Reading the Comics, August 23, 2019: Basics of Logic Edition

While there were a good number of comic strips to mention mathematics this past week, there were only a few that seemed substantial to me. This works well enough. This probably is going to be the last time I keep the Reading the Comics post until after Sunday, at least until the Fall 2019 A To Z is finished.

And I’m still open to topics for the first third of the alphabet. If you’d like to see my try to understand a thing of your choice please nominate one or more concepts over at this page. You might be the one to name a topic I can’t possibly summarize!

Gordon Bess’s Redeye rerun for the 18th is a joke building on animals’ number sense. And, yeah, about dumb parents too. Horses doing arithmetic have a noteworthy history. But more in the field of understanding how animals learn, than in how they do arithmetic. In particular in how animals learn to respond to human cues, and how slight a cue has to be to be recognized and acted on. I imagine this reflects horses being unwieldy experimental animals. Birds — pigeons and ravens, particularly — make better test animals.

Gordon Bess’s Redeye rerun for the 18th of August, 2019. It originally ran the 1st of April, 1973. Essays with mention of Redeye are at this link. This seems to be the first time in over a year the strip has included an actual image and not just a casual “oh, this also mentioned mathematics” line.

Art Sansom and Chip Sansom’s The Born Loser for the 18th gives a mental arithmetic problem. It’s a trick question, yes. But Brutus gives up too soon on what the problem is supposed to be. Now there’s no calculating, in your head, exactly how many seconds are in a year; that’s just too much work. But an estimate? That’s easy.

At least it’s easy if you remember one thing: a million seconds is about eleven and a half days. I find this easy to remember because it’s one of the ideas used all the time to express how big a million, a billion, and a trillion are. A million seconds are about eleven and a half days. A billion seconds are a little under 32 years. A trillion seconds are about 32,000 years, which is about how long it’s been since the oldest known domesticated dog skulls were fossilized. I’m sure that gives everyone a clear idea of how big a trillion is. The important thing, though, is that a million seconds is about eleven and a half days.

Art Sansom and Chip Sansom’s The Born Loser for the 18th of August, 2019. Appearances by The Born Loser should be at this link.

So. Think of the year. There are — as the punch line to Hattie’s riddle puts it — twelve 2nd’s in the year. So there are something like a million seconds spent each year on days that are the 2nd of the month. There about a million seconds spent each year on days that are the 1st of the month, too. There are about a million seconds spent each year on days that are the 3rd of the month. And so on. So, there’s something like 31 million seconds in the year.

You protest. There aren’t a million seconds in twelve days; there’s a million seconds in eleven and a half days. True. Also there aren’t 31 days in every month; there’s 31 days in seven months of the year. There’s 30 days in four months, and 28 or 29 in the remainder. That’s fine. This is mental arithmetic. I’m undercounting the number of seconds by supposing that a million seconds makes twelve days. I’m overcounting the number of seconds by supposing that there are twelve months of 31 days each. I’m willing to bet this undercount and this overcount roughly balance out. How close do I get?

There are 31,536,000 seconds in a common year. That is, a non-leap-year. So “31 million” is a bit low. But it’s not bad for working without a calculator.

Ryan North’s Dinosaur Comics for the 19th of August, 2019. This and other essays with Dinosaur Comics under discussion should be at this link.

Ryan North’s Dinosaur Comics for the 19th lays on us the Eubulides Paradox. It’s traced back to the fourth century BCE. Eubulides was a Greek philosopher, student of “Not That” Euclid of Megara. We know Eubulides for a set of paradoxes, including the Sorites paradox. As T-Rex’s friends point out, we’ve all heard this paradox. We’ve all gone on with our lives, knowing that the person who said it wanted us to say they were very clever. Fine.

But if we take this seriously we find … this keeps not being simple. We can avoid the problem by declaring self-referential statements exist outside of truth or falsity. This forces us to declare the sentence “this sentence is true” can’t be true. This seems goofy. We can avoid the problem by supposing there are things that are neither true nor false. That solves our problem here at the mere cost of ruining our ability to prove stuff by contradiction. There’s a lot of stuff we prove by contradiction. It’s hard to give that all up for this (Although, so far as I’m aware, anything that can be proved by contradiction can also be proven by a direct line of reasoning. The direct line may just be tedious.) We can solve this problem by saying that our words are fuzzy imprecise things. This is true enough, as see any time my love and I debate how many things are in “a couple of things”. But declaring that we just can’t express the problem well enough to answer it seems like running away from the question. We can resolve things by accepting there are limits to what can be proved by logic. Gödel’s Incompleteness Theorem shows that any interesting enough logic system has statements that are true but unprovable. A version of this paradox helps us get to this interesting conclusion.

So this is one of those things it should be easy to laugh off, but why it should be easy is hard.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st of August, 2019. In case I ever mention Saturday Morning Breakfast Cereal in an essay you’ll see it at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st is about the other great logic problem of the 20th century. The Halting Problem here refers to Turing Machines. This is the algorithmic model for computing devices. It’s rather abstract, so the model won’t help you with your C++ homework, but nothing will. But it turns out we can represent a computer running a program as a string of cells. Each cell holds one of a couple possible values. The program is a series of steps. Each step starts at one cell. The program resets the value of that cell to something dictated by the algorithm. Then, the program moves focus to another cell, again as the algorithm dictates. Do enough of this and you get SimCity 2000. I don’t know all the steps in-between.

So. The Halting Program is this: take a program. Run it. What happens in the long run? Well, it does something or other, yes. But there’s three kinds of things it can do. It can run for a while and then finish, that is, ‘halt’. It can run for a while and then get into a repeating loop, after which it repeats things forever. It can run forever without repeating itself. (Yeah, I see the structural resemblance to terminating decimals, repeating decimals, and irrational numbers too, but I don’t know of any link there.) The Halting Problem asks, if all we know is the algorithm, can we know what happens? Can we say for sure the program will always end, regardless of what the data it works on are? Can we say for sure the program won’t end if we feed it the right data to start?

If the program is simple enough — and it has to be extremely simple — we can say. But, basically, if the program is complicated enough to be even the least bit interesting, it’s impossible to say. Even just running the program isn’t enough: how do you know the difference between a program that takes a trillion seconds to finish and one that never finishes?

For human needs, yes, a program that needs a trillion seconds might as well be one that never finishes. Which is not precisely the joke Weinersmith makes here, but is circling around similar territory.

Mark Anderson’s Andertoons for the 23rd of August, 2019. Tune in to all the times Wavehead says something to a teacher with the Andertoons-based essays at this link.

Mark Anderson’s Andertoons for the 23rd is the Mark Anderson’s Andertoons for the week. And it teases my planned post for Thursday, available soon at this link. Thanks for reading.

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Reading the Comics, August 10, 2019: In Security Edition

There were several more comic strips last week worth my attention. One of them, though, offered a lot for me to write about, packed into one panel featuring what comic strip fans call the Wall O’ Text.

Bea R’s In Security for the 9th is part of a storyline about defeating an evil “home assistant”. The choice of weapon is Michaela’s barrage of questions, too fast and too varied to answer. There are some mathematical questions tossed in the mix. The obvious one is “zero divided by two equals zero, but why’z two divided by zero called crazy town?” Like with most “why” mathematics questions there are a range of answers.

Bea R’s In Security for the 9th of August, 2019. This is a new comic strip for these parts. So this essay and any future ones which explore topics raised by In Security are to be be at this link.

The obvious one, I suppose, is to appeal to intuition. Think of dividing one number by another by representing the numbers with things. Start with a pile of the first number of things. Try putting them into the second number of bins. How many times can you do this? And then you can pretty well see that you can fill two bins with zero things zero times. But you can fill zero bins with two things — well, what is filling zero bins supposed to mean? And that warns us that dividing by zero is at least suspicious.

That’s probably enough to convince a three-year-old, and probably most sensible people. If we start getting open-mined about what it means to fill no containers, we might say, well, why not have two things fill the zero containers zero times over, or once over, or whatever convenient answer would work? And here we can appeal to mathematical logic. Start with some ideas that seem straightforward. Like, that division is the inverse of multiplication. That addition and multiplication work like you’d guess from the way integers work. That distribution works. Then you can quickly enough show that if you allow division by zero, this implies that every number equals every other number. Since it would be inconvenient for, say, “six” to also equal “minus 113,847,506 and three-quarters” we say division by zero is the problem.

This is compelling until you ask what’s so great about addition and multiplication as we know them. And here’s a potentially fruitful line of attack. Coming up with alternate ideas for what it means to add or to multiply are fine. We can do this easily with modular arithmetic, that thing where we say, like, 5 + 1 equals 0 all over again, and 5 + 2 is 1 and 5 + 3 is 2. This can create a ring, and it can offer us wild ideas like “3 times 2 equals 0”. This doesn’t get us to where dividing by zero means anything. But it hints that maybe there’s some exotic frontier of mathematics in which dividing by zero is good, or useful. I don’t know of one. But I know very little about topics like non-standard analysis (where mathematicians hypothesize non-negative numbers that are not zero, but are also smaller than any positive number) or structures like surreal numbers. There may be something lurking behind a Quanta Magazine essay I haven’t read even though they tweet about it four times a week. (My twitter account is, for some reason, not loading this week.)

Michaela’s questions include a couple other mathematically-connected topics. “If infinity is forever, isn’t that crazy, too?” Crazy is a loaded word and probably best avoided. But there are infinity large sets of things. There are processes that take infinitely many steps to complete. Please be kind to me in my declaration “are”. I spent five hundred words on “two divided by zero”. I can’t get into that it means for a mathematical thing to “exist”. I don’t know. In any event. Infinities are hard and we rely on them. They defy our intuition. Mathematicians over the 19th and 20th centuries worked out fairly good tools for handling these. They rely on several strategies. Most of these amount to: we can prove that the difference between “infinitely many steps” and “very many steps” can be made smaller than any error tolerance we like. And we can say what “very many steps” implies for a thing. Therefore we can say that “infinitely many steps” gives us some specific result. A similar process holds for “infinitely many things” instead of “infinitely many steps”. This does not involve actually dealing with infinity, not directly. It involves dealing with large numbers, which work like small numbers but longer. This has worked quite well. There’s surely some field of mathematics about to break down that happy condition.

And there’s one more mathematical bit. Why is a ball round? This comes around to definitions. Suppose a ball is all the points within a particular radius of a center. What shape that is depends on what you mean by “distance”. The common definition of distance, the “Euclidean norm”, we get from our physical intuition. It implies this shape should be round. But there are other measures of distance, useful for other roles. They can imply “balls” that we’d say were octahedrons, or cubes, or rounded versions of these shapes. We can pick our distance to fit what we want to do, and shapes follow.

I suspect but do not know that it works the other way, that if we want a “ball” to be round, it implies we’re using a distance that’s the Euclidean measure. I defer to people better at normed spaces than I am.

Mark Anderson’s Andertoons for the 10th of August, 2019. The handful of times that I’ve mentioned explore Andertoons around here can be found at this link.

Mark Anderson’s Andertoons for the 10th is the Mark Anderson’s Andertoons for the week. It’s also a refreshing break from talking so much about In Security. Wavehead is doing the traditional kid-protesting-the-chalkboard-problem. This time with an electronic chalkboard, an innovation that I’ve heard about but never used myself.

Bob Scott’s Bear With Me for the 10th of August, 2019. Appearances by Bear With Me should be at this link. This strip originally ran the 15th of October, 2015, when the comic was titled Molly and the Bear.

Bob Scott’s Bear With Me for the 10th is the Pi Day joke for the week.

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And that last one seemed substantial enough to highlight. There were even slighter strips. Among them: Mark Anderson’s Andertoons for the 4th features latitude and longitude, the parts of spherical geometry most of us understand. At least feel we understand. Jim Toomey’s Sherman’s Lagoon for the 8th mentions mathematics as the homework parents most dread helping with. Larry Wright’s Motley rerun for the 10th does a joke about a kid being bad at geography and at mathematics.

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And that’s this past week’s mathematics comics. Reading the Comics essays should all be gathered at this link. Thanks for reading this far.

Reading the Comics, August 3, 2019: Summer Trip Edition

I was away from home most of last week. Comic Strip Master Command was kind and acknowledged this. There wasn’t much for me to discuss. There’s not even many comics too slight to discuss. I thank them for their work in not overloading me. But if you wondered why Sunday’s post was what it was, you now know. I suspect you didn’t wonder.

Mark Anderson’s Andertoons for the 29th of July is a comfortable and familiar face for these Reading the Comics posts. I’m glad to see it. The joke is built on negative numbers, and Wavehead’s right to say this is kind of the reason people hate mathematics. At least, that mathematicians will become comfortable with something that has a clear real-world intuitive meaning, such as that adding things together gets you a bigger thing. And then for good reasons of logic get to counter-intuitive things, such as adding things together to get a lesser thing. Negative numbers might be the first of these intuition-breaking things that people encounter. That or fractions. I encounter stories of people who refuse to accept that, say, is smaller than , although I’ve never seen it myself.

Mark Anderson’s Andertoons for the 29th of July, 2019. Essays with some mention of Andertoons are common enough, and are at this link.

So why do mathematicians take stuff like “adding” and break it? Convenience, I suppose, is the important reason. Having negative numbers lets us treat “having a quantity” and “lacking a quantity” using the same mechanisms. So that’s nice to have. If we have positive and negative numbers, then we can treat “adding” and “subtracting” using the same mechanisms. That’s nice to do. The trouble is then knowing, like, “if -3 times 4 is greater than -16, is -3 times -4 greater than 16? Or less than? Why?”

Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 31st of July, 2019. Fewer essays mention Yaffle, but those that do are at this link.

Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 31st of July uses the blackboard-full-of-mathematics as shorthand for deep thought about topics. The equations don’t mean much of anything, individually or collectively. I’m curious whether Caulfield and Ponshock mean, in the middle there, for that equation to be π times y² equalling z³, or whether it’s π times x times y² that is. Doens’t matter either way. It’s just decoration.

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And then there are the most marginal comic strips for the week. And if that first Yaffle didn’t count as too marginal to mention, think what that means for the others. Yaffle on the 28th of July features a mention of sudoku as the sort of thing one struggles to solve. Tony Rubino and Gary Markstein’s Daddy’s Home for the 1st of August mentions mathematics as the sort of homework a parent can’t help with. Jim Toomey’s Sherman’s Lagoon for the 2nd sets up a math contest. It’s mentioned as the sort of things the comic strip’s regular cast can’t hope to do.

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And there we go. I’m ready now for August. Around Sunday I should have a fresh Reading the Comics page here. And it does seem like I’m missing my other traditional post here, doesn’t it? Have to work on that.

Reading the Comics, July 22, 2019: Mathematics Education Edition

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.

Jason Chatfield’s Ginger Meggs for the 21st of July, 2019. Essays which mention Ginger Meggs are at this link.

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. P_j 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.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st of July, 2019. It’s not quite every Reading the Comics post with some mention of this comic. Those which do explore Saturday Morning Breakfast Cereal are at this link.

The second part is hard. You have to show that whenever P_j is true, this implies that P_{j + 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 of July, 2019. It’s not quite every Reading the Comics post with some mention of this comic. Those which do explore Andertoons are at this link.

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 rerun for the 22nd of July, 2019. It originally ran sometime in 2014, based on the copyright notice. Essays mentioning Herb and Jamaal in some way are at this link. Also, what’s the cheaper but more fun snark: observing the genericness of “that new rap song” or the slightly out-of-date nature of a kid listening to the radio?

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.

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This was a busy week for comic strips. I hope to have another Reading the Comics post around Tuesday, and at this link. There might even be another one this week. Please check back in.

Reading the Comics, June 20, 2019: Old Friends Edition

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.

Samson’s Dark Side of the Horse for the 17th of June, 2019. Some of the many essays inspired by Dark Side of the Horse are at this link.

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.

Mark Anderson’s Andertoons for the 18th sees Wavehead react to the terminology of the “improper fraction”. “Proper” and “improper” as words carry a suggestion of … well, decency. Like there’s something faintly immoral about having an improper fraction. “Proper” and “improper”, as words, attach to many mathematical concepts. Several years ago I wrote that “proper” amounted to “it isn’t boring”. This is a fair way to characterize, like, proper subsets or proper factors or the like. It’s less obvious that is a boring fraction.

Mark Anderson’s Andertoons for the 18th of June, 2019. Essays with some mention of a topic from Andertoons are at this link.

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.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of June, 2019. Anyone who’s been reading these for a couple weeks knows, but, Saturday Morning Breakfast Cereal features in essays at this link. Hey, every essay is somebody’s first.

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.

Bill Amend’s FoxTrot Classics for the 20th of June, 2019. It originally ran the 3rd of July, 1997. Essays based on FoxTrot, either the current-run Sundays, newspaper-rerun 2000s strips, or 90s-run Classics, are at this link.

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.

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And that’s about half of last week’s comics. A fresh Reading the Comics post should be at this link later this week. Thank you for reading along.

Reading the Comics, June 15, 2019: School Is Out? Edition

This has not been the slowest week for mathematically-themed comic strips. The slowest would be the week nothing on topic came up. But this was close. I admit this is fine as I have things disrupting my normal schedule this week. I don’t need to write too many essays too.

On-topic enough to discuss, though, were:

Lalo Alcaraz’s La Cucaracha for the 9th features a teacher trying to get ahead of student boredom. The idea that mathematics is easier to learn if it’s about problems that seem interesting is a durable one. It agrees with my intuition. I’m less sure that just doing arithmetic while surfing is that helpful. My feeling is that a problem being interesting is separate from a problem naming an intersting thing. But making every problem uniquely interesting is probably too much to expect from a teacher. A good pop-mathematics writer can be interesting about any problem. But the pop-mathematics writer has a lot of choice about what she’ll discuss. And doesn’t need to practice examples of a problem until she can feel confident her readers have learned a skill. I don’t know that there is a good answer to this.

Lalo Alcaraz’s La Cucaracha for the 9th of June, 2019. I had thought I’d mentioned this comic at least a couple times in the past, and seem to be wrong. So this is a new tag and that’s always nice to have. Any future essays which mention something inspired by La Cucaracha should be at this link.

Also part of me feels that “eight sick waves times eight sick waves” has to be “sixty-four sick-waves-squared”. This is me worrying about the dimensional analysis of a joke. All right, but if it were “eight inches times eight inches” and you came back with “sixty-four inches” you’d agree something was off, right? But it’s easy to not notice the units. That we do, mechanically, the same thing in multiplying (oh) three times $1.20 or three times 120 miles or three boxes times 120 items per box as we do multiplying three times 120 encourages this. But if we are using numbers to measure things, and if we are doing calculations about things, then the units matter. They carry information about the kinds of things our calculations represent. It’s a bad idea to misuse or ignore those tools.

Paul Trap’s Thatababy for the 14th is roughly the anthropomorphized geometry cartoon of the week. It does name the three ways to group triangles based on how many sides have the same length. Or if you prefer, how many interior angles have the same measure. So it’s probably a good choice for your geometry tip sheet. “Scalene” as a word seems to have entered English in the 1730s. Its origin traces to Late Latin “scalenus”, from the Greek “skalenos” and meaning “uneven” or “crooked”.

Paul Trap’s Thatababy for the 14th of June, 2019. Now, this strip I thought I featured more around here. It doesn’t seem to have gotten an appearance in over a year, though. Still, other appearances by Thatababy should be in essays at this link.

“Isosceles” also goes to Late Latin and, before that, the Greek “isoskeles”, with “iso” the prefix meaning “equal” and “skeles” meaning “legs”. The curious thing to me is “Isosceles”, besides sounding more pleasant, came to English around 1550. Meanwhile, “equilateral” — a simple Late Latin for “equal sides” — appeared around 1570. I don’t know what was going on that it seemed urgent to have a word for triangles with two equal sides first, and a generation later triangles with three equal sides. And then triangles with no two equal sides went nearly two centuries without getting a custom term.

But, then, I’m aware of my bias. There might have been other words for these concepts, recognized by mathematicians of the year 1600, that haven’t come to us. Or it might be that scalene triangles were thought to be so boring there wasn’t any point giving them a special name. It would take deeper mathematics history knowledge than I have to say.

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Those are all the mathematically-themed comic strips I can find something to discuss from the past week. There were some others with mentions of mathematics, though. These include:

Tony Rubino and Gary Markstein’s Daddy’s Home for the 9th, in which mathematics is the last class of the school year. Francesco Marciuliano and Jim Keefe’s Sally Forth for the 11th has a study session with “math charades” mentioned. Mark Andersons Andertoons for the 11th wants in on some of my sweet Thatababy exposition. Harley Schwadron’s 9 to 5 for the 14th is trying to become the default pie chart joke around here. It won’t beat out Randolph Itch, 2 am without a stronger punch line. And Mark Tatulli’s Heart of the City for the 15th sees Dean mention hiding sleeping in algebra class.

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This closes out a week’s worth of comic strips. My next Reading the Comics post should be at this link next Sunday. And now I need to think of something to post for the Thursday and, if I can, Tuesday publication dates.

Reading the Comics, May 20, 2019: I Guess I Took A Week Off Edition

I’d meant to get back into discussing continuous functions this week, and then didn’t have the time. I hope nobody was too worried.

Bill Amend’s FoxTrot for the 19th is set up as geometry or trigonometry homework. There are a couple of angles that we use all the time, and they do correspond to some common unit fractions of a circle: a quarter, a sixth, an eighth, a twelfth. These map nicely to common cuts of circular pies, at least. Well, it’s a bit of a freak move to cut a pie into twelve pieces, but it’s not totally out there. If someone cuts a pie into 24 pieces, flee.

Bill Amend’s FoxTrot for the 19th of May, 2019. Essays featuring FoxTrot, either the current (Sunday-only) strips or the 1990s-vintage reruns, should be at this link.

Tom Batiuk’s vintage Funky Winkerbean for the 19th of May is a real vintage piece, showing off the days when pocket electronic calculators were new. The sales clerk describes the calculator as having “a floating decimal”. And here I must admit: I’m poorly read on early-70s consumer electronics. So I can’t say that this wasn’t a thing. But I suspect that Batiuk either misunderstood “floating-point decimal”, which would be a selling point, or shortened the phrase in order to make the dialogue less needlessly long. Which is fine, and his right as an author. The technical detail does its work, for the setup, by existing. It does not have to be an actual sales brochure. Reducing “floating point decimal” to “floating decimal” is a useful artistic shorthand. It’s the dialogue equivalent to the implausibly few, but easy to understand, buttons on the calculator in the title panel.

Tom Batiuk’s vintage Funky Winkerbean for the 19th of May, 2019. The strip originally ran the 17th of June, 1973. Comics Kingdom is printing both the current Funky Winkerbean strips and early-70s reprints. Essays that mention Funky Winkerbean, old or new, should appear at this link.

Floating point is one of the ways to represent numbers electronically. The storage scheme is much like scientific notation. That is, rather than think of 2,038, think of 2.038 times 10³. In the computer’s memory are stored the 2.038 and the 3, with the “times ten to the” part implicit in the storage scheme. The advantage of this is the range of numbers one can use now. There are different ways to implement this scheme; a common one will let one represent numbers as tiny as 10^-308 or as large as 10³⁰⁸, which is enough for most people’s needs.

The disadvantage is that floating point numbers aren’t perfect. They have only around (commonly) sixteen digits of significance. That is, the first sixteen or so nonzero numbers in the number you represent mean anything; everything after that is garbage. Most of the time, that trailing garbage doesn’t hurt. But most is not always. Trying to add, for example, a tiny number, like 10^-20, to a huge number, like 10²⁰ won’t get the right answer. And there are numbers that can’t be represented correctly anyway, including such exotic and novel numbers as . A lot of numerical mathematics is about finding ways to compute that avoid these problems.

Back when I was a grad student I did have one casual friend who proclaimed that no real mathematician ever worked with floating point numbers, because of the limitations they impose. I could not get him to accept that no, in fact, mathematicians are fine with these limitations. Every scheme for representing numbers on a computer has limitations, and floating point numbers work quite well. At some point, you have to suspect some people would rather fight for a mistaken idea they already have than accept something new.

Mac King and Bill King’s Magic in a Minute for the 19th of May, 2019. So far as I know all these panels are new ones, although they do reuse gimmicks now and then. But the arithmetic and logic tricks featured in Magic In A Minute get discussed at this link, when they get mention from me at all.

Mac King and Bill King’s Magic in a Minute for the 19th does a bit of stage magic supported by arithmetic: forecasting the sum of three numbers. The trick is that all eight possible choices someone would make have the same sum. There’s a nice bit of group theory hidden in the “Howdydoit?” panel, about how to do the trick a second time. Rotating the square of numbers makes what looks, casually, like a different square. It’s hard for human to memorize a string of digits that don’t have any obvious meaning, and the longer the string the worse people are at it. If you’ve had a person — as directed — black out the rows or columns they didn’t pick, then it’s harder to notice the reused pattern.

The different directions that you could write the digits down in represent symmetries of the square. That is, geometric operations that would replace a square with something that looks like the original. This includes rotations, by 90 or 180 or 270 degrees clockwise. Mac King and Bill King don’t mention it, but reflections would also work: if the top row were 4, 9, 2, for example, and the middle 3, 5, 7, and the bottom 8, 1, 6. Combining rotations and reflections also works.

If you do the trick a second time, your mark might notice it’s odd that the sum came up 15 again. Do it a third time, even with a different rotation or reflection, and they’ll know something’s up. There are things you could do to disguise that further. Just double each number in the square, for example: a square of 4/18/8, 14/10/6, 12/2/16 will have each row or column or diagonal add up to 30. But this loses the beauty of doing this with the digits 1 through 9, and your mark might grow suspicious anyway. The same happens if, say, you add one to each number in the square, and forecast a sum of 18. Even mathematical magic tricks are best not repeated too often, not unless you have good stage patter.

Mark Anderson’s Andertoons for the 20th of May, 2019. Always glad to discuss Andertoons, as you can see from these essays.

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead’s marveling at what seems at first like an asymmetry, about squares all being rhombuses yet rhombuses not all being squares. There are similar results with squares and rectangles. Still, it makes me notice something. Nobody would write a strip where the kid marvelled that all squares were polygons but not all polygons were squares. It seems that the rhombus connotes something different. This might just be familiarity. Polygons are … well, if not a common term, at least something anyone might feel familiar. Rhombus is a more technical term. It maybe never quite gets familiar, not in the ways polygons do. And the defining feature of a rhombus — all four sides the same length — seems like the same thing that makes a square a square.

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There should be another Reading the Comics post this coming week, and it should appear at this link. I’d like to publish it Tuesday but, really, Wednesday is more probable.

Reading the Comics, May 8, 2019: Strips With Art I Like Edition

Of course I like all the comics. … Well, that’s not literally true; but I have at least some affection for nearly all of the syndicated comics. This essay I bring up some strips, partly, because I just like them. This is my content hole. If you want a blog not filled with comic strips, go start your own and don’t put these things on it.

Mark Anderson’s Andertoons for the 5th is the Mark Anderson’s Andertoons for the week. Also a bit of a comment on the ability of collective action to change things. Wavehead is … well, he’s just wrong about making the number four plus the number four equal to the number seven. Not based on the numbers we mean by the words “four” and “seven”, and based on the operation we mean by “plus” and the relationship we mean by “equals”. The meaning of those things is set by, ultimately, axioms and deductive reasoning and the laws of deductive reasoning and there’s no changing the results.

Mark Anderson’s Andertoons for the 5th of May, 2019. Essays mentioning Andertoons are at this link and also at nearly every Reading the Comics post, it feels like.

But. The thing we’re referring to when we say “seven”? Or when we write the symbol “7”? That is convention. That is a thing we’ve agreed on as a reference for this concept. And that we can change, if we decide we want to. We’ve done this. Look at a thousand-year-old manuscript and the symbol that looks like ‘4’ may represent the number we call five. And the names of numbers are just common words. They’re subject to change the way every other common word is. Which is, admittedly, not very subject. It would be almost as much bother to change the word ‘four’ as it would be to change the word ‘mom’. But that’s not impossible. Just difficult.

Juba’s Viivi and Wagner for the 5th of May, 2019. I don’t often have chances to talk about Viivi and Wagner but when I do, it’s here.

Juba’s Viivi and Wagner for the 5th is a bit of a percentage joke. The characters also come to conclude that a thing either happens or it does not; there’s no indefinite states. This principle, the “excluded middle”, is often relied upon for deductive logic, and fairly so. It gets less clear that this can be depended on for predictions of the future, or fears for the future. And real-world things come in degrees that a mathematical concept might not. Like, your fear of the home catching fire comes true if the building burns down. But it’s also come true if a quickly-extinguished frying pan fire leaves the wall scorched, embarrassing but harmless. Anyway, relaxing someone else’s anxiety takes more than a quick declaration of statistics. Show sympathy.

Harry Bliss and Steve Martin’s Bliss for the 6th of May, 2019. Yes, by the way, it’s the Steve Martin you know and love from Looney Tunes: Back In Action and from the 1996 Sergeant Bilko movie. Anyway I haven’t had chance to write about this strip before but this and future appearances of Bliss should be here.

Harry Bliss and Steve Martin’s Bliss for the 6th is a cute little classroom strip, with arithmetic appearing as the sort of topic that students feel overwhelmed and baffled by. It could be anything, but mathematics uses the illustration space efficiently. The strip may properly be too marginal to include, but I like Bliss’s art style and want more people to see it.

Will Henry’s Wallace the Brave for the 7th of May, 2019. This is one of the comic strips I’m most excited about, the last several years. Wallace the Brave appears in essays at this link.

Will Henry’s Wallace the Brave for the 7th puts up what Spud calls a sadistic math problem. And, well, it is a story problem happening in their real life. You could probably turn this into an actual exam problem without great difficulty.

Rick Detorie’s One Big Happy for the 8th of May, 2019. There are two strings of One Big Happy available for daily reading. Appearances by the current or the several-years-old GoComics prints of One Big Happy should be at this link.

Rick Detorie’s One Big Happy for the 8th is a bit of wordplay built around geometry, as Ruthie plays teacher. She’s a bit dramatic, but she always has been.

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I’ll read some more comics for later in this week. That essay, and all similar comic strip talk, should appear at this link. Thank you.

Reading the Comics, April 24, 2019: Mic Drop Edition Edition

I can’t tell you how hard it is not to just end this review of last week’s mathematically-themed comic strips after the first panel here. It really feels like the rest is anticlimax. But here goes.

John Deering’s Strange Brew for the 20th is one of those strips that’s not on your mathematics professor’s office door only because she hasn’t seen it yet. The intended joke is obvious, mixing the tropes of the Old West with modern research-laboratory talk. “Theoretical reckoning” is a nice bit of word juxtaposition. “Institoot” is a bit classist in its rendering, but I suppose it’s meant as eye-dialect.

John Deering’s Strange Brew for the 20th of April, 2019. Other appearances by Strange Brew, including ones less diligent about making the blackboard stuff sensible, are at this link.

What gets it a place on office doors is the whiteboard, though. They’re writing out mathematics which looks legitimate enough to me. It doesn’t look like mathematics though. What’s being written is something any mathematician would recognize. It’s typesetting instructions. Mathematics requires all sorts of strange symbols and exotic formatting. In the old days, we’d handle this by giving the typesetters hazard pay. Or, if you were a poor grad student and couldn’t afford that, deal with workarounds. Maybe just leave space in your paper and draw symbols in later. If your university library has old enough papers you can see them. Maybe do your best to approximate mathematical symbols using ASCII art. So you get expressions that look something like this:

  / 2 pi  
 |   2
 |  x cos(theta) dx - 2 F(theta) == R(theta)
 |
/ 0

This gets old real fast. Mercifully, Donald Knuth, decades ago, worked out a great solution. It uses formatting instructions that can all be rendered in standard, ASCII-available text. And then by dark incantations and summoning of Linotype demons, re-renders that as formatted text. It handles all your basic book formatting needs — much the way HTML, used for web pages, will — and does mathematics much more easily. For example, I would enter a line like:

\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)

And this would be rendered in print as:

There are many, many expansions available to this, to handle specialized needs, hardcore mathematics among them.

Anyway, the point that makes me realize John Deering was aiming at everybody with an advanced degree in mathematics ever with this joke, using a string of typesetting instead of the usual equations here?

The typesetting language is named TeX.

Mark Anderson’s Andertoons for the 21st of April, 2019. When do I ever not discuss this comic? All the essays at this link are about Andertoons, at least in part.

Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for the week. It’s about one of those questions that nags at you as a kid, and again periodically as an adult. The perimeter is the boundary around a shape. The circumference is the boundary around a circle. Why do we have two words for this? And why do we sound all right talking about either the circumference or the perimeter of a circle, while we sound weird talking about the circumference of a rhombus? We sound weird talking about the perimeter of a rhombus too, but that’s the rhombus’s fault.

The easy part is why there’s two words. Perimeter is a word of Greek origin; circumference, of Latin. Perimeter entered the English language in the early 15th century; circumference in the 14th. Why we have both I don’t know; my suspicion is either two groups of people translating different geometry textbooks, or some eager young Scholastic with a nickname like ‘Doctor Magnifico Triangulorum’ thought Latin sounded better. Perimeter stuck with circules early; every etymology I see about why we use the symbol π describes it as shorthand for the perimeter of the circle. Why `circumference’ ended up the word for circles or, maybe, ellipses and ovals and such is probably the arbitrariness of language. I suspect that opening “circ” sound cues people to think of it for circles and circle-like shapes, in a way that perimeter doesn’t. But that is my speculation and should not be mistaken for information.

Steve McGarry’s KidTown for the 21st of April, 2019. It’s rare that this panel is on-topic enough for me to bring up, but at least a few KidTown panels are discussed here.

Steve McGarry’s KidTown for the 21st is a kids’s information panel with a bit of talk about representing numbers. And, in discussing things like how long it takes to count to a million or a billion, or how long it would take to type them out, it gets into how big these numbers can be. Les Stewart typed out the English names of numbers, in words, by the way. He’d also broken the Australian record for treading water, and for continuous swimming.

Gary Delainey and Gerry Rasmussen’s Betty for the 24th of April, 2019. I don’t seem to have discussed this comic before. This and future appearances by Betty should be at this link.

Gary Delainey and Gerry Rasmussen’s Betty for the 24th is a sudoku comic. Betty makes the common, and understandable, conflation of arithmetic with mathematics. But she’s right in identifying sudoku as a logical rather than an arithmetic problem. You can — and sometimes will see — sudoku type puzzles rendered with icons like stars and circles rather than numerals. That you can make that substitution should clear up whether there’s arithmetic involved. Commenters at GoComics meanwhile show a conflation of mathematics with logic. Certainly every mathematician uses logic, and some of them study logic. But is logic mathematics? I’m not sure it is, and our friends in the philosophy department are certain it isn’t. But then, if something that a recognizable set of mathematicians study as part of their mathematics work isn’t mathematics, then we have a bit of a logic problem, it seems.

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Come Sunday I should have a fresh Reading the Comics essay available at this link.

Reading the Comics, April 10, 2019: Grand Avenue and Luann Want My Attention Edition

So this past week has been a curious blend for the mathematically-themed comics. There were many comics mentioning some mathematical topic. But that’s because Grand Advenue and Luann Againn — reprints of early 90s Luann comics — have been doing a lot of schoolwork. There’s a certain repetitiveness to saying, “and here we get a silly answer to a story problem” four times over. But we’ll see what I do with the work.

Mark Anderson’s Andertoons for the 7th is Mark Anderson’s Andertoons for the week. Very comforting to see. It’s a geometry-vocabulary joke, with Wavehead noticing the similar ends of some terms. I’m disappointed that I can’t offer much etymological insight. “Vertex”, for example, derives from the Latin for “highest point”, and traces back to the Proto-Indo-European root “wer-”, meaning “to turn, to bend”. “Apex” derives from the Latin for “summit” or “extreme”. And that traces back to the Proto-Indo-European “ap”, meaning “to take, to reach”. Which is all fine, but doesn’t offer much about how both words ended up ending in “ex”. This is where my failure to master Latin by reading a teach-yourself book on the bus during my morning commute for three months back in 2002 comes back to haunt me. There’s probably something that might have helped me in there.

Mark Anderson’s Andertoons for the 7th of March, 2019. I write about this strip a lot. Essays mentioning Andertoons are at this link.

Mac King and Bill King’s Magic in a Minute for the 7th is an activity puzzle this time. It’s also a legitimate problem of graph theory. Not a complicated one, but still, one. Graph theory is about sets of points, called vertices, and connections between points, called edges. It gives interesting results for anything that’s networked. That shows up in computers, in roadways, in blood vessels, in the spreads of disease, in maps, in shapes.

Mac King and Bill King’s Magic in a Minute for the 7th of March, 2019. I should have the essays mentioning Magic In A Minute at this link.

One common problem, found early in studying graph theory, is about whether a graph is planar. That is, can you draw the whole graph, all its vertices and edges, without any lines cross each other? This graph, with six vertices and three edges, is planar. There are graphs that are not. If the challenge were to connect each number to a 1, a 2, and a 3, then it would be nonplanar. That’s a famous non-planar graph, given the obvious name K_{3, 3}. A fun part of learning graph theory — at least fun for me — is looking through pictures of graphs. The goal is finding K_{3, 3} or another one called K₅, inside a big messy graph.

Mike Thompson’s Grand Avenue for the 8th of March, 2019. I’m not always cranky about this comic strips. Examples of when I’m not are at this link, as are the times I’m annoyed with Grand Avenue.

Mike Thompson’s Grand Avenue for the 8th has had a week of story problems featuring both of the kid characters. Here’s the start of them. Making an addition or subtraction problem about counting things is probably a good way of making the problem less abstract. I don’t have children, so I don’t know whether they play marbles or care about them. The most recent time I saw any of my niblings I told them about the subtleties of industrial design in the old-fashioned Western Electric Model 2500 touch-tone telephone. They love me. Also I’m not sure that this question actually tests subtraction more than it tests reading comprehension. But there are teachers who like to throw in the occasional surprisingly easy one. Keeps students on their toes.

Greg Evans’s Luann Againn for the 10th of March, 2019. This strip originally ran the 10th of March, 1991. Essays which include some mention of Luann, either current or 1990s reprints, are at this link.

Greg Evans’s Luann Againn for the 10th is part of a sequence showing Gunther helping Luann with her mathematics homework. The story started the day before, but this was the first time a specific mathematical topic was named. The point-slope form is a conventional way of writing an equation which corresponds to a particular line. There are many ways to write equations for lines. This is one that’s convenient to use if you know coordinates for one point on the line and the slope of the line. Any coordinates which make the equation true are then the coordinates for some point on the line.

Doug Savage’s Savage Chickens for the 10th of March, 2019. And when I think of something to write about Savage Chickens the results are at this link.

Doug Savage’s Savage Chickens for the 10th tosses in a line about logical paradoxes. In this case, using a classic problem, the self-referential statement. Working out whether a statement is true or false — its “truth value” — is one of those things we expect logic to be able to do. Some self-referential statements, logical claims about themselves, are troublesome. “This statement is false” was a good one for baffling kids and would-be world-dominating computers in science fiction television up to about 1978. Some self-referential statements seem harmless, though. Nobody expects even the most timid world-dominating computer to be bothered by “this statement is true”. It takes more than just a statement being about itself to create a paradox.

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And a last note. The blog hardly needs my push to help it out, but, sometimes people will miss a good thing. Ben Orlin’s Math With Bad Drawings just ran an essay about some of the many mathematics-themed comics that Hilary Price and Rina Piccolo’s Rhymes With Orange has run. The comic is one of my favorites too. Orlin looks through some of the comic’s twenty-plus year history and discusses the different types of mathematical jokes Price (with, in recent years, Piccolo) makes.

Myself, I keep all my Reading the Comics essays at this link, and those mentioning some aspect of Rhymes With Orange at this link.

Reading the Comics, March 19, 2019: Average Edition

This time around, averages seem important.

Mark Anderson’s Andertoons for the 18th is the Mark Anderson’s Andertoons for the week. This features the kids learning some of the commonest terms in descriptive statistics. And, as Wavehead says, the similarity of names doesn’t help sorting them out. Each is a kind of average. “Mean” usually is the arithmetic mean, or the thing everyone including statisticians calls “average”. “Median” is the middle-most value, the one that half the data is less than and half the data is greater than. “Mode” is the most common value. In “normally distributed” data, these three quantities are all the same. In data gathered from real-world measurements, these are typically pretty close to one another. It’s very easy for real-world quantities to be normally distributed. The exceptions are usually when there are some weird disparities, like a cluster of abnormally high-valued (or low-valued) results. Or if there are very few data points.

Mark Anderson’s Andertoons for the 18th of March, 2019. Essays which discuss topics raised by Andertoons can be found at this link. Also at this link, nearly enough.

The word “mean” derives from the Old French “meien”, that is, “middle, means”. And that itself traces to the Late Latin “medianus”, and the Latin “medius”. That traces back to the Proto-Indo-European “medhyo”, meaning “middle”. That’s probably what you might expect, especially considering that the mean of a set of data is, if the data is not doing anything weird, likely close to the middle of the set. The term appeared in English in the middle 15th century.

The word “median”, meanwhile, follows a completely different path. That one traces to the Middle French “médian”, which traces to the Late Latin “medianus” and Latin “medius” and Proto-Indo-European “medhyo”. This appeared as a mathematical term in the late 19th century; Etymology Online claims 1883, but doesn’t give a manuscript citation.

The word “mode”, meanwhile, follows a completely different path. This one traces to the Old French “mode”, itself from the Latin “modus”, meaning the measure or melody or style. We get from music to common values by way of the “style” meaning. Think of something being done “á la mode”, that is, “in the [ fashionable or popular ] style”. I haven’t dug up a citation about when this word entered the mathematical parlance.

So “mean” and “median” don’t have much chance to do anything but alliterate. “Mode” is coincidence here. I agree, it might be nice if we spread out the words a little more.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th of March, 2019. I’ve been talking about this strip a lot lately, it seems to me. Essays where I do discuss Edison Lee are at this link.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th has Edison introduce a sequence to his grandfather. Doubling the number of things for each square of a checkerboard is an ancient thought experiment. The notion, with grains of wheat rather than cookies, seems to be first recorded in 1256 in a book by the scholar Ibn Khallikan. One story has it that the inventor of chess requested from the ruler that many grains of wheat as reward for inventing the game.

If we followed Edison Lee’s doubling through all 64 squares we’d have, in total, need for 2⁶³-1 or 18,446,744,073,709,551,615 cookies. You can see why the inventor of chess didn’t get that reward, however popular the game was. It stands as a good display of how exponential growth eventually gets to be just that intimidatingly big.

Edison, like many a young nerd, is trying to stagger his grandfather with the enormity of this. I don’t know that it would work. Grandpa ponders eating all that many cookies, since he’s a comical glutton. I’d estimate eating all that many cookies, at the rate of one a second, eight hours a day, to take something like eighteen billion centuries. If I’m wrong? It doesn’t matter. It’s a while. But is that any more staggering than imagining a task that takes a mere ten thousand centuries to finish?

Greg Cravens’s The Buckets for the 19th of March, 2019. It also seems like I discuss The Buckets more these days, as seen at this link.

Greg Cravens’s The Buckets for the 19th sees Toby surprised by his mathematics homework. He’s surprised by how it turned out. I know the feeling. Everyone who does mathematics enough finds that. Surprise is one of the delights of mathematics. I had a great surprise last month, with a triangle theorem. Thomas Hobbes, the philosopher/theologian, entered his frustrating sideline of mathematics when he found the Pythagorean Theorem surprising.

Mathematics is, to an extent, about finding interesting true statements. What makes something interesting? That depends on the person surprised, certainly. A good guideline is probably “something not obvious before you’ve heard it, thatlooks inevitable after you have”. That is, a surprise. Learning mathematics probably has to be steadily surprising, and that’s good, because this kind of surprise is fun.

If it’s always a surprise there might be trouble. If you’re doing similar kinds of problems you should start to see them as pretty similar, and have a fair idea what the answers should be. So, from what Toby has said so far … I wouldn’t call him stupid. At most, just inexperienced.

Eric the Circle for the 19th of March, 2019, this one by Janka. This and other essays with Eric the Circle, by any artist, should be at this link.

Eric the Circle for the 19th, by Janka, is the Venn Diagram joke for the week. Properly any Venn Diagram with two properties has an overlap like this. We’re supposed to place items in both circles, and in the intersection, to reflect how much overlap there is. Using the sizes of each circle to reflect the sizes of both sets, and the size of the overlap to represent the size of the intersection, is probably inevitable. The shorthand calls on our geometric intuition to convey information, anyway.

Tony Murphy’s It’s All About You for the 19th of March, 2019. And the occasional time I discuss something from It’s All About You are here.

Tony Murphy’s It’s All About You for the 19th has a bunch of things going on. The punch line calls “algebra” what’s really a statistics problem, calculating the arithmetic mean of four results. The work done is basic arithmetic. But making work seem like a more onerous task is a good bit of comic exaggeration, and algebra connotes something harder than arithmetic. But Murphy exaggerates with restraint: the characters don’t rate this as calculus.

Then there’s what they’re doing at all. Given four clocks, what’s the correct time? The couple tries averaging them. Why should anyone expect that to work?

There’s reason to suppose this might work. We can suppose all the clocks are close to the correct time. If they weren’t, they would get re-set, or not looked at anymore. A clock is probably more likely to be a little wrong than a lot wrong. You’d let a clock that was two minutes off go about its business, in a way you wouldn’t let a clock that was three hours and 42 minutes off. A clock is probably as likely to show a time two minutes too early as it is two minutes too late. This all suggests that the clock errors are normally distributed, or something like that. So the error of the arithmetic mean of a bunch of clock measurements we can expect to be zero. Or close to zero, anyway.

There’s reasons this might not work. For example, a clock might systematically run late. My mantle clock, for example, usually drifts about a minute slow over the course of the week it takes to wind. Or the clock might be deliberately set wrong: it’s not unusual to set an alarm clock to five or ten or fifteen minutes ahead of the true time, to encourage people to think it’s later than it really is and they should hurry up. Similarly with watches, if their times aren’t set by Internet-connected device. I don’t know whether it’s possible to set a smart watch to be deliberately five minutes fast, or something like that. I’d imagine it should be possible, but also that the people programming watches don’t see why someone might want to set their clock to the wrong time. From January to March 2018, famously, an electrical grid conflict caused certain European clocks to lose around six minutes. The reasons for this are complicated and technical, and anyway The Doctor sorted it out. But that sort of systematic problem, causing all the clocks to be wrong in the same way, will foil this take-the-average scheme.

Murphy’s not thinking of that, not least because this comic’s a rerun from 2009. He was making a joke, going for the funnier-sounding “it’s 8:03 and five-eights” instead of the time implied by the average, 8:04 and a half. That’s all right. It’s a comic strip. Being amusing is what counts.

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There were just enough mathematically-themed comic strips this past week for one more post. When that is ready, it should be at this link. I’ll likely post it Tuesday.

Reading the Comics, March 12, 2019: Back To Sequential Time Edition

Since I took the Pi Day comics ahead of their normal sequence on Sunday, it’s time I got back to the rest of the week. There weren’t any mathematically-themed comics worth mentioning from last Friday or Saturday, so I’m spending the latter part of this week covering stuff published before Pi Day. It’s got me slightly out of joint. It’ll all be better soon.

Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for this week. That’s nice to have. It’s built on the concept of story problems. That there should be “stories” behind a problem makes sense. Most actual mathematics, even among mathematicians, is done because we want to know a thing. Acting on a want is a story. Wanting to know a thing justifies the work of doing this calculation. And real mathematics work involves looking at some thing, full of the messiness of the real world, and extracting from it mathematics. This would be the question to solve, the operations to do, the numbers (or shapes or connections or whatever) to use. We surely learn how to do that by doing simple examples. The kid — not Wavehead, for a change — points out a common problem here. There’s often not much of a story to a story problem. That is, where we don’t just want something, but someone else wants something too.

Mark Anderson’s Andertoons for the 11th of March, 2019. Essays which discuss topics raised by Andertoons can be found at this link. Also at this link, nearly enough.

Parker and Hart’s The Wizard of Id for the 11th is a riff on the “when do you use algebra in real life” snark. Well, no one disputes that there are fields which depend on advanced mathematics. The snark comes in from supposing that a thing is worth learning only if it’s regularly “useful”.

Parker and Hart’s The Wizard of Id for the 11th of March, 2019. This is part of the current run of The Wizard of Id, such as you might find in a newspaper. Both the current and 1960s-vintage reruns get discussion at this link.

Rick Detorie’s One Big Happy for the 12th has Joe stalling class to speak to “the guy who invented zero”. I really like this strip since it’s one of those cute little wordplay jokes that also raises a legitimate point. Zero is this fantastic idea and it’s hard to imagine mathematics as we know it without the concept. Of course, we could say the same thing about trying to do mathematics without the concept of, say, “twelve”.

Rick Detorie’s One Big Happy for the 12th of March, 2019. This is part of the current run of One Big Happy, such as you might find in a newspaper. At GoComics.com strips from several years ago are reprinted. Both runs of One Big Happy get their discussion at this link.

We don’t know who’s “the guy” who invented zero. It’s probably not all a single person, though, or even a single group of people. There are several threads of thought which merged together to zero. One is the notion of emptiness, the absense of a measurable thing. That probably occurred to whoever was the first person to notice a thing wasn’t where it was expected. Another part is the notion of zero as a number, something you could add to or subtract from a conventional number. That is, there’s this concept of “having nothing”, yes. But can you add “nothing” to a pile of things? And represent that using the addition we do with numbers? Sure, but that’s because we’re so comfortable with the idea of zero that we don’t ponder whether “2 + 1” and “2 + 0” are expressing similar ideas. You’ll occasionally see people asking web forums whether zero is really a number, often without getting much sympathy for their confusion. I admit I have to think hard to not let long reflex stop me wondering what I mean by a number and why zero should be one.

And then there’s zero, the symbol. As in having a representation, almost always a circle, to mean “there is a zero here”. We don’t know who wrote the first of that. The oldest instance of it that we know of dates to the year 683, and was written in what’s now Cambodia. It’s in a stone carving that seems to be some kind of bill of sale. I’m not aware whether there’s any indication from that who the zero was written for, or who wrote it, though. And there’s no reason to think that’s the first time zero was represented with a symbol. It’s the earliest we know about.

Darrin Bell’s Candorville for the 12th of March, 2019. Essays featuring some topic raised by of Candorville should appear here.

Darrin Bell’s Candorville for the 12th has some talk about numbers, and favorite numbers. Lemont claims to have had 8 as his favorite number because its shape, rotated, is that of the infinity symbol. C-Dog disputes Lemont’s recollection of his motives. Which is fair enough; it’s hard to remember what motivated you that long ago. What people mostly do is think of a reason that they, today, would have done that, in the past.

The ∞ symbol as we know it is credited to John Wallis, one of that bunch of 17th-century English mathematicians. He did a good bit of substantial work, in fields like conic sections and physics and whatnot. But he was also one of those people good at coming up with notation. He developed what’s now the standard notation for raising a number to a power, that stuff, and showed how to define raising a number to a rational-number power. Bunch of other things. He also seems to be the person who gave the name “continued fraction” to that concept.

Wallis never explained why he picked ∞ as a shape, of all the symbols one could draw, for this concept. There’s speculation he might have been varying the Roman numeral for 1,000, which we’ve simplified to M but which had been rendered as (|) or () and I can see that. (Well, really more of a C and a mirror-reflected C rather than parentheses, but I don’t have the typesetting skills to render that.) Conflating “a thousand” with “many” or “infinitely many” has a good heritage. We do the same thing when we talk about something having millions of parts or costing trillions of dollars or such. But, Wallis never explained (so far as we’re aware), so all this has to be considered speculation and maybe mnemonic helps to remembering the symbol.

Terry Laban and Patty LaBan’s Edge City rerun for the 12th of March, 2019. (The strip has ended.) This comic originally ran in 2004. Edge City inspires discussions in the essays at this link.

Terry LaBan and Patty LaBan’s Edge City for the 12th is another story problem joke. Curiously the joke seems to be simply that the father gets confused following the convolutions of the story. The specific story problem circles around the “participation awards are the WORST” attitude that newspaper comics are surprisingly prone to. I think the LaBans just wanted the story problem to be long and seem tedious enough that our eyes glazed over. Anyway you could not pay me to read whatever the comments on this comic are. Sorry not sorry.

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I figure to have one more Reading the Comics post this week. When that’s posted it should be available at this link. Thanks for being here.

Reading the Comics, February 2, 2019: Not The February 1, 2019 Edition

The last burst of mathematically-themed comic strips last week nearly all came the 1st of the month. But the count fell just short. I can only imagine what machinations at Comic Strip Master Command went wrong, that we couldn’t get a full four comics for the same day. Well, life is messy and things will happen.

Stephen Bentley’s Herb and Jamaal for the 1st is a rerun. I discussed it last time I noticed it too. I’d previously taken Herb to be gloating about not using the calculus he’d studied. I may be reading too much into what seems like a smirk in the final panel, though. Could be he’s thinking of the strangeness that something which, at the time, is challenging and difficult and all-consuming turns out to not be such a big deal. Which could be much of high school.

Stephen Bentley’s Herb and Jamaal for the 1st of February, 2019. It previously ran the 25th of November, 2013. Essays that mention Herb and Jamaal should be at this link. Although, it happens, the last time this ran was before I started tagging comic strips by name. Whoops?

But my first instinct is still to read this as thinking of the “uselessness” of calculus. It betrays the terrible attitude that education is about job training. It should be about letting people be literate in the world’s great thoughts. Mathematics seems to get this attitude a lot, but I’m aware I may feel a confirmation bias. If I had become a French major perhaps I’d pay attention to all the comic strips where someone giggles about how they never use the foreign languages they learned in high school either.

Jon Rosenberg’s Scenes from a Multiverse for the 1st of February, 2019. Essays with some discussion sparked by Scenes from a Multiverse are at this link.

Jon Rosenberg’s Scenes from a Multiverse for the 1st is set in a “Mathpinion City”, showing people arguing about mathematical truths. It seems to me a political commentary, about the absurdity of rejecting true things over perceived insults. The 1+1=3 partisans aren’t even insisting they’re right, just that the other side is obnoxious. Arithmetic here serves as good source for things that can’t be matters of opinion, at least provided we’ve agreed on what’s meant by ideas like ‘1’ and ‘3’.

Mathematics is a human creation, though. What we decide to study, and what concepts we think worth interesting, are matters of opinion. It’s difficult to imagine people who think 1+1=2 a statement so unimportant they don’t care whether it’s true or false. At least not ones who reason anything like we do. But that is our difficulty, not a constraint on what life could think.

Neil Kohney’s The Other End for the 1st of February, 2019. I thought this might be a new tag, but no. I’ve discussed The Other End at some essays linked here.

Neil Kohney’s The Other End for the 1st has a mathematics cameo. It’s the subject of a quiz so difficult that the kid begs for God’s help sorting it out. The problems all seem to be simplifying expressions. It’s a skill worth having. There are infinitely many ways to write the same quantity. Some of them are more convenient than others. Brief expressions, for example, are often easier to understand. But a longer expression might let us tease out relationships that are good to know. Many analysis proofs end up becoming simpler when you multiply by one — that is, multiplying by and dividing by the same quantity, but using the numerator to reduce one part of the expression and the denominator to reduce some other. Or by adding zero, in which you add and subtract a quantity and use either side to simplify other parts of the expression. So, y’know, just do the work. It’s better that way.

Mark Anderson’s Andertoons for the 2nd of February, 2019. I’ve discussed Andertoons in so many essays like this you might think I was Mark Anderson’s publicity agent, but that I had an extremely narrow focus of what I thought marketable.

Mark Anderson’s Andertoons for the 2nd is the Mark Anderson’s Andertoons for the week. Wavehead’s learning about invertible operations: that a particular division can undo a multiplication. Or, presumably, that a particular multiplication can undo a division. Fair to wonder why you’d want to do that, though. Most of the operations we use in arithmetic have inverses, or come near it. (There’s one thing you can multiply by which you can’t divide out.) The term used in group theory for this is to say the real numbers are a “field”. This is a ring in which not just does addition have an inverse, but so does multiplication. And the operations commute; dividing by four and multiplying by four is as good as multiplying by for and dividing by four. You can build interesting mathematical structures that don’t have some of these properties. Elementary-school division, where you might describe (say) 26 divided by 4 as “6 with a remainder of 2” is one of them.

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And that covers the comic strips. Come Sunday should be the next of this series, and it should be at this link.

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

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

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

Jonathan Lemon and Joey Alison Sayers’s Alley Oop for the 20th of January, 2019. I don’t seem to have had much cause to mention Alley Oop before; the only previous reference I can find is in this cameo in a different comic. Well, this and any other essays that write about Alley Oop at any length should be at this link. I don’t figure to differentiate between the weekday Alley Oop strip and the Sunday Little Oop line in using this tag.

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

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

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of January, 2019. Pretty much every Reading the Comics brings up this strip. But the essays that specifically mention Saturday Morning Breakfast Cereal should be at this link.

The catch — well, one catch — is that this tempts one to reward a process. Once it’s taken for granted the process works, then whether it’s actually doing what you want gets forgotten. And once everyone knows what’s being measured it becomes possible to game the system. Famously, in the mid-1960s the United States tried to judge its progress in the 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.

Mark Anderson’s Andertoons for the 21st of January, 2019. Pretty much every Reading the Comics brings up this strip, at least since last fall’s weird gap has ended. But the essays that specifically mention Andertoons should be at this link.

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

Charles Brubaker’s Ask A Cat for the 21st of January, 2019. Although I’ve mentioned this comic before, it wasn’t when I was putting up tags naming each comic. I’ll fix that now. This and future essays mentioning Ask A Cat (or its Fuzzy Princess sideline, as long as that hasn’t got its own GoComics presence) should appear at this link. And this strip previously ran the 19th of June, 2016.

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

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

Benita Epstein’s Six Chix for the 25th of January, 2019. The essays wherein I mention Six Chix, from any of the shared strip’s authors, should appear at this link.

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

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

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

Reading the Comics, January 12, 2019: A Edition

As I said Sunday, last week was a slow one for mathematically-themed comic strips. Here’s the second half of them. They’re not tightly on point. But that’s all right. They all have titles starting with ‘A’. I mean if you ignore the article ‘the’, the way we usually do when alphabetizing titles.

Tony Cochran’s Agnes for the 11th is basically a name-drop of mathematics. The joke would be unchanged if the teacher asked Agnes to circle all the adjectives in a sentence, or something like that. But there are historically links between religious thinking and mathematics. The Pythagoreans, for example, always a great and incredible starting point for any mathematical topic or just some preposterous jokes that might have nothing to do with their reality, were at least as much a religious and philosophical cult. For a long while in the Western tradition, the people with the time and training to do advanced mathematics work were often working for the church. Even as people were more able to specialize, a mystic streak remained. It’s easy to understand why. Mathematics promises to speak about things that are universally true. It encourages thinking about the infinite. It encourages thinking about the infinitely tiny. It courts paradoxes as difficult as any religious Mystery. It’s easy to snark at someone who takes numerology seriously. But I’m not sure the impulse that sees magic in arithmetic is different to the one that sees something supernatural in a “transfinite” item.

Tony Cochran’s Agnes for the 11th of January, 2019. I’m always glad to have the chance to write about Agnes, and you can see things I’ve written about it at this link.

Scott Hilburn’s The Argyle Sweater for the 11th is another mistimed Pi Day joke. π is, famously, an irrational number. But so is every number, except for a handful of strange ones that we’ve happened to find interesting. That π should go on and on follows from what an irrational number means. It’s a bit surprising the 4 didn’t know all this before they married.

Scott Hilburn’s The Argyle Sweater for the 11th of January, 2019. It’s not quite the most common strip mentioned here. But you can find other topics raised by The Argyle Sweater at this link.

I appreciate the secondary joke that the marriage counselor is a “Hugh Jripov”, and the counselor’s being a ripoff is signaled by being a ÷ sign. It suggests that maybe successful reconciliation isn’t an option. I’m curious why the letters ‘POV’ are doubled, in the diploma there. In a strip with tighter drafting I’d think it was suggesting the way a glass frame will distort an image. But Hilburn draws much more loosely than that. I don’t know if it means anything.

Mark Anderson’s Andertoons for the 12th of January, 2019. There’s more essays with Andertoons at this link.

Mark Anderson’s Andertoons for the 12th is the Mark Anderson’s Andertoons for the essay. I’m so relieved to have a regular stream of these again. The teacher thinks Wavehead doesn’t need to annotate his work. And maybe so. But writing down thoughts about a problem is often good practice. If you don’t know what to do, or you aren’t sure how to do what you want? Absolutely write down notes. List the things you’d want to do. Or things you’d want to know. Ways you could check your answer. Ways that you might work similar problems. Easier problems that resemble the one you want to do. You find answers by thinking about what you know, and the implications of what you know. Writing these thoughts out encourages you to find interesting true things.

And this was too marginal a mention of mathematics even for me, even on a slow week. But Georgia Dunn’s Breaking Cat News for the 12th has a cat having a nightmare about mathematics class. And it’s a fun comic strip that I’d like people to notice more.

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And that’s as many comics as I have to talk about from last week. Sunday, I should have another Reading the Comics post and it’ll be at this link.

Reading the Comics, January 9, 2018: I Go On About Johnny Appleseed Edition

This was a slow week for mathematically-themed comic strips. Such things happen. I put together a half-dozen that see on-topic enough to talk about, but I stretched to do it. You’ll see.

Mark Anderson’s Andertoons for the 6th mentions addition as one of the things you learn in an average day of elementary school. I can’t help noticing also the mention of Johnny Appleseed, who’s got a weird place in my heart as he and I share a birthday. He got to it first. Although Johnny Appleseed — John Champan — is legendary for scattering apple seeds, that’s not what he mostly did. He would more often grow apple-tree nurseries, from which settlers could buy plants and demonstrate they were “improving” their plots. He was also committed to spreading the word of Emanuel Swedenborg’s New Church, one of those religious movements that you somehow don’t hear about. But there was this like 200-year-long stretch where a particular kind of idiosyncratic thinker was Swedenborgian, or at least influenced by that. I don’t know offhand of any important Swedenborgian mathematicians, I admit, but I’m glad to hear if someone has news.

Mark Anderson’s Andertoons for the 6th of January, 2019. Andertoons often appears in these essays. You can see the proof of that Andertoons claim at this link.

Justin Thompson’s MythTickle rerun for the 9th mentions “algebra” as something so dreadful that even being middle-aged is preferable. Everyone has their own tastes, yes, although it would be the same joke if it were “gym class” or something. (I suppose that’s not one word. “Dodgeball” would do, but I never remember playing it. It exists just as a legendarily feared activity, to me.) Granting, though, that I had a terrible time with the introduction to algebra class I had in middle school.

Justin Thompson’s MythTickle rerun for the 9th of January, 2019. MythTickle has only barely appeared before in these essays. You can see the proof of that MythTickle claim at this link.

Tom Wilson’s Ziggy for the 9th is a very early Pi Day joke, so, there’s that. There’s not much reason a take-a-number dispenser couldn’t give out π, or other non-integer numbers. What the numbers are doesn’t matter. It’s just that the dispensed numbers need to be in order. It should be helpful if there’s a clear idea how uniformly spaced the numbers are, so there’s some idea how long a wait to expect between the currently-serving number and whatever number you’ve got. But that only helps if you have a fair idea of how long an order should on average take.

Tom Wilson’s Ziggy for the 9th of January, 2019. Ziggy has turned up once or twice in these essays. You can see the proof of that Ziggy claim at this link.

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I’ll close out last week’s comics soon. The next Reading the Comics post, like all the earlier ones, should be at this link.

Reading the Comics, January 5, 2019: Start of the Year Edition

With me wrapping up the mathematically-themed comic strips that ran the first of the year, you can see how far behind I’m falling keeping everything current. In my defense, Monday was busier than I hoped it would be, so everything ran late. Next week is looking quite slow for comics, so maybe I can catch up then. I will never catch up on anything the rest of my life, ever.

Scott Hilburn’s The Argyle Sweater for the 2nd is a bit of wordplay about regular and irregular polygons. Many mathematical constructs, in geometry and elsewhere, come in “regular” and “irregular” forms. The regular form usually has symmetries that make it stand out. For polygons, this is each side having the same length, and each interior angle being congruent. Irregular is everything else. The symmetries which constrain the regular version of anything often mean we can prove things we otherwise can’t. But most of anything is the irregular. We might know fewer interesting things about them, or have a harder time proving them.

Scott Hilburn’s The Argyle Sweater for the 2nd of January, 2019. The many appearances of Argyle Sweater in these pages are at this link.

I’m not sure what the teacher would be asking for in how to “make an irregular polygon regular”. I mean if we pretend that it’s not setting up the laxative joke. I can think of two alternatives that would make sense. One is to draw a polygon with the same number of sides and the same perimeter as the original. The other is to draw a polygon with the same number of sides and the same area as the original. I’m not sure of the point of either. I suppose polygons of the same area have some connection to quadrature, that is, integration. But that seems like it’s higher-level stuff than this class should be doing. I hate to question the reality of a comic strip but that’s what I’m forced to do.

Bud Fisher’s Mutt and Jeff rerun for the 4th of January, 2019. The several appearances of Mutt and Jeff in these pages are at this link.

Bud Fisher’s Mutt and Jeff rerun for the 4th is a gambler’s fallacy joke. Superficially the gambler’s fallacy seems to make perfect sense: the chance of twelve bad things in a row has to be less than the chance of eleven bad things in a row. So after eleven bad things, the twelfth has to come up good, right? But there’s two ways this can go wrong.

Suppose each attempted thing is independent. In this case, what if each patient is equally likely to live or die, regardless of what’s come before? And in that case, the eleven deaths don’t make it more likely that the next will live.

Suppose each attempted thing is not independent, though. This is easy to imagine. Each surgery, for example, is a chance for the surgeon to learn what to do, or not do. He could be getting better, that is, more likely to succeed, each operation. Or the failures could reflect the surgeon’s skills declining, perhaps from overwork or age or a loss of confidence. Impossible to say without more data. Eleven deaths on what context suggests are low-risk operations suggest a poor chances of surviving any given surgery, though. I’m on Jeff’s side here.

Mark Anderson’s Andertoons for the 5th of January, 2019. The amazingly many appearances of Andertoons in these pages are at this link.

Mark Anderson’s Andertoons for the 5th is a welcome return of Wavehead. It’s about ratios. My impression is that ratios don’t get much attention in themselves anymore, except to dunk on stupid Twitter comments. It’s too easy to jump right into fractions, and division. Ratios underlie this, at least historically. It’s even in the name, ‘rational numbers’.

Wavehead’s got a point in literally comparing apples and oranges. It’s at least weird to compare directly different kinds of things. This is one of those conceptual gaps between ancient mathematics and modern mathematics. We’re comfortable stripping the units off of numbers, and working with them as abstract entities. But that does mean we can calculate things that don’t make sense. This produces the occasional bit of fun on social media where we see something like Google trying to estimate a movie’s box office per square inch of land in Australia. Just because numbers can be combined doesn’t mean they should be.

Larry Wright’s Motley rerun for the 5th of January, 2019. The occasional appearances of Motley in these pages are at this link.

Larry Wright’s Motley rerun for the 5th has the form of a story problem. And one timely to the strip’s original appearance in 1987, during the National Football League players strike. The setup, talking about the difference in weekly pay between the real players and the scabs, seems like it’s about the payroll difference. The punchline jumps to another bit of mathematics, the point spread. Which is an estimate of the expected difference in scoring between teams. I don’t know for a fact, but would imagine the scab teams had nearly meaningless point spreads. The teams were thrown together extremely quickly, without much training time. The tools to forecast what a team might do wouldn’t have the data to rely on.

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The at-least-weekly appearances of Reading the Comics in these pages are at this link.

Reading the Comics, December 19, 2018: Andertoons Is Back Edition

I had not wanted to mention, for fear of setting off a panic. But Mark Anderson’s Andertoons, which I think of as being in every Reading the Comics post, hasn’t been around lately. If I’m not missing something, it hasn’t made an appearance in three months now. I don’t know why, and I’ve been trying not to look too worried by it. Mostly I’ve been forgetting to mention the strange absence. This even though I would think any given Tuesday or Friday that I should talk about the strip not having anything for me to write about. Fretting about it would make a great running theme. But I have never spotted a running theme before it’s finished. In any event the good news is that the long drought has ended, and Andertoons reappears this week. Yes, I’m hoping that it won’t be going to long between appearances this time.

Jef Mallett’s Frazz for the 16th of December, 2018. Other essays discussing topics raised by Frazz are at this link.

Jef Mallett’s Frazz for the 16th talks about probabilities. This in the context of assessing risks. People are really bad at estimating probabilities. We’re notoriously worse at assessing risks, especially when it’s a matter of balancing a present cost like “fifteen minutes waiting while the pharmacy figures out whether insurance will pay for the flu shot” versus a nebulous benefit like “lessened chance of getting influenza, or at least having a less severe influenza”. And it’s asymmetric, too. We view improbable but potentially enormous losses differently from the way we view improbable but potentially enormous gains. And it’s hard to make the rationally-correct choice reliably, not when there are so many choices of this kind every day.

Tak Bui’s PC and Pixel for the 16th of December, 2018. This and other essays, when they’re written, inspired by PC and Pixel should be at this link. It’s a new tag, which surprises me.

Tak Bui’s PC and Pixel for the 16th features a wall full of mathematical symbols, used to represent deep thought about a topic. The symbols are gibberish, yes. I’m not sure that an actual “escape probability” could be done in a legible way, though. Or even what precisely Professor Phillip might be calculating. I imagine it would be an estimate of the various ways he might try to escape, and what things might affect that. This might be for the purpose of figuring out what he might do to maximize his chances of a successful escape. Although I wouldn’t put it past the professor to just be quite curious what the odds are. There’s a thrill in having a problem solved, even if you don’t use the answer for anything.

Ruben Bolling’s Super-Fun-Pak Comix for the 18th of December, 2018. Essays based on Super-Fun-Pak Comix are at this link. (Amazing Yet Tautological is one of the features that turns up in Super-Fun-Pak Comix, which is why it doesn’t rate a tag on its own)

Ruben Bolling’s Super-Fun-Pak Comix for the 18th has a trivia-panel-spoof dubbed Amazing Yet Tautological. One could make an argument that most mathematics trivia fits into this category. At least anything about something that’s been proven. Anyway, whether this is a tautological strip depends on what the strip means by “average” in the phrase “average serving”. There’s about four jillion things dubbed “average” and each of them has a context in which they make sense. The thing intended here, and the thing meant if nobody says anything otherwise, is the “arithmetic mean”. That’s what you get from adding up everything in a sample (here, the amount of egg salad each person in America eats per year) and dividing it by the size of the sample (the number of people in America that year). Another “average” which would make sense, but would break this strip, would be the median. That would be the amount of egg salad that half of all Americans eat more than, and half eat less than. But whether every American could have that big a serving really depends on what that median is. The “mode”, the most common serving, would also be a reasonable “average” to expect someone to talk about.

Mark Anderson’s Andertoons for the 19th of December, 2018. The many essays which discuss Andertoons are at this link.

Mark Anderson’s Andertoons for the 19th is that strip’s much-awaited return to my column here. It features solid geometry, which is both an important part of geometry and also a part that doesn’t get nearly as much attention as plane geometry. It’s reductive to suppose the problem is that it’s harder to draw solids than planar figures. I suspect that’s a fair part of the problem, though. Mathematicians don’t get much art training, not anymore. And while geometry is supposed to be able to rely on pure reasoning, a good picture still helps. And a bad picture will lead us into trouble.

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Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

Reading the Comics, September 17, 2018: Hard To Credit Edition

Two of the four comic strips I mean to feature here have credits that feel unsatisfying to me. One of them is someone’s pseudonym and, yeah, that’s their business. One is Dennis the Menace, for which I find an in-strip signature that doesn’t match the credentials on Comics Kingdom’s web site, never mind Wikipedia. I’ll go with what’s signed in the comic as probably authoritative. But I don’t like it.

R Ferdinand and S Ketcham’s Dennis the Menace for the 16th is about calculation. One eternally surprising little thing about calculators and computers is that they don’t do anything you can’t do by hand. Or, for that matter, in your head. They do it faster, typically, and more reliably. They can seem magical. But the only difference between what they do and what we do is the quantity with which they do this work. You can take this as humbling or as inspirational, as fits your worldview.

R Ferdinand and S Ketcham’s Dennis the Menace for the 16th of September, 2018. I know it’s a standard bit of Dennis the Menace snarking to rate Dennis’s actual menacing nature. But forcing his father to show that he’s lost his proficiency at arithmetic in the guise of helping him? … It’s kind of a long-game bit of menace, I suppose.

Ham’s Life on Earth for the 16th is a joke about the magical powers we attribute to mathematics. It’s also built on one of our underlying assumptions of the world, that it must be logically consistent. If one has an irrefutable logical argument that something isn’t so, then that thing must not be so. It’s hard to imagine how an illogical world would work. But it is hard not to wonder if there’s some arrogance involved in supposing the world has to square with the rules of logic that we find sensible. And to wonder whether we perceive world consistent with that logic because our expectations frame what we’re able to perceive.

Ham’s Life on Earth for the 16th of September, 2018. Raise your hand if you’ve been there. Hah! You’re fibbing.

In any case, as we frame logic, an argument’s validity shouldn’t depend on the person making the argument. Or even whether the argument has been made. So it’s hard to see how simply voicing the argument that one doesn’t exist could have that effect. Except that mathematics has got magical connotations, and vice-versa. That’ll be good for building jokes for a while yet.

Mark Anderson’s Andertoons for the 17th of September, 2018. So is this a substitute or does Wavehead just have a new mathematics teacher?

Mark Anderson’s Andertoons for the 17th is the Mark Anderson’s Andertoons for the week. It’s wordplay, built on the connotation that division is a bad thing. It seems less dire if we think of division as learning how to equally share something that’s been held in common, though. Or if we think of it as learning what to multiply a thing by to get a particular value. Most mathematical operations can be taken to mean many things. Surely division has some constructive and happy interpretations.

Paul Gilligan’s Pooch Cafe for the 17th of September, 2018. For my money the cat hasn’t walked across the keyboard right if it hasn’t got you Freakazoid powers.

Paul Gilligan’s Pooch Cafe for the 17th is a variation of the monkeys-on-keyboards joke. If what you need is a string of nonsense characters then … well, a cat on the keys is at least famous for producing some gibberish. It’s likely not going to be truly random, though. If a cat’s paw has stepped on, say, the ‘O’, there’s a good chance the cat is also stepping on ‘P’ or ‘9’. It also suggests that if the cat starts from the right, they’re more likely to have a character like ‘O’ early in the string of characters and less likely at the end. A completely random string would be as likely to have an ‘O’ at the start as at the end of the string.

And even if a cat on the keyboard did produce good-quality randomness, well. How likely a randomly-generated string of characters is to match a thing depends on the length of the thing. If the meaning of the symbols doesn’t matter, then ‘Penny Lane’ is as good as ‘*2ft,2igFIt’. This is not to say you can just use, say, ‘asdfghjkl’ as your password, at least not for anything that would hurt you if it were cracked. If everyone picked all passwords with no regard for what the symbols meant, these would be. But passwords that seem easy to think get used more often than they should be. It’s not that they’re easier to guess, but that guessing them is more likely to be correct.

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Later this week I’ll host this month’s Playful Mathematics Blog Carnival! If you know of any mathematics that teaches or delights or both please share it with me, and we’ll let the world know. Also this week I should finally start my 2018 Mathematics A To Z, explaining words from mathematics one at a time.

And there’ll be another Reading the Comics Post before next Sunday. It and all my other Reading the Comics posts should be at this tag. Other appearances of Dennis the Menace should be at this link. This and other essays mentioning Life On Earth are at this link. The many appearances of Andertoons are at this link And other essays with Pooch Cafe should be at this link. Thanks for reading along.

Reading the Comics, September 11, 2018: 60% Reruns Edition

Three of the five comic strips I review today are reruns. I think that I’ve only mentioned two of them before, though. But let me preface all this with a plea I’ve posted before: I’m hosting the Playful Mathematics Blog Carnival the last week in September. Have you run across something mathematical that was educational, or informative, or playful, or just made you glad to know about? Please share it with me, and we can share it with the world. It can be for any level of mathematical background knowledge. Thank you.

Tom Batiuk’s Funky Winkerbean vintage rerun for the 10th is part of an early storyline of Funky attempting to tutor football jock Bull Bushka. Mathematics — geometry, particularly — gets called on as a subject Bull struggles to understand. Geometry’s also well-suited for the joke because it has visual appeal, in a way that English or History wouldn’t. And, you know, I’ll take “pretty” as a first impression to geometry. There are a lot of diagrams whose beauty is obvious even if their reasons or points or importance are obscure.

Tom Batiuk’s Funky Winkerbean vintage rerun for the 10th of September, 2018. It originally ran the 28th of September, 1972.

Dan Collins’s Looks Good on Paper for the 10th is about everyone’s favorite non-orientable surface. The first time this strip appeared I noted that the road as presented isn’t a Möbius strip. The opossums and the car are on different surfaces. Unless there’s a very sudden ‘twist’ in the road in the part obscured from the viewer, anyway. If I’d drawn this in class I would try to save face by saying that’s where the ‘twist’ is, but none of my students would be convinced. But we’d like to have it that the car would, if it kept driving, go over all the pavement.

Dan Collins’s Looks Good on Paper for the 10th of September, 2018. It originally ran the 10th of September, 2016.

Bud Fisher’s Mutt and Jeff for the 10th is a joke about story problems. The setup suggests that there’s enough information in what Jeff has to say about the cop’s age to work out what it must be. Mutt isn’t crazy to suppose there is some solution possible. The point of this kind of challenge is realizing there are constraints on possible ages which are not explicit in the original statements. But in this case there’s just nothing. We would call the cop’s age “underdetermined”. The information we have allows for many different answers. We’d like to have just enough information to rule out all but one of them.

Bud Fisher’s Mutt and Jeff for the 10th of September, 2018. No guessing when this originally ran, and the lettering has been re-done. This is probably why Jeff’s word balloons in the first and second panel don’t quite low logically together.

John Rose’s Barney Google and Snuffy Smith for the 11th is here by popular request. Jughead hopes that a complicated process of dubious relevance will make his report card look not so bad. Loweezey makes a New Math joke about it. This serves as a shocking reminder that, as most comic strip characters are fixed in age, my cohort is now older than Snuffy and Loweezey Smith. At least is plausibly older than them.

John Rose’s Barney Google and Snuffy Smith for the 11th of September, 2018. Not a rerun! As far as I know. But see how Henry fooled me about its rerun cycle lately. Not answered: a report card in early-to-mid September?

Anyway it’s also a nice example of the lasting cultural reference of the New Math. It might not have lasted long as an attempt to teach mathematics in ways more like mathematicians do. But it’s still, nearly fifty years on, got an unshakable and overblown reputation for turning mathematics into doubletalk and impossibly complicated rules. I imagine it’s the name; “New Math” is a nice, short, punchy name. But the name also looks like what you’d give something that was being ruined, under the guise of improvement. It looks like that terrible moment of something familiar being ruined even if you don’t know that the New Math was an educational reform movement. Common Core’s done well in attracting a reputation for doing problems the complicated way. But I don’t think its name is going to have the cultural legacy of the New Math.

Mark Anderson’s Andertoons for the 11th of September, 2018. All right, but another problem they have: no chalk.

Mark Anderson’s Andertoons for the 11th is another kid-resisting-the-problem joke. Wavehead’s obfuscation does hit on something that I have wondered, though. When we describe things, we aren’t just saying what we think of them. We’re describing what we think our audience should think of them. This struck me back around 1990 when I observed to a friend that then-current jokes about how hard VCRs were to use failed for me. Everyone in my family, after all, had no trouble at all setting the VCR to record something. My friend pointed out that I talked about setting the VCR. Other people talk about programming the VCR. Setting is what you do to clocks and to pots on a stove and little things like that; an obviously easy chore. Programming is what you do to a computer, an arcane process filled with poor documentation and mysterious problems. We framed our thinking about the task as a simple, accessible thing, and we all found it simple and accessible. Mathematics does tend to look at “problems”, and we do, especially in teaching, look at “finding solutions”. Finding solutions sounds nice and positive. But then we just go back to new problems. And the most interesting problems don’t have solutions, at least not ones that we know about. What’s enjoyable about facing these new problems?

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One thing that’s not a problem: finding other Reading the Comics posts. They should all appear at this link. Appearances by the current-run and the vintage Funky Winkerbean are at this link. Essays with a mention of Looks Good On Paper are at this link. Meanwhile, essays with Mutt and Jeff in the are at this link. Other appearances by Barney Google and Snuffy Smith — current and vintage, if vintage ever does something on-topic — are at this link. And the many appearances by Andertoons are at this link, or just use any Reading the Comics post, really. Thank you.