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.

Evil Alexa: 'I ordered a spanking for you: express.' Sedine: 'DIE!' Michaela: 'How 'we defeat this evil genius? (To the home-assistant) What's the diffrence between wrong and right? Who's got better fries, McD or BK? Why's a ball round? Is a wingless fly a 'walk'? Why'z all this communism so capitalistic? If Jeff Bezos is so rich why'zint he abel to own a toupee? Zero divded by two equals zero, but why'z two divided by zero called crazy town? So if infinity is forever, isn't that crazy too? If reality is a human construck why does my mommy act so normal? Tell me!' Sputtering Alexia: 'I - I must compute!'
Bea R’s In Security for the 9th of August, 2019. This is a new comic strip for these parts. So this essay and any future ones which explore topics raised by In Security are to be be at this link.

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

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

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

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

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

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

Wavehead, standing in front of a digital blackboard which has the problem 3 + 5 on it: 'I'm just saying, with all the computing power in this electronic board, I bet it could take care of this itself.'
Mark Anderson’s Andertoons for the 10th of August, 2019. The handful of times that I’ve mentioned explore Andertoons around here can be found at this link.

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

Molly: 'We'll play after I finish my homework. I'm studying pi.' Bear: (Panel filled with the word GUSH! His mouth dangles open, and he drools.) 'You said pie!!'
Bob Scott’s Bear With Me for the 10th of August, 2019. Appearances by Bear With Me should be at this link. This strip originally ran the 15th of October, 2015, when the comic was titled Molly and the Bear.

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

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, February 13, 2019: Light Geometry Edition

Comic Strip Master Command decided this would be a light week, with about six comic strips worth discussing. I’ll go into four of them here, and in a day or two wrap up the remainder. There were several strips that didn’t quite rate discussion, and I’ll share those too. I never can be sure what strips will be best taped to someone’s office door.

Alex Hallatt’s Arctic Circle for the 10th was inspired by a tabular iceberg that got some attention in October 2018. It looked surprisingly rectangular. Smoother than we expect natural things to be. My first thought about this strip was to write about crystals. The ways that molecules can fit together may be reflected in how the whole structure looks. And this gets us to studying symmetries.

Ed Penguin 'Did you see the perfectly rectangular iceberg?' Lenny Lemming: 'Yes, but I've seen perfect triangles, rhomboids, and octagons, too.' (Oscar Penguin is startled. He walks over to Frank, who is chiseling out some kind of octagonal prism.) Oscar: 'Ok, Frank, I know I said you needed a hobby ... ' Frank: 'Let's see them explain THIS one with science.'
Alex Hallatt’s Arctic Circle for the 10th of February, 2019. Essays in which I discuss Arctic Circle should be at this link.

But I got to another thought. We’re surprised to see lines in nature. We know what lines are, and understand properties of them pretty well. Even if we don’t specialize in geometry we can understand how we expect them to work. I don’t know how much of this is a cultural artifact: in the western mathematics tradition lines and polygons and circles are taught a lot, and from an early age. My impression is that enough different cultures have similar enough geometries, though. (Are there any societies that don’t seem aware of the Pythagorean Theorem?) So what is it that has got so many people making perfect lines and circles and triangles and squares out of crooked timbers?

Broom Hilda: 'I'm a winner! I won $18 in the lottery!' Gaylord: 'How much did you spend on tickets?' Hilda: '$20.' Gaylord: 'So you're actually a loser!' Hilda: 'Well, I guess you could say that, but I wish you hadn't!'
Russell Myers’s Broom Hilda for the 13th of February, 2019. Essays inspired by Broom Hilda should be gathered at this link.

Russell Myers’s Broom Hilda for the 13th is a lottery joke. Also, really, an accounting joke. Most of the players of a lottery will not win, of course. Nearly none of them will win more than they’ve paid into the lottery. If they didn’t, there would be an official inquiry. So, yes, nearly all people, even those who win money at the lottery, would have had more money if they skipped playing altogether.

Where it becomes an accounting question is how much did Broom Hilda expect to have when the week was through? If she planned to spend $20 on lottery tickets, and got exactly that? It seems snobbish to me to say that’s a dumber way to spend twenty bucks than, say, buying twenty bucks worth of magazines that you’ll throw away in a month would be. Or having dinner at a fast-casual place. Or anything else that you like doing even though it won’t leave you, in the long run, any better off. Has she come out ahead? That depends where she figures she should be.

Caption: Transcendental Eric achieves a higher plane. It shows a shaded, spherical Eric in a three-dimensional space, while below him a square asks some other polygon, 'Where's Eric gone to?'
Eric the Circle for the 13th of February, 2019, this one by Alabama_Al. Appearances by Eric the Circle, whoever the writer, should be at this link.

Eric the Circle for the 13th, this one by Alabama_Al, is a plane- and solid-geometry joke. This gets it a bit more solidly on-topic than usual. But it’s still a strip focused on the connotations of mathematically-connected terms. There’s the metaphorical use of the ‘plane’ as in the thing people perceive as reality. There’s conflation between the idea of a ‘higher plane’ and ‘higher dimensions’. Also somewhere in here is the idea that ‘higher’ and ‘more’ dimensions of space are the same thing. ‘Transcendental’ here is used in the common English sense of surpassing something. ‘Transcendental’ has a mathematical definition too. That one relates to polynomials, because everything in mathematics is about polynomials. And, of course, one of the two numbers we know to be transcendental, and that people have any reason to care about, is π, which turns up all over circles.

Joey: 'Mom, can I have a cookie?' Mom: 'Joey, you had two cookies this morning, three at lunch, and one an hour ago! Now how many is that?' Joey: 'I changed my mind.' (Thinking) 'No cookie is worth a pop quiz in math.'
Larry Wright’s Motley for the 13th of February, 2019. It originally ran in 1988, I believe on the same date. When I have written about Motley the results should appear at this link. In transcribing the strip for the alt-text here I was getting all ready to grumble that I didn’t know the kid’s name, and the strip is so old and minor that nobody has a cast list on it. Then I noticed, oh, yes, Mom says what the kid’s name is.

Larry Wright’s Motley for the 13th riffs on the form of a story problem. Joey’s mother does ask something that seems like a plausible addition problem. I’m a bit surprised he hadn’t counted all the day’s cookies already, but perhaps he doesn’t dwell on past snacks.

This and all my Reading the Comics posts should appear at this link. Thanks for looking at my comments.

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

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

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

Teacher: 'Well, class, who'd like to show Mr Hoffmeyer how to correctly make an irregular polygon regular?' On the blackboard is an irregular pentagon and, drawn by Mr Hoffmeyer, a box of Ex-Lax.
Scott Hilburn’s The Argyle Sweater for the 2nd of January, 2019. The many appearances of Argyle Sweater in these pages are at this link.

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

Mutt, to Jeff in the hospital bed: 'Don't be afraid! Surgery on the tonsils is very simple!' Doctor: 'Don't you worry about the results!' Jeff: 'How do you know I'll be all right?' Doctor: 'Well, I lost my last eleven patients! So if the law of probabilities doesn't lie, you'll be all right! May I do something for you before I begin?' Jeff: 'Oh, yes, Doc! Help me put on my trousers and my jacket!'
Bud Fisher’s Mutt and Jeff rerun for the 4th of January, 2019. The several appearances of Mutt and Jeff in these pages are at this link.

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

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

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

On the blackboard: 'Ratios: Apples 9, Oranges 6'. Wavehead, to teacher: 'Technically the ratio is 3:2, but as a practical matter we shouldn't even really be considering this.'
Mark Anderson’s Andertoons for the 5th of January, 2019. The amazingly many appearances of Andertoons in these pages are at this link.

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

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

Kid: 'Dad, I need help with a math problem. If striking NFL players who get $35,000 a game are replaced by scab players who get $1,000 a game ... what will be the point spread in a game between the Lions and the Packers?'
Larry Wright’s Motley rerun for the 5th of January, 2019. The occasional appearances of Motley in these pages are at this link.

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

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

Reading the Comics, October 6, 2018: Square Root of 144 Edition

And I have three last strips from last week to talk about. For those curious, I have ten comics for this week that I flagged for mention, at least before reading the Saturday GoComics pages. So that will probably be two or three installments next week. It’ll depend how many Saturday GoComics strips raise a point I feel like discussing.

Jim Toomey’s Sherman’s Lagoon for the 5th uses arithmetic as the archetypical homework problem that’s short enough to fit in a panel but also too hard for an adult to do. And, neatly, easy for a computer to do. Were I either shark here I’d have reasoned out the square root of 144 something like this: they’re not getting homework where they’d be asked the square root of something that wasn’t a perfect square. So it’s got to be a whole number. 144 is between 100 and 400, so it’s got to be the square root of something between 10 and 20. 144 is pretty close to 100, so 144’s square root is probably close to 10. The square of 1 is 1, so 11 squared has to be one-hundred-something-and-one. The square of 2 is 4, so 12 squared has to be one-hundred-something-and-four. The square of 3 is 9, so 13 squared has to be one-hundred-something-and-nine. The square of 4 is 16, so 14 squared has to be at least one-hundred-something-and-six. And by then we’re getting pretty far from 10. So the only plausible candidate is 12. Test that out and, what do you know, there it is.

Herman: 'Dad, can you help me with my math homework?' Sherman: 'Highly doubtful. But I'm sure Alexa can. Ask her anything.' Herman: 'Alexa, what's the square root of 144?' Alexa: 'The square root of 144 is 12.' Herman: 'Wow. She's good. has Mom finally decided to replace you?' Sherman: 'Should I be worried?'
Jim Toomey’s Sherman’s Lagoon for the 5th of October, 2018. And if you’re wondering how an Alexa eavesdropping device is working underwater, go away. The strip’s not for you, and that’s your loss, because it’s nicely low-key weird and teaches me more about ocean biology than I ever imagined I’d know.

Greg Cravens’s The Buckets for the 6th is a riff on the monkeys-at-keyboards joke. Well, what keeps monkeys-at-typewriters from writing interesting things is that they don’t have any selection. They just produce text to no end, in principle. Picking out characters and words that carry narrative is what makes essayists and playwrights. … That said, I think every instructor has faced the essay that is, somehow, worse than gibberish. The process is to try to find anything that could be credited, even if it’s just including at least one of the words from the topic of the essay, and move briskly on.

Greg Cravens’s The Buckets for the 6th of October, 2018. I’m mostly sure the guy with all the hair is Eddie but, again, character lists. Please.

Larry Wright’s Motley for the 6th is a riff on the idea tips are impossibly complicated to calculate. And that any mathematics might as well be algebra. My question: what the heck calculation is Debbie describing here? There are different ways to find a 15 percent tip. One two-step one is to divide the bill by ten, which is easy and gets you 10 percent. Then divide that by two, which is not-hard, and gets you 5 percent. Add together the 10 percent and 5 percent and you get 15 percent. A one-step method is to just divide by six. This gets you a bit under 17 percent, but that’s close enough. It just requires an ability to divide by six.

Debbie, on the phone: '... multiply by two and move the decimal over one digit ... then divide by four and subtract that answer from the first answer.' Toady: 'Were you helping someone with their algebra?' Debbie: 'No, Dad's at the restaurant and wanted to leave a 15 percent tip.'
Larry Wright’s Motley for the 6th of October, 2018. This strip originally ran in 1987, although I can’t make out when. Also see earlier comments about cast lists; I had to look through about a month’s worth of comics to find both characters’ names here.

There’s other ways to go about it, surely. There are many ways to do any calculation you like. Some of them have the advantages of requiring fewer steps. Some require more steps, but hopefully easier steps. Debbie is, obviously, just describing a nonsensically complicated calculation, to fit the needs of the joke. I’m just trying to think of what a plausible process would lead into the first panel and still get the right answer.

My many Reading the Comics posts are at this link. Essays which mention Sherman’s Lagoon should be at this link. Other essays with The Buckets should appear at this link. And other essays discussing Motley Classics should be here.








































Reading the Comics, May 31, 2017: Feast Week Edition









