## Reading the Comics, May 16, 2019: Two and Two Edition

It might be more fair to call this a blackboard edition, as three of the strips worth discussing feature that element. But I think I’ve used that name recently. And two of the strips feature specifically 2 + 2, so I’ll use that instead.

And here’s a possible movie heads-up. Turner Classic Movies, United States feed, is showing Monday at 9:30 am (Eastern/Pacific) All-American Chump. All I know about this 1936 movie is from its Leonard Maltin review:

[ Stuart ] Erwin is funny, in his usual country bumpkin way, as a small-town math whiz known as “the human adding machine” who is exploited by card sharks and hustlers. Fairly diverting double-feature item.

People with great powers of calculation were — and still are — with us. Before calculating machines were common they were, pop mathematicians tell us, in demand for doing the kinds of arithmetic mathematicians and engineers need a lot of. They’d also have value in performing, if they can put together some good patter. And, sure, gambling is just another field that needs calculation done well. I have no idea the quality of the film (it’s rated two and a half stars, but Leonard Maltin rates many things two and a half stars). But it’s there if you’re curious. The film also stars Robert Armstrong. I assume it’s not the guy I know but, you know? We live in a strange world. Now on to the comics.

Glenn McCoy and Gary McCoy’s The Flying McCoys for the 13th uses the image of a blackboard full of mathematics symbols to represent deep thought. The equations on the board are mostly nonsense, although some, like $E = mc^2$, have obvious meaning. Many of the other symbols have some meaning to them too. In the upper-right corner, for example, is what looks like $E = \hbar \omega$. This any physics major would recognize: it’s the energy of a photon, which is equal to Planck’s constant (that $\hbar$ stuff) times its frequency.

And there are other physics-relevant symbols. In the bottom center is a line that starts $\oint \vec{B}$. The capital B is commonly used to represent a magnetic field. The arrow above the capital B is a warning that this is a vector, which magnetic fields certainly are. (Mathematicians see vectors as a quite abstract concept. Physicists are more likely to see them as an intensity and direction, like forces, and the fields that make fields.) The $\oint$ symbol comes from vector calculus. It represent an integral taken along a closed loop, a shape that goes out along some path and comes back to where it started without crossing itself. This turns out to be useful all the time in dynamics problems. So the McCoys drew something that doesn’t mean anything, but looks ready to mean things.

“Overthinking this” is a problem common to mathematicians, even at an advanced level. Real problems don’t make clear what their boundaries are, the things that are important and the things that aren’t and the things that are convenient but not essential. Making mistakes picking them out, and working too hard on the wrong matters, will happen.

Graham Harrop’s Ten Cats for the 14th sees the cats pondering the counts of vast things. These are famous problems. Archimedes composed a text, The Sand Reckoner, which tried to estimate how much sand there could be in the universe. To work on the question he had to think of new ways to represent numbers. Grains of sand become numerous by being so tiny. Stars become numerous by the universe being so vast. Comparing the two quantities is a good challenge. For both numbers we have to make estimates. The volume of beaches in the world. The typical size of a grain of sand. The number of galaxies in the universe. The typical number of stars in a galaxy. There’s room to dispute all these numbers; we really have to come up with a range of possible values, with maybe some idea of what seems more likely.

Thaves’s Frank and Ernest for the 15th has the student bringing authority to his answer. The mathematician is called on to prove an answer is “technically” correct. I’m not sure whether the kid is meant to be prefacing the answer he’s about to give, or whether his answer was rewriting the horizontal “2 + 2 = ” in a vertical form.

Brant Parker and Johnny Hart’s The Wizard of Id Classics for the 15th is built around the divisibility of whole numbers, and of relative primes. Setting the fee as some simple integer fraction of the whole has practicality to it. It likely seemed even more practical in the days before currencies decimalized. The common £sd style currency Europeans used before decimals could be subdivided many ways evenly, with one-third of a pound (livre, Reichsgulden, etc) becoming 80 pence (deniers, Pfennig, etc). Unit fractions, and combinations of unit fractions, could offer interesting ways to slice up anything to a desired amount.

Jim Unger’s Herman for the 16th is a student-talking-back-to-the-teacher strip. It also uses the 2 + 2 problem. It’s a common thing for teachers to say they learn from their students. It’s even true, although I son’t know that people ever quite articulate how teachers learn. A good mistake is a great chance to learn. A good mistake shows off a kind of brilliant twist. That the student has understood some but not all of the idea, and has filled in the misunderstood parts with something plausible enough one has to think about why it’s wrong. And why someone would think the wrong idea might be right. There is a kind of mistake that inspires you to think closely about what “right” has to be, and students who know how to make those mistakes are treasures.

And for comic strips that aren’t quite worth a paragraph. Julia Kaye’s Up and Out for the 13th uses mathematics as stand-in for the sort of general education that anybody should master. David Waisglass and Gordon Coulthart’s Farcus for the 17th I don’t think is trying to be a mathematics joke. It’s sufficient joke that the painter’s spelled ‘sign’ wrong. But it did hit on the spelling that would encourage mathematics teachers to notice the strip. Patrick Roberts’s Todd the Dinosaur for the 18th mentions sudoku.

And with that I am caught up on the past week’s mathematically-themed comic strips. My next Reading the Comics post should be next Sunday, and at this link. Oh, I could have made the edition name something bragging about being on time.

## The Arthur Christmas Season

I don’t know how you spend your December, but part of it really ought to be done watching the Aardman Animation film Arthur Christmas. It inspired me to ponder a mathematical-physics question that got into some heady territory and this is a good time to point people back to that.

The first piece is Could Arthur Christmas’ Happen In Real Life? At one point in the movie Arthur and Grand-Santa are stranded on a Caribbean island while the reindeer and sleigh, without them, go flying off in a straight line. This raises the question of what is a straight line if you’re on the surface of something spherical like the Earth. Also, Grand-Santa is such a fantastic idea for the Santa canon it’s hard to believe that Rankin-Bass never did it.

Returning To Arthur Christmas was titled that because I’d left the subject for a couple weeks. You know how it gets. Here the discussion becomes more spoiler-y. And it has to address the question of what kind of straight line the reindeer might move in. There’s several possible answers and they’re all interesting.

Arthur Christmas And The Least Common Multiple supposes that reindeer move as way satellites do. By making some assumptions about the speed of the reindeer and the path they’re taking, I get to see how long Arthur and Grand-Santa would need to wait before the reindeer and sled are back if they’re lucky enough to be waiting on the equator.

Six Minutes Off makes the problem of Arthur and Grand-Santa waiting for the return of flying reindeer more realistic. This involves supposing that they’re not on the equator, which makes meeting up the reindeer a much nastier bit of timing. If they get unlucky it could make their rescue take over five thousand years, which would complicate the movie’s plot some.

And finally Arthur Christmas and the End of Time gets into one of those staggering thoughts. This would be recurrence, an idea that weaves into statistical mechanics and that seems to require that we accept how the conservation of energy and the fact of entropy are, together, a paradox. So we get into considerations of the long-term fate of the universe. Maybe.

## A Leap Day 2016 Mathematics A To Z: Energy

Another of the requests I got for this A To Z was for energy. It came from Dave Kingsbury, of the A Nomad In Cyberspace blog. He was particularly intersted in how E = mc2 and how we might know that’s so. But we ended up threshing that out tolerably well in the original Any Requests post. So I’ll take the energy as my starting point again and go in a different direction.

## Energy.

When I was in high school, back when the world was new, our physics teacher presented the class with a problem inspired by an Indiana Jones movie. There’s a scene where Jones is faced with dropping to sure death from the rope. He cuts the bridge instead, swinging on it to the cliff face and safety. He asked: would that help any?

It’s easy to understand a person dropping the fifty feet we supposed it was. A high school physics class can do the mathematics involved and say how fast Jones would hit the water below. You don’t even need the little bit of calculus we could do then. At least if you’re willing to ignore air resistance. High school physics classes always are.

Swinging on the rope bridge, though — that’s harder. We could model it all right. We could pretend Jones was a weight on the end of a rigid pendulum. And we could describe what the forces accelerating this weight on a pendulum are going through as it swings its arc down. But we looked at the integrals we would have to work out to say how fast he would hit the cliff face. It wasn’t pretty. We had no idea how to even look up how to do these.

He spared us this work. His point in this was to revive our interest in physics by bringing in pop culture and to introduce the conservation of energy. We can ignore all these forces and positions and the path of a falling thing. We can look at the potential energy, the result of gravity, at the top of the bridge. Then look at how much less there is at the bottom. Where does that energy go? It goes into kinetic energy, increasing the momentum of the falling body. We can get what we are interested in — how fast Jones is moving at the end of his fall — with a lot less work.

Why is this less work? I doubt I can explain the deep philosophical implications of that well enough. I can point to the obvious. Forces and positions and velocities and all are vectors. They’re ordered sets of numbers. You have to keep the ordering consistent. You have to pay attention to paths. You have to keep track of the angles between, say, what direction gravity accelerates Jones, and where Jones is relative his starting point, and in what direction he’s moving. We have notation that makes all this easy to follow. But there’s a lot of work hiding behind the symbols.

Energy, though … well, that’s just a number. It’s even a constant number, if energy is conserved. We can split off a potential energy. That’s still just a number. If it changes, we can tell how much it’s changed by subtraction. We’re comfortable with that.

Mathematicians call that a scalar. That just means that it’s a real number. It happens to represent something interesting. We can relate the scalar representing potential energy to the vectors of forces that describe how things move. (Spoiler: finding the relationship involves calculus. We go from vectors to a scalar by integration. We go from the scalar to the vector by a gradient, which is a kind of vector-valued derivative.) Once we know this relationship we have two ways of describing the same stuff. We can switch to whichever one makes our work easier. This is often the scalar. Solitary numbers are just so often easier than ordered sets of numbers.

The energy, or the potential energy, of a physical system isn’t the only time we can change a vector problem into a scalar. And we can’t always do that anyway. If we have to keep track of things like air resistance or energy spent, say, melting the ice we’re staking over, then the change from vectors to a scalar loses information we can’t do without. But the trick often works. Potential energy is one of the most familiar ways this is used.

I assume Jones got through his bridge problem all right. Happens that I still haven’t seen the movies, but I have heard quite a bit about them and played the pinball game.

## Only Fractions

My love and I saw Only Yesterday recently. It’s a 1991 Studio Ghibli film, directed by Isao Takahata. It hasn’t had a United States release before, which is a pity; it’s quite good. The movie is about a woman, Taeko, reflecting on her childhood as she considers changing her life. One of the many wonderfully-realized scenes is about ten-year-old Taeko’s struggles with arithmetic. You probably guessed that, as otherwise the movie would seem outside the remit of this blog.

In the scene Taeko has had a disastrous arithmetic test. Her older sister is trying to coach her through how to divide fractions. It goes lousy. Her older sister insists it’s just a matter of inverting and multiplying. This is a useful tip if you understand how to divide fractions and need to keep straight what you’re doing. If you don’t understand, then it’s whatever the modern equivalent is for instructions on how to set a VCR.

Taeko tries to understand one problem. $\frac{2}{3} \div \frac{4}{1}$. She pictures it as an apple and draws a circle, blacking out a third of it. She cuts the rest into four equally-sized pieces and concludes that you could fit six slices into the original apple. Her sister stammers over this and fumes. She declares “that’s multiplication!”. She complains her sister isn’t doing the right thing, she’s not inverting and multiplying. I recognize her sister’s panic. It’s the bluster of someone trying to explain something not actually understood, on watching someone going far off the script.

The scene’s filled with irony. Taeko has a better understanding of what she’s doing than her sister has, but never knows it. Her sister understands a procedure but not what fractions dividing signifies. She can’t say why one wants to invert anything or multiply something. Taeko knows what the question she’s asked means, but not how to relate that to what she’s asked to do.

I don’t want to undervalue learning procedures. They’re worth knowing. They are, once you master them, efficient ways to compute. But there are many ways to master a procedure. I can’t believe there is one way to learn anything that works for everyone. One of many challenges teachers face is exploring the different ways their students best learn. Another is getting close enough to how they best learn that most of the students can understand something. It’s a pity when real people akin to Taeko can’t get that little bridge to connect their drawings of an apple to the page of fractions to be worked out.

## The Arthur Christmas Problem

Since it’s the season for it I’d like to point new or new-wish readers to a couple of posts I did in 2012-13, based on the Aardman Animation film Arthur Christmas, which was just so very charming in every way. It also puts forth some good mathematical and mathematical-physics questions.

Opening the scene is “Could Arthur Christmas’ Happen In Real Life?” which begins with a scene in the movie: Arthur and Grand-Santa are stranded on a Caribbean island while the reindeer and sleigh, without them, go flying off in a straight line. This raises the question of what is a straight line if you’re on the surface of something spherical like the Earth.

“Returning To Arthur Christmas” was titled that because I’d left the subject for a couple weeks, as is my wont, and it gets a little bit more spoiler-y since the film seems to come down on the side of the reindeer moving on a path called a Great Circle. This forces us to ask another question: if the reindeer are moving around the Earth, are they moving with the Earth’s rotation, like an airplane does, or freely of it, like a satellite does?

“Arthur Christmas And The Least Common Multiple” starts by supposing that the reindeer are moving the way satellites do, independent of the Earth’s rotation, and on making some assumptions about the speed of the reindeer and the path they’re taking, works out how long Arthur and Grand-Santa would need to wait before the reindeer and sled are back if they’re lucky enough to be waiting on the equator.

“Six Minutes Off” shifts matters a little, by supposing that they’re not on the equator, which makes meeting up the reindeer a much nastier bit of timing. If they’re willing to wait long enough the reindeer will come as close as they want to their position, but the wait can be impractically long, for example, eight years, or over five thousand years, which would really slow down the movie.

And finally “Arthur Christmas and the End of Time” wraps up matters with a bit of heady speculation about recurrence: the way that a physical system can, if the proper conditions are met, come back either to its starting point or to a condition arbitrarily close to its starting point, if you wait long enough. This offers some dazzling ideas about the really, really long-term fate of the universe, which is always a heady thought. I hope you enjoy.

## December 2013’s Statistics

There’s a hopeful trend in my readership statistics for December 2013 around these parts: according to WordPress, my number of readers grew from 308 in November to 352 and the number of unique visitors grew from 158 to 176. Even the number of views per visitor grew, from 1.95 to 2.00. None of these are records, but the fact of improvement is a good one.

I can’t figure exactly how to get the report on most popular articles for the exact month of December, and was too busy with other things to check the past-30-day report on New Year’s Eve, but at least the most popular articles for the 30 days ending today were:

The countries sending me the most readers were the United States, Canada, Denmark and Austria (tied, and hi again, Elke), and the United Kingdom. Sending me just one viewer each were a slew of nations: Bangladesh, Cambodia, India, Japan, Jordan, Malaysia, Norway, Romania, Slovenia, South Africa, Spain, Sweden, Turkey, and Viet Nam. On that list last month were Jordan and Slovenia, so I’m also marginally interesting to a different group of people this time around.

This has all caused me to realize that I failed to promote my string of articles inspired by Arthur Christmas and getting to the recurrence theorem and the existential dread of the universe’s end during the Christmas season. Maybe next year, then.

## Arthur Christmas and the End of Time

In working out my little Arthur Christmas-inspired problem, I argued that if the reindeer take some nice rational number of hours to complete one orbit of the Earth, eventually they’ll meet back up with Arthur and Grand-Santa stranded on the ground. And if the reindeer take an irrational number of hours to make one orbit, they’ll never meet again, although if they wait long enough, they’ll get pretty close together, eventually.

So far this doesn’t sound like a really thrilling result: the two parties, moving on their own paths, either meet again, or they don’t. Doesn’t sound quite like I earned the four-figure income I got from mathematics work last year. But here’s where I get to be worth it: if the reindeer and Arthur don’t meet up again, but I can accept their being very near one another, then they will get as close as I like. I only figured how long it would take for the two to get about 23 centimeters apart, but if I wanted, I could wait for them to be two centimeters apart, or two millimeters, or two angstroms if I wanted. I’d pay for this nearer miss with a longer wait. And this gives me my opening to a really stunning bit of mathematics.

## Six Minutes Off

Let me return, reindeer-like, to my problem, pretty well divorced from the movie at this point, of the stranded Arthur Christmas and Grand-Santa, stuck to wherever they happen to be on the surface of the Earth, going around the Earth’s axis of rotation every 86,164 seconds, while their reindeer and sleigh carry on orbiting the planet’s center once every $\sqrt{2}$ hours. That’s just a touch more than every 5,091 seconds. This means, sadly, that the reindeer will never be right above Arthur again, or else the whole system of rational and irrational numbers is a shambles. Still, they might come close.

After all, one day after being stranded, Arthur and Grand-Santa will be right back to the position where they started, and the reindeer will be just finishing up their seventeenth loop around the Earth. To be more nearly exact, after 86,164 seconds the reindeer will have finished just about 16.924 laps around the planet. If Arthur and Grand-Santa just hold out for another six and a half minutes (very nearly), the reindeer will be back to their line of latitude, and they’ll just be … well, how far away from that spot depends on just where they are. Since this is my problem, I’m going to drop them just a touch north of 30 degrees north latitude, because that means they’ll be travelling a neat 400 meters per second due to the Earth’s rotation and I certainly need some nice numbers here. Any nice number. I’m putting up with a day of 86,164 seconds, for crying out loud.

## How Fast Is The Earth Spinning?

To get to my next point about Arthur Christmas I needed to know how fast an arbitrary point on the Earth is moving, as the Earth rotates. This required me getting out a sheet of paper and doing some sketches, so, I figured it’s worth a side article to explain what I was doing.

The first thing was that I simplified stuff. In particular, I decided the Earth is near enough a sphere that I’m not bothering with the fact that it isn’t. The difference between an actual sphere and the geoid is not worth bothering with unless you’re timing the retrofire for a ballistically-reentering space capsule. That’s … actually fairly close to the problem I want, about how long it might take the reindeer and sleigh to get back to Arthur Christmas and Grand-Santa, but that’s also too much work for the improvement in the answer I’d get.

## Reading the Comics, January 21, 2013

Feast or famine, as I said. It’s not a week since the last comics roundup and I have eight comic strips that have enough mathematical content for me to discuss. Well, they’re fun essays to write, and people seem to quite like them, so why not another so soon?

Well, because I’m overlooking all the King Features Syndicate comics. I’m not actually overlooking them — I’m keeping track of just which ones have something I could write about — but they haven’t had a nice, archive-friendly way to point people to the strips being discussed. (Most newspaper web sites that have King Features comics have links to those pages expire in a measly 28 days.) Based on the surprising number of people who come to my site by searching for Norm Feuti’s Retail comic strip, they certainly deserve to be talked about. I’ll have something worked out about that soon, I promise.

## Arthur Christmas and the Least Common Multiple

I left Arthur Christmas and Grand-Santa in a hypothetical puzzle, inspired by the movie, with them stranded on a tiny island while their team of flying reindeer and sleigh carried on in a straight line without them. I am assuming for the sake of an interesting problem that this means the reindeer are carrying on the Great Circle route, favored by airplanes and satellites, and that the reindeer are in an orbit more like the satellite’s than the reindeers — that is, they keep to a circle in a plane which isn’t rotating while the Earth does, since otherwise, Arthur and Grand-Santa have to wait only for the reindeer to finish one lap around the planet and somehow get up to flying altitude to be picked up. If the reindeer aren’t rotating the with the Earth, then, when the reindeer finish one circuit our heroes are going to be … well, maybe east, maybe west, of the reindeer; the problem is, they’re going to be away.

## Returning to Arthur Christmas

As promised, since I’ve got the chance, I want to return to the question of the reindeer behavior as shown in the Aardman movie Arthur Christmas, and what would ultimately happen to them if the reindeer carry on as Grand-Santa claims they will. (Again, this does require spoiling a plot point of the film and so I tuck the rest behind a cut.)

## Could “Arthur Christmas” Happen In Real Life?

If you haven’t seen the Aardman Animation movie Arthur Christmas, first, shame on you as it’s quite fun. But also you may wish to think carefully before reading this entry, and a few I project to follow, as it takes one plot point from the film which I think has some interesting mathematical implications, reaching ultimately to the fate of the universe, if I can get a good running start. But I can’t address the question without spoiling a suspense hook, so please do consider that. And watch the film; it’s a grand one about the Santa family.

The premise — without spoiling more than the commercials did — starts with Arthur, son of the current Santa, and Grand-Santa, father of the current fellow, and a linguistic construct which perfectly fills a niche I hadn’t realized was previously vacant, going off on their own to deliver a gift accidentally not delivered to one kid. To do this they take the old sleigh, as pulled by the reindeer, and they’re off over the waters when something happens and there I cut for spoilers.