## Reading the Comics, January 30, 2019: Interlude Edition

I think there are just barely enough comic strips from the past week to make three essays this time around. But one of them has to be a short group, only three comics. That’ll be for the next essay when I can group together all the strips that ran in February. One strip that I considered but decided not to write at length about was Ed Allison’s dadaist Unstrange Phenomena for the 28th. It mentions Roman Numerals and the idea of sneaking message in through them. But that’s not really mathematics. I usually enjoy the particular flavor of nonsense which Unstrange Phenomena uses; you might, too.

John McPherson’s Close to Home for the 29th uses an arithmetic problem as shorthand for an accomplished education. The problem is solvable. Of course, you say. It’s an equation with quadratic polynomial; it can hardly not be solved. Yes, fine. But McPherson could easily have thrown together numbers that implied x was complex-valued, or had radicals or some other strange condition. This is one that someone could do in their heads, at least once they practiced in mental arithmetic.

I feel reasonably confident McPherson was just having a giggle at the idea of putting knowledge tests into inappropriate venues. So I’ll save the full rant. But there is a long history of racist and eugenicist ideology that tried to prove certain peoples to be mentally incompetent. Making an arithmetic quiz prerequisite to something unrelated echoes that. I’d have asked McPherson to rework the joke to avoid that.

(I’d also want to rework the composition, since the booth, the swinging arm, and the skirted attendant with the clipboard don’t look like any tollbooth I know. But I don’t have an idea how to redo the layout so it’s more realistic. And it’s not as if that sort of realism would heighten the joke.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th riffs on the problem of squaring the circle. This is one of three classical problems of geometry. The lecturer describes it just fine: is it possible to make a square that’s got the same area as a given circle, using only straightedge and compass? There are shapes it’s easy to do this for, such as rectangles, parallelograms, triangles, and (why not?) this odd crescent-moon shaped figure called the lune. Circles defied all attempts. In the 19th century mathematicians found ways to represent the operations of classical geometry with algebra, and could use the tools of algebra to show squaring the circle was impossible. The squaring would be equivalent to finding a polynomial, with integer coefficients, that has $\sqrt{\pi}$ as a root. And we know from the way algebra works that this can’t be done. So squaring the circle can’t be done.

The lecturer’s hack, modifying the compass and straightedge, lets you in principle do whatever you want. The hack isn’t new either. Modifying the geometric tools changes what you can and can’t do. The Ancient Greeks recognized that adding some specialized tools would make the problem possible. But that falls outside the scope of the problem.

Which feeds to the secondary joke, of making the philosophers sad. Often philosophy problems test one’s intuition about an idea by setting out a problem, often with unpleasant choices. A common problem with students that I’m going ahead and guessing are engineers is then attacking the setup of the question, trying to show that the problem couldn’t actually happen. You know, as though there were ever a time significant numbers of people were being tied to trolley tracks. (By the way, that thing about silent movie villains tying women to railroad tracks? Only happened in comedies spoofing Victorian melodramas. It’s always been a parody.) Attacking the logic of a problem may make for good movie drama. But it makes for a lousy student and a worse class discussion.

Ted Shearer’s Quincy rerun for the 30th uses a bit of mathematics and logic talk. It circles the difference between the feeling one can have about the rational meaning of a situation and how the situation feels to someone. It seems like a jump that Quincy goes from being asked about logic to talking about arithmetic. Possibly Quincy’s understanding of logic doesn’t start from the sort of very abstract concept that makes arithmetic hard to get to, though.

There should be another Reading the Comics post this week. It should be here, when it appears. There should also be one on Sunday, as usual.

## 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”.

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

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

The catch — well, one catch — is that this tempts one to reward a process. Once it’s taken for granted the process works, then whether it’s actually doing what you want gets forgotten. And once everyone knows what’s being measured it becomes possible to game the system. Famously, in the mid-1960s the United States tried to judge its progress in the Vietname War by counting the number of enemy soldiers killed. There was then little reason to care about who was killed, or why. And reason to not care whether actual enemy soldiers were being killed. There’s good to be said about testing whether the things you try to do work. There’s great danger in thinking that the thing you can measure guarantees success.

Mark Anderson’s Andertoons for the 21st is a bit of fun with definitions. Mathematicians rely on definitions. It’s hard to imagine a proof about something undefined. But definitions are hard to compose. We usually construct a definition because we want a common term to describe a collection of things, and to exclude another collection of things. And we need people like Wavehead who can find edge cases, things that seem to satisfy a definition while breaking its spirit. This can let us find unstated assumptions that we should pay attention to. Or force us to accept that the definition is so generally useful that we’ll tolerate it having some counter-intuitive implications.

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

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

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

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

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.

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 16, 2019: Young People’s Mathematics Edition

Today’s quartet of mathematically-themed comic strips doesn’t have an overwhelming theme. There’s some bits about the mathematics that young people do, so, that’s enough to separate this from any other given day’s comics essay.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is built on a bit of mathematical folklore. As Weinersmith’s mathematician (I don’t remember that we’ve been given her name) mentions, there is a belief that “revolutionary” mathematics is done by young people. That isn’t to say that older mathematicians don’t do great work. But the stereotype is that an older mathematician will produce masterpieces in already-established fields. It’s the young that establish new fields. Indeed, one of mathematics’s most prestigious awards, the Fields Medal, is only awarded to mathematicians under the age of forty. I was cheated of mine. Long story.

There’s intuitive appeal in the idea that revolutions in thinking are for the young. We think that people get set in their ways as they develop their careers. We have a couple dramatic examples, most notably Évariste Galois, who developed what we now see as foundations of group theory and died at twenty. While the idea is commonly held, I don’t know that it’s actually true. That is, that it holds up to scrutiny. It seems hard to create a definition for “revolutionary mathematics” that could be agreed upon by two people. So it would be difficult to test at what age people do their most breathtaking work, and whether it is what they do when young or when experienced.

Is there harm to believing an unprovable thing? If it makes you give up on trying, yes. My suspicion is that true revolutionary work happens when a well-informed, deep thinker comes to a field that hasn’t been studied in that way before. And when it turns out to be a field well-suited to study that way. That doesn’t require youth. It requires skill in one field, and an understanding that there’s another field ready to be studied that way.

Will Henry’s Wallace the Brave for the 14th is a mathematics anxiety joke. Wallace tries to help by turning an abstract problem into a concrete one. This is often a good way to approach a problem. Even in more advanced mathematics, one can often learn the way to solve a general problem by trying a couple of specific examples. It’s almost as though there’s only a certain amount of abstraction people can deal with, and you need to re-cast problems so they stay within your limits.

Yes, the comments turn to complaining about Common Core. I’m not sure what would help Spud work through this problem (or problems in general). But thinking of alternate problems that estimated or approached what he really wanted might help. If he noticed, for example, that 10 + 12 has to be a little more than 10 + 10, and he found 10 + 10 easy, then he’d be close to a right answer. If he noticed that 10 + 12 had to be 10 + 10 + 2, and he found 10 + 10 easy, then he might find 20 + 2 easy as well. Maybe Spud would be better off thinking of ways to rewrite a problem without changing the result.

Wiley Miller’s Non Sequitur for the 15th mentions calculus. It’s more of a probability joke. To speak of a calculated risk is to speak of doing something that’s not certain, but that has enough of a payoff to be worth the cost of failure. But one problem with this attitude is that people are very, very bad at estimating probabilities. We have terrible ideas of how likely losses are and how uncertain rewards can be. But even if we allow that the risks and rewards are calculated right, there’s a problem with things you only do once. Or only can do once. You can get into a good debate about whether there’s even a meaningful idea of probability for things that happen only the one time. Life’s among them.

Bob Weber Sr’s Moose and Molly for the 16th is a homework joke. It does actually depend on being mathematics homework, though, or there’d be no grounds for Moose’s kid to go to the savings and loan clerk who’ll help with “money problems”.

I think there’s one more batch of comic strips to discuss this week. When I’ve published it, you should find the essay at this link. And then there’ll be Sunday again.

## Reading the Comics, November 29, 2018: Closing Out November Edition

Today, I get to wrap up November’s suggested discussion topics as prepared by Comic Strip Master Command.

Mark Tatulli’s Lio for the 28th features a cameo for mathematics. At least mathematics class. It’s painted as the most tedious part of the school day. I’m not sure this is quite right for Lio as a character. He’s clever in a way that I think harmonizes well with how mathematics brings out universal truths. But there is a difference between mathematics and mathematics class, of course.

Tom Toles’s Randolph Itch, 2am for the 28th shows how well my resolution to drop the strip from my rotation here has gone. I don’t seem to have found it worthy of mention before, though. It plays on the difference between a note of money, the number of units of currency that note represents, and between “zero” and “nothing”. Also I’m enchanted now by the idea that maybe some government might publish a zero-dollar bill. At least for the sake of movie and television productions that need realistic-looking cash.

In the footer joke Randolph mentions how you can never have enough zeroes. Yes, but I’d say that’s true of twenties, too. There is a neat sense in which this is true for working mathematicians, though. At least for those doing analysis. One of the reliable tricks that we learn to do in analysis is to “add zero” to a quantity. This is, literally, going from some expression that might be, say, “a – b” to “a + 0 – b”, which of course has the same value. The point of doing that is that we know other things equal to zero. For example, for any number L, “-L + L” is zero. So we get the original expression from “a + 0 – b” over to “a – L + L – b”. And that becomes useful is you picked L so that you know something about “a – L” and about “L – b”. Because then it tells you something about “a – b” that you didn’t know before. Picking that L, and showing something true about “a – L” and “L – b”, is the tricky part.

Dan Collins’s Looks Good On Paper for the 29th is back with another Möbius Strip comic strip. Last time it was presented as the “Möbius Trip”, a looping journey. This time it’s a comic strip proper. If this particular Looks Good On Paper has run before I don’t seem to have mentioned it. Unlike the “Möbius Trip” comic, this one looks more clearly like it actually is a Möbius strip.

The Dumpties in the comic strip are presented as getting nauseated at the strange curling around. It’s good sense for the comic-in-the-comic, which just has to have something happen and doesn’t really need to make sense. But there is no real way to answer where a Möbius strip wraps around itself. I mean, we can declare it’s at the left and right ends of the strip as we hold it, sure. But this is an ad hoc placement. We can roll the belt along a little bit, not changing its shape, but changing the points where we think of the strip as turning over.

But suppose you were a flat creature, wandering a Möbius strip. Would you have any way to tell that you weren’t on the plane? You could, but it takes some subtle work. Like, you could try drawing shapes. These let you count a thing called the Euler Characteristic, which relates the numer of vertices, edges, and faces of a polyhedron. The Euler Characteristic for a Möbius strip is the same as that for a Klein bottle, a cylinder, or a torus. You could try drawing regions, and coloring them in, calling on the four-color map theorem. (Here I want just to mention the five-color map theorem, which is as these things go easy to prove.) A map on the plane needs at most four colors to have no neighboring territories share a color along an edge. (Territories here are contiguous, and we don’t count territories meeting at only a point as sharing an edge.) Same for a sphere, which is good for we folks who have the job of coloring in both globes and atlases. It’s also the same for a cylinder. On a Möbius strip, this number is six. On a torus, it’s seven. So we could tell, if we were on a Möbius strip, that we were. It can be subtle to prove, is all.

All of my regular Reading the Comics posts should all be at this link. The next in my Fall 2018 Mathematics A To Z glossary should be posted Tuesday. I’m glad for it if you do come around and read again.

## Reading the Comics, November 24, 2018: Origins Edition

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

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

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

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

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

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

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

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

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

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

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

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

## Reading the Comics, November 16, 2018: The Rest Of The Week Edition

After that busy start last Sunday, Comic Strip Master Command left only a few things for the rest of the week. Here’s everything that seemed worthy of some comment to me:

Alex Hallatt’s Arctic Circle for the 12th is an arithmetic cameo. It’s used as the sort of thing that can be tested, with the straightforward joke about animal testing to follow. It’s not a surprise that machines should be able to do arithmetic. We’ve built machines for centuries to do arithmetic. Literally; Wilhelm Gottfried Leibniz designed and built a calculating machine able to add, subtract, multiply, and divide. This accomplishment from one of the founders of integral calculus is a potent reminder of how much we can accomplish if we’re supposed to be writing instead. (That link is to Robert Benchley’s classic essay “How To Get Things Done”. It is well worth reading, both because it is funny and because it’s actually good, useful advice.)

But it’s also true that animals do know arithmetic. At least a bit. Not — so far as we know — to the point they ponder square roots and such. But certainly to count, to understand addition and subtraction roughly, to have some instinct for calculations. Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics is a fascinating book about this. I’m only wary about going deeper into the topic since I don’t know a second (and, better, third) pop book touching on how animals understand mathematics. I feel more comfortable with anything if I’ve encountered it from several different authors. Anyway it does imply the possibility of testing a polar bear’s abilities at arithmetic, only in the real world.

Berkeley Breathed’s Bloom County rerun for the 13th has another mathematics cameo. Geometry’s a subject worthy of stoking Binkley’s anxieties, though. It has a lot of definitions that have to be carefully observed. And while geometry reflects the understanding we have of things from moving around in space, it demands a precision that we don’t really have an instinct for. It’s a lot to worry about.

Terry Border’s Bent Objects for the 15th is our Venn Diagram joke for the week. I like this better than I think the joke deserves, probably because it is done in real materials. (Which is the Bent Objects schtick; it’s always photographs of objects arranged to make the joke.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th is a joke on knowing how far to travel but not what direction. Normal human conversations carry contextually reasonable suppositions. Told something is two miles away, it’s probably along the major road you’re on, or immediately nearby. I’d still ask for clarification told something was “two miles away”. Two blocks, I’d let slide, on the grounds that it’s no big deal to correct a mistake.

Still, mathematicians carry defaults with them too. They might be open to a weird, general case, certainly. But we have expectations. There’s usually some obvious preferred coordinate system, or directions. If it’s important that we be ready for alternatives we highlight that. We specify the coordinate system we want. Perhaps we specify we’re taking that choice “without loss of generality”, that is, without supposing some other choice would be wrong.

I noticed the mathematician’s customized plate too. “EIPI1” is surely a reference to the expression $e^{\imath \pi} + 1$. That sum, it turns out, equals zero. It reflects this curious connection between exponentiation, complex-valued numbers, and the trigonometric functions. It’s a weird thing to know is true, and it’s highly regarded in certain nerd circles for that weirdness.

Hilary Price’s Rhymes With Orange for the 16th features a what-are-the-odds sort of joke, this one about being struck by a bolt from the sky. Lightning’s the iconic bolt to strike someone, and be surprising about it. Fabric would be no less surprising, though. And there’s no end of stories of weird things falling from the skies. It’s easier to get stuff into the sky than you might think, and there are only a few options once that’s happened.

And as ever, all my Reading the Comics posts should all be at this link.

Through the end of December my Fall 2018 Mathematics A To Z continues. I’m still open for topics to discuss from the last half-dozen letters of the alphabet. Even if someone’s already given a word for some letter, suggest something anyway. You might inspire me in good ways.

## Reading the Comics, November 9, 2018: Standing For Things Edition

There was something in common in two of the last five comic strips worth attention from last week. That’s good enough to give the essay its name.

Greg Cravens’s The Buckets for the 8th showcases Toby discovering the point of letters in algebra. It’s easy to laugh at him being ignorant. But the use of letters this way is something it’s easy to miss. You need first to realize that we don’t need to have a single way to represent a number. Which is implicit in learning, say, that you can write ‘7’ as the Roman numeral ‘VII’ or so, but I’m not sure that’s always clear. And realizing that you could use any symbol to write out ‘7’ if you agree that’s what the symbol means? That’s an abstraction tossed onto people who often aren’t really up for that kind of abstraction. And that we can have a symbol for “a number whose identity we don’t yet know”? Or even “a number whose identity we don’t care about”? Don’t blame someone for rearing back in confusion at this.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th talks about vectors and scalars. And about the little ways that instructors in one subject can sabotage one another. In grad school I was witness to the mathematics department feeling quite put-upon by the engineering departments, who thought we were giving their students inadequate calculus training. Meanwhile we couldn’t figure out what they were telling students about calculus except that it was screwing up their understanding.

To a physicist, a vector is a size and a direction together. (At least until they get seriously into mathematical physics when they need a more abstract idea.) A scalar is a number. Like, a real-valued number such as ‘4’. Maybe a complex-valued number such as ‘4 + 6i’. Vectors are great because a lot of physics problems become easier when thought of in terms of directions and amounts in that direction.

A mathematician would start out with vectors and scalars like that. But then she’d move into a more abstract idea. A vector, to a mathematician, is a thing you can add to another vector and get a vector out. A scalar is something that’s not a vector but that, multiplied by a vector, gets you a vector out. This sounds circular. But by defining ‘vector’ and ‘scalar’ in how they interact with each other we get a really sweet flexibility. We can use the same reasoning — and the same proofs — for lots of things. Directions, yes. But also matrices, and continuous functions, and probabilities of events, and more. That’s a bit much to give the engineering student who’s trying to work out some problem about … I don’t know. Whatever they do over in that department. Truss bridges or electrical circuits or something.

Mark Leiknes’s Cow and Boy for the 9th is really about misheard song lyrics, a subject that will never die now that we don’t have the space to print lyrics in the album lining anymore, or album linings. But it has a joke resonant with that of The Buckets, in supposing that algebra is just some bunch of letters mixed up with numbers. And Cow and Boy was always a strip I loved, as baffling as it might be to a casual reader. It had a staggering number of running jokes, although not in this installment.

Greg Evans’s Luann Againn for the 9th shows Brad happy to work out arithmetic when it’s for something he’d like to know. The figure Luan gives is ridiculously high, though. If he needs 500 hairs, and one new hair grows in each week, then that’s a little under ten years’ worth of growth. Nine years and a bit over seven months to be exact. If a moustache hair needs to be a half-inch long, and it grows at 1/8th of an inch per month, then it takes four months to be sufficiently long. So in the slowest possible state it’d be nine years, eleven months. I can chalk Luann’s answer up to being snidely pessimistic about his hair growth. But his calculator seems to agree and that suggests something went wrong along the way.

John Zakour and Scott Roberts’s Maria’s Day for the 9th is a story problem joke. It looks to me like a reasonable story problem, too: the distance travelled and the speed are reasonable, and give sensible numbers. The two stops add a bit of complication that doesn’t seem out of line. And the kid’s confusion is fair enough. It takes some experience to realize that the problem splits into an easy part, a hard part, and an easy part. The first easy part is how long the stops take all together. That’s 25 minutes. The hard part is realizing that if you want to know the total travel time it doesn’t matter when the stops are. You can find the total travel time by adding together the time spent stopped with the time spent driving. And the other easy part is working out how long it takes to go 80 miles if you travel at 55 miles per hour. That’s just a division. So find that and add to it the 25 minutes spent at the two stops.

The various Reading the Comics posts should all be at this link. Essays which discuss The Buckets are at this link. The incredibly many essays mentioning Saturday Morning Breakfast Cereal are at this link. Essays which mention Cow and Boy are at this link. Essays inspired in part by Luann, both the current-day and the vintage 1990 run, are at this link. The credibly many essays mentioning Maria’s Day are at this link.

And through the end of December my Fall 2018 Mathematics A-To-Z should have two new posts a week. You might like some of them.

## Reading the Comics, November 5, 2018: November 5, 2018 Edition

This past week included one of those odd days that’s so busy I get a column’s worth of topics from a single day’s reading. And there was another strip (the Cow and Boy rerun) which I might have roped in had the rest of the week been dead. The Motley rerun might have made the cut too, for a reference to $E = mc^2$.

Jason Chatfield’s Ginger Meggs for the 5th is a joke about resisting the story problem. I’m surprised by the particulars of this question. Turning an arithmetic problem into counts of some number of particular things is common enough and has a respectable history. But slices of broccoli quiche? I’m distracted by the choice, and I like quiche. It’s a weird thing for a kid to have, and a weird amount for anybody to have.

JC Duffy’s Lug Nuts for the 5th uses mathematics as a shorthand for intelligence. And it particularly uses π as shorthand for mathematics. There’s a lot of compressed concepts put into this. I shouldn’t be surprised if it’s rerun come mid-March.

Tom Toles’s Randolph Itch, 2 am for the 5th I’ve highlighted before. It’s the pie chart joke. It will never stop amusing me, but I suppose I should take Randolph Itch, 2 am out of my rotation of comics I read to include here.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th is a logic puzzle joke. And a set theory joke. Dad is trying to argue he can’t be surprised by his gift because it’ll belong to one of two sets of things. And he receives nothing. This ought to defy his expectations, if we think of “nothing” as being “the empty set”. The empty set is an indispensable part of set theory. It’s a set that has no elements, has nothing in it. Then suppose we talk about what it means for one set to be contained in another. Take what seems like an uncontroversial definition: set A is contained in set B if there’s nothing in A which is not also in B. Then the empty set is contained inside every set. So Dad, having supposed that he can’t be surprised, since he’d receive either something that is “socks” or something that is “not-socks”, does get surprised. He gets the one thing that is both “socks” and “not-socks” simultaneously.

I hate to pull this move a third time in one week (see here and here), but the logic of the joke doesn’t work for me. I’ll go along with “nothing” as being “the empty set” for these purposes. And I’ll accept that “nothing” is definitely “not-socks”. But to say that “nothing” is also “socks” is … weird, unless you are putting it in the language of set theory. I think the joke would be saved if it were more clearly established that Dad should be expecting some definite thing, so that no-thing would defy all expectations.

“Nothing” is a difficult subject to treat logically. I have been exposed a bit to the thinking of professional philosophers on the subject. Not enough that I feel I could say something non-stupid about the subject. But enough to say that yeah, they’re right, we have a really hard time describing “nothing”. The null set is better behaved. I suppose that’s because logicians have been able to tame it and give it some clearly defined properties.

Mike Shiell’s The Wandering Melon for the 5th felt like a rerun to me. It wasn’t. But Shiell did do a variation on this joke in August. Both are built on the same whimsy of probability. It’s unlikely one will win a lottery. It’s unlikely one will die in a particular and bizarre way. What are the odds someone would have both things happen to them?

This and every Reading the Comics post should be at this link. Essays that include Ginger Meggs are at this link. Essays in which I discuss Lug Nuts are at this link. Essays mentioning Randolph Itch, 2 am, should be at this link. The many essays with a mention of Saturday Morning Breakfast Cereal are at this link. And essays where I’m inspired by something in The Wandering Melon should be at this link. And, what the heck, when I really discuss Cow and Boy it’s at this link. Real discussions of Motley are at this link. And my Fall 2018 Mathematics A-To-Z averages two new posts a week, now and through December. Thanks again for reading.

## Reading the Comics, October 27, 2018: Surprise Rerun Edition

While putting together the last comics from a week ago I realized there was a repeat among them. And a pretty recent repeat too. I’m supposing this is a one-off, but who can be sure? We’ll get there. I figure to cover last week’s mathematically-themed comics in posts on Wednesday and Thursday, subject to circumstances.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th is a joking reminder that educational texts, including in mathematics, don’t have to be boring. We can have narrative thrust and energy. It’s a good reminder.

As fits the joke, the bit of calculus in this textbook paragraph is wrong. $\int \sqrt{x^2 + x} dx$ does not equal $\left(x^2 + x\right)^{-\frac12}$. This is even ignoring that we should expect, with an indefinite integral like this, a constant of integration. An indefinite integral like this is equal to a family of related functions. But it’s common shorthand to write out one representative function. But the indefinite integral of $\sqrt{x^2 + x}$ is not $\left(x^2 + x\right)^{-\frac12}$. You can confirm that by differentiating $\left(x^2 + x\right)^{-\frac12}$. The result is nothing like $\sqrt{x^2 + x}$. Differentiating an indefinite integral should get the original function back. Here are the rules you need to do that for yourself.

As I make it out, a correct indefinite integral would be:

$\int{\sqrt{x^2 + x} dx} = \frac{1}{4}\left( \left(2x + 1\right)\sqrt{x^2 + x} + \log \left|\sqrt{x} + \sqrt{x + 1} \right| \right)$

Plus that “constant of integration” the value of which we can’t tell just from the function we want to indefinitely-integrate. I admit I haven’t double-checked that I’m right in my work here. I trust someone will tell me if I’m not. I’m going to feel proud enough if I can get the LaTeX there to display.

Stephen Beals’s Adult Children for the 27th has run already. It turned up in late March of this year. Michael Spivak’s Calculus is a good choice for representative textbook. Calculus holds its terrors, too. Even someone who’s gotten through trigonometry can find the subject full of weird, apparently arbitrary rules. And formulas like those in the above paragraph.

Rob Harrell’s Big Top for the 27th is a strip about the difficulties of splitting a restaurant bill. And they’ve not even got to calculating the tip. (Maybe it’s just a strip about trying to push the group to splitting the bill a way that lets you off cheap. I haven’t had to face a group bill like this in several years. My skills with it are rusty.)

Dave Whamond’s Reality Check for the 27th is a Pi Day joke shifted to the Halloween season.

And I have more Reading the Comics post at this link. Since it’s not true that every one of these includes a Saturday Morning Breakfast Cereal mention, you can find those that have one at this link. Essays discussing Adult Children, including the first time this particular strip appeared, are at this link. Essays with a mention of Big Top are at this link. And essays with a mention of Reality Check are at this link. Furthermore, this month and the rest of this year my Fall 2018 Mathematics A-To-Z should continue. And it is open for requests for more of the alphabet.

## Reading the Comics, October 19, 2018: More Short Things Edition

At least, I’d thought the last half of last week’s comics were mostly things I could discuss quickly. Then Frank and Ernest went and sprawled on me. Such will happen.

Before I get to that, I did want to mention that Gregory Taylor’s paneling for votes for the direction his mathematics-inspired serial takes:

You may enjoy; at least, give it a try.

Thaves’s Frank and Ernest for the 18th is a bit of wordplay. There’s something interesting culturally about phrasing “lots of math, but no chemistry”. Algorithms as mathematics makes sense. Much of mathematics is about finding processes to do interesting things. Algorithms, and the mathematics which justifies them, can at least in principle be justified with deductive logic. And we like to think that the universe must make deductive-logical sense. So it is easy to suppose that something mathematical simply must make logical sense.

Chemistry, though. It’s a metaphor for whatever the difference is between a thing’s roster of components and the effect of the whole. The suggestion is that it is mysterious and unpredictable. It’s an attitude strange to actual chemists, who have a rather good understanding of why most things happen. My suspicion is that this sense of chemistry is old, dating to before we had a good understanding of why chemical bonds work. We have that understanding thanks to quantum mechanics, and its mathematical representations.

But we can still allow for things that happen but aren’t obvious. When we write about “emergent properties” we describe things which are inherent in whatever we talk about. But they only appear when the things are a large enough mass, or interact long enough. Some things become significant only when they have enough chance to be seen.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is about mathematicians’ favorite Ancient Greek philosopher they haven’t actually read. (In fairness, Zeno is hard to read, even for those who know the language.) Zeno’s famous for four paradoxes, the most familiar of which is alluded to here. To travel across a space requires travelling across half of it first. But this applies recursively. To travel any distance requires accomplishing infinitely many partial-crossings. How can you do infinitely many things, each of which take more than zero time, in less than an infinitely great time? But we know we do this; so, what aren’t we understanding? A callow young mathematics major would answer: well, pick any tiny interval of time you like. All but a handful of the partial-crossings take less than your tiny interval time. This seems like a sufficient answer and reason to chuckle at philosophers. Fine; an instant has zero time elapse during it. Nothing must move during that instant, then. So when does movement happen, if there is no movement during all the moments of time? Reconciling these two points slows the mathematician down.

Patrick Roberts’s Todd the Dinosaur for the 19th mentions fractions. It’s only used to list a kind of mathematics problem a student might feign unconsciousness rather than do. And takes quite little space in the word balloon to describe. It’d be the same joke if Todd were asked to come up and give a ten-minute presentation on the Battle of Bunker Hill.

Julie Larson’s The Dinette Set for the 19th mentions the Rubik’s Cube. Sometime I should do a proper essay about its mathematics. Any Rubik’s Cube can be solved in at most 20 moves. And it’s apparently known there are some cube configurations that take at least 20 moves, so, that’s nice to have worked out. But there are many approaches to solving a cube, none of which I am competent to do. Some algorithms are, apparently, easier for people to learn, at the cost of taking more steps. And that’s fine. You should understand something before you try to do it efficiently.

John Atkinson’s Wrong Hands for the 19th is the Venn Diagram joke for the week. Good to have one around.

## Reading the Comics, October 11, 2018: Under Weather Edition

I ended up not finding more comics on-topic on GoComics yesterday. So this past week’s mathematically-themed strips should fit into two posts well. I apologize for any loss of coherence in this essay, as I’m getting a bit of a cold. I’m looking forward to what this cold does for the A To Z essays coming Tuesday and Friday this week, too.

Stephen Beals’s Adult Children for the 7th uses Albert Einstein’s famous equation as shorthand for knowledge. I’m a little surprised it’s written out in words, rather than symbols. This might reflect that $E = mc^2$ is often understood just as this important series of sounds, rather than as an equation relating things to one another. Or it might just reflect the needs of the page composition. It could be too small a word balloon otherwise.

Julie Larson’s The Dinette Set for the 9th continues the thread of tip-calculation jokes around here. I have no explanation for this phenomenon. In this case, Burl is doing the calculation correctly. If the tip is supposed to be 15% of the bill, and the bill is reduced 10%, then the tip would be reduced 10%. If you already have the tip calculated, it might be quicker to figure out a tenth of that rather than work out 15% of the original bill. And, yes, the characters are being rather unpleasantly penny-pinching. That was just the comic strip’s sense of humor.

Todd Clark’s Lola for the 9th take the form of your traditional grumbling about story problems. It also shows off the motif of updating of the words in a story problem to be awkwardly un-hip. The problem seems to be starting in a confounding direction anyway. The first sentence isn’t out and it’s introducing the rate at which Frank is shedding social-media friends over time and the rate at which a train is travelling, some distancer per time. Having one quantity with dimensions friends-per-time and another with dimensions distance-per-time is begging for confusion. Or for some weird gibberish thing, like, determining something to be (say) ninety mile-friends. There’s trouble ahead.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th proposes naming a particular kind of series. A series is the sum of a sequence of numbers. It doesn’t have to be a sequence with infinitely many numbers in it, but it usually is, if it’s to be an interesting series. Properly, a series gets defined by something like the symbols in the upper caption of the panel:

$\sum_{i = 1}^{\infty} a_i$

Here the ‘i’ is a “dummy variable”, of no particular interest and not even detectable once the calculation is done. It’s not that thing with the square roots of -1 in thise case. ‘i’ is specifically known as the ‘index’, since it indexes the terms in the sequence. Despite the logic of i-index, I prefer to use ‘j’, ‘k’, or ‘n’. This avoids confusion with that square-root-of-minus-1 meaning for i. The index starts at some value, the one to the right of the equals sign underneath the capital sigma; in this case, 1. The sequence evaluates whatever the formula described by $a_i$ is, for each whole number between that lowest ‘i’, in this case 1, and whatever the value above the sigma is. For the infinite series, that’s infinitely large. That is, work out $a_i$ for every counting number ‘i’. For the first sum in the caption, that highest number is 4, and you only need to evaluate four terms and add them together. There’s no rule given for $a_i$ in the caption; that just means that, in this case, we don’t yet have reason to care what the formula is.

This is the way to define a series if we’re being careful, and doing mathematics properly. But there are shorthands, and we fall back on them all the time. On the blackboard is one of them: $24 + 12 + 6 + 3 + \cdots$. The $\cdots$ at the end of a summation like this means “carry on this pattern for infinitely many terms”. If it appears in the middle of a summation, like $2 + 4 + 6 + 8 + \cdots + 20$ it means “carry on this pattern for the appropriate number of terms”. In that case, it would be $10 + 12 + 14 + 16 + 18$.

The flaw with this “carry on this pattern” is that, properly, there’s no such thing as “the” pattern. There are infinitely many ways to continue from whatever the start was, and they’re all equally valid. What lets this scheme work is cultural expectations. We expect the difference between one term and the next to follow some easy patterns. They increase or decrease by the same amount as we’ve seen before (an arithmetic progression, like 2 + 4 + 6 + 8, increasing by two each time). They increase or decrease by the same ratio as we’ve seen before (a geometric progression, like 24 + 12 + 6 + 3, cutting in half each time). Maybe the sign alternates, or changes by some straightforward rule. If it isn’t one of these, then we have to fall back on being explicit. In this case, it would be that $a_i = 24 \cdot \left(\frac{1}{2}\right)^{i - 1}$.

The capital-sigma as shorthand for “sum” traces to Leonhard Euler, because of course. I’m finding it hard, in my copy of Florian Cajori’s History of Mathematical Notations, to find just where the series notation as we use it got started. Also I’m not finding where ellipses got into mathematical notation either. It might reflect everybody realizing this was a pretty good way to represent “we’re not going to write out the whole thing here”.

Norm Feuti’s Retail for the 11th riffs on how many people, fundamentally, don’t know what percentages are. I think it reflects thinking of a percentage as some kind of unit. We get used to measurements of things, like, pounds or seconds or dollars or degrees or such that are fixed in value. But a percentage is relative. It’s a fraction of some original quantity. A difference of (say) two pounds in weight is the same amount of weight whatever the original was; why wouldn’t two percent of the weight behave similarly? … Gads, yes, I feel for the next retailer who gets these customers.

I think I’ve already used the story from when I worked in the bookstore about the customer concerned whether the ten-percent-off sticker applied before or after sales tax was calculated. So I’ll only share if people ask to hear it. (They won’t ask.)

When I’m not getting a bit ill, I put my Reading the Comics posts at this link. Essays which mention Adult Children are at this link. Essays with The Dinette Set discussions should be at this link. The essays inspired by Lola are at this link. There’s some mention of Saturday Morning Breakfast Cereal in essays at link, or pretty much every Reading the Comics post. And Retail gets discussed at this link.

## Reading the Comics, September 22, 2018: Last Chance Edition

I plan tomorrow to have another of my Mathematics A To Z posts. This weekend I’ll publish this month’s Playful Mathematics Blog Carnival. So if you’ve seen any web site, blog, video, podcast, or other reference that had something that delighted and taught you something, this is your last chance to let me know, and let my audience know about it. Please leave a comment if you know about anything I ought to see. Thank you.

Mark Tatulli’s Lio for the 20th is a numerals and a wordplay joke. It is not hard to make numerals tattooed on a person an alarming thing. But when done with (I trust) the person’s consent, and done whimsically like this, it’s more a slightly odd bit of play.

Tony Cochrane’s Agnes for the 21st is ultimately a strip about motivating someone to learn arithmetic. Agnes’s reasoning is sound, though. If the only reason to learn this unpleasant chore is because your job may need it, why not look at another job? We wouldn’t try to convince someone who didn’t want to learn French that they’ll need it for their job as … a tour guide in Quebec? There’s plenty of work that doesn’t need that. I suspect kids don’t buy “this is good for your future job” as a reason. Even if it were, general education should not be job training either.

Juba’s Viivi and Wagner for the 21st gives Wagner a short-lived ambition to be a wandering mathematician. The abacus serves as badge of office. There are times and places that his ambition wouldn’t be completely absurd. Before the advent of electric and electronic computing, people who could calculate were worth hiring for their arithmetic. In 18th Century London there was a culture of “penny universities”, people with academic training making a living by giving lectures and courses to whatever members of the public cared to come to their talk, often in coffee-houses or barns.

Mathematicians learn that there used to be public spectacles, mathematicians challenging one another to do problems, with real cash or jobs on the line. They learn this because one such challenge figures in to the story of Gerolamo Cardano and Niccolò Fontana, known as Tartaglia. It’s about how we learned formulas to solve some kinds of polynomials. You may sense uncertainty in my claim there. It’s because it turns out it’s hard to find clear records of this sort of challenge outside the Cardano-Tartaglia match. That isn’t to say these things weren’t common. It’s just that I’ve been slowly learning to be careful about my claims.

(I’m aided here by a startling pair of episodes of The History of Philosophy Without Any Gaps podcast. This pair — “Trivial Pursuits: Fourteenth Century Logic” and “Sara Uckleman on Obligations” — describe a fascinating logic game that sounds like it would still be a great party game, for which there’s numerous commentaries and rule sets and descriptions of how to play. But no records of people actually ever playing it, or talking about games they had played, or complaining about being cheated out of a win or stuff like that. It’s a strong reminder to look closely at what your evidence does support.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 22nd is the comforting return of Zach Weinersmith to these essays. And yes, it’s horrible parenting to promise something fun and have it turn out to be a mathematics lecture, but that’s part of the joke.

Karl Weierstrass was a real person, and a great mathematician best known for giving us a good, rigorous idea of what a limit is. We need limits because, besides their being nice things to have, calculus depends on them. At least, calculus depends on thinking about calculations on infinitely many things. Or on things infinitesimally small. Trying to do this works pretty well, much of the time. But you can also start calculating like this and get nonsense. How to tell whether your particular calculation works out or is nonsense?

Weierstrass worked out a good, rigorous idea for what we mean by a limit. It mostly tracks with what we’d intuitively expect. And it avoids all the dangerous spots we’ve noticed so far. Particularly, it doesn’t require us to ever look at anything that’s infinitely vast, or infinitesimally small. Anything we calculate on is done with regular arithmetic, that we’re quite confident in. But it lets us draw conclusions about the infinitely numerous or tiny. It’s brilliant work. When it’s presented to someone in the start of calculus, it leaves them completely baffled but they can maybe follow along with the rules. When it’s presented to mathematics majors in real analysis, it leaves them largely baffled but they can maybe follow along with the reasons. Somewhere around grad school I got comfortable with it, even excited. Weierstrass’s sort of definition turns up all over the place in real and in functional analysis. So at the least you get very comfortable with it.

So it is part of Weinersmith’s joke that this is way above that kid’s class level. As a joke, that fails for me. The luchador might as well be talking complete nonsense and the kid would realize that right away. There’s not the threat that this is something he ought to be able to understand. But it will probably always be funny to imagine mathematician wrestlers. Can count on that. I didn’t mean that as a joke, but you’ll notice I’m letting it stand.

And with that, you know what I figure to post on Sunday. It and my other Reading the Comics posts should be at this tag. Other appearances of Lio should be at this link. The mentions of Agnes should be at this link. Essays with some mention of Viivi and Wagner will be at this link, although it’s a new tag, so who knows how long it’ll take for the next to appear? And other essays with Saturday Morning Breakfast Cereal will be at this link when there’s any to mention.

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

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

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

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

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

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

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

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

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

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

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

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

## Reading the Comics, August 16, 2018: Recursive Edition

This edition of Reading the Comics can be found at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is a fractals joke. Benoit Mandelbrot became the centerpiece of the big fractals boom in pop mathematics in the 80s and 90s. This was thanks to a fascinating property of complex-valued numbers that he discovered and publicized. The Mandelbrot set is a collection of complex-valued numbers. It’s a border, properly, between two kinds of complex-valued numbers. This boundary has this fascinating shape that looks a bit like a couple kidney beans surrounded by lightning. That’s neat enough.

What’s amazing, and makes this joke anything, is what happens if you look closely at this boundary. Anywhere on it. In the bean shapes or in the lightning bolts. You find little replicas of the original shape. Not precisely the original shape. No two of these replicas are precisely identical (except for the “complex conjugate”, that is, something near the number $-1 + 1 \imath$ has a mirror image near $-1 - 1 \imath$). None of these look precisely like the original shape. But they look extremely close to one another. They’re smaller, yes, and rotated relative to the original, and to other copies. But go anywhere on this boundary and there it is: the original shape, including miniature imperfect copies, all over again.

The Mandelbrot Set itself — well, there are a bunch of ways to think of it. One is in terms of something called the Julia Set, named for Gaston Julia. In 1918 he published a massive paper about the iteration of rational functions. That is, start with some domain and a function rule; what’s the range? Now if we used that range as the domain again, and used the same rule for the function, what’s the range of that range? If we use the range-of-that-range as the domain for the same function rule, what’s the range-of-the-range-of-the-range? The particular function here has one free parameter, a single complex-valued number. Depending on what it is, the range-of-the-range-of-the-range-etc becomes a set that’s either one big blob or a bunch of disconnected blobs. The Mandelbrot Set is the locus of parameters separating the one-big-blob from the many-disconnected-blob outcomes.

By the way, yes, Julia published this in 1918. The work was amazing. It was also forgotten. You can study this stuff analytically, but it’s hard. To visualize it you need to do incredible loads of computation. So this is why so much work lay fallow until the 1970s, when Mandelbrot could let computers do incredible loads of computation, and even draw some basic pictures.

Doug Savage’s Savage Chickens for the 14th is another instance of the monkeys-at-typewriters joke. I’ve written about this and the history of the monkeys-at-typewriters bit recently enough to feel comfortable pointing people there. It’s interesting that monkeys should have a connotation of reliably random typewriting, while cats would be reliably not doing something. But that’s a cultural image that’s a little too far from being mathematics for me to spend 800 words discussing.

Thom Bleumel’s Birdbrains for the 15th is a calendars joke. Numbers come into play since, well, it seems odd to try tracking large numbers of dates without some sense of arithmetic. Also, likely, without some sense of geometry. Calendars are much used to forecast coming events, such as New and Full Moons or the seasons. That takes basic understanding of how to locate things in the sky to do at all. It takes sophisticated understanding of how to locate things in the sky to do well.

Scott Hilburn’s The Argyle Sweater for the 16th is the first anthropomorphic-numerals joke around here in like three days. Certainly, the scandalous thing is supposed to be these numbers multiplying out in public where anyone might see them. I wonder if any part of the scandal should be that multiplication like this has to include three partners: the 4, the 7, and the x. In algebra we get used to a convention in which we do without the ‘x’. Just placing one term next to another carries an implicit multiplication: ‘4a’ for ‘4 times a’. But that convention fails disastrously with numerals; what should we make of ’47’? We might write 4(7), or maybe (4)(7), to be clear. Or we might put a little centered dot between the two, $4 \cdot 7$. The ‘x’ by that point is reserved for “some variable whose value isn’t specified”. And it would be weird to write ‘4 times x times 7’. It wouldn’t be wrong; it’d just look weird. It would suggest you were trying to emphasize a point. I’ve probably done it in one of my long derivation-happy posts.

Other essays about comic strips are at this link. When I’ve talked about Saturday Morning Breakfast Cereal I’ve tried to make sure it turns up at this link. Essays in which I’ve discussed Savage Chickens should be at this link. The times I’ve discussed Birdbrains should be at this link. And other essays describing The Argyle Sweater are at this link.

## Reading the Comics, August 4, 2018: August 4, 2018 Edition

And finally, at last, there’s a couple of comics left over from last week and that all ran the same day. If I hadn’t gone on forever about negative Kelvin temperatures I might have included them in the previous essay. That’s all right. These are strips I expect to need relatively short discussions to explore. Watch now as I put out 2,400 words explaining Wavehead misinterpreting the teacher’s question.

Dave Whamond’s Reality Check for the 4th is proof that my time spent writing about which is better, large numbers or small last week wasn’t wasted. There I wrote about four versus five for Beetle Bailey. Here it’s the same joke, but with compound words. Well, that’s easy to take care of.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is driving me slightly crazy. The equation on the board looks like an electrostatics problem to me. The ‘E’ is a common enough symbol for the strength of an electric field. And the funny-looking K’s look to me like the Greek kappa. This often represents the dielectric constant. That measures how well an electric field can move through a material. The upside-down triangles, known in the trade as Delta, describe — well, that’s getting complicated. By themselves, they describe measuring “how much the thing right after this changes in different directions”. When there’s a x symbol between the Delta and the thing, it measures something called the “curl”. This roughly measures how much the field inspires things caught up in it to turn. (Don’t try passing this off to your thesis defense committee.) The Delta x Delta x E describes the curl of the curl of E. Oh, I don’t like visualizing that. I don’t blame you if you don’t want to either.

Anyway. So all this looks like it’s some problem about a rod inside an electric field. Fine enough. What I don’t know and can’t work out is what the problem is studying exactly. So I can’t tell you whether the equation, so far as we see it, is legitimately something to see in class. Envisioning a rod that’s infinitely thin is a common enough mathematical trick, though. Three-dimensional objects are hard to deal with. They have edges. These are fussy to deal with. Making sure the interior, the boundary, and the exterior match up in a logically consistent way is tedious. But a wire? A plane? A single point? That’s easy. They don’t have an interior. You don’t have to match up the complicated stuff.

For real world problems, yeah, you have to deal with the interior. Or you have to work out reasons why the interiors aren’t important in your problem. And it can be that your object is so small compared to the space it has to work in that the fact it’s not infinitely thin or flat or smooth just doesn’t matter. Mathematical models, such as give us equations, are a blend of describing what really is there and what we can work with.

Mike Shiell’s The Wandering Melon for the 4th is a probability joke, about two events that nobody’s likely to experience. The chance any individual will win a lottery is tiny, but enough people play them that someone wins just about any given week. The chance any individual will get struck by lightning is tiny too. But it happens to people. The combination? Well, that’s obviously impossible.

In July of 2015, Peter McCathie had this happen. He survived a lightning strike first. And then won the Atlantic Lotto 6/49. This was years apart, but the chance of both happening the same day, or same week? … Well, the world is vast and complicated. Unlikely things will happen.

And that’s all that I have for the past week. Come Sunday I should have my next Reading the Comics post, and you can find it and other essays at this link. Other essays that mention Reality Check are at this link. The many other essays which talk about Saturday Morning Breakfast Cereal are at this link. And other essays about The Wandering Melon are at this link. Thanks.

## Reading the Comics, July 7, 2018: Mutt and Jeff Relettering Scandal Edition

I apologize for not having a more robust introduction here. My week’s been chopped up by concern with the health of the older of our rabbits. Today’s proved to be less alarming than we had feared, but it’s still a lot to deal with. I appreciate your kind thoughts. Thank you.

Meanwhile the comics from last week have led me to discover something really weird going on with the Mutt and Jeff reruns.

Charles Schulz’s Peanuts Classics for the 6th has the not-quite-fully-formed Lucy trying to count the vast. She’d spend a while trying to count the stars and it never went well. It does inspire the question of how to count things when doing a simple tally is too complicated. There are many mathematical approaches. Most of them are some kind of sampling. Take a small enough part that you can tally it, and estimate the whole based on what your sample is. This can require ingenuity. For example, when estimating our goldfish population, it was impossible to get a good sample at one time. When tallying the number of visible stars in the sky, we have the problem that the Galaxy has a shape, and there are more stars in some directions than in others. This is why we need statisticians.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th looks initially like it’s meant for a philosophy blog’s Reading the Comics post. It’s often fruitful in the study of ethics to ponder doing something that is initially horrible, but would likely have good consequences. Or something initially good, but that has bad effects. These questions challenge our ideas about what it is to do good or bad things, and whether transient or permanent effects are more important, and whether it is better to be responsible for something (or to allow something) by action or inaction.

It comes to mathematics in the caption, though, and with an assist from the economics department. Utilitarianism seems to offer an answer to many ethical problems. It posits that we need to select a primary good of society, and then act so as to maximize that good. This does have an appeal, I suspect even to people who don’t thrill of the idea of finding the formula that describes society. After all, if we know the primary good of society, why should we settle for anything but the greatest value of that good? It might be difficult in practice, say, to discount the joy a musician would bring over her lifetime with her performances fairly against the misery created by making her practice the flute after school when she’d rather be playing. But we can imagine working with a rough approximation, at least. Then the skilled thinkers point out even worse problems and we see why utilitarianism didn’t settle all the big ethical questions, even in principle.

The mathematics, though. As Weinersmith’s caption puts it, we can phrase moral dilemmas as problems of maximizing evil. Typically we pose them as ones of maximizing good. Or at least of minimizing evil. But if we have the mechanism in place to find where evil is maximized, don’t we have the tools to find where good is? If we can find the set of social parameters x, y, and z which make E(x, y, z) as big as possible, can’t we find where -E(x, y, z) is as big, too? And isn’t that then where E(x, y, z) has to be smallest?

And, sure. As long as the maximum exists, or the minimum exists. Maybe we can tell whether or not there is one. But this is why when you look at the mathematics of finding maximums you realize you’re also doing minimums, or vice-versa. Pretty soon you either start referring to what you find as extremums. Or you stop worrying about the difference between a maximum and a minimum, at least unless you need to check just what you have found. Or unless someone who isn’t mathematically expert looks at you wondering if you know the difference between positive and negative numbers.

Bud Fisher’s Mutt and Jeff for the 7th has run here before. Except that was before they redid the lettering; it was a roast beef in earlier iterations. I was thinking to drop Mutt and Jeff from my Reading the Comics routine before all these mysteries in the lettering turned up. Anyway. The strip’s joke starts with a work-rate problems. Given how long some people take to do a thing, how long does it take a different number of people to do a thing? These are problems that demand paying attention to units, to the dimensions of a thing. That seems to be out of fashion these days, which is probably why these questions get to be baffling. But if eating a ham takes 25 person-minutes to do, and you have ten persons eating, you can see almost right away how long to expect it to take. If the ham’s the same size, anyway.

Olivia Jaimes’s Nancy for the 7th is built on a spot of recreational mathematics. Also on the frustration one can have when a problem looks like it’s harmless innocent fun and turns out to take just forever and you’re never sure you have the answers just right. The commenters on GoComics.com have settled on 18. I’m content with that answer.

Care for more of this? You can catch all my Reading the Comics posts at this link. Essays with Saturday Morning Breakfast Cereal content are at this link. Essays with Peanuts are at this link. Those with Mutt and Jeff are at this link. And those with Nancy are here. Thank you.

## Reading the Comics, June 29, 2018: Chuckle and Breakfast Cereal Edition

The last half of last week was not entirely the work of Chuckle Brothers and Saturday Morning Breakfast Cereal. It seemed like it, though. Let’s review.

Patrick Roberts’s Todd the Dinosaur for the 28th is a common sort of fear-of-mathematics joke. In this case the fear of doing arithmetic even when it is about something one would really like to know. I think the question got away from Todd, though. If they just wanted to know whether they had enough money, well, they need twelve dollars and have seven. Subtracting seven from twelve is only needed if they want to know how much more they need. Which they should want to know, but wasn’t part of the setup.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 28th uses mathematics as the sine qua non of rocket science. As in, well, the stuff that’s hard and takes some real genius to understand. It’s not clear to me that the equations are actually rocket science. There seem to be a shortage of things in exponentials to look quite right to me. But I can’t zoom in on the art, so, who knows just what might be in there.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a set theory joke. Or a logic joke, anyway. It refers to some of the mathematics/logic work of Bertrand Russell. Among his work was treating seriously the problems of how to describe things defined in reference to themselves. These have long been a source of paradoxes, sometimes for fun, sometimes for fairy-tale logic, and sometimes to challenge our idea of what we mean by definitions of things. Russell made a strong attempt at describing what we mean when we describe a thing by reference to itself. The iconic example here was the “set of all sets not members of themselves”.

Russell started out by trying to find some way to prove Georg Cantor’s theorems about different-sized infinities wrong. He worked out a theory of types, and what kinds of rules you can set about types of things. Most mathematicians these days prefer to solve the paradox with a particular organization of set theory. But Russell’s type theory still has value, particularly as part of the logic behind lambda calculus. This is an approach to organizing relationships between things that can do wonderful things, including in computer programming. It lets one write code that works extremely efficiently and can never be explained to another person, modified, or debugged ever. I may lack the proper training for the uses I’ve made of it.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 29th is a lottery joke. It does happen that more than one person wins a drawing; sometimes three or even four people do, for the larger prizes. The chance that there’s a million winners? Frightfully unlikely unless something significant went wrong with the lottery mechanism.

So what are the chances of a million lottery winners? If I’m not mistaken the only way to do this is to work out a binomial distribution. The binomial distribution is good for cases where you have many attempts at doing a thing, where each thing can either succeed or fail, and the likelihood of success or failure is independent of all the other attempts. In this case each lottery ticket is an attempt; it winning is success and it losing is failure. Each ticket has the same chance of winning or losing, and that chance doesn’t depend on how many wins or losses there are. What is that chance? … Well, if each ticket has one chance in a million of winning, and there are a million tickets out there, the chance of every one of them winning is about one-millionth raised to the millionth power. Which is so close to zero it might as well be nothing. … And yet, for all that it’s impossible, there’s not any particular reason it couldn’t happen. It just won’t.

Jef Mallet’s Frazz for the 29th is a less dire take on what-you-learned-this-year. In this case it’s trivia, but it’s a neat sort of trivia. Once you understand how it works you can understand how to make all sorts of silly little divisibility rules. The threes rule — and the nines rule — work by the same principle. Suppose you have a three-digit number. Let me call ‘a’ the digit in the hundreds column, ‘b’ the digit in the tens column, and ‘c’ the digit in the ones column. Then the number is equal to $100\cdot a + 10\cdot b + 1\cdot c$. And, well, that’s equal to $99\cdot a + 1\cdot a + 9 \cdot b + 1 \cdot b + 1 \cdot c$. Which is $99\cdot a + 9 \cdot b + a + b + c$. 99 times any whole number is a multiple of 9, and also of 3. 9 times any whole number is a multiple of 9, and also of 3. So whether the original number is divisible by 9, or by 3, depends on whether $a + b + c$ is. And that’s why adding the digits up tells you whether a number is a whole multiple of three.

This has only proven anything for three-digit numbers. But with that proof in mind, you probably can imagine what the proof looks like for two- or four-digit numbers, and would believe there’s one for five- and for 500-digit numbers. Or, for that matter, the proof for an arbitrarily long number. So I’ll skip actually doing that. You can fiddle with it if you want a bit of fun yourself.

Also maybe it’s me, or the kind of person who gets into mathematics. But I find silly little rules like this endearing. It’s a process easy to understand that anyone can do and it tells you something not obvious from when you start. It feels like getting let in on a magic trick. That seems like the sort of thing that endears people to mathematics.

Mike Thompson’s Grand Avenue for the 29th is trying to pick its fight with me again. I can appreciate someone wanting to avoid kids losing their mathematical skills over summer. It’s just striking how Thompson has consistently portrayed their grandmother as doing this in a horrible, joy-crushing manner.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th gets into a philosophy-of-mathematics problem. Also a pure philosophy problem. It’s a problem of what things you can know independently of experience. There are things it seems as though are true, and that seem independent of the person who is aware of them, and what culture that person comes from. All right. Then how can these things be relevant to the specifics of the universe that we happen to be in just now? If ‘2’ is an abstraction that means something independent of our universe, how can there be two books on the table? There’s something we don’t quite understand yet, and it’s taking our philosophers and mathematicians a long while to work out what that is.

And as ever, if you’d like to see more Reading the Comics posts, please look to this page. For essays with Todd the Dinosaur in them, look here. For essays with the Chuckle Brothers, here you go. For some of the many, many essays with Saturday Morning Breakfast Cereal, follow this link. For more talk about Frazz, look here. And for the Grand Avenue comics, try this link please.

## Reading the Comics, June 27, 2018: Stitch Day Edition

For a while I thought this essay would include only the mathematically-themed strips which Comic Strip Master Command sent out through to June 26th, which is picking up the nickname Stitch Day (for 6-26, the movie character’s experiment number). And then I decided some from last Sunday weren’t on-point enough (somehow), and there were enough that came later in the week that I couldn’t do a June 26th Only edition. Which is my longwinded way of saying this one doesn’t have a nonsense name. It just has a name that’s only partially on point.

Mike Baldwin’s Cornered for the 26th is the Rubik’s Cube/strange geometry joke for the week. It seems to me I ought to be able to make some link between the number of various ways to arrange a Rubik’s Cube — which pieces can and which ones cannot be neighbors to a red piece, say, no matter how one scrambles the cube — and the networking between people that you can get from an office where people have to see each other. But I’m not sure that I can make that metaphor work. I’m blaming the temperature, both mine (I have a cold) and the weather’s (it’s a heat wave).

Mark Leiknes’s Cow and Boy for the 26th makes literal the trouble some people have with the phrase “110 percent”. Read uncharitably, yes, “110 of a hundred” doesn’t make sense, if 100 percent is all that could conceivably be of the thing. But if we can imagine, say, the number of cars passing a point on the highway being 90 percent of the typical number, surely we can imagine the number of cars also being 110 percent. To give an example of why I can’t side with pedants in objecting to the phrase.

Jef Mallett’s Frazz for the 26th is just itching for a fight. From me and from the Creative Writing department. Yes, mathematics rewards discipline. All activities do. At the risk of making a prescription: if you want to do something well, spend time practicing the boring parts. For arithmetic, that’s times tables and regrouping calculations and factoring and long division. For writing, that’s word choice and sentence structure and figuring how to bring life to describing dull stuff. Do the fun stuff too, yes, but because it is fun. Getting good at the boring stuff makes you an expert. When you discover that the boring stuff is also kinda fun, you will do the fun stuff masterfully.

But to speak of mathematics as pursuing a single right answer — well, perhaps. In an elementary-school problem there is typically just the one right answer, and the hope is that students learn how to get there efficiently. But if the subject is something well-worn, then there are many ways to do any problem. All are legitimate and the worst one can say of a method is maybe it’s not that efficient, or maybe it’s good here but doesn’t generally work. If the subject is on the edge of what mathematics we know, there may be only one way to get there. But there are many things to find, including original ways to understand what we have already found. To not see that mathematics is creative is to not see mathematics. Or, really, any field of human activity.

Samson’s Dark Side of the Horse for the 27th edges up to being the anthropomorphic numerals joke for the week. I need a good name for this sort of joke about mathematical constructs made tangible, even if they aren’t necessarily characters.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th I hope makes sense if you just know the words “graph” and “drunk”, and maybe “McNugget”. That’s all you truly need to understand why this contains a joke. But there is some good serious mathematical terminology at work here.

So. A “graph” is a thing that’s turned up in my A To Z serieses. In this context a graph is a collection of points, called “vertices”, and a collection of “edges” that connect vertices. Often the vertices represent something of interest and the edges ways to turn one thing into another. Sometimes the edges are the thing of interest and the vertices are just there to be manipulated in some way by edges. It’s a way to make visual the studying of how stuff is connected, and how things can pass from one to another.

A “stochastic process” is about random variables. Random variables are some property about a system. And you know some things about that variable’s value. You know maybe the range of possible values it could have. You know whether some values are more likely than others. But you do not know what the value is at any particular moment. Consider, say, the temperature outside where you live at a particular time of day. You may have no idea what that is. But you can say, for example, whether today it is more likely to be 90 degrees Fahrenheit or 60 degrees Fahrenheit or 20 degrees Fahrenheit. For a stochastic process we have some kind of index. We can say, for example, which values of temperature are more likely today, the 1st of July, and which ones will be more likely the 1st of August, and which ones will be less likely the 1st of December. Calling it a “process”, to my intuition, makes it sound like we expect something to happen that causes the likelihood of some temperatures to change. And many processes are time-indexed. They study problems where something interesting changes in time, predictable in aggregate but not in detail.

So a graph like this, representing a stochastic process, is a shorthand. Each vertex is a state that something might be in. Each edge is a way to get from one state to another when — something — happens. Doesn’t matter what thing.

A “drunk walk”, or as it’s known to tenderer writers a “random walk”, is a term of art. Not a deep one. It’s meant to evoke the idea of a severely drunk person who yes, can move, but has no control over which way. Thus he wanders around, reaching any point only by luck. Many things look like random walks, in which there is no overall direction, just an unpredictable shuffling around. A drunk walk on this graph would be, well, start at any of the vertices. Then follow edges, chosen randomly. If you start at the uppermost point of the triangle on top, for example, there’s two places to go on the second step: the lower-left or the center-right vertex on the upper triangle. Suppose you go to the center-right vertex. On the next step, you might go right back where you started. You might go to the lower-left vertex on the triangle. You might drop down that bridge to the top of that quadrilateral. And so on, for another step.

Do that some presumably big number of times. Where are you? … Anywhere, of course. But are there vertices you’re more likely to be on? Ones you’re less likely to be on? How does the shape of the graph affect that likelihood? How does how long you spend walking affect that? These tell us things about the process, and are why someone would draw this graph and talk about a random walk on it.

If you’d like to read more of my comic-strip review posts please do! They all should be available at this link, listed in reverse chronological order.

To read more of the individual comics? Here are essays with Cornered in them. These are Cow and Boy comics at this link. Frazz strips are here. Essays including Dark Side of the Horse are here. And Saturday Morning Breakfast Cereal, which is threatening to take over “being the majority of my blog” from Andertoons, I have at that link.

## Reading the Comics, March 9, 2018: Some Old Lines Edition

To close out last week’s comics I got a bunch of strips that were repeats, or that touch on topics I’ve discussed quite a bit around these parts already. I’m pretty sure all the words I have here are new in their specific organization. The words themselves are pretty old.

Maria Scrivan’s Half Full for the 4th is the Rubik’s Cube joke for the week. I ought to write up a proper description of the algebra of Rubik’s Cubes. The real stuff is several books’ worth of material, yes. But a couple hundred words about what’s interesting should be doable. … Or I could just ask folks if they’ve read good descriptions of the group theory that cubes show off. I’m always open to learning other people have said stuff better than me. This is part of why I’ve never published an essay about Cantor’s Diagonal Proof; many people have written such essays and I couldn’t add anything useful to that heap of words.

Ryan North’s Dinosaur Comics for the 5th is about the heap paradox. Or the sorites paradox, depending on what book you’ve been reading from. The problem is straightforward enough. As God, in the strip says, a big pile of sand is clearly a heap. One or two grains of sand is clearly not. If you remove grains from the heap, eventually, you lose the heap-ness. T-Rex suggests solving the question of when that happens by statistical survey, finding what people on average find to be the range where things shift over.

As with many attempts to apply statistical, or experimental, methods to philosophical questions it misses the point. There are properties that things seem to have only as aggregations. Where do they come from? How can there be something true about a collection of things that isn’t true about any part of the thing? This is not just about messy real-world properties either; we can say stuff about groups of mathematical objects that aren’t true about individual objects within the set. For example, suppose we want to draw a real number at random, uniformly, from the continuous interval 0 to 10. There’s a 50% chance we’ll draw a number greater than 5. The chance of drawing any specific number greater than 5, though, is zero. But we can always draw one. Something weird is happening here, as often happens with questions we’ve been trying to answer for thousands of years.

Norm Feuti’s Retail for the 6th is a new strip, although the joke’s appeared before. There’s some arithmetic calculations that are easy to do, or that become easy because you do them a lot. Or because you see them done a lot and learn what the patterns are. A handful of basic tricks — like that 80 percent off is 20 percent of something, or that 20 percent of a thing is one-fifth the original thing — can be stunning. Stage magicians find the same effect.

John Zakour and Scott Roberts’s Working Daze for the 6th is another chance for me to talk about the supposed folly of giving 110 percent. Or point you to where I did already. I’m forgiving of the use of the phrase.

Bob Shannon’s Tough Town for the 7th is the anthropomorphized abacus joke of the week. Been a while since we had one of those. I suppose an adding machine would be at least as good a representative of the abstract concept of doing arithmetic, but it’s likely harder to draw too. This is just tiring to draw.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th presents the old complaint about mathematics’s utility, here in an ancient setting. I’m intereste that the caveman presents counting in terms of matching up other things to his fingers. We use this matching of one set of things to another even today. It gets us to ordinal and cardinal numbers, and the to what we feel pretty sure about with infinitely large sets. An idea can be ancient and basic and still be vital.

Steve Sicula’s Home and Away for the 9th is about the hatred people profess for mathematics. Some of that is more hatred of how it’s taught, which is too often as a complicated and apparently pointless activity. Some of that is hatred of how it’s used, since it turns up in a lot of jobs. And for some reason we’ve designed society so that we do jobs we don’t like. I don’t know why we think that’s a good idea. We should work on that.

## Reading the Comics, June 4, 2018: Weezer’s Africa Edition

Once again the name of this Reading the Comics edition has nothing to do with any of the strips. I’m just aware that Weezer’s cover of Africa is quite popular right now and who am I to deny people things they want? (I like the cover, but it’s not different enough for me to feel satisfied by it. I tend to like covers that highlight something minor in the original, or that go in a strange direction. Shifting a peppy song into a minor key doesn’t count anymore. But bear in mind, I’m barely competent at listening to music. Please now enjoy my eight hours of early electronica in which various beeps and whistles are passed off as music.)

Samson’s Dark Side of the Horse for the 3rd is the Roman numerals joke for the week. And a welcome return for Dark Side of the Horse. It feels like it’s been gone a while. I wouldn’t try counting by Roman numerals to lull myself to sleep; it seems like too much fussy detail work. But I suppose if you’ve gotten good at it, it’s easy.

Jef Mallett’s Frazz for the 3rd builds on removing statistics from their context. It’s a common problem. It’s possible to measure so very many things. Without a clear idea of what we should expect as normal the measurement doesn’t tell us much. And it can be hard to know what the right context for something even is. Let me deconstruct Caulfield’s example. We’re supposed to reflect on and consider that 40% of all weekdays are Monday and Friday too. But it’s not only weekdays that people work. Even someone working a Sunday might take a sick day. Monday and Friday are a bit over 28% of the whole week. But more people do work Monday-to-Friday than do Saturdays and Sundays, so the Sunday sick day is surely rarer than the Monday. So even if we grant Caulfield’s premise, what does it tell us?

Jason Chatfield’s Ginger Meggs for the 3rd is a bit of why-learn-mathematics propaganda. Megg’s father has a good answer. But it does shift the question back one step. Also I see in the top row that Meggs has one of those comic-strip special editions where the name of the book is printed on the back cover instead. (I’m also skeptical of the photo and text layout on the newspaper Megg’s father is reading. But I don’t know the graphic design style of Australian, as opposed to United States, newspapers.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd may belong on some philosopher’s Reading the Comics blog instead. No matter. There’s some mathematical-enough talk going on here. There’s often many ways to approach the same problem. For example, approaching a system as a handful of items. Or as a huge number of them. Or as infinitely many things. Or as a continuum of things. There are advantages each way. A handful of things, for example, we can often model as interactions between pairs of things. We can model a continuum as a fluid. A vast number of things can let one’s computer numerically approximate a fluid. Or infinitely many particles if that’s more convenient.

To describe all these different models as sharing an “ontology-space” is good mathematical jargon too. In this context the “-space” would mean the collection of all these things that are built by the same plan but with different values of whichever parameter matters.

Bud Blake’s Tiger for the 6th of August, 1965 features Einstein’s famous equation. I suppose it’s showing how well-informed Julian is, that he knows and can present such a big result. There is beauty in mathematics (and physics). Mathematicians (and physicists) find the subject beautiful to start with, and try to find attractive results. I’m curious what the lay reader makes of mathematical symbols, though, just as pieces of art. I remember as a child finding this beauty in a table of integrals in the front of one of my mother’s old college textbooks. All those parallel rows of integral symbols drew me in though nothing I’d seen in mathematics had prepared me to even read it. I still find that beautiful, but I can’t swear that I would even if I hadn’t formed that impression early in life. Are lay and professional readers’ views of mathematical-expression beauty similar? How are they different?