As though to reinforce how nothing was basically wrong, Comic Strip Master Command sent a normal number of mathematically themed comics around this past week. They bunched the strips up in the first half of the week, but that will happen. It was a fun set of strips in any event.

Rob Harrell’s Adam @ Home for the 11th tells of a teacher explaining division through violent means. I’m all for visualization tools and if we are going to use them, the more dramatic the better. But I suspect Mrs Clark’s students will end up confused about what exactly they’ve learned. If a doll is torn into five parts, is that communicating that one divided by five is five? If the students were supposed to identify the mass of the parts of the torn-up dolls as the result of dividing one by five, was that made clear to them? Maybe it was. But there’s always the risk in a dramatic presentation that the audience will misunderstand the point. The showier the drama the greater the risk, it seems to me. But I did only get the demonstration secondhand; who knows how well it was done?

Greg Cravens’ The Buckets for the 11th has the kid, Toby, struggling to turn a shirt backwards and inside-out without taking it off. As the commenters note this is the sort of problem we get into all the time in topology. The field is about what can we say about shapes when we don’t worry about distance? If all we know about a shape is the ways it’s connected, the number of holes it has, whether we can distinguish one side from another, what else can we conclude? I believe Gocomics.com commenter Mike is right: take one hand out the bottom of the shirt and slide it into the other sleeve from the outside end, and proceed from there. But I have not tried it myself. I haven’t yet started wearing long-sleeve shirts for the season.

Bill Amend’s FoxTrot for the 11th — a new strip — does a story problem featuring pizzas cut into some improbable numbers of slices. I don’t say it’s unrealistic someone might get this homework problem. Just that the story writer should really ask whether they’ve ever seen a pizza cut into sevenths. I have a faint memory of being served a pizza cut into tenths by same daft pizza shop, which implies fifths is at least possible. Sevenths I refuse, though.

Mark Tatulli’s Heart of the City for the 12th plays on the show-your-work directive many mathematics assignments carry. I like Heart’s showiness. But the point of showing your work is because nobody cares what (say) 224 divided by 14 is. What’s worth teaching is the ability to recognize what approaches are likely to solve what problems. What’s tested is whether someone can identify a way to solve the problem that’s likely to succeed, and whether that can be carried out successfully. This is why it’s always a good idea, if you are stumped on a problem, to write out how you think this problem should be solved. Writing out what you mean to do can clarify the steps you should take. And it can guide your instructor to whether you’re misunderstanding something fundamental, or whether you just missed something small, or whether you just had a bad day.

Norm Feuti’s Gil for the 12th, another rerun, has another fanciful depiction of showing your work. The teacher’s got a fair complaint in the note. We moved away from tally marks as a way to denote numbers for reasons. Twelve depictions of apples are harder to read than the number 12. And they’re terrible if we need to depict numbers like one-half or one-third. Might be an interesting side lesson in that.

Brian Basset’s Red and Rover for the 14th is a rerun and one I’ve mentioned in these parts before. I understand Red getting fired up to be an animator by the movie. It’s been a while since I watched Donald Duck in Mathmagic Land but my recollection is that while it was breathtaking and visually inventive it didn’t really get at mathematics. I mean, not at noticing interesting little oddities and working out whether they might be true always, or sometimes, or almost never. There is a lot of play in mathematics, especially in the exciting early stages where one looks for a thing to prove. But it’s also in seeing how an ingenious method lets you get just what you wanted to know. I don’t know that the short demonstrates enough of that.

Bud Blake’s Tiger rerun for the 15th gives Punkinhead the chance to ask a question. And it’s a great question. I’m not sure what I’d say arithmetic is, not if I’m going to be careful. Offhand I’d say arithmetic is a set of rules we apply to a set of things we call numbers. The rules are mostly about how we can take two numbers and a rule and replace them with a single number. And these turn out to correspond uncannily well with the sorts of things we do with counting, combining, separating, and doing some other stuff with real-world objects. That it’s so useful is why, I believe, arithmetic and geometry were the first mathematics humans learned. But much of geometry we can see. We can look at objects and see how they fit together. Arithmetic we have to infer from the way the stuff we like to count works. And that’s probably why it’s harder to do when we start school.

What’s not good about that as an answer is that it actually applies to a lot of mathematical constructs, including those crazy exotic ones you sometimes see in science press. You know, the ones where there’s this impossibly complicated tangle with ribbons of every color and a headline about “It’s Revolutionary. It’s 46-Dimensional. It’s Breaking The Rules Of Geometry. Is It The Shape That Finally Quantizes Gravity?” or something like that. Well, describe a thing vaguely and it’ll match a lot of other things. But also when we look to new mathematical structures, we tend to look for things that resemble arithmetic. Group theory, for example, is one of the cornerstones of modern mathematical thought. It’s built around having a set of things on which we can do something that looks like addition. So it shouldn’t be a surprise that many groups have a passing resemblance to arithmetic. Mathematics may produce universal truths. But the ones we see are also ones we are readied to see by our common experience. Arithmetic is part of that common experience.

Also Jerry Scott and Jim Borgman’s Zits for the 14th I think doesn’t really belong here. It’s just got a cameo appearance by the concept of mathematics. Dave Whamond’s Reality Check for the 17th similarly just mentions the subject. But I did want to reassure any readers worried after last week that Pierce recovered fine. Also that, you know, for not having a stomach for mathematics he’s doing well carrying on. Discipline will carry one far.

## Bringing Up Arthur Christmas Again

Since it’s the week for this, I would like to remind folks they could be watching the Aardman Animation film Arthur Christmas. Also, I was able to spin out a couple of mathematical and physics questions from one scene in the film. Last year I collected links to the essays — there’s five of them — into a single cover page. I hope you’ll consider them.

## Reblog: Animated Sieve of Eratosthenes

The Math Less Traveled has a lovely video here, animating the Sieve of Eratosthenes, one of the classic methods of finding all of the prime numbers one wants. I suppose it won’t eliminate writing out and crossing off numbers for extra credit on a math test. I actually remember that being one test I had in, I believe, seventh grade, for reasons that I don’t think I ever got. Possibly the teacher wanted to have an easy time grading, or was giving everyone a break from too much computation by shifting to evaluation of our crossing-out abilities.

Here’s something I made yesterday! (Note, I strongly suggest watching it fullscreen, in HD if you have the bandwidth for it.)

Can you figure out what’s going on? The source code for the animation is here; I was inspired by Jason Davies’ visualization which was in turn inspired by this.

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## 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.

## 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.