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

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

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

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

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

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

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

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

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

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

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

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

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

## Reading the Comics, August 9, 2019: Venn Diagrams Edition

Thanks for sticking around as I finally got to the past week’s comic strips. There were just enough for me to divide them into two chunks and not feel like I’m cheating anyone of my sparkling prose.

Sandra Bell-Lundy’s Between Friends for the 4th is another entry in this strip’s string of not-quite-Venn-Diagram jokes. As will happen, the point of the diagram seems clear enough even if it doesn’t quite parse. And it isn’t a proper Venn diagram, of course; a Venn diagram for five propositions has to have 31 regions, representing all the possible ways five things can combine or be excluded. They can be beautiful to look at, but start losing their value as ways to organize thought. This is again a Euclid diagram, which doesn’t need to show every possible overlap.

Michael Jantze’s The Norm 4.0 for the 5th is the other Venn Diagram joke for the week. Again properly the first one, showing the complete lack of overlap between two positions, is an Euler rather than a Venn diagram. The second, the “Amity Venn diagram on planet X”, is a Venn diagram and showing the intersection of blue and yellow regions as green is a nice way to show that. (I’m not fond of the gender stereotyping here, nor of the conflation of gender and chromosomes. But the comic strip does have to rely on shorthands or there’s just not going to be the space to compose a joke.)

Harry Bliss’s Bliss for the 6th name-checks tetrahedrons. These are the shapes the rest of us would probably call pyramids or perhaps d4. It’s a bit silly to suppose a hairball should be a tetrahedron. But natural processes will form particular shapes. The obvious example is the hexagonal prisms of honeycombs, which come about for reasons … I’m not sure biologists are completely agreed on. Hexagons do seem to be efficient ways to encompass a lot of volume with a minimum of material, at least. But even the classic hairball looks like that for reasons, related to how it’s created and how it’s expelled from the cat. They just don’t usually have corners.

Niklas Eriksson’s Carpe Diem for the 9th has you common blackboard full of symbols to represent mathematical work. It also evokes a well-worn joke that defines a mathematician as a mechanism for turning coffee into theorems. The explosion of creativity though is true to mathematicians, though. When inspiration is flowing the notes will get abundant and start going in many different wild directions. The symbols in the comic strip don’t mean anything. But that’s not inauthentic. The notes written during an inspired burst will be nonsensical. The great idea needs to be preserved. It can be cleaned up and, one hopes, made presentable later.

This and other Reading the Comics posts are at this link. I should have a fresh one on Thursday, wrapping up the past week.

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

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

Mark Anderson’s Andertoons for the 29th of July is a comfortable and familiar face for these Reading the Comics posts. I’m glad to see it. The joke is built on negative numbers, and Wavehead’s right to say this is kind of the reason people hate mathematics. At least, that mathematicians will become comfortable with something that has a clear real-world intuitive meaning, such as that adding things together gets you a bigger thing. And then for good reasons of logic get to counter-intuitive things, such as adding things together to get a lesser thing. Negative numbers might be the first of these intuition-breaking things that people encounter. That or fractions. I encounter stories of people who refuse to accept that, say, $\frac16$ is smaller than $\frac13$, although I’ve never seen it myself.

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

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

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

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

## Reading the Comics, July 27, 2019: July 27, 2019 Edition

Last week was busy enough in mathematically themed comic strips. Some of these are pretty slight topics. But including them lets me do one of my favorite things, to have an essay that’s all comics from a single day. It’s my blog, I can use it to amuse myself.

Marcus Hamilton and Ron Ferdinand’s Dennis the Menace for the 27th shows the kind of slightness I’m dealing with. ‘Statistic’ has some nasty connotations in this sense. It suggests something dehumanizing has happened. But the word was maybe doomed to that. The word came about in the 18th century, to describe the systematic collection and study of information about whole populations. They started out being the gathering of information about the state.

But gathering information about a whole state implies, first, that the thing one finds interesting about a people are some measured and recorded aspect. Not the whole of their person-hood. Second, it implies that you wish to approximate the diversity of a whole people with some smaller set of numbers. There’s compelling reasons for a state to want to have statistics. They make it more plausible to know what the state can do. They make it plausible to forecast the results of a policy. Ideally, this encourages wisdom in policy-making. If the tools are used well.

Jan Eliot’s Stone Soup Classics for the 27th is the slightest of the comic strips I’m featuring this week. Really it should have been just a mention, but I wanted to have at least three comics shown for today’s essay. Making and counting change is constantly held up as the supreme purpose of teaching arithmetic. This though most any shop has a cash register that will calculate change faster and more accurately than even someone skilled in arithmetic will. I understand the crankiness of people who give the cashier $15.13 for their$12.38 bill, and get the thirteen cents handed back to them before it’s rung up. It’s not evidence that civilization is collapsing. It’s loose change.

Paul Trap’s Thatababy for the 27th continues the strip’s thread of turning geometry figures into jokes. This one is less useful than the comic featured Tuesday, which might help one remember what a scalene triangle or a rhombus looks like. Still might be fun.

And with that, last week’s mathematically-themed comic strips are fully discussed. This week’s comics will get discussion at an essay linked from here. Please visit soon and we’ll see what I have to say, and about what.

## Reading the Comics, July 26, 2019: Children With Mathematics Edition

Three of the strips I have for this installment feature kids around mathematics talk. That’s enough for a theme name.

Gary Delainey and Gerry Rasmussen’s Betty for the 23rd is a strip about luck. It’s easy to form the superstitious view that you have a finite amount of luck, or that you have good and bad lucks which offset each other. It feels like it. If you haven’t felt like it, then consider that time you got an unexpected $200, hours before your car’s alternator died. If events are independent, though, that’s just not so. Whether you win$600 in the lottery this week has no effect on whether you win any next week. Similarly whether you’re struck by lightning should have no effect on whether you’re struck again.

Except that this assumes independence. Even defines independence. This is obvious when you consider that, having won $600, it’s easier to buy an extra twenty dollars in lottery tickets and that does increase your (tiny) chance of winning again. If you’re struck by lightning, perhaps it’s because you tend to be someplace that’s often struck by lightning. Probability is a subtler topic than everyone acknowledges, even when they remember that it is such a subtle topic. It sure seems like this strip wants to talk about lottery winners struck by lightning, doesn’t it? Darrin Bell’s Candorville for the 23rd jokes about the uselessness of arithmetic in modern society. I’m a bit surprised at Lemont’s glee in not having to work out tips by hand. The character’s usually a bit of a science nerd. But liking science is different from enjoying doing arithmetic. And bad experiences learning mathematics can sour someone on the subject for life. (Which is true of every subject. Compare the number of people who come out of gym class enjoying physical fitness.) If you need some Internet Old, read the comments at GoComics, which include people offering dire warnings about what you need in case your machine gives the wrong answer. Which is technically true, but for this application? Getting the wrong answer is not an immediately awful affair. Also a lot of cranky complaining about tipping having risen to 20% just because the United States continues its economic punishment of working peoples. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is some wordplay. Mathematicians often need to find minimums of things. Or maximums of things. Being able to do one lets you do the other, as you’d expect. If you didn’t expect, think about it a moment, and then you expect it. So min and max are often grouped together. Paul Trap’s Thatababy for the 26th is circling around wordplay, turning some common shape names into pictures. This strip might be aimed at mathematics teachers’ doors. I’d certainly accept these as jokes that help someone learn their shapes. And you know what? I hope to have another Reading the Comics post around Thursday at this link. And that’s not even thinking what I might do for this coming Sunday. ## Reading the Comics, July 22, 2019: Mathematics Education Edition There were a decent number of mathematically-themed comic strips this past week. This figures, because I’ve spent this past week doing a lot of things, and look to be busier this coming week. Nothing to do but jump into it, then. Jason Chatfield’s Ginger Meggs for the 21st is your usual strip about the student resisting the story problem. Story problems are hard to set. Ideally, they present problems like mathematicians actually do, proposing the finding of something it would be interesting to learn. But it’s hard to find different problems like this. You might be fairly interested in how long it takes a tub filling with water to overflow, but the third problem of this kind is going to look a lot like the first two. And it’s also hard to find problems that allow for no confounding alternate interpretations, like this. Have some sympathy and let us sometimes just give you an equation to solve. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st is a pun built on two technical definitions for “induction”. The one used in mathematics, and logic, is a powerful tool for certain kinds of proof. It’s hard to teach how to set it up correctly, though. It’s a way to prove an infinitely large number of logical propositions, though. Let me call those propositions P1, P2, P3, P4, and so on. Pj for every counting number j. The first step of the proof is showing that some base proposition is true. This is usually some case that’s really easy to do. This is the fun part of a proof by induction, because it feels like you’ve done half the work and it amounts to something like, oh, showing that 1 is a triangular number. The second part is hard. You have to show that whenever Pj is true, this implies that Pj + 1 is also true. This is usually a step full of letters representing numbers rather than anything you can directly visualize with, like, dots on paper. This is usually the hard part. But put those two halves together? And you’ve proven that all your propositions are true. Making things line up like that is so much fun. Mark Anderson’s Andertoons for the 22nd is the Mark Anderson’s Andertoons for the week. It’s again your student trying to get out of not really knowing mathematics in class. Longtime readers will know, though, that I’m fond of rough drafts in mathematics. I think most mathematicians are. If you are doing something you don’t quite understand, then you don’t know how to do it well. It’s worth, in that case, doing an approximation of what you truly want to do. This is for the same reason writers are always advised to write something and then edit later. The rough draft will help you find what you truly want. In thinking about the rough draft, you can get closer to the good draft. Stephen Bentley’s Herb and Jamaal for the 22nd is one lost on me. I grew up when Schoolhouse Rock was a fun and impossible-to-avoid part of watching Saturday Morning cartoons. So there’s a lot of simple mathematics that I learned by having it put to music and played often. Still, it’s surprising Herb can’t think of why it might be easier to remember something that’s fun, that’s put to a memory-enhancing tool like music, and repeated often, than it is to remember whether 8 times 7 is 54. Arithmetic gets easier to remember when you notice patterns, and find them delightful. Even fun. It’s a lot like everything else humans put any attention to, that way. This was a busy week for comic strips. I hope to have another Reading the Comics post around Tuesday, and at this link. There might even be another one this week. Please check back in. ## Reading the Comics, July 20, 2019: What Are The Chances Edition The temperature’s cooled. So let me get to the comics that, Saturday, I thought were substantial enough to get specific discussion. It’s possible I was overestimating how much there was to say about some of these. These are the risks I take. Paige Braddock’s Jane’s World for the 15th sees Jane’s niece talk about enjoying mathematics. I’m glad to see. You sometimes see comic strip characters who are preposterously good at mathematics. Here I mean Jason and Marcus over in Bill Amend’s FoxTrot. But even they don’t often talk about why mathematics is appealing. There is no one answer for all people. I suspect even for a single person the biggest appeal changes over time. That mathematics seems to offer certainty, though, appeals to many. Deductive logic promises truths that can be known independent of any human failings. (The catch is actually doing a full proof, because that takes way too many boring steps. Mathematicians more often do enough of a prove to convince anyone that the full proof could be produced if needed.) Alexa also enjoys math for there always being a right answer. Given her age there probably always is. There are mathematical questions for which there is no known right answer. Some of these are questions for which we just don’t know the answer, like, “is there an odd perfect number?” Some of these are more like value judgements, though. Is Euclidean geometry or non-Euclidean geometry more correct? The answer depends on what you want to do. There’s no more a right answer to that question than there is a right answer to “what shall I eat for dinner”. Jane is disturbed by the idea of there being a right answer that she doesn’t know. She would not be happy to learn about “existence proofs”. This is a kind of proof in which the goal is not to find an answer. It’s just to show that there is an answer. This might seem pointless. But there are problems for which there can’t be an answer. If an answer’s been hard to find, it’s worth checking whether there are answers to find. Art Sansom and Chip Sansom’s The Born Loser for the 16th builds on comparing the probability of winning the lottery to that of being hit by lightning. This comparison’s turned up a couple of times, including in Mister Boffo and The Wandering Melon, when I learned that Peter McCathie had both won the lottery and been hit by lightning. Pab Sungenis’s New Adventures of Queen Victoria for the 17th is maybe too marginal for full discussion. It’s just reeling off a physics-major joke. The comedy is from it being a pun: Planck’s Constant is a number important in many quantum mechanics problems. It’s named for Max Planck, one of the pioneers of the field. The constant is represented in symbols as either $h$ or as $\hbar$. The constant $\hbar$ is equal to $\frac{h}{2 \pi}$ and might be used even more often. It turns out $\frac{h}{2 \pi}$ appears all over the place in quantum mechanics, so it’s convenient to write it with fewer symbols. $\hbar$ is maybe properly called the reduced Planck’s constant, although in my physics classes I never encountered anyone calling it “reduced”. We just accepted there were these two Planck’s Constants and trusted context to make clear which one we wanted. It was $\hbar$. Planck’s Constant made some news among mensuration fans recently. The International Bureau of Weights and Measures chose to fix the value of this constant. This, through various physics truths, thus fixes the mass of the kilogram in terms of physical constants. This is regarded as better than the old method, where we just had a lump of metal that we used as reference. Jonathan Lemon’s Rabbits Against Magic for the 17th is another probability joke. If a dropped piece of toast is equally likely to land butter-side-up or butter-side-down, then it’s quite unlikely to have it turn up the same way twenty times in a row. There’s about one chance in 524,288 of doing it in a string of twenty toast-flips. (That is, of twenty butter-side-up or butter-side-down in a row. If all you want is twenty butter-side-up, then there’s one chance in 1,048,576.) It’s understandable that Eight-Ball would take Lettuce to be quite lucky just now. But there’s problems with the reasoning. First is the supposition that toast is as likely to fall butter-side-up as butter-side-down. I have a dim recollection of a mid-2000s pop physics book explaining why, given how tall a table usually is, a piece of toast is more likely to make half a turn — to land butter-side-down — before falling. Lettuce isn’t shown anywhere near a table, though. She might be dropping toast from a height that makes butter-side-up more likely. And there’s no reason to suppose that luck in toast-dropping connects to any formal game of chance. Or that her luck would continue to hold: even if she can drop the toast consistently twenty times there’s not much reason to think she could do it twenty-five times, or even twenty-one. And then there’s this, a trivia that’s flawed but striking. Suppose that all seven billion people in the world have, at some point, tossed a coin at least twenty times. Then there should be seven thousand of them who had the coin turn up tails every single one of the first twenty times they’ve tossed a coin. And, yes, not everyone in the world has touched a coin, much less tossed it twenty times. But there could reasonably be quite a few people who grew up just thinking that every time you toss a coin it comes up tails. That doesn’t mean they’re going to have any luck gambling. Thanks for waiting for me. The weather looks like I should have my next Reading the Comics post at this link, and on time. I’ll let you know if circumstances change. ## Reading the Comics, July 20, 2019: Heat Wave Marginalia Edition So, it has been hot around here. Extremely hot. Like, hot to the point that there’s nothing to do but form hyperbolic statements about the heat. This does not help anyone feel cooler, but it does help us feel like we’re doing something relevant to the weather. The result is that I haven’t had time to think about my comic strip reading. I’ve been very busy trying to pop my head off and leave it in the freezer. This has not worked. Our refrigerator’s dying and we have a replacement scheduled to arrive this week. The consequence is that I haven’t had time to write my paragraphs about the comic strips that mention mathematical issues of substance. To not be a complete void, though, let me give you the marginalia. These are the comics that mentioned mathematics in some way so slight that I don’t think them worth further discussion. I’ll get to substantial stuff Tuesday. Thank you. Tony Rubino and Gary Markstein’s Daddy’s Home for the 15th has a kid doing remarkably well in a mathematics exam. It’s treated as extraordinary. This is the traditional use of mathematics as the hard subject. Percy Crosby’s Skippy for the 7th of March, 1932, and reprinted the 16th of July has Skippy talk about arithmetic lessons. Here, again, it could be any subject, but mathematics has the reputation for being a subject one wants to avoid. Jon Rosenberg’s Scenes from a Multiverse rerun for the 17th shows off a girl talking about her father’s ability to help with mathematics homework. There is a theme developing in the past week’s mentions. Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 17th has a ‘Fake Maths’ textbook, the falseness of it proven by the arithmetic being wrong. So that uses a different part of mathematics’ reputation, that of giving us things we can know are certainly true, or certainly false. The weather should be much nicer the next few days. Trusting that it is, I’ll have an essay at this link Tuesday with a new Reading the Comics post. Thank you for understanding. It’s quite hard to do anything when it’s so hot you realize your couch is melting. ## Reading the Comics, July 13, 2019: Marginal Supplemental Edition So last week there were only a handful of comic strips which mentioned mathematics in any detail. That is, that brought up some point that I could go on about for a paragraph or so. There were more that had some marginal mathematics content. I gather them here for the interested. Gordon Bess’s Redeye rerun for the 7th mentions mathematics as the homework that the chief is helping his son with. It could be any subject, but arithmetic is easy to fit into one panel of comic strip. And it’s also easy to establish that the work is on a low level. The comic originally ran the 18th of February, 1973. Bob Shannon’s Tough Town for the 7th has an appearance by a Rubik’s Cube. I’m always going on about that as a group theory artifact. Tough Town on the 9th also mentioned algebra as a tough subject for students. John Allen’s Nest Heads for the 10th mentions sudoku. Also the trouble with accounting. John McPherson’s Close to Home for the 11th mentions percentages. The joke’s built on doing a meaningless calculation. And a bit of convention, in which the label has been reduced to the point people could mis-read it. You just know this guy would tell the “scanner didn’t pick it up, it must be free” joke if he thought of it that fast. Paige Braddock’s Jane’s World for the 11th is part of a sequence from 2002 in which Jane concludes the problems in her life came from the introduction of algebra. Her niece is having fun with algebra, a thing I understand. Algebra can be a more playful, explorative kind of mathematics than you get with, like, long division. For some people it’s liberating. This one’s a new tag, so I’m sure to be surprised that I have ever mentioned Jane’s World sometime in the future. Wiley Miller’s Non Sequitur for the 11th presents a Sphere of Serenity. Or, as Danae’s horse points out, a Cube of Serenity. There are ways that the difference between a sphere and a cube becomes nothing. If the cube and the sphere have infinitely great extent, for example, then there’s no observable difference between the shapes. Or if we use certain definitions of distance then the sphere — as in, the points all an equal distance from a center — can be indistinguishable from a cube. That’s not what the comic is going for. There were no comic strips with any mathematical content last Saturday, it turns out. There have already been a couple comic strips I think I can discuss. One comic strip, anyway. I should have my essay about it for eager readers on Sunday. Thanks for your patience. ## Reading the Comics, July 12, 2019: Ricci Tensor Edition So a couple days ago I was chatting with a mathematician friend. He mentioned how he was struggling with the Ricci Tensor. Not the definition, not exactly, but its point. What the Ricci Tensor was for, and why it was a useful thing. He wished he knew of a pop mathematics essay about the thing. And this brought, slowly at first, to my mind that I knew of one. I wrote such a pop-mathematics essay about the Ricci Tensor, as part of my 2017 A To Z sequence. In it, I spend several paragraphs admitting that I’m not sure I understand what the Ricci tensor is for, and why it’s a useful thing. Daniel Beyer’s Long Story Short for the 11th mentions some physics hypotheses. These are ideas about how the universe might be constructed. Like many such cosmological thoughts they blend into geometry. The no-boundary proposal, also known as the Hartle-Hawking state (for James Hartle and Stephen Hawking), is a hypothesis about the … I want to write “the start of time”. But I am not confident that this doesn’t beg the question. Well, we think we know what we mean by “the start of the universe”. A natural question in mathematical physics is, what was the starting condition? At the first moment that there was anything, what did it look like? And this becomes difficult to answer, difficult to even discuss, because part of the creation of the universe was the creation of spacetime. In this no-boundary proposal, the shape of spacetime at the creation of the universe is such that there just isn’t a “time” dimension at the “moment” of the Big Bang. The metaphor I see reprinted often about this is how there’s not a direction south of the south pole, even though south is otherwise a quite understandable concept on the rest of the Earth. (I agree with this proposal, but I feel like analogy isn’t quite tight enough.) Still, there are mathematical concepts which seem akin to this. What is the start of the positive numbers, for example? Any positive number you might name has some smaller number we could have picked instead, until we fall out of the positive numbers altogether and into zero. For a mathematical physics concept there’s absolute zero, the coldest temperature there is. But there is no achieving absolute zero. The thermodynamical reasons behind this are hard to argue. (I’m not sure I could put them in a two-thousand-word essay, not the way I write.) It might be that the “moment of the Big Bang” is similarly inaccessible but, at least for the correct observer, incredibly close by. The Weyl Curvature is a creation of differential geometry. So it is important in relativity, in describing the curve of spacetime. It describes several things that we can think we understand. One is the tidal forces on something moving along a geodesic. Moving along a geodesic is the general-relativity equivalent of moving in a straight line at a constant speed. Tidal forces are those things we remember reading about. They come from the Moon, sometimes the Sun, sometimes from a black hole a theoretical starship is falling into. Another way we are supposed to understand it is that it describes how gravitational waves move through empty space, space which has no other mass in it. I am not sure that this is that understandable, but it feels accessible. The Weyl tensor describes how the shapes of things change under tidal forces, but it tracks no information about how the volume changes. The Ricci tensor, in contrast, tracks how the volume of a shape changes, but not the shape. Between the Ricci and the Weyl tensors we have all the information about how the shape of spacetime affects the things within it. Ted Baum, writing to John Baez, offers a great piece of advice in understanding what the Weyl Tensor offers. Baum compares the subject to electricity and magnetism. If one knew all the electric charges and current distributions in space, one would … not quite know what the electromagnetic fields were. This is because there are electromagnetic waves, which exist independently of electric charges and currents. We need to account for those to have a full understanding of electromagnetic fields. So, similarly, the Weyl curvature gives us this for gravity. How is a gravitational field affected by waves, which exist and move independently of some source? I am not sure that the Weyl Curvature is truly, as the comic strip proposes, a physics hypothesis “still on the table”. It’s certainly something still researched, but that’s because it offers answers to interesting questions. But that’s also surely close enough for the comic strip’s needs. Dave Coverly’s Speed Bump for the 11th is a wordplay joke, and I have to admit its marginality. I can’t say it’s false for people who (presumably) don’t work much with coefficients to remember them after a long while. I don’t do much with French verb tenses, so I don’t remember anything about the pluperfect except that it existed. (I have a hazy impression that I liked it, but not an idea why. I think it was something in the auxiliary verb.) Still, this mention of coefficients nearly forms a comic strip synchronicity with Mike Thompson’s Grand Avenue for the 11th, in which a Math Joke allegedly has a mistaken coefficient as its punch line. Mike Thompson’s Grand Avenue for the 12th is the one I’m taking as representative for the week, though. The premise has been that Gabby and Michael were sent to Math Camp. They do not want to go to Math Camp. They find mathematics to be a bewildering set of arbitrary and petty rules to accomplish things of no interest to them. From their experience, it’s hard to argue. The comic has, since I started paying attention to it, consistently had mathematics be a chore dropped on them. And not merely from teachers who want them to solve boring story problems. Their grandmother dumps workbooks on them, even in the middle of summer vacation, presenting it as a chore they must do. Most comic strips present mathematics as a thing students would rather not do, and that’s both true enough and a good starting point for jokes. But I don’t remember any that make mathematics look so tedious. Anyway, I highlight this one because of the Math Camp jokes it, and the coefficients mention above, are the most direct mention of some mathematical thing. The rest are along the lines of the strip from the 9th, asserting that the “Math Camp Activity Board” spelled the last word wrong. The joke’s correct but it’s not mathematical. So I had to put this essay to bed before I could read Saturday’s comics. Were any of them mathematically themed? I may know soon! And were there comic strips with some mention of mathematics, but too slight for me to make a paragraph about? What could be even slighter than the mathematical content of the Speed Bump and the Grand Avenue I did choose to highlight? Please check the Reading the Comics essay I intend to publish Tuesday. I’m curious myself. ## Reading the Comics, July 2, 2019: Back On Schedule Edition I hoped I’d get a Reading the Comics post in for Tuesday, and even managed it. With this I’m all caught up to the syndicated comic strips which, last week, brought up some mathematics topic. I’m open for nominations about what to publish here Thursday. Write in quick. Hilary Price’s Rhymes With Orange for the 30th is a struggling-student joke. And set in summer school, so the comic can be run the last day of June without standing out to its United States audience. It expresses a common anxiety, about that point when mathematics starts using letters. It superficially seems strange that this change worries students. Students surely had encountered problems where some term in an equation was replaced with a blank space and they were expected to find the missing term. This is the same work as using a letter. Still, there are important differences. First is that a blank line (box, circle, whatever) has connotations of “a thing to be filled in”. A letter seems to carry meaning in to the problem, even if it’s just “x marks the spot”. And a letter, as we use it in English, always stands for the same thing (or at least the same set of things). That ‘x’ may be 7 in one problem and 12 in another seems weird. I mean weird even by the standards of English orthography. A letter might represent a number whose value we wish to know; it might represent a number whose value we don’t care about. These are different ideas. We usually fall into a convention where numbers we wish to know are more likely x, y, and z, while those we don’t care about are more likely a, b, and c. But even that’s no reliable rule. And there may be several letters in a single equation. It’s one thing to have a single unknown number to deal with. To have two? Three? I don’t blame people fearing they can’t handle that. Mark Leiknes’s Cow and Boy for the 30th has Billy and Cow pondering the Prisoner’s Dilemma. This is one of the first examples someone encounters in game theory. Game theory sounds like the most fun part of mathematics. It’s the study of situations in which there’s multiple parties following formal rules which allow for gains or losses. This is an abstract description. It means many things fit a mathematician’s idea of a game. The Prisoner’s Dilemma is described well enough by Billy. It’s built on two parties, each — separately and without the ability to coordinate — having to make a choice. Both would be better off, under interrogation, to keep quiet and trust that the cops can’t get anything significant on them. But both have the temptation that if they rat out the other, they’ll get off free while their former partner gets screwed. And knowing that their partner has the same temptation. So what would be best for the two of them requires them both doing the thing that maximizes their individual risk. The implication is unsettling: everyone acting in their own best interest is supposed to produce the best possible result for society. And here, for the society of these two accused, it breaks down entirely. Jason Poland’s Robbie and Bobby for the 1st is a rerun. I discussed it last time it appeared, in November 2016, which was before I would routinely include the strips under discussion. The strip’s built on wordplay, using the word ‘power’ in its connotations for might and for exponents. Exponents have been written as numbers in superscript following a base for a long while now. The notation developed over the 17th century. I don’t know why mathematicians settled on superscripts, as opposed to the many other ways a base and an exponent might fit together. It’s a good mnemonic to remember, say, “z raised to the 10th” is z with a raised 10. But I don’t know the etymology of “raised” in a mathematical context well enough. It’s plausible that we say “raised” because that’s what the notation suggests. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 2nd argues for the beauty of mathematics as a use for it. It’s presented in a brutal manner, but saying brutal things to kids is a comic motif with history to it. Well, in an existentialist manner, but that gets pretty brutal quickly. The proof of the Pythagorean Theorem is one of the very many known to humanity. This one is among the family of proofs that are wordless. At least nearly wordless. You can get from here to $a^2 + b^2 = c^2$ with very little prompting. If you do need prompting, it’s this: there are two expressions for how much area of the square with sides a-plus-b. One of these expressions uses only terms of a and b. The other expression uses terms of a, b, and c. If this doesn’t get a bit of a grin out of you, don’t worry. There’s, like, 2,037 other proofs we already know about. We might ask whether we need quite so many proofs of the Pythagorean theorem. It doesn’t seem to be under serious question most of the time. And then a couple comic strips last week just mentioned mathematics. Morrie Turner’s Wee Pals for the 1st of July has the kids trying to understand their mathematics homework. Could have been anything. Mike Thompson’s Grand Avenue for the 5th started a sequence with the kids at Math Camp. The comic is trying quite hard to get me riled up. So far it’s been the kids agreeing that mathematics is the worst, and has left things at that. Hrmph. Whether or not I have something for Thursday, by Sunday I should have anotherReading the Comics post. It, as well as my back catalogue of these essays, should be at this link. Thanks for worrying about me. ## Reading the Comics, June 29, 2019: Pacing Edition These are the last of the comics from the final full week of June. Ordinarily I’d have run this on Tuesday or Thursday of last week. But I also had my monthly readership-report post and that bit about a particle physics simulator also to post. It better fit a posting schedule of something every two or three days to move this to Sunday. This is what I tell myself is the rationale for not writing things up faster. Ernie Bushmiller’s Nancy Classics for the 27th uses arithmetic as an economical way to demonstrate intelligence. At least, the ability to do arithmetic is used as proof of intelligence. Which shouldn’t surprise. The conventional appreciation for Ernie Bushmiller is of his skill at efficiently communicating the ideas needed for a joke. That said, it’s a bit surprising Sluggo asks the dog “six times six divided by two”; if it were just showing any ability at arithmetic “one plus one” or “two plus two” would do. But “six times six divided by two” has the advantage of being a bit complicated. That is, it’s reasonable Sluggo wouldn’t know it right away, and would see it as something only the brainiest would. But it’s not so complicated that Sluggo wouldn’t plausibly know the question. Eric the Circle for the 28th, this one by AusAGirl, uses “Non-Euclidean” as a way to express weirdness in shape. My first impulse was to say that this wouldn’t really be a non-Euclidean circle. A non-Euclidean geometry has space that’s different from what we’re approximating with sheets of paper or with boxes put in a room. There are some that are familiar, or roughly familiar, such as the geometry of the surface of a planet. But you can draw circles on the surface of a globe. They don’t look like this mooshy T-circle. They look like … circles. Their weirdness comes in other ways, like how the circumference is not π times the diameter. On reflection, I’m being too harsh. What makes a space non-Euclidean is … well, many things. One that’s easy to understand is to imagine that the space uses some novel definition for the distance between points. Distance is a great idea. It turns out to be useful, in geometry and in analysis, to use a flexible idea of of what distance is. We can define the distance between things in ways that look just like the Euclidean idea of distance. Or we can define it in other, weirder ways. We can, whatever the distance, define a “circle” as the set of points that are all exactly some distance from a chosen center point. And the appearance of those “circles” can differ. There are literally infinitely many possible distance functions. But there is a family of them which we use all the time. And the “circles” in those look like … well, at the most extreme, they look like squares. Others will look like rounded squares, or like slightly diamond-shaped circles. I don’t know of any distance function that’s useful that would give us a circle like this picture of Eric. But there surely is one that exists and that’s enough for the joke to be certified factually correct. And that is what’s truly important in a comic strip. Sandra Bell-Lundy’s Between Friends for the 29th is the Venn Diagram joke for the week. Formally, you have to read this diagram charitably for it to parse. If we take the “what” that Maeve says, or doesn’t say, to be particular sentences, then the intersection has to be empty. You can’t both say and not-say a sentence. But it seems to me that any conversation of importance has the things which we choose to say and the things which we choose not to say. And it is so difficult to get the blend of things said and things unsaid correct. And I realize that the last time Between Friends came up here I was similarly defending the comic’s Venn Diagram use. I’m a sympathetic reader, at least to most comic strips. And that was the conclusion of comic strips through the 29th of June which mentioned mathematics enough for me to write much about. There were a couple other comics that brought up something or other, though. Wulff and Morgenthaler’s WuMo for the 27th of June has a Rubik’s Cube joke. The traditional Rubik’s Cube has three rows, columns, and layers of cubes. But there’s no reason there can’t be more rows and columns and layers. Back in the 80s there were enough four-by-four-by-four cubes sold that I even had one. Wikipedia tells me the officially licensed cubes have gotten only up to five-by-five-by-five. But that there was a 17-by-17-by-17 cube sold, with prototypes for 22-by-22-by-22 and 33-by-33-by-33 cubes. This seems to me like a great many stickers to peel off and reattach. And two comic strips did ballistic trajectory calculation jokes. These are great introductory problems for mathematical physics. They’re questions about things people can observe and so have a physical intuition for, and yet involve mathematics that’s not too subtle or baffling. John Rose’s Barney Google and Snuffy Smith mentioned the topic the 28th of June. Doug Savage’s Savage Chickens used it the 28th also, because sometimes comic strips just line up like that. This and other Reading the Comics posts should be at this link. This includes, I hope, the strips of this past week, that is, the start of July, which should be published Tuesday. Thanks for reading at all. ## Reading the Comics, June 27, 2019: Closing A Slow Month Edition Some months stretch my pop-mathematics writing skills, tasking me with finding new insights into the things I thought I understood and new ways to present them. Some months I’ve written about comic strips a lot. This was one of the latter. Here, let me nearly finish writing about the comic strips of June 2019 that had some mathematical content. Jonathan Lemon’s Rabbits Against Magic for the 23rd is the Venn Diagram meta-joke for the week. Properly speaking, yes, Eight-Ball hasn’t drawn a Venn Diagram here. Representing two sets in a Venn Diagram, by the proper rules, requires two circles with one overlap. Indicating that both sets have the same elements means noting that there are no elements outside the intersection of these circles. One point of a Venn Diagram is showing all the possible logical relations between sets and maybe then marking off the ones that happen to be relevant to the problem. What Eight-Ball is drawing is an Euler Diagram, which has looser requirements. There’s no sense fighting this terminology battle, though. It makes cleaner pictures to draw a Venn Diagram modified to only show the relations that actually exist. If the goal is to communicate information, clarity counts. A joke counts as information. Eight-Ball’s propositions are … well, a bit muddled. His first set is “people who like to think they are good at math”. His second set is “which of those people like Venn Diagrams”. This implies the second set can’t be anything but a subset of the first. So this we’d represent as one circle inside another, at least if we allow that there exists at least one person who likes to think they’re good at math, but still doesn’t like Venn Diagrams. It’s fine for the purposes of comic hyperbole to claim there is no such thing, of course, and I don’t quarrel with that. Why not have the second group be “people who like Venn Diagrams”, without the restriction that they already think they’re good at math? Here I think there is a serious logical constraint. My suspicion is that Venn Diagrams are liked by people who don’t think they’re good at math. Also by people who aren’t good at math. Venn Diagrams are a wonderful tool because they present the relationships of sets in a way that uses our spatial intuitions. They wouldn’t make a good Internet joke format if they were liked only by people who think they’re good at math. Which is why Jonathan Lemon had to write the joke that way. It’s plausible comic hyperbole to say everyone who thinks they’re good at math likes Venn Diagrams. But there are too many people who react to explicit mathematics content with a shudder, but who like Venn Diagram jokes, to make “everyone who likes Venn Diagrams thinks they’re good at math” plausible. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 23rd is a lying-with-statistics joke. The median is an average of a data set. It’s “an” average because, in English, we mean several different things by “average”. Translated into mathematics these different things are, really, completely unrelated. The “median” is the midpoint of the ordered list of the data set. So, as the Man In Black says, half the data in the set is below that value, and half is above. This can be a better measure of “average” than the arithmetic mean is. It tells us a slight something about the distribution, about how the data is arranged. Not much, but then, it’s just one number. What do you want? It has an advantage over the arithmetic mean, which is the thing normal people intend when they say “average”. That advantage is that it’s relatively insensitive to outliers. One or two really large, or tiny, data points can throw the mean way off. The classic example we use these days is to look at the average wealth of twenty people in the room. If Bill Gates enters the room, the mean jumps way up. The median? Doesn’t alter much. (Bill Gates is the figure I see used these days, but it could be anyone impossibly wealthy. I imagine there are versions where it’s Jeff Bezos entering the room. I imagine a century ago, the proposition would be to imagine J P Morgan entering the room, except that a century ago he had been dead six years.) Steve Skelton’s 2 Cows and a Chicken for the 26th shows off a counting chicken as a wonder. Animals do have some sense of mathematics. We know in some detail how well crows and ravens can count, and do simple arithmetic. This is partly because we know good ways to test crow and raven arithmetic skills. And we’ve come to appreciate their intelligence as deep and surprising. Chickens, to my knowledge, have gotten less study. But I would expect they’ve got skills. If nothing else, I would expect chickens to have a good understanding of the transitive property. This is the rule that if ‘a’ is greater than ‘b’, and ‘b’ is greater than ‘c’, then it follows that ‘a’ is greater than ‘c’. Chickens have a pecking order, and animals with that kind of hierarchy tend to know transitivity. I don’t know that the reasons for that link have been proven, but, c’mon. And animals doing arithmetic, like the cook says, have been good sideshow attractions or performances for a long while. They’ve also been good starts for scientific study, as people try to work out questions like how intelligence formed, and what other ways it might have formed. Greg Cravens’s The Buckets for the 27th is a joke about the representation of numbers. Cravens has a good observation here about learning the differences between representations, and of not being able to express just what representation you want. I love Eddie’s horrified face as his mother (Sarah) tries to spell out the word. There’s probably a good exercise to be done in thinking of as many ways to represent fifteen as possible. Etymologically, “fifteen” has exactly the origin you would say if you were dragged out of a sound sleep by someone demanding the history of the word RIGHT NOW, THERE’S NO TIME TO EXPLAIN. In Old English it was “fiftyne”, with “fif” meaning “five” and “tyne” meaning “ten more than”. This construction, pretty much five-and-ten, has fallen out of favor in English. Once we get past nineteen we more commonly write out, like, “twenty-one” and “thirty-five” and such. The alternate construction, which would be, like, one-and-twenty, or nine-and-sixty, or such, seems to have fallen out of use except as a more poetic way to express the idea. I don’t know why, say, five-and-twenty would have shifted to twenty-five while the equivalent five-and-ten didn’t shift to … teenfive(?). I would make an uninformed guess that words used more commonly tend to be more stable, and we tend to need smaller numbers more than bigger ones. I’ll have some more comic strips for you later in the week. Before then should be a statistics review, as I figure out whether anyone is reading this blog after a month when I wrote basically nothing. The next Reading the Comics post should be at this link probably on Thursday. Thank you for reading any of this. ## Reading the Comics, June 21, 2019: I Have An Anecdote Edition A couple years back we needed to patch a bunch of weak spots in the roof. We found all the spots that needed shoring up and measured how long they were, and went to buy some wood and get it cut to fit. I turned over the list of sizes and the guy told us we’d have to buy more than one of the standard-size sheets of plywood to do it. I thought, wait, no, that can’t be, and sketched out possible ways to cut the wood and fit pieces together. Finally I concluded that, oh, yes, the guy whose job it was to figure out how much wood was needed for particular tasks knew what he was talking about. His secret? I don’t know. What finally convinced me was adding up the total area of the wood we’d need, and finding that it was more than what one sheet would be. Dave Blazek’s Loose Parts for the 19th uses a whiteboard full of mathematics as visual shorthand for “some really complicated subject”. It’s a good set of mathematics symbols on the whiteboard. They don’t mean anything in the combination shown, though. It’s just meant to bewilder. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st is bewildering, unless you know what the mathematics principle the joke intends to present. This is what I’m here for. The key is the Mover’s claim that he can look at any amount of stuff and tell you whether it fits in the moving bins. Working out something like this is a version of the knapsack problem. The knapsack problem is … well, the problem you imagine it might be, if someone told you “some mathematicians study a thing called the knapsack problem”? That’s about right. Formally, it’s about selecting from a set of things of different value. How hard is it to pick a subset of things with exactly that value? Or find that there is no such subset? Well, in a sense, not hard at all. You can just keep trying combinations. Eventually you’ll either find a set that works, or you’ll try every possibility and find none of them work. This is known as “exhaustion”, and correctly. If there are ten things, there are 3,628,800 possibilities. Then it gets really bad. If there are twenty things, there are 2,432,902,008,176,640,000 possibilities. Finding the one that works? That could take a while. So being able to tell whether a collection of things can fit within a particular space? That’s a form of the knapsack problem. Being able to always solve that any faster than just “try out every combination until you find one that works”? That would be incredible. The problem is hard. That’s a technical term. It means what you imagine it means, but more precisely. So why the mathematician’s response? It’s because the problem of hacking the common Internet security algorithms is also hard. (I am discussing here how difficult hacking would be if the algorithm were implemented perfectly. There are many hacking techniques available because of bugs. Programs are not written perfectly. Compilers do not translate them to computer code perfectly. Computers are not built perfectly. These and more flaws make hacking more possible than it should be.) It’s the same kind of hard as this knapsack problem. I mean “the same” more technically than you might imagine. If you had a method to quickly solve this knapsack problem, then, you could use this to break computer encryption quickly. And, it turns out, vice-versa, so at least there’s some fairness to things. So if the the Mover can, truly, always instantly tell whether a set of things fit in the moving bins, then hacking e-mails should be possible to. The Mover would have to team up with a mathematician who studies computational problems like this. I don’t know how to do it, myself. I think about the how to do this and feel lost, myself. So is the Mover full of it? Let’s put this more nicely. Is he at least unduly optimistic about his claims? Nah. What makes the knapsack problem hard is that you have to find a solution that quickly finds answers for every possible set of things. But the Mover doesn’t have to deal with that. Most of the stuff is in boxes. It’s in mostly simple polygonal shapes. There’s not, like, 400 million items, each the size of a Cheerio. The Mover may plausibly have never encountered a set of things to move where he couldn’t tell whether it fits. And, yes, there’s selection bias. Suppose he declared that no, this load had to fit into two vans. But that actually a sufficiently clever arrangement would have let it fit in one. Who would ever know he was wrong? He’d only ever know his intuition was wrong if he declared something would fit in one van and, in fact, it couldn’t. Percy Crosby’s Skippy for the 21st is a student-at-the-board problem. It’s using the punch line that “I don’t know” might be a true answer to any problem. There are many real mathematics problems for which nobody really knows an answer. But Miss Larkin has good advice here. Maybe you don’t know the final answer. But do you know anything? Write it down. It’s good for partial credit, at least. Working out a part of the problem might also be useful, too. Often you can work out how to do a hard problem by looking at a similar but simpler problem. If Skippy is lost at 8 + 4 + 7 + 5, could he do at least 8 + 4 + 7? Could he do 8 + 4? Maybe this wouldn’t help him get to the ultimate answer. Often a difficult problem turns out to be solved by solving a circle of simple problems, that starve out the hard. Samson’s Dark Side of the Horse for the 21st is the Roman Numerals joke for this time around. I’m not sure this whether this is a repeat. The strip does a lot of Roman Numerals jokes, and counting-sheep jokes. Our roof patches held up for their need, which was just to last a couple months while we contracted for a replacement roof. And, happily, the roof replacement got done speedily and during a week that did not rain. (Back in grad school the apartment I was in had its roof replaced on a day that, it turns out, would get a spontaneous downpour halfway through. My apartment was on the top floor. This made for an exciting afternoon.) This wraps up the past week’s comics. There weren’t any that mentioned mathematics more fleetingly than Dark Side of the Horse did. A new Reading the Comics post should be at this link on Sunday. Thank you for reading along. ## Reading the Comics, June 20, 2019: Old Friends Edition We continue to be in the summer vacation doldrums for mathematically-themed comic strips. But there’ve been a couple coming out. I could break this week’s crop into two essays, for example. All of today’s strips are comics that turn up in my essays a lot. It’s like hanging out with a couple of old friends. Samson’s Dark Side of the Horse for the 17th uses the motif of arithmetic expressions as “difficult” things. The expressions Samson quotes seem difficult for being syntactically weird: What does the colon under the radical sign mean in $\sqrt{9:}33$? Or they’re difficult for being indirect, using a phrase like “50%” for “half”. But with some charity we can read this as Horace talking about 3:33 am to about 6:30 am. I agree that those are difficult hours. It also puts me in mind of a gift from a few years back. An aunt sent me an Irrational Watch, with a dial that didn’t have the usual counting numbers on it. Instead there were various irrational numbers, like the Golden Ratio or the square root of 50 or the like. Also the Euler-Mascheroni Constant, a number that may or may not be irrational. Nobody knows. It’s likely that it is irrational, but it’s not proven. It’s a good bit of fun, although it does make it a bit harder to use the watch for problems like “how long is it until 4:15?” This isn’t quite what’s going on here — the square root of nine is a noticeably rational number — but it seems in that same spirit. Mark Anderson’s Andertoons for the 18th sees Wavehead react to the terminology of the “improper fraction”. “Proper” and “improper” as words carry a suggestion of … well, decency. Like there’s something faintly immoral about having an improper fraction. “Proper” and “improper”, as words, attach to many mathematical concepts. Several years ago I wrote that “proper” amounted to “it isn’t boring”. This is a fair way to characterize, like, proper subsets or proper factors or the like. It’s less obvious that $\frac{13}{12}$ is a boring fraction. I may need to rewrite that old essay. An “improper” form satisfies all the required conditions for the term. But it misses some of the connotation of the term. It’s true that, say, the new process takes “a fraction of the time” of the old, if the old process took one hour and the new process takes fourteen years. But if you tried telling someone that they would assume you misunderstood something. The ordinary English usage of “fraction” carries the connotation of “a fraction between zero and one”, and that’s what makes a “proper fraction”. In practical terms, improper fractions are fine. I don’t know of any mathematicians who seriously object to them, or avoid using them. The hedging word “seriously” is in there because of a special need. That need is: how big is, say, $\frac{75}{14}$? Is it bigger than five? Is it smaller than six? An improper fraction depends on you knowing, in this case, your fourteen-times tables to tell. Switching that to a mixed fraction, $5 + \frac{5}{14}$, helps figure out what the number means. That’s as far as we have to worry about the propriety of fractions. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses the form of a Fermi problem for its joke. Fermi problems have a place in mathematical modeling. The idea is to find an estimate for some quantity. We often want to do this. The trick is to build a simple model, and to calculate using a tiny bit of data. The Fermi problem that has someone reached public consciousness is called the Fermi paradox. The question that paradox addresses is, how many technologically advanced species are there in the galaxy? There’s no way to guess. But we can make models and those give us topics to investigate to better understand the problem. (The paradox is that reasonable guesses about the model suggest there should be so many aliens that they’d be a menace to air traffic. Or that the universe should be empty except for us. Both alternatives seem unrealistic.) Such estimates can be quite wrong, of course. I remember a Robert Heinlein essay in which he explained the Soviets were lying about the size of Moscow, his evidence being he didn’t see the ship traffic he expected when he toured the city. I do not remember that he analyzed what he might have reasoned wrong when he republished this in a collection of essays he didn’t seem to realize were funny. So the interview question presented is such a Fermi problem. The job applicant, presumably, has not committed to memory the number of employees at the company. But there would be clues. Does the company own the whole building it’s in, or just a floor? Just an office? How large is the building? How large is the parking lot? Are there people walking the hallways? How many desks are in the offices? The question could be answerable. The applicant has a pretty good chain of reasoning too. Bill Amend’s FoxTrot Classics for the 20th has several mathematical jokes in it. One is the use of excessively many decimal points to indicate intelligence. Grant that someone cares about the hyperbolic cosines of 15.2. There is no need to cite its wrong value to nine digits past the decimal. Decimal points are hypnotic, though, and listing many of them has connotations of relentless, robotic intelligence. That is what Amend went for in the characters here. That and showing how terrible nerds are when they find some petty issue to rage over. Eugene is correct about the hyperbolic cosine being wrong, there, though. He’s not wrong to check that. It’s good form to have some idea what a plausible answer should be. It lets one spot errors, for one. No mathematician is too good to avoid making dumb little mistakes. And computing tools will make mistakes too. Fortunately they don’t often, but this strip originally ran a couple years after the discovery of the Pentium FDIV bug. This was a glitch in the way certain Pentium chips handled floating-point division. It was discovered by Dr Thomas Nicely, at Lynchberg College, who found inconsistencies in some calculations when he added Pentium systems to the computers he was using. This Pentium bug may have been on Amend’s mind. Eugene would have spotted right away that the hyperbolic cosine was wrong, though, and didn’t need nine digits for it. The hyperbolic cosine is a function. Its domain is the real numbers. It range is entirely numbers greater than or equal to one, or less than or equal to minus one. A 0.9 something just can’t happen, not as the hyperbolic cosine for a real number. And what is the hyperbolic cosine? It’s one of the hyperbolic trigonometric functions. The other trig functions — sine, tangent, arc-sine, and all that — have their shadows too. You’ll see the hyperbolic sine and hyperbolic tangent some. You will never see the hyperbolic arc-cosecant and anyone trying to tell you that you need it is putting you on. They turn up in introductory calculus classes because you can differentiate them, and integrate them, the way you can ordinary trig functions. They look just different enough from regular trig functions to seem interesting for half a class. By the time you’re doing this, your instructor needs that. The ordinary trig functions come from the unit circle. You can relate the Cartesian coordinates of a point on the circle described by $x^2 + y^2 = 1$ to the angle made between that point and the center of the circle and the positive x-axis. Hyperbolic trig functions we can relate the Cartesian coordinates of a point on the hyperbola described by $x^2 - y^2 = 1$ to angles instead. The functions … don’t have a lot of use at the intro-to-calculus level. Again, other than that they let you do some quite testable differentiation and integration problems that don’t look exactly like regular trig functions do. They turn up again if you get far enough into mathematical physics. The hyperbolic cosine does well in describing catenaries, that is, the shape of flexible wires under gravity. And the family of functions turn up in statistical mechanics, often, in the mathematics of heat and of magnetism. But overall, these functions aren’t needed a lot. A good scientific calculator will offer them, certainly. But it’ll be harder to get them. There is another oddity at work here. The cosine of 15.2 degrees is about 0.965, yes. But mathematicians will usually think of trigonometric functions — regular or hyperbolic — in terms of radians. This is just a different measure of angles. A right angle, 90 degrees, is measured as $\frac{1}{2}\pi$ radians. The use of radians makes a good bit of other work easier. Mathematicians get to accustomed to using radians that to use degrees seems slightly alien. The cosine of 15.2 radians, then, would be about -0.874. Eugene has apparently left his calculator in degree mode, rather than radian mode. If he weren’t so worked up about the hyperbolic cosine being wrong he might have noticed. Perhaps that will be another exciting error to discover down the line. This strip was part of a several-months-long story Bill Amend did, in which Jason has adventures at Math Camp. I don’t remember the whole story. But I do expect the strip to have several more appearances here this summer. And that’s about half of last week’s comics. A fresh Reading the Comics post should be at this link later this week. Thank you for reading along. ## Reading the Comics, June 15, 2019: School Is Out? Edition This has not been the slowest week for mathematically-themed comic strips. The slowest would be the week nothing on topic came up. But this was close. I admit this is fine as I have things disrupting my normal schedule this week. I don’t need to write too many essays too. On-topic enough to discuss, though, were: Lalo Alcaraz’s La Cucaracha for the 9th features a teacher trying to get ahead of student boredom. The idea that mathematics is easier to learn if it’s about problems that seem interesting is a durable one. It agrees with my intuition. I’m less sure that just doing arithmetic while surfing is that helpful. My feeling is that a problem being interesting is separate from a problem naming an intersting thing. But making every problem uniquely interesting is probably too much to expect from a teacher. A good pop-mathematics writer can be interesting about any problem. But the pop-mathematics writer has a lot of choice about what she’ll discuss. And doesn’t need to practice examples of a problem until she can feel confident her readers have learned a skill. I don’t know that there is a good answer to this. Also part of me feels that “eight sick waves times eight sick waves” has to be “sixty-four sick-waves-squared”. This is me worrying about the dimensional analysis of a joke. All right, but if it were “eight inches times eight inches” and you came back with “sixty-four inches” you’d agree something was off, right? But it’s easy to not notice the units. That we do, mechanically, the same thing in multiplying (oh) three times$1.20 or three times 120 miles or three boxes times 120 items per box as we do multiplying three times 120 encourages this. But if we are using numbers to measure things, and if we are doing calculations about things, then the units matter. They carry information about the kinds of things our calculations represent. It’s a bad idea to misuse or ignore those tools.

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

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

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

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

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

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

## Reading the Comics, June 6, 2019: Not The Slowest Week Edition

Comic Strip Master Command started the summer vacation early this year. There have been even slower weeks for mathematically-themed comics, but not many, and not much slower. Well, it’s looking like a nice weekend anyway. We can go out and do something instead.

And I’m doing a little experiment to see what happens if I publish posts a bit earlier in the day. My suspicion is nothing that reaches statistical significance. But statistical significance isn’t everything. I can devote a month or two to a lark.

Piers Baker’s Ollie and Quentin for the 2nd is a rerun. The strip ended several years ago, and has not been one of those formerly syndicated comics gone to web-only publication. And it’s one that I’ve discussed before, in a 2014 repeat and briefly in 2015. I don’t know why it reran six months apart. Having a particular daily strip repeat so often is usually a sign I should retire the strip from this blog. Likely I won’t retire it from my reading. I like its style a bit too much.

The joke is built on Quentin hearing that only 50% of people are not happy. And as he is happy, and he and Ollie are two people, it follows Ollie can’t be. The joke builds on the logic of the gambler’s fallacy. This is the idea that the probability of some independent event depends on what has recently happened. Here “event” means what it does to statisticians, what it turns out something is. This can be the result of a coin toss. This can be finding out whether a person is happy or not. The gambler’s fallacy has a hard-to-resist logic to it. We know it is unlikely that a coin tossed fairly ten times will come up tails each time. We also know it is even more unlikely that a coin tossed fairly eleven times will turn up tails every time. So if the coin has already come up tails ten times? It’s easy in the abstract to sneer at people who make this mistake. But at some point or other we all think some unpredictable event is “due”.

There is a catch here, though. The gambler’s fallacy covers independent events. One coin’s toss does not affect whether the next toss should be heads or tails. But personal happiness? That is something affected by other people. Perhaps not dramatically. But one person’s mood can certainly alter another’s, just as the strip demonstrates. In past appearances of this strip I’ve written about it as though the mathematical comedy element were obvious. Now I realize I may have under-explored what is happening here.

Harley Schwadron’s 9 to 5 for the 3rd is a student-at-the-blackboard joke. And a joke about the uselessness of learning arithmetic if there are computing devices around. There have always been computing devices around, though. I’d prefer them for tedious problems, or for problems in which mistakes have serious consequences. But I think it’s worth knowing at least what to do. But I like mathematics. Of course I would.

Mike Baldwin’s Cornered for the 6th is another student-at-the-blackboard joke. This one has the student excusing his wrong answer, a number too high, as a tip. In the student’s defense, I’ll say being able to come up with a decent approximate answer, even one you know is a little too high, is worth it. Often an important step in a problem is knowing about what a reasonable answer is. This can involve mental-mathematics tricks. For example, remembering that 7 times 7 is just under fifty, which would help with a problem like 7 times 6.

And that’s all the comic strips I found worth any mention last week. There weren’t even any that rated a “there’s a comic that said ‘math class’, so here you go” aside. This bodes well for an interesting week of content around here. My next Reading the Comics post should appear next Sunday at this link. All the past comic strip discussion should, too. If you should find a comics essay that doesn’t appear in those archives please let me know. I’ll fix it.

## Reading the Comics, June 1, 2019: More Than I Thought Edition

When I collected last week’s mathematically-themed comic strips I thought this set an uninspiring one. That changed sometime while I wrote. That’s the sort of week I like to have.

Richard Thompson’s Richard’s Poor Almanac for the 28th is a repeat; all these strips are. And I’ve featured it here before too. But never before in color, so I’ll take this chance to show it one last time. One of the depicted plants is the “Non-Euclidean Creeper”, which “ignores the geometry of the space-time continuum”. Non-Euclidean is one of those few geometry-related words that people recognize — maybe even only learn — in their adulthood. It has connotations of the bizarre and the weird and the wrong.

And it is a bit weird. While we live in a non-Euclidean space, we never really notice. Euclidean space is the geometry we’re used to from drawing shapes on paper and putting boxes in the corners of basements. And from this we’ve given “non-Euclidean” this sinister reputation. We credit it with defying common sense and even logic itself, although it’s geometry. It can’t defy logic. It can defy intuition. Non-Euclidean geometries have the idea that there are no such things as parallel lines. Or the idea that there are too many parallel lines. And it can get to weird results, particularly if we look at more than three dimensions of space. Those also tax the imagination. It will get a weed a bad reputation.

Chen Weng’s Messycow Comics for the 30th is about a child’s delight in learning how to count. I don’t remember ever being so fascinated by counting that it would distract me permanently. I do remember thinking it was amazing that once a pattern was established it kept on, with no reason to ever stop, or even change. My recollection is I thought this somehow unfair to the alphabet, which had a very sudden sharp end.

The counting numbers — counting in general — seem to be things we’ve evolved to understand. Other animals know how to count. Here I recommend again Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics, which describes some of the things we know about how animals do mathematics. It also describes how children come to understand it.

Samson’s Dark Side of the Horse for the 31st is a bit of play with arithmetic. Horace simplifies his problem by catching all the numerals with loops in them — the zeroes and the eights — and working with what’s left. Evidently he’s already cast out all the nines. (This is me making a joke. Casting out nines is a simple checksum that you can do which can guard against some common arithmetic mistakes. It doesn’t catch everything. But it is simple enough to do that it can be worth using.)

The part that disappoints me is that to load the problem up with digits with loops, we get a problem that’s not actually hard: 100 times anything is easy. If the problem were, say, 189 times 80008005 then you’d have a problem someone might sensibly refuse to do. But without those zeroes at the start it’d be harder to understand what Horace was doing. Maybe if it were 10089 times 800805 instead.

Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st is the anthropomorphic numerals joke for the week. Also the anthropomorphic letters joke. The capital B sees occasional use in mathematics. It can represent the ball, that is, the set of all points that represent the interior of a sphere of a set radius. Usually a radius of 1. It also sometimes appears in equations as a parameter, a number whose value is fixed for the length of the problem but whose value we don’t care about. I had thought there were a few other roles for B alone, such as a label to represent the Bessel functions. These are a family of complicated-looking polynomials with some nice properties it’s too great a diversion for me to discuss just now. But they seem to more often be labelled with a capital J for reasons that probably seemed compelling at the time. It’ll also get used in logic, where B might stand for the second statement of some argument. 4, meanwhile, is that old familiar thing.

And there were a couple of comics which I like, but which mentioned mathematics so slightly that I couldn’t put a paragraph into them. Henry Scarpelli and Craig Boldman’s Archie rerun for the 27th, for example, mentions mathematics class as one it’s easy to sleep through. And Tony Cochrane’s Agnes for the 28th also mentions mathematics class, this time as one it’s hard to pay attention to.

This clears out last week’s comic strips. This present week’s strips should be at this link on Sunday. I haven’t yet read Friday or Saturday’s comics, so perhaps there’s been a flood, but this has been a slow week so far.

## Reading the Comics, May 30, 2019: Catching Out Tiger Mode

So this has been a week full of plans and machinations. But along the way, I made a discovery about Tiger. Curious? Of course you are. Who would not be? Read on and learn what my discovery is.

Hector D. Cantú and Carlos Castellanos’s Baldo for the 26th has Gracie counting by mathematical expressions. This kind of thing can be fun, at least for someone who enjoys doing arithmetic. Several years ago someone gave me a calendar in which every day was designated by an expression. As a mental exercise it wasn’t much, to my tastes. If you know that this is the second of the month, it’s no great work to figure out what $\cos(0) + \sin(\frac{\pi}{2})$ should be. But there is the fun in coming up with different ways to express a number. And here let me mention an old piece about how Paul Dirac worked out an expression for every counting number, using exactly four 2’s.

John Graziano’s Ripley’s Believe It or Not for the 26th mentions several fairly believable things. The relevant part is about naming the kind of surface that a Pringles chip represents. That is, the surface a Pringles chip would be if it weren’t all choppy and irregular, and if it continued indefinitely.

The shape is, as Graziano’s Ripley’s claims, a hypberbolic paraboloid. It’s a shape you get to know real well if you’re a mathematics major. They turn up in multivariable calculus and, if you do mathematical physics, in dynamical systems. It’s also a shape mathematics majors get to calling a “saddle shape”, because it looks enough like a saddle if you’re not really into horses.

The shape is one of the “quadratic surfaces”. These are shapes which can be described as the sets of Cartesian coordinates that make a quadratic equation true. Equations in Cartesian coordinates will have independent variables x, y, and z, unless there’s a really good reason. A quadratic equation will be the sum of some constant times x, and some constant times x2, and some constant times y, and some constant times y2, and some constant times z, and some constant times z2. Also some constant times xy, and some constant times yz, and some constant times xz. No xyz, though. And it might have some constant added to the mix at the end of all this.

There are seventeen different kinds of quadratic surfaces. Some of them are familiar, like ellipsoids or cones. Some hardly seem like they could be called “quadratic”, like intersecting planes. Or parallel planes. Some look like mid-century modern office lobby decor, like elliptic cylinders. And some have nice, faintly science-fictional shapes, like hyperboloids or, as in here, hyperbolic paraboloids. I’m not a judge of which ones would be good snack shapes.

Samson’s Dark Side of the Horse for the 26th is a funny-answer-to-a-story-problem joke. I had thought these had all switched over to apples, rather than candy bars. But that would make the punch line less believable.

Bud Blake’s Tiger for the 31st is a rerun, of course. Blake died in 2005 and no one else drew his comic strip. It’s a funny-answer-to-a-story-problem joke. And, more, it’s a repeat of a Tiger strip I’ve already run here. I admit a weird pride when I notice a comic strip doing a repeat. It gives me some hope that I might still be able to remember things. But this is also a special Tiger repeat. It’s the strip which made me notice Bud Blake redrawing comics he had already used. This one is not a third iteration of the strip which reran in April 2015 and June 2016. It’s a straight repeat of the June 2016 strip.

The mystery to me now is why King Features apparently has less than three years’ worth of reruns in the bank for Tiger. The comic ran from 1965 to 2003, and it’s not as though the strip made pop culture references or jokes ripped from the headlines. Even if the strip changed its dimensions over the decades, to accommodate shrinking newspapers, there should be a decade at least of usable strips to rerun.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 31st uses a chart to tease mathematicians, both in the comic and in the readership. The joke is in the format of the graph. The graph is supposed to argue that the Mathematician’s pedantry is increasing with time, and it does do that. But it is customary in this sort of graph for the independent variable to be the horizontal axis and the dependent variable the vertical. So, if the claim is that the pedantry level rises as time goes on, yes, this is a … well, I want to say wrong way to arrange the axes. This is because the chart, as drawn, breaks a convention. But convention is a tool to help people’s comprehension. We are right to ignore convention if doing so makes the chart better serve its purpose. Which, the punch line is, this does.

There’s just enough comics for me to do another essay this coming week. That next Reading the Comics post should be at this link around Thursday. That would be Tuesday except I need to fit my monthly readership report in sometime, don’t I? I think I need to, anyway.

## Reading the Comics, May 25, 2019: Slighter Comics Edition.

It turned out to be Thursday. These things happen. The comics for the second half of last week were more marginal

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a joke about holographic cosmology, proving that there are such things as jokes about holographic cosmology. Cosmology is about the big picture stuff, like, why there is a universe and why it looks like that. It’s a rather mathematical field, owing to the difficulty of doing controlled experiments. Holograms are that same technology used back in the 80s to put shoddy three-dimensional-ish pictures of eagles on credit cards. (In the United States. I imagine they were other animals in other countries.) Holograms, at least when they’re well-made, encode the information needed to make a three-dimensional image in a two-dimensional surface. (Please pretend that anything made of matter is two-dimensional like that.)

Holographic cosmology is a mathematical model for the universe. It represents the things in a space with a description of information on the boundary of this space. This seems bizarre and it won’t surprise you that key inspiration was in the strange physics of black holes. Properties of everything which falls into a black hole manifest in the event horizon, the boundary between normal space and whatever’s going on inside the black hole. The black hole is this three-dimensional volume, but in some way everything there is to say about it is the two-dimensional edge.

Dr Leonard Susskind did much to give this precise mathematical form. You didn’t think the character name was just a bit of whimsy, did you? Susskind’s work showed how the information of a particle falling into a black hole — information here meaning stuff like its position and momentum — turn into oscillations in the event horizon. The holographic principle argues this can be extended to ordinary space, the whole of the regular universe. Is this so? It’s hard to say. It’s a corner of string theory. It’s difficult to run experiments that prove very much. And we are stuck with an epistemological problem. If all the things in the universe and their interactions are equally well described as a three-dimensional volume or as a two-dimensional surface, which is “real”? It may seem intuitively obvious that we experience a three-dimensional space. But that intuition is a way we organize our understanding of our experiences. That’s not the same thing as truth.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 22nd is a joke about power, and how it can coerce someone out of truth. Arithmetic serves as an example of indisputable truth. It could be any deductive logic statement, or for that matter a definition. Arithmetic is great for the comic purpose needed here, though. Anyone can understand, at least the simpler statements, and work out their truth or falsity. And need very little word balloon space for it.

Bill Griffith’s Zippy the Pinhead for the 25th also features a quick mention of algebra as the height of rationality. Also as something difficult to understand. Most fields are hard to understand, when you truly try. But algebra works well for this writing purpose. Anyone who’d read Zippy the Pinhead has an idea of what understanding algebra would be like, the way they might not have an idea of holographic cosmology.

Teresa Logan’s Laughing Redhead Comics for the 25th is the Venn diagram joke for the week, this one with a celebrity theme. Your choice whether the logic of the joke makes sense. Ryan Reynolds and John Krasinski are among those celebrities that I keep thinking I don’t know, but that it turns out I do know. Ryan Gosling I’m still not sure about.

And then there are a couple strips too slight even to appear in this collection. Dean Young and John Marshall’s Blondie on the 22nd did a lottery joke, with discussion of probability along the way. (And I hadn’t had a tag for ‘Blondie’ before, so that’s an addition which someday will baffle me.) Bob Shannon’s Tough Town for the 23rd mentions mathematics teaching. It’s in service of a pun.

And now I’ve had the past week covered. The next Reading the Comics post should be at this link come Sunday.