments, particularly.)
The catch — well, one catch — is that this tempts one to reward a process. Once it’s taken for granted the process works, then whether it’s actually doing what you want gets forgotten. And once everyone knows what’s being measured it becomes possible to game the system. Famously, in the mid-1960s the United States tried to judge its progress in the Vietname War by counting the number of enemy soldiers killed. There was then little reason to care about who was killed, or why. And reason to not care whether actual enemy soldiers were being killed. There’s good to be said about testing whether the things you try to do work. There’s great danger in thinking that the thing you can measure guarantees success.
Mark Anderson’s Andertoons for the 21st is a bit of fun with definitions. Mathematicians rely on definitions. It’s hard to imagine a proof about something undefined. But definitions are hard to compose. We usually construct a definition because we want a common term to describe a collection of things, and to exclude another collection of things. And we need people like Wavehead who can find edge cases, things that seem to satisfy a definition while breaking its spirit. This can let us find unstated assumptions that we should pay attention to. Or force us to accept that the definition is so generally useful that we’ll tolerate it having some counter-intuitive implications.
My favorite counter-intuitive implication is in analysis. The field has a definition for what it means that a function is continuous. It’s meant to capture the idea that you could draw a curve representing the function without having to lift the pen that does it. The best definition mathematicians have settled on allows you to count a function that’s continuous at a single point in all of space. Continuity seems like something that should need an interval to happen. But we haven’t found a better way to define “continuous” that excludes this pathological case. So we embrace the weirdness in exchange for general usefulness.
Charles Brubaker’s Ask A Cat for the 21st is a guest appearance from Brubaker’s other strip, The Fuzzy Princess. It’s a rerun and I did discuss it earlier. Soap bubbles make for great mathematics. They’re easy to play with, for one thing. That’s good for capturing imagination. And the mathematics behind them is deep, and led to important results analytically and computationally. It happens when this strip first ran I’d encountered a triplet of essays about the mathematics of soap bubbles and wireframe surfaces. My introduction to those essays is here.
Benita Epstein’s Six Chix for the 25th I wasn’t sure I’d include. But Roy Kassinger asked about it, so that tipped the scales. The dog tries to blame his bad behavior on “the algorithm”, bringing up one of the better monsters of the last couple years. An algorithm is just the procedure by which you do something. Mathematically, that’s usually to solve a problem. That might be finding some interesting part of the domain or range of a function. That might be putting a collection of things in order. that might be any of a host of things. And then we go make a decision based on the results of the algorithm.
What earns The Algorithm its deserved bad name is mindlessness. The idea that once you have an algorithm that a problem is solved. Worse, that once an algorithm is in place it would be irrational to challenge it. I have seen the process termed “mathwashing”, by analogy with whitewashing, and it’s a good one. The notion that because something is done by computer it must be done correctly is absurd. We knew it was absurd before there were computers as we knew them, as see anyone for the past century who has spoken of a “Kafkaesque” interaction with a large organization. It’s impossible to foresee all the outcomes of any reasonably complicated process, much less to verify that all the outcomes are handled correctly. This is before we consider that there will always be mistakes made in the handling of data. Or in the carrying out of the process. And that’s before we consider bad actors. I’m sure there must be research into algorithms designed to handle gaming of the system. I don’t know that there are any good results yet, though. We certainly need them.
Today’s quartet of mathematically-themed comic strips doesn’t have an overwhelming theme. There’s some bits about the mathematics that young people do, so, that’s enough to separate this from any other given day’s comics essay.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is built on a bit of mathematical folklore. As Weinersmith’s mathematician (I don’t remember that we’ve been given her name) mentions, there is a belief that “revolutionary” mathematics is done by young people. That isn’t to say that older mathematicians don’t do great work. But the stereotype is that an older mathematician will produce masterpieces in already-established fields. It’s the young that establish new fields. Indeed, one of mathematics’s most prestigious awards, the Fields Medal, is only awarded to mathematicians under the age of forty. I was cheated of mine. Long story.
There’s intuitive appeal in the idea that revolutions in thinking are for the young. We think that people get set in their ways as they develop their careers. We have a couple dramatic examples, most notably Évariste Galois, who developed what we now see as foundations of group theory and died at twenty. While the idea is commonly held, I don’t know that it’s actually true. That is, that it holds up to scrutiny. It seems hard to create a definition for “revolutionary mathematics” that could be agreed upon by two people. So it would be difficult to test at what age people do their most breathtaking work, and whether it is what they do when young or when experienced.
Is there harm to believing an unprovable thing? If it makes you give up on trying, yes. My suspicion is that true revolutionary work happens when a well-informed, deep thinker comes to a field that hasn’t been studied in that way before. And when it turns out to be a field well-suited to study that way. That doesn’t require youth. It requires skill in one field, and an understanding that there’s another field ready to be studied that way.
Will Henry’s Wallace the Brave for the 14th is a mathematics anxiety joke. Wallace tries to help by turning an abstract problem into a concrete one. This is often a good way to approach a problem. Even in more advanced mathematics, one can often learn the way to solve a general problem by trying a couple of specific examples. It’s almost as though there’s only a certain amount of abstraction people can deal with, and you need to re-cast problems so they stay within your limits.
Yes, the comments turn to complaining about Common Core. I’m not sure what would help Spud work through this problem (or problems in general). But thinking of alternate problems that estimated or approached what he really wanted might help. If he noticed, for example, that 10 + 12 has to be a little more than 10 + 10, and he found 10 + 10 easy, then he’d be close to a right answer. If he noticed that 10 + 12 had to be 10 + 10 + 2, and he found 10 + 10 easy, then he might find 20 + 2 easy as well. Maybe Spud would be better off thinking of ways to rewrite a problem without changing the result.
Wiley Miller’s Non Sequitur for the 15th mentions calculus. It’s more of a probability joke. To speak of a calculated risk is to speak of doing something that’s not certain, but that has enough of a payoff to be worth the cost of failure. But one problem with this attitude is that people are very, very bad at estimating probabilities. We have terrible ideas of how likely losses are and how uncertain rewards can be. But even if we allow that the risks and rewards are calculated right, there’s a problem with things you only do once. Or only can do once. You can get into a good debate about whether there’s even a meaningful idea of probability for things that happen only the one time. Life’s among them.
Bob Weber Sr’s Moose and Molly for the 16th is a homework joke. It does actually depend on being mathematics homework, though, or there’d be no grounds for Moose’s kid to go to the savings and loan clerk who’ll help with “money problems”.
Today, I get to wrap up November’s suggested discussion topics as prepared by Comic Strip Master Command.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th mentions along its way the Liar Paradox and Zeno’s Paradoxes. Both are ancient problems. The paradoxes arise from thinking with care and rigor about things we seem to understand intuitively. For the Liar Paradox it’s about what we mean to declare a statement true or false. For Zeno’s Paradoxes it’s about whether we think space (and time) are continuous or discrete. And, as the strip demonstrates, there is a particular kind of nerd that declares the obvious answer is the only possible answer and that it’s foolish to think deeper. To answer a question’s literal words while avoiding its point is a grand old comic tradition, of course, predating even the antijoke about chickens crossing roads. Which is what gives these answers the air of an old stage comedian.
Mark Tatulli’s Lio for the 28th features a cameo for mathematics. At least mathematics class. It’s painted as the most tedious part of the school day. I’m not sure this is quite right for Lio as a character. He’s clever in a way that I think harmonizes well with how mathematics brings out universal truths. But there is a difference between mathematics and mathematics class, of course.
Tom Toles’s Randolph Itch, 2am for the 28th shows how well my resolution to drop the strip from my rotation here has gone. I don’t seem to have found it worthy of mention before, though. It plays on the difference between a note of money, the number of units of currency that note represents, and between “zero” and “nothing”. Also I’m enchanted now by the idea that maybe some government might publish a zero-dollar bill. At least for the sake of movie and television productions that need realistic-looking cash.
In the footer joke Randolph mentions how you can never have enough zeroes. Yes, but I’d say that’s true of twenties, too. There is a neat sense in which this is true for working mathematicians, though. At least for those doing analysis. One of the reliable tricks that we learn to do in analysis is to “add zero” to a quantity. This is, literally, going from some expression that might be, say, “a – b” to “a + 0 – b”, which of course has the same value. The point of doing that is that we know other things equal to zero. For example, for any number L, “-L + L” is zero. So we get the original expression from “a + 0 – b” over to “a – L + L – b”. And that becomes useful is you picked L so that you know something about “a – L” and about “L – b”. Because then it tells you something about “a – b” that you didn’t know before. Picking that L, and showing something true about “a – L” and “L – b”, is the tricky part.
The Dumpties in the comic strip are presented as getting nauseated at the strange curling around. It’s good sense for the comic-in-the-comic, which just has to have something happen and doesn’t really need to make sense. But there is no real way to answer where a Möbius strip wraps around itself. I mean, we can declare it’s at the left and right ends of the strip as we hold it, sure. But this is an ad hoc placement. We can roll the belt along a little bit, not changing its shape, but changing the points where we think of the strip as turning over.
But suppose you were a flat creature, wandering a Möbius strip. Would you have any way to tell that you weren’t on the plane? You could, but it takes some subtle work. Like, you could try drawing shapes. These let you count a thing called the Euler Characteristic, which relates the numer of vertices, edges, and faces of a polyhedron. The Euler Characteristic for a Möbius strip is the same as that for a Klein bottle, a cylinder, or a torus. You could try drawing regions, and coloring them in, calling on the four-color map theorem. (Here I want just to mention the five-color map theorem, which is as these things go easy to prove.) A map on the plane needs at most four colors to have no neighboring territories share a color along an edge. (Territories here are contiguous, and we don’t count territories meeting at only a point as sharing an edge.) Same for a sphere, which is good for we folks who have the job of coloring in both globes and atlases. It’s also the same for a cylinder. On a Möbius strip, this number is six. On a torus, it’s seven. So we could tell, if we were on a Möbius strip, that we were. It can be subtle to prove, is all.
I’m not sure there is a theme to the back half of last week’s mathematically-based comic strips. If there is, it’s about showing some origins of things. I’ll go with that title, then.
Bill Holbrook’s On The Fastrack for the 21st is another in the curious thread of strips about Fi talking about mathematics. She’s presented as doing a good job inspiring kids to appreciate mathematics as a fun, exciting, interesting thing to think about. It’s good work. And I hope this does not sound like I am envious of a more successful, if fictional, mathematics popularizer. But I don’t see much in the strip of her doing this side job well. That is, of making the case that mathematics is worth the time spent on it. That’s a lot to ask given the confines of a syndicated daily newspaper comic strip, yes. What we can expect is some hint of what the actual good argument would look like. But this particular day’s strip rings false to me, for example. I don’t see how “here’s some pizza — but first, here’s a pop quiz” makes mathematics look as something other than a chore.
Pizza area offers many ways into mathematical ideas. How the area depends on the size of the pizza, for example. How the area depends on the shape, even independently of the size. How to slice a pizza fairly, especially if it’s not to be between four or six or eight people. What is the strangest shape you could make that would give people equal areas? Just the way slices intersect at angles inspires neat little geometry problems. How you might arrange toppings opens up symmetries and tilings, which are surprisingly big areas of mathematics. Setting problems on a pizza gives them a tangibility that could help capture young minds, surely. But I can’t make myself believe that this is a conversation to have when the pizza is entering the room.
Mike Peters’s Mother Goose and Grimm for the 22nd is a lottery joke. So if we suppose this was written about the last time the Powerball jackpot reached a half-billion dollars we can work out how far ahead of publication Mike Peters is working. One solid argument against ever buying a lottery ticket is, as Grimm notes, that you have zero chance of winning. (I’m open to an argument based on expectation value. And even more, I don’t object to people spending a reasonable bit of disposable income “foolishly”.) Mother Goose argues that her chances are vastly worse if she doesn’t buy a ticket. This is true. Are her chances “astronomically” worse? … That depends. A one in three hundred million chance (to use, roughly, the Powerball odds) is so small that it won’t happen to you. Is that any different than a zero in three hundred million chance [*]? Or than a six in three hundred million chance? In any case it won’t happen to you.
[*] Do you actually have zero chance of winning if you don’t have a ticket? I say no, you don’t. Someone might give you a winning ticket. Maybe you find one as a bookmark in a library book. Maybe you find it on the street and figure, what the heck, I’ll check. Unlikely? Sure. But impossible? Hardly.
Johnny Hart’s Back to BC for the 22nd has the form of the world’s oldest story problem. It could also be a joke about the discovery of the concept of zero and the struggle to understand it as a number. Given that clams are used as currency in the BC setting it also shows how finance has driven mathematical development. So the strip actually packs a fair bit of stuff into two panels. … And I’ll admit I’m not quite sure the joke parses, but if you read it quickly it looks like a good enough joke.
Johnny Hart’s Back to BC for the 24th is a more obvious joke. And it’s built on the learning abilities of animals, and the number sense of animals. A large animal stomping a foot evokes, to me at least, Clever Hans. This is a horse presented in the early 20th century as being able to actually do arithmetic. The horse would be given a question and would stop his hoof enough times to get to the right answer. However good the horse’s number sense might be, he had quite good behavioral sense. It turned out — after brilliant and pioneering work in animal cognition — that Hans was observing his trainer’s body language. When Wilhelm von Osten was satisfied that there’d been the right number of stomps, the horse stopped. This is sometimes presented as Hans `merely’ taking subconscious cues from his trainer. But consider how carefully the horse must be observing an animal with a very different body, and how it must have understood cues of satisfaction. I can’t call that `mere’. And the work of tracking down a signal that von Osten himself did not know he was sending (and, apparently, never accepted that he did) is also amazing. It serves as a reminder how hard biologists and zoologists have to work.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th gives a bit of Dad History about perspective. And, particularly, why artists didn’t seem to use it much before the 16th century. It gets more blatantly tied to mathematics by pointing out how it took ten thousand years of civilization to get Cartesian coordinates. We can argue about how many years civilization has been around. But it does seem strange that we went along for certainly the majority of that time without Cartesian coordinates. They seem so obvious it’s almost hard to not think of them. Many good ideas have such a legacy.
It’s easy to say why older pictures didn’t use perspective, though. For the most part, artists didn’t think perspective gave them something they wanted to show. Ancient peoples knew of perspective. It’s not as if ancient peoples were any dumber than we are, or any less able to look at square tiles held at different angles and at different distances. But we can convey information about the importance of things, or the flow of action of things, using position and relative size. That can be more important than showing that yes, an artist is aware that a square building far away looks small.
I’m less sure what I know about the history of coordinate systems, though, and particularly why it took until René Descartes to describe them. We have a legend of Descartes laying in bed, watching a fly on the tiled ceiling, and realizing he could describe where the fly was by what row and column of tile it was on. (In the past I have written this as though it happened. In writing this essay I went looking for a primary source and found nobody seems to have one. I shall try not to pass it on again without being very clear that it is just a legend.) But there have been tiled floors and walls and ceilings for a very long time. There have been flies even longer. Why didn’t anyone notice this?
One answer may be that they did. We just haven’t heard about it, because it was found by someone who didn’t catch the interest of a mathematical community. There’s likely a lot of such lost mathematics out there. But still, why not? Wouldn’t anyone with a mathematical inclination see that this is plainly a great discovery? And maybe not. What made Cartesian coordinates great was the realization that arithmetic and geometry, previously seen as separate liberal arts, were duals. A problem in one had an expression as a problem in the other. If you don’t make that connection, then Cartesian coordinates don’t solve any problems you have. They’re just a new way to index things you didn’t need indexed. So that would slow down using them any.
After that busy start last Sunday, Comic Strip Master Command left only a few things for the rest of the week. Here’s everything that seemed worthy of some comment to me:
Alex Hallatt’s Arctic Circle for the 12th is an arithmetic cameo. It’s used as the sort of thing that can be tested, with the straightforward joke about animal testing to follow. It’s not a surprise that machines should be able to do arithmetic. We’ve built machines for centuries to do arithmetic. Literally; Wilhelm Gottfried Leibniz designed and built a calculating machine able to add, subtract, multiply, and divide. This accomplishment from one of the founders of integral calculus is a potent reminder of how much we can accomplish if we’re supposed to be writing instead. (That link is to Robert Benchley’s classic essay “How To Get Things Done”. It is well worth reading, both because it is funny and because it’s actually good, useful advice.)
But it’s also true that animals do know arithmetic. At least a bit. Not — so far as we know — to the point they ponder square roots and such. But certainly to count, to understand addition and subtraction roughly, to have some instinct for calculations. Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics is a fascinating book about this. I’m only wary about going deeper into the topic since I don’t know a second (and, better, third) pop book touching on how animals understand mathematics. I feel more comfortable with anything if I’ve encountered it from several different authors. Anyway it does imply the possibility of testing a polar bear’s abilities at arithmetic, only in the real world.
Berkeley Breathed’s Bloom County rerun for the 13th has another mathematics cameo. Geometry’s a subject worthy of stoking Binkley’s anxieties, though. It has a lot of definitions that have to be carefully observed. And while geometry reflects the understanding we have of things from moving around in space, it demands a precision that we don’t really have an instinct for. It’s a lot to worry about.
Terry Border’s Bent Objects for the 15th is our Venn Diagram joke for the week. I like this better than I think the joke deserves, probably because it is done in real materials. (Which is the Bent Objects schtick; it’s always photographs of objects arranged to make the joke.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th is a joke on knowing how far to travel but not what direction. Normal human conversations carry contextually reasonable suppositions. Told something is two miles away, it’s probably along the major road you’re on, or immediately nearby. I’d still ask for clarification told something was “two miles away”. Two blocks, I’d let slide, on the grounds that it’s no big deal to correct a mistake.
Still, mathematicians carry defaults with them too. They might be open to a weird, general case, certainly. But we have expectations. There’s usually some obvious preferred coordinate system, or directions. If it’s important that we be ready for alternatives we highlight that. We specify the coordinate system we want. Perhaps we specify we’re taking that choice “without loss of generality”, that is, without supposing some other choice would be wrong.
I noticed the mathematician’s customized plate too. “EIPI1” is surely a reference to the expression . That sum, it turns out, equals zero. It reflects this curious connection between exponentiation, complex-valued numbers, and the trigonometric functions. It’s a weird thing to know is true, and it’s highly regarded in certain nerd circles for that weirdness.
Through the end of December my Fall 2018 Mathematics A To Z continues. I’m still open for topics to discuss from the last half-dozen letters of the alphabet. Even if someone’s already given a word for some letter, suggest something anyway. You might inspire me in good ways.
There was something in common in two of the last five comic strips worth attention from last week. That’s good enough to give the essay its name.
Greg Cravens’s The Buckets for the 8th showcases Toby discovering the point of letters in algebra. It’s easy to laugh at him being ignorant. But the use of letters this way is something it’s easy to miss. You need first to realize that we don’t need to have a single way to represent a number. Which is implicit in learning, say, that you can write ‘7’ as the Roman numeral ‘VII’ or so, but I’m not sure that’s always clear. And realizing that you could use any symbol to write out ‘7’ if you agree that’s what the symbol means? That’s an abstraction tossed onto people who often aren’t really up for that kind of abstraction. And that we can have a symbol for “a number whose identity we don’t yet know”? Or even “a number whose identity we don’t care about”? Don’t blame someone for rearing back in confusion at this.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th talks about vectors and scalars. And about the little ways that instructors in one subject can sabotage one another. In grad school I was witness to the mathematics department feeling quite put-upon by the engineering departments, who thought we were giving their students inadequate calculus training. Meanwhile we couldn’t figure out what they were telling students about calculus except that it was screwing up their understanding.
To a physicist, a vector is a size and a direction together. (At least until they get seriously into mathematical physics when they need a more abstract idea.) A scalar is a number. Like, a real-valued number such as ‘4’. Maybe a complex-valued number such as ‘4 + 6i’. Vectors are great because a lot of physics problems become easier when thought of in terms of directions and amounts in that direction.
A mathematician would start out with vectors and scalars like that. But then she’d move into a more abstract idea. A vector, to a mathematician, is a thing you can add to another vector and get a vector out. A scalar is something that’s not a vector but that, multiplied by a vector, gets you a vector out. This sounds circular. But by defining ‘vector’ and ‘scalar’ in how they interact with each other we get a really sweet flexibility. We can use the same reasoning — and the same proofs — for lots of things. Directions, yes. But also matrices, and continuous functions, and probabilities of events, and more. That’s a bit much to give the engineering student who’s trying to work out some problem about … I don’t know. Whatever they do over in that department. Truss bridges or electrical circuits or something.
Mark Leiknes’s Cow and Boy for the 9th is really about misheard song lyrics, a subject that will never die now that we don’t have the space to print lyrics in the album lining anymore, or album linings. But it has a joke resonant with that of The Buckets, in supposing that algebra is just some bunch of letters mixed up with numbers. And Cow and Boy was always a strip I loved, as baffling as it might be to a casual reader. It had a staggering number of running jokes, although not in this installment.
Greg Evans’s Luann Againn for the 9th shows Brad happy to work out arithmetic when it’s for something he’d like to know. The figure Luan gives is ridiculously high, though. If he needs 500 hairs, and one new hair grows in each week, then that’s a little under ten years’ worth of growth. Nine years and a bit over seven months to be exact. If a moustache hair needs to be a half-inch long, and it grows at 1/8th of an inch per month, then it takes four months to be sufficiently long. So in the slowest possible state it’d be nine years, eleven months. I can chalk Luann’s answer up to being snidely pessimistic about his hair growth. But his calculator seems to agree and that suggests something went wrong along the way.
John Zakour and Scott Roberts’s Maria’s Day for the 9th is a story problem joke. It looks to me like a reasonable story problem, too: the distance travelled and the speed are reasonable, and give sensible numbers. The two stops add a bit of complication that doesn’t seem out of line. And the kid’s confusion is fair enough. It takes some experience to realize that the problem splits into an easy part, a hard part, and an easy part. The first easy part is how long the stops take all together. That’s 25 minutes. The hard part is realizing that if you want to know the total travel time it doesn’t matter when the stops are. You can find the total travel time by adding together the time spent stopped with the time spent driving. And the other easy part is working out how long it takes to go 80 miles if you travel at 55 miles per hour. That’s just a division. So find that and add to it the 25 minutes spent at the two stops.
Jason Chatfield’s Ginger Meggs for the 5th is a joke about resisting the story problem. I’m surprised by the particulars of this question. Turning an arithmetic problem into counts of some number of particular things is common enough and has a respectable history. But slices of broccoli quiche? I’m distracted by the choice, and I like quiche. It’s a weird thing for a kid to have, and a weird amount for anybody to have.
JC Duffy’s Lug Nuts for the 5th uses mathematics as a shorthand for intelligence. And it particularly uses π as shorthand for mathematics. There’s a lot of compressed concepts put into this. I shouldn’t be surprised if it’s rerun come mid-March.
Tom Toles’s Randolph Itch, 2 am for the 5th I’ve highlighted before. It’s the pie chart joke. It will never stop amusing me, but I suppose I should take Randolph Itch, 2 am out of my rotation of comics I read to include here.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th is a logic puzzle joke. And a set theory joke. Dad is trying to argue he can’t be surprised by his gift because it’ll belong to one of two sets of things. And he receives nothing. This ought to defy his expectations, if we think of “nothing” as being “the empty set”. The empty set is an indispensable part of set theory. It’s a set that has no elements, has nothing in it. Then suppose we talk about what it means for one set to be contained in another. Take what seems like an uncontroversial definition: set A is contained in set B if there’s nothing in A which is not also in B. Then the empty set is contained inside every set. So Dad, having supposed that he can’t be surprised, since he’d receive either something that is “socks” or something that is “not-socks”, does get surprised. He gets the one thing that is both “socks” and “not-socks” simultaneously.
I hate to pull this move a third time in one week (see here and here), but the logic of the joke doesn’t work for me. I’ll go along with “nothing” as being “the empty set” for these purposes. And I’ll accept that “nothing” is definitely “not-socks”. But to say that “nothing” is also “socks” is … weird, unless you are putting it in the language of set theory. I think the joke would be saved if it were more clearly established that Dad should be expecting some definite thing, so that no-thing would defy all expectations.
“Nothing” is a difficult subject to treat logically. I have been exposed a bit to the thinking of professional philosophers on the subject. Not enough that I feel I could say something non-stupid about the subject. But enough to say that yeah, they’re right, we have a really hard time describing “nothing”. The null set is better behaved. I suppose that’s because logicians have been able to tame it and give it some clearly defined properties.
While putting together the last comics from a week ago I realized there was a repeat among them. And a pretty recent repeat too. I’m supposing this is a one-off, but who can be sure? We’ll get there. I figure to cover last week’s mathematically-themed comics in posts on Wednesday and Thursday, subject to circumstances.
As fits the joke, the bit of calculus in this textbook paragraph is wrong. does not equal . This is even ignoring that we should expect, with an indefinite integral like this, a constant of integration. An indefinite integral like this is equal to a family of related functions. But it’s common shorthand to write out one representative function. But the indefinite integral of is not . You can confirm that by differentiating . The result is nothing like . Differentiating an indefinite integral should get the original function back. Here are the rules you need to do that for yourself.
As I make it out, a correct indefinite integral would be:
Plus that “constant of integration” the value of which we can’t tell just from the function we want to indefinitely-integrate. I admit I haven’t double-checked that I’m right in my work here. I trust someone will tell me if I’m not. I’m going to feel proud enough if I can get the LaTeX there to display.
Stephen Beals’s Adult Children for the 27th has run already. It turned up in late March of this year. Michael Spivak’s Calculus is a good choice for representative textbook. Calculus holds its terrors, too. Even someone who’s gotten through trigonometry can find the subject full of weird, apparently arbitrary rules. And formulas like those in the above paragraph.
Rob Harrell’s Big Top for the 27th is a strip about the difficulties of splitting a restaurant bill. And they’ve not even got to calculating the tip. (Maybe it’s just a strip about trying to push the group to splitting the bill a way that lets you off cheap. I haven’t had to face a group bill like this in several years. My skills with it are rusty.)