I’m writing this on Thursday, because I’m expecting to be busy Friday and Saturday. It might be a good policy if I planned the deadline for all these Reading the Comics posts to be a couple days before publishing. But it’ll probably ever come to that. I am not yet begun resisting treating this blog like a professional would. Well, what’s been interesting this week so far have been comic strips presenting or about story problems. That’s enough for a theme.
Olivia Jaimes’s Nancy for the 13th makes the familiar complaint that story problems aren’t “useful”. Perhaps not; she can cut the Gordion knot for most common setups. Well, students aren’t likely to get problems for which there’s no other way to find a solution. I suppose what’s happening is that many mathematical puzzles come from questions like, what’s the least amount of information you need to deduce something? Or what’s an indirect way to find that? Mathematicians are often drawn to questions like this. At least Nancy has found there are problems she’s legitimately interested in, questions about how to do a thing she finds important.
Mort Walker and Dik Browne’s vintage Hi and Lois for the 17th has Chip’s new arithmetic book trying to be more relevant. Chip’s still bored by the problem, which chooses to be about foreign aid. (In the 50s and 60s comics discovered it was very funny that the United States would just give money to other countries and not get anything back out of it except maybe their economies staying stable or their countries not going to war too much.)
Today, I’m just listing the comics from last week that mentioned mathematics, but which didn’t raise a deep enough topic to be worth discussing. You know what a story problem looks like. I can’t keep adding to that.
Hector D. Cantú and Carlos Castellanos’s Baldo for the 10th quotes René Descartes, billing him as a “French mathematician”. Which is true, but the quote is one about living properly. That’s more fairly a philosophical matter. Descartes has some reputation for his philosophical work, I understand.
Comic Strip Master Command hoped to give me an easy week, one that would let me finally get ahead on my A-to-Z essays and avoid the last-minute rush to complete tasks. I showed them, though. I can procrastinate more than they can give me breaks. This essay alone I’m writing about ten minutes after you read it.
Eric the Circle for the 7th, by Shoy, is one of the jokes where Eric’s drawn as something besides a circle. I can work with this, though, because the cube is less far from a circle than you think. It gets to what we mean by “a circle”. If it’s all the points that are exactly a particular distance from a given center? Or maybe all the points up to that particular distance from a given center? This seems too reasonable to argue with, so you know where the trick is.
The trick is asking what we mean by distance? The ordinary distance that normal people use has a couple names. The Euclidean distance, often. Or Euclidean metric. Euclidean norm. It has some fancier names that can wait. Give two points. You can find this distance easily if you have their coordinates in a Cartesian system. (There’s infinitely many Cartesian systems you could use. You can pick whatever one you like; the distance will be the same whatever they are.) That’s that thing about finding the distance between corresponding coordinates, squaring those distances, adding that up, and taking the square root. And that’s good.
That’s not our only choice, though. We can make a perfectly good distance using other rules. For example, take the difference between corresponding coordinates, take the absolute value of each, and add all those absolute values up. This distance even has real-world application. It’s how far it is to go from one place to another on a grid of city squares, where it’s considered poor form to walk directly through buildings. There’s another. Instead of adding those absolute values up? Just pick the biggest of the absolute values. This is another distance. In it, circles look like squares. Or, in three dimensions, spheres look like cubes.
Ryan North’s Dinosaur Comics for the 9th builds on a common science fictional premise, that contact with an alien intelligence is done through mathematics first. It’s a common supposition in science fiction circles, and among many scientists, that mathematics is a truly universal language. It’s hard to imagine a species capable of communication with us that wouldn’t understand two and two adding up to four. Or about the ratio of a circle circumference to its diameter being independent of that diameter. Or about how an alternating knot for which the minimum number of crossing points is odd can’t ever be amphicheiral.
All right, I guess I can imagine a species that never ran across that point. Which is one of the things we suppose in using mathematics as a universal language. Its truths are indisputable, if we allow the rules of logic and axioms and definitions that we use. And I agree I don’t know that it’s possible not to notice basic arithmetic and basic geometry, not if one lives in a sensory world much like humans’. But it does seem to me at least some of mathematics is probably idiosyncratic. In representation at least; certainly in organization. I suspect there may be trouble in using universal and generically true things to express something local and specific. I don’t know how to go from deductive logic to telling someone when my birthday is. Well, I’m sure our friends in the philosophy department have considered that problem and have some good thoughts we can use, if there were only some way to communicate with them.
Bill Whitehead’s Free Range for the 12th is your classic blackboard-full-of-symbols. I like the beauty of the symbols used. I mean, the whole expression doesn’t parse, but many of the symbols do and are used in reasonable ways. Long trailing strings of arrows to extend one line to another are common and reasonable too. In the middle of the second line is , which doesn’t make sense, but which doesn’t make sense in a way that seems authentic to working out an idea. It’s something that could be cleaned up if the reasoning needed to be made presentable.
John Hambrock’s The Brilliant Mind of Edison Lee for the 1st of October is a calendar joke. Well, many of the months used to have names that denoted their count. Month names have changed more than you’d think. For a while there every Roman Emperor was renaming months after himself. Most of these name changes did not stick. Lucius Aurelius Commodus, who reined from 177 to 192, gave all twelve months one or another of his names.
Several of the mathematically-themed comic strips from last week featured the fine art of calculation. So that was set to be my title for this week. Then I realized that all the comics worth some detailed mention were published last Sunday, and I do like essays that are entirely one-day affairs. There are a couple of other comic strips that mentioned mathematics tangentially and I’ll list those later this week.
John Hambrock’s The Brilliant Mind of Edison lee for the 29th has Edison show off an organic computer. This is a person, naturally enough. Everyone can do some arithmetic in their heads, especially if we allow that sometimes approximate answers are often fine. People with good speed and precision have always been wonders, though. The setup may also riff on the ancient joke of mathematicians being ways to turn coffee into theorems. (I would imagine that Hambrock has heard that joke. But it is enough to suppose that he’s aware many adult humans drink coffee.)
John Kovaleski’s Daddy Daze for the 29th sees Paul, the dad, working out the calculations his son (Angus) proposed. It’s a good bit of arithmetic that Paul’s doing in his head. The process of multiplying an insubstantial thing by many, many times until you get something of moderate size happens all the time. Much of integral calculus is based on the idea that we can add together infinitely many infinitesimal numbers, and from that get something understandable on the human scale. Saving nine seconds every other day is useless for actual activities, though. You need a certain fungibility in the thing conserved for the bother to be worth it.
Dan Thompson’s Harley for the 29th gets us into some comic strips not drawn by people named John. The comic has some mathematics in it qualitatively. The observation that you could jump a motorcycle farther, or higher, with more energy, and that you can get energy from rolling downhill. It’s here mostly because of the good fortune that another comic strip did a joke on the same topic, and did it quantitatively. That comic?
Bill Amend’s FoxTrot for the 29th. Young prodigies Jason and Marcus are putting serious calculation into their Hot Wheels track and working out the biggest loop-the-loop possible from a starting point. Their calculations are right, of course. Bill Amend, who’d been a physics major, likes putting authentic mathematics and mathematical physics in. The key is making sure the car moves fast enough in the loop that it stays on the track. This means the car experiencing a centrifugal force that’s larger than that of gravity. The centrifugal force on something moving in a circle is proportional to the square of the thing’s speed, and inversely proportional to the radius of the circle. This for a circle in any direction, by the way.
So they need to know, if the car starts at the height A, how fast will it go at the top of the loop, at height B? If the car’s going fast enough at height B to stay on the track, it’s certainly going fast enough to stay on for the rest of the loop.
The hard part would be figuring the speed at height B. Or it would be hard if we tried calculating the forces, and thus acceleration, of the car along the track. This would be a tedious problem. It would depend on the exact path of the track, for example. And it would be a long integration problem, which is trouble. There aren’t many integrals we can actually calculate directly. Most of the interesting ones we have to do numerically or work on approximations of the actual thing. This is all right, though. We don’t have to do that integral. We can look at potential energy instead. This turns what would be a tedious problem into the first three lines of work. And one of those was “Kinetic Energy = Δ Potential Energy”.
But as Peter observes, this does depend on supposing the track is frictionless. We always do this in basic physics problems. Friction is hard. It does depend on the exact path one follows, for example. And it depends on speed in complicated ways. We can make approximations to allow for friction losses, often based in experiment. Or try to make the problem one that has less friction, as Jason and Marcus are trying to do.
The second half of last week’s mathematically-themed comic strips had an interesting range of topics. Two of them seemed to circle around the making of models. So that’s my name for this installment.
Ryan North’s Dinosaur Comics for the 26th has T-Rex trying to build a model. In this case, it’s to project how often we should expect to see a real-life Batman. T-Rex is building a simple model, which is fine. Simple models, first, are usually easier to calculate with. How they differ from reality can give a guide to how to make a more complex model. Or they can indicate the things that have to be learned in order to make a more complex model. The difference between a model’s representation and the observed reality (or plausibly expected reality) can point out problems in one’s assumptions, too.
For example, T-Rex supposes that a Batman needs to have billionaire parents. This makes for a tiny number of available parents. But surely what’s important is that a Batman be wealthy enough he doesn’t have to show up to any appointments he doesn’t want to make. Having a half-billion dollars, or a “mere” hundred million, would allow that. Even a Batman who had “only” ten million dollars would be about as free to be a superhero. Similarly, consider the restriction to Olympic athletes. Astronaut Ed White, who on Gemini IV became the first American to walk in space, was not an Olympic athlete; but he certainly could have been. He missed by a split-second in the 400 meter hurdles race. Surely someone as physically fit as Ed White would be fit enough for a Batman. Not to say that “Olympic athletes or NASA astronauts” is a much bigger population than “Olympic athletes”. (And White was unusually fit even for NASA astronauts.) But it does suggest that merely counting Olympic athletes is too restrictive.
But that’s quibbling over the exact numbers. The process is a good rough model. List all the factors, suppose that all the factors are independent of one another, and multiply how likely it is each step happens by the population it could happen to. It’s hard to imagine a simpler model, but it’s a place to start.
Greg Wallace’s Nothing Is Not Something for the 26th is a bit of a geometry joke. It’s built on the idiom of the love triangle, expanding it into more-sided shapes. Relationships between groups of people like this can be well-represented in graph theory, with each person a vertex, and each pair of involved people an edge. There are even “directed graphs”, where each edge contains a direction. This lets one represent the difference between requited and unrequited interests.
Brian Anderson’s Dog Eat Doug for the 27th has Sophie the dog encounter some squirrels trying to disprove a flat Earth. They’re not proposing a round Earth either; they’ve gone in for a rhomboid. Sophie’s right to point out that drilling is a really hard way to get through the Earth. That’s a practical matter, though.
Is it possible to tell something about the shape of a whole thing from a small spot? In the terminology, what kind of global knowledge can we get from local information? We can do some things. For example, we can draw a triangle on the surface of the Earth and measure the interior angles to see what they sum to. If this could be done perfectly, finding that the interior angles add up to more than 180 degrees would show the triangle’s on a spherical surface. But that also has practical limitations. Like, if we find that locally the planet is curved then we can rule out it being entirely flat. But it’s imaginable that we’d be on the one dome of an otherwise flat planet. At some point you have to either assume you’re in a typical spot, or work out ways to find what’s atypical. In the Conspiracy Squirrels’ case, that would be the edge between two faces of the rhomboid Earth. Then it becomes something susceptible to reason.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th has the mathematician making another model. And this is one of the other uses of a model: to show a thing can’t happen, show that it would have results contrary to reason. But then you have to validate the model, showing that its premises do represent reality so well that its conclusion should be believed. This can be hard. There’s some nice symbol-writing on the chalkboard here, although I don’t see that they parse. Particularly, the bit on the right edge of the panel, where the writing has a rotated-by-180-degrees ‘E’ followed by an ‘x’, a rotated-by-180-degrees ‘A’, and then a ‘z’, is hard to fit inside an equation like this. The string of symbols mean “there exists some x for which, for all z, (something) is true”. This fits at the start of a proof, or before an equation starts. It doesn’t make grammatical sense in the middle of an equation. But, in the heat of writing out an idea, mathematicians will write out ungrammatical things. As with plain-text writing, it’s valuable to get an idea down, and edit it into good form later.
Tom Batiuk’s Funky Winkerbean Vintage for the 28th sees the school’s Computer explaining the nature of its existence, and how it works. Here the Computer claims to just be filled with thousands of toes to count on. It’s silly, but it is the case that there’s no operation a computer does that isn’t something a human can do, manually. If you had the paper and the time you could do all the steps of a Facebook group chat, a game of SimCity, or a rocket guidance computer’s calculations. The results might just be impractically slow.
I trust nobody’s too upset that I postponed the big Reading the Comics posts of this week a day. There’s enough comics from last week to split them into two essays. Please enjoy.
Scott Shaw! and Stan Sakai’s Popeye’s Cartoon Club for the 22nd is one of a yearlong series of Sunday strips, each by different cartoonists, celebrating the 90th year of Popeye’s existence as a character. And, I’m a Popeye fan from all the way back when Popeye was still a part of the pop culture. So that’s why I’m bringing such focus to a strip that, really, just mentions the existence of algebra teachers and that they might present a fearsome appearance to people.
Lincoln Pierce’s Big Nate for the 22nd has Nate seeking an omen for his mathematics test. This too seems marginal. But I can bring it back to mathematics. One of the fascinating things about having data is finding correlations between things. Sometimes we’ll find two things that seem to go together, including apparently disparate things like basketball success and test-taking scores. This can be an avenue for further research. One of these things might cause the other, or at least encourage it. Or the link may be spurious, both things caused by the same common factor. (Superstition can be one of those things: doing a thing ritually, in a competitive event, can help you perform better, even if you don’t believe in superstitions. Psychology is weird.)
But there are dangers too. Nate shows off here the danger of selecting the data set to give the result one wants. Even people with honest intentions can fall prey to this. Any real data set will have some points that just do not make sense, and look like a fluke or some error in data-gathering. Often the obvious nonsense can be safely disregarded, but you do need to think carefully to see that you are disregarding it for safe reasons. The other danger is that while two things do correlate, it’s all coincidence. Have enough pieces of data and sometimes they will seem to match up.
Norm Feuti’s Gil rerun for the 22nd has Gil practicing multiplication. It’s really about the difficulties of any kind of educational reform, especially in arithmetic. Gil’s mother is horrified by the appearance of this long multiplication. She dubs it both inefficient and harder than the way she learned. She doesn’t say the way she learned, but I’m guessing it’s the way that I learned too, which would have these problems done in three rows beneath the horizontal equals sign, with a bunch of little carry notes dotting above.
Gil’s Mother is horrified for bad reasons. Gil is doing exactly the same work that she was doing. The components of it are just written out differently. The only part of this that’s less “efficient” is that it fills out a little more paper. To me, who has no shortage of paper, this efficiency doens’t seem worth pursuing. I also like this way of writing things out, as it separates cleanly the partial products from the summations done with them. It also means that the carries from, say, multiplying the top number by the first digit of the lower can’t get in the way of carries from multiplying by the second digits. This seems likely to make it easier to avoid arithmetic errors, or to detect errors once suspected. I’d like to think that Gil’s Mom, having this pointed out, would drop her suspicions of this different way of writing things down. But people get very attached to the way they learned things, and will give that up only reluctantly. I include myself in this; there’s things I do for little better reason than inertia.
People will get hung up on the number of “steps” involved in a mathematical process. They shouldn’t. Whether, say, “37 x 2” is done in one step, two steps, or three steps is a matter of how you’re keeping the books. Even if we agree on how much computation is one step, we’re left with value judgements. Like, is it better to do many small steps, or few big steps? My own inclination is towards reliability. I’d rather take more steps than strictly necessary, if they can all be done more surely. If you want speed, my experience is, it’s better off aiming for reliability and consistency. Speed will follow from experience.
We can find the path by using the Lagrangian. Particularly, integrate the Lagrangian over every possible curve that connects the starting point and the ending point. This is every possible way to match start and end. The path that the system actually follows will be an extremum. The actual path will be one that minimizes (or maximizes) this integral, compared to all the other paths nearby that it might follow. Yes, that’s bizarre. How would the particle even know about those other paths?
This seems bad enough. But we can ignore the problem in classical mechanics. The extremum turns out to always match the path that we’d get from taking derivatives of the Lagrangian. Those derivatives look like calculating forces and stuff, like normal.
Then in quantum mechanics the problem reappears and we can’t just ignore it. In the quantum mechanics view no particle follows “a” “path”. It instead is found more likely in some configurations than in others. The most likely configurations correspond to extreme values of this integral. But we can’t just pretend that only the best-possible path “exists”.
Thus the strip’s point. We can represent mechanics quite well. We do this by pretending there are designated starting and ending conditions. And pretending that the system selects the best of every imaginable alternative. The incautious pop physics writer, eager to find exciting stuff about quantum mechanics, will describe this as a particle “exploring” or “considering” all its options before “selecting” one. This is true in the same way that we can say a weight “wants” to roll down the hill, or two magnets “try” to match north and south poles together. We should not mistake it for thinking that electrons go out planning their days, though. Newtonian mechanics gets us used to the idea that if we knew the positions and momentums and forces between everything in the universe perfectly well, we could forecast the future and retrodict the past perfectly. Lagrangian mechanics seems to invite us to imagine a world where everything “perceives” its future and all its possible options. It would be amazing if this did not capture our imaginations.
Bil Keane and Jeff Keane’s Family Circus for the 24th has young Billy amazed by the prospect of algebra, of doing mathematics with both numbers and letters. I’m assuming Billy’s awestruck by the idea of letters representing numbers. Geometry also uses quite a few letters, mostly as labels for the parts of shapes. But that seems like a less fascinating use of letters.
There were more mathematically-themed comic strips last week than I had time to deal with. This is in part because of something Saturday which took several more hours than I had expected. So let me start this week with some of the comics that, last week, mentioned mathematics in a marginal enough way there’s nothing to say about them besides yeah, that’s a comic strip which mentioned mathematics.
Jef Mallett’s Frazz for the 27th has a kid wondering why they need in-person instruction for arithmetic. (I’d agree that rehearsing arithmetic skills is very easy to automate. You can make practice problems pretty near without limit. How much this has to do with mathematics is a point of debate.)
This is almost all a post about some comics that don’t need more than a mention. You know, strips that just have someone in class not buying the word problem. These are the rest of last week’s.
Before I get there, though, I want to share something. I ran across an essay by Chris K Caldwell and Yeng Xiong: What Is The Smallest Prime? The topic is about 1, and whether that should be a prime number. Everyone who knows a little about mathematics knows that 1 is generally not considered a prime number. But we’re also a bit stumped to figure out why, since the idea of “a prime number is divisible by 1 and itself” seems to fit this, even if the fit is weird. And we have an explanation for this: 1 used to be thought of as prime, but it made various theorems more clumsy to present. So it was either cut 1 out of the definition or add the equivalent work to everything, and mathematicians went for the solution that was less work. I know that I’ve shared this story around here. (I’m surprised to find I didn’t share it in my Summer 2017 A-to-Z essay about prime numbers.)
The truth is more complicated than that. The truth of anything is always more complicated than its history. Even an excellent history’s. It’s not that the short story has things wrong, precisely. But that that matters are more complicated than that. The history includes things we forget were ever problems, like, the question of whether 1 should be a number. And that the question of whether mathematicians “used to” consider 1 a number is built on the supposition that mathematicians were a lot more uniform in their thinking than they were. Even to the individual: people were inconsistent in what they themselves wrote, because most mathematicians turn out to be people.
Tim Rickard’s Brewster Rockit for the 17th mentions entropy, which is so central to understanding statistical mechanics and information theory. It’s in the popular understanding of entropy, that of it being a thing which makes stuff get worse. But that’s of mathematical importance too.
There were a couple more comic strips than made a good fit in yesterday’s recap. Here’s the two that I had much to write about.
Jason Poland’s Robbie and Bobby for the 18th is another rerun. I mentioned it back in December of 2016. Zeno’s Paradoxical Pasta plays on the most famous of Zeno’s Paradoxes, about how to get to a place one has to get halfway there, but to get halfway there requires getting halfway to halfway. This goes on in infinite regression. The paradox is not a failure to understand that we can get to a place, or finish swallowing a noodle.
Norm Feuti’s Gil did not last long enough in syndication. This is a shame. The characters were great, the humor in a mode I like, and young Gil’s fascination with shows about the paranormal was eerily close to my own young self. But it didn’t last; my understanding is newspapers were reluctant to bring in a comic strip starring an impoverished family. This is a many-faceted shame, not least because the eternal tension between Gil’s fantasy life and his reality made it one of the few strips to reproduce the most vital element of Calvin and Hobbes. But Feuti decided to resume drawing Sunday strips, and I choose to include that in my Reading the Comics reading, because this is my blog and I can make the rules here, at least.
So here’s Norm Feuti’s Gil for the 15th. A couple days ago I saw someone amazed at finally learning where sunflower seeds come from. They’re the black part in the center of a sunflower, the part that makes the big yellow flower stand out in such contrast. People were giving the poster a hard time, asking, where did he think they came from? And the answer is just, he hadn’t thought about it. Why would he? It’s quite reasonable to go through life never encountering a sunflower seed except as a snack or as part of bird or squirrel food. Where on the sunflower plant it’d even be just doesn’t come up. If you want to make this a dire commentary on society losing its sense of where things come from, all right, I won’t stop you. But I think it’s more that there are a billion things to notice in the world, and so many things have names that are fanciful or allusive or ironic, that it’s normal not to realize that a phrase might literally represent its content.
So Gil having so associated a quarter with 25 cents, rather than one-fourth of a something, makes sense to me. (Especially given, as noted, that he and his mother are poor, and so he grows up attentive to cash.)
Isaac Asimov, prolific writer of cozy mysteries, had one short story built on the idea that a person might misremember 5:50, seen on a digital clock, as half-past five. I mention this to show how the difference between a quarter of a hundred of things, and the quarter of sixty things, will get mixed together.
Greg Evans’s Luann Againn for the 15th sees Luann struggling with algebra. And thinking of ways to at least get the answers. One advantage mathematics instructors have which many other subjects don’t is that you can create more problems easily. If for some reason isn’t usable anymore, you can make it and still be testing the same skills. But if you want to (as is reasonable) stick to what’s in a published text, yeah, you’re vulnerable to this.
And you can’t always just change a problem arbitrarily. For example, the expression in the second panel of the top row — — I notice factors into . I don’t know the objective of Luann’s homework, but it would probably be messed up if the problem were just changed to . Not that this couldn’t be worked, but that the work would involve annoying and complicated expressions instead of nice whole numbers or reasonable fractions.
Paul Trap’s Thatababy for the 15th presents Thatabay’s first counting-exponentially book, with the number of rabbits doubling every time. I admire the work Trap put in to drawing — in what we see here — 255 bunnies. I’m trusting there’s 128 in the last bunny panel; I’m not counting. At any rate he drew enough bunnies to not make it obvious to me where he repeats figures.
The traditional ever-increasing bunny spiral is the Fibonacci series. But in that, each panel would on average have only about three-fifths more bunnies than the one before it. That’s good, but it isn’t going to overwhelm as fast as the promise of 256 bunnies on the next page will.
The “dual” here is a mathematical term. Many mathematical things have duals. Polyhedrons have a commonly defined dual shape, though. Start with a polyhedron like, oh, the cube. The dual is a new polyhedron. The vertices of the dual are at the centers of the faces of the original polyhedron. And if two faces of the original polyhedron meet at an edge, then there’s an edge connecting the vertices at the centers of those faces. If several faces meet at a vertex in the original polyhedron, then in the dual there’s a face connecting the vertices dual to the original faces. Work all this out and you get, as you might expect, that the shape that’s dual to a cube is the octahedron we’re told just walked into the bar. The dual to the octahedron, meanwhile … well, that is a cube, which is nice and orderly. You might get a bit of a smile working out what the dual to a tetrahedron is.
Duals are useful, generically, because usually if you can prove something about a dual then you can prove it about the original thing. And we may find that something is easier to prove for the dual than for the original. This isn’t guaranteed, especially for geometric shapes like this, where it’s hard to say that either shape is harder to work with than the other. But it’s one of the tools we have to try sliding between the problem we need to do and the problem we can do.
Olivia Jaimes’s Nancy for the 17th has claims about the usefulness of arithmetic. And Nancy skeptical of them, as you expect for a kid facing mathematics in a comic strip. I admit I’ve never needed to do much arithmetic when I cooked. The most would be figuring out how to adjust the cooking time when two things need very different temperatures. But I always do that by winging it. Now I’m curious whether there are good references for suggested alternate times.
The past week included another Friday the 13th. Several comic strips found that worth mention. So that gives me a theme by which to name this look over the comic strips.
Charles Schulz’s Peanuts rerun for the 12th presents a pretty wordy algebra problem. And Peppermint Patty, in the grips of a math anxiety, freezing up and shutting down. One feels for her. Great long strings of words frighten anyone. The problem seems a bit complicated for kids Peppermint Patty’s and Franklin’s age. But the problem isn’t helping. One might notice, say, that a parent’s age will be some nice multiple of a child’s in a year or two. That in ten years a man’s age will be 14 greater than the combined age of their ages then? What imagination does that inspire?
Grant Peppermint Patty her fears. The situation isn’t hopeless. It helps to write out just what know, and what we would like to know. At least what we would like to know if we’ve granted the problem worth solving. What we would like is to know the man’s age. That’s some number; let’s call it M. What we know are things about how M relates to his daughter’s and his son’s age, and how those relate to one another. Since we know several things about the daughter’s age and the son’s age it’s worth giving those names too. Let’s say D for the daughter’s age and S for the son’s.
So. We know the son is three years older than the daughter. This we can write as . We know that in one year, the man will be six times as old as the daughter is now. In one year the man will be M + 1 years old. The daughter’s age now is D; six times that is 6D. So we know that . In ten years the man’s age will be M + 10; the daughter’s age, D + 10; the son’s age, S + 10. In ten years, M + 10 will be 14 plus D + 10 plus S + 10. That is, . Or if you prefer, . Or even, .
So this is a system of three equation, all linear, in three variables. This is hopeful. We can hope there will be a solution. And there is. There are different ways to find an answer. Since I’m grading this, you can use the one that feels most comfortable to you. The problem still seems a bit advanced for Peppermint Patty and Franklin.
Julie Larson’s The Dinette Set rerun for the 13th has a bit of talk about a mathematical discovery. The comic is accurate enough for its publication. In 2008 a number known as M43112609 was proven to be prime. The number, 243,112,609 – 1, is some 12,978,189 digits long. It’s still the fifth-largest known prime number (as I write this).
Prime numbers of the form 2N – 1 for some whole number N are known as Mersenne primes. These are named for Marin Mersenne, a 16th century French friar and mathematician. They’re a neat set of numbers. Each Mersenne prime matches some perfect number. Nobody knows whether there are finite or infinitely many Mersenne primes. Every even perfect number has a form that matches to some Mersenne prime. It’s unknown whether there are any odd perfect numbers. As often happens with number theory, the questions are easy to ask but hard to answer. But all the largest known prime numbers are Mersenne primes; they’re of a structure we can test pretty well. At least that electronic computers can test well; the last time the largest known prime was found by mere mechanical computer was 1951. The last time a non-Mersenne was the largest known prime was from 1989 to 1992, and before that, 1951.
T Shepherd’s Snow Sez for the 13th finishes off the unlucky-13 jokes. It observes that whatever a symbol might connote generally, your individual circumstances are more important. There are people for whom 13 is a good omen, or for whom Mondays are magnificent days, or for whom black cats are lucky.
These are all the comics I can write paragraphs about. There were more comics mentioning mathematics last week. Here were some of them:
There were a healthy number of comic strips with at least a bit of mathematical content the past week. Enough that I would maybe be able to split them across three essays in all. This conflicts with my plans to post two A-To-Z essays, and two short pieces bringing archived things back to some attention, when you consider the other thing I need to post this week. Well, I’ll work out something, this week at least. But if Comic Strip Master Command ever sends me a really busy week I’m going to be in trouble.
Bud Blake’s Tiger rerun for the 7th has Punkinhead ask one of those questions so basic it ends up being good and deep. What is arithmetic, exactly? Other than that it’s the mathematics you learn in elementary school that isn’t geometry? — an answer that’s maybe not satisfying but at least has historical roots. The quadrivium, four of the seven liberal arts of old, were arithmetic, geometry, astronomy, and music. Each of these has a fair claim on being a mathematics study, though I’d agree that music is a small part of mathematics these days. (I first wrote a “minor” piece, and didn’t want people to think I was making a pun, but you’ll notice I’m sharing it anyway.) I can’t say what people who study music learn about mathematics these days. Still, I’m not sure I can give a punchy answer to the question.
Mathworld offers the not-quite-precise definition that arithmetic is the field of mathematics dealing with integers or, more generally, numerical computation. But then it also offers a mnemonic for the spelling of arithmetic, which I wouldn’t have put in the fourth sentence of an article on the subject. I’m also not confident in that limitation to integers. Arithmetic certainly is about things we do on the integers, like addition and subtraction, multiplication and division, powers, roots, and factoring. So, yes, adding five and two is certainly arithmetic. But would we say that adding one-fifth and two is not arithmetic? Most other definitions I find allow that it can be about the rational numbers, or the real numbers. Some even accept the complex-valued numbers. The core is addition and subtraction, multiplication and division.
Arithmetic blends almost seamlessly into more complicated fields. One is number theory, which is the posing of problems that anyone can understand and that nobody can solve. If you ever run across a mathematical conjecture that’s over 200 years old and that nobody’s made much progress on besides checking that it’s true for all the whole numbers below 21,000,000,000 – 1, it’s probably number theory. Another is group theory, in which we think about structures that look like arithmetic without necessarily having all its fancy features like, oh, multiplication or the ability to factor elements. And it weaves into computing. Most computers rely on some kind of floating-point arithmetic, which approximates a wide range of the rational numbers that we’d expect to actually need.
So arithmetic is one of those things so fundamental and universal that it’s hard to take a chunk and say that this is it.
John Zakour and Scott Roberts’s Maria’s Day for the 8th has Maria fretting over what division means for emotions. I was getting ready to worry about Maria having the idea division means getting less of something. Five divided by one-half is not less than either five or one-half. My understanding is this unsettles a great many people learning division. But she does explicitly say, divide two, which I’m reading as “divide by two”. (I mean to be charitable in my reading of comic strips. It’s only fair.)
Still, even division into two things does not necessarily make things less. One of the fascinating and baffling discoveries of the 20th century was the Banach-Tarski Paradox. It’s a paradox only in that it defies intuition. According to it, one ball can be divided into as few as five pieces, and the pieces reassembled to make two whole balls. I would not expect Maria’s Dad to understand this well enough to explain.
There were naturally comic strips with too marginal a mention of mathematics to rate paragraphs. Among them the past week were these.
Stephen Bentley’s Herb and Jamaal rerun for the 11th portrays the aftermath of realizing a mathematics problem is easier than it seemed. Realizing this after a lot of work should feel good, as discovering a clever way around tedious work is great. But the lost time can still hurt.
Part of my work reading lots of comic strips is to report the ones that mention mathematics, even if the mention is so casual there’s no building an essay around them. Here’s the minor mathematics mentions of last week.
Bob Scott’s Bear With Me for the 3rd has Molly and the Bear in her geometry class. Bear’s shown as surprised the kids are still learning Euclidean geometry, which is your typical joke about the character with a particularly deep knowledge of a narrow field.
Gary Brookins’s Pluggers for the 5th is the old joke about how one never uses algebra in real life. The strip is not dated as a repeat. But I’d be surprised if this joke hasn’t run in Pluggers before. I didn’t have a tag for Pluggers before, but there was a time I wasn’t tagging the names of comic strips.
One thing to worry about during an A To Z sequence is how busy Comic Strip Master Command will decide I need to be. I’m glad to say that this first week, it wasn’t too overly busy. Even the comic strips that are most on topic are not ones that need too much explanation. They’re also all reruns from their original publication, although I don’t know the dates that any of these first ran. A casual search doesn’t find that I said anything about these in their previous appearances.
Mac King and Bill King’s Magic in a Minute for the 1st is a rerun printed without the editor reading the thing. If they had, they’d have edited the 13 to be a 19. As the explanation at the bottom of the page almost makes clear, the ‘magic number’ produced by this will be the last two digits of the current year. After all, your age (at the end of this year) will be this year minus the year of your birth.
That this can be used for a magic trick relies on two things. One is that while, yes, anyone who thinks about it sees the relationship between their birth year, their age, and the current year, the magic trick is done before they can do that thinking. They’re too busy calculating, and then counting out cards and trying to see where this is going. Calculating without thinking about why this calculation is dangerous for mathematics. But it allows for some recreational fun. the other thing this trick depends on is showmanship: the purpose of the calculation is meant to be surprising enough, and delightful enough, that people won’t care to deconstruct its logic.
Morrie Turner’s Wee Pals rerun for the 1st is your basic joke about the kid subverting a word problem. But it also shows a bit why mathematicians get trained to make as explicit as possible their assumptions. This saves us from dumb mistakes, but at the cost of putting a prologue to anything we do want to ask. But it’s a legitimate part of mathematics to look at the questions someone else has asked and find their unstated assumptions, the things that could be true and would make their claims wrong.
Ryan North’s Dinosaur Comics rerun for the 2nd presents Utahraptor struggling with a mathematics problem. This is in character for him and for the comic. The particular problem is a classic recreational mathematics puzzle. Given a balance that can only give relative weights, and that you can use up to three times, find the one ball out of twelve which is of a different weight. It’s also a classic information theory problem. We know we can solve it, though. Each weighing gives us information about which of the twelve balls might, or might not, be abnormal. There is enough information in these three weighings to pick out which ball is the unusual one.
Granted, though, just knowing three weighings are enough doesn’t tell us what to weigh, or in what order. I haven’t looked at the GoComics comments. But there are likely at least three people who’ve explained some way to do it. It’s worth playing with the problem a while to see if you have any good ideas. You can use coins if you want to play with possibilities.
Ryan North’s Dinosaur Comics rerun for the 6th is, as the last panels suggest, a sequel to a comic rerun in mid-August. The question of whether the word ‘heterological’ is itself heterological is a recasting of one of Eubulides’s paradoxes. It’s the problem of working out whether a self-referential statement can be true. Or false. It shouldn’t surprise us that common language statements can defy being called true or false. But definitions are so close to logical structures that it’s hard to see why these refuse to fit. The problem is silly, but why it’s silly is hard to say.
Each week Comic Strip Master Command sends out some comics that mention mathematics, but that aren’t substantial enough to write miniature essays about. This past week, too. Here are the comics that just mention mathematics. You may like them; there’s just not more to explain is all.
Dan Collins’s Looks Good On Paper rerun for the 27th uses a blackboard of mathematics — geometry-related formulas — to stand in for all classwork. This strip also ran in 2017 and in 2015. I haven’t checked 2013. I know the strip is still in original production, as it’ll include strips referring to current events, so I’ll keep reading it a while yet.
Ernie Bushmiller’s Nancy Classics for the 29th, which originally ran the 23rd of November, 1949, is a basic cheating-in-class joke. It works for mathematics in a way it wouldn’t for, say, history. Mathematics has enough symbols that don’t appear in ordinary writing that you could copy them upside-down without knowing that you transcribe something meaningless. Well, not realizing an upside-down 4 isn’t anything is a bit odd, but anyone can get pretty lost in symbols.
And so the Reading the Comics posts have returned to Sunday after a month in exile to Tuesdays. I’m curious whether Sunday is actually the best day to post my signature series of essays, since everybody is usually doing stuff on the weekends. Tuesdays more people are at work and looking for other things to think about. But at least for the duration of the A to Z series there’s not a good time to schedule them besides Sundays. So Sundays it is and I’ll possibly think things over again in December, if all goes well.
Ralph Hagen’s The Barn for the 27th poses a question that’s ridiculous when you look at it. Why should being twenty times as old as your newborn (sic) when you’re twenty years old imply you’d be twenty times as old as the newborn when you’re sixty? Age increases linearly. The ratios between ages, though, those decrease, in a ratio asymptotically approaching 1. So as far as that goes, this strip isn’t much of anything.
But I do like how it captures the way a mathematics puzzle can come from nowhere. Often interesting ones seem to generate themselves. You notice a pattern and wonder whether it reaches some interesting point. If you convince yourself it does, you wonder when it does. If it does not, you wonder why it can’t. This is the fun sort of mathematics, and you create it by looking at the two separate tile patterns in the kitchen or, as here, thinking about the ages of parent and child. Anything that catches the imagination of a bored mind. It’s fun being there.
Rory (the sheep) makes a common enough slip. Saying a twenty-year-old with a newborn is twenty times as old as the newborn is, implicitly, saying the newborn is one year old. This kind of error is so common it’s got a folksy name, the “fencepost error”. It has a more respectable name, for its LinkedIn profile, the “off-by-one error”. But you see the problem. Say that your birthday is the 1st of September. How many times were you alive on the 1st of September by the time you’re ten years old? Eleven times, the first one being the one you were born on, with one more counted up each year you’d lived. This was probably more clear before I explained it.
John Rose’s Barney Google and Snuffy Smith for the 27th has Mis Prunelly complimenting Jughaid’s creativity, but not wanting it in arithmetic. There is creativity in mathematics. And there is great value in calculating something in an original way. There’s value in calculating things wrong, too, if it’s an approximate calculation. Knowing whether your answer is nearer 10 or 20 is of some value, and it might be all that you in fact want. That’s being wrong in a productive way, though.
Harry Bliss and Steve Martin’s Bliss for the 27th uses a string of mathematical symbols as emblem of genius. Most of the symbols look just near enough meaningful that I wonder if Bliss and Martin got a mathematician friend of theirs to give them some scraps. Why I say mathematician rather than, say, physicist is because some of the lines look more mathematician than physicist.
The most distinctive one, to me, is right above Dumbo’s pencil and trunk there: . This is the kind of equation you’ll see all the time in group theory. It’s an important field of mathematics, the one studying sets that work like arithmetic does. This starts with groups, which have a set of things and a binary operation between those things. Think of it as either addition or multiplication. You notice that already looks like multiplication. ‘g’ and ‘h’ serve, for group theory, the roles that ‘x’ and ‘y’ do in (high school) algebra. ‘x’ and ‘y’ mean some number, whose value we might or might not care about. Similarly, ‘g’ and ‘h’ are some elements, things in the set for our group. We might or might not care which ones they are. means the identity element, the thing which won’t change the value of the other partner in an operation. The thing that works like zero for addition, or like one for multiplication. And means the inverse of : the thing which, added (or multiplied) to gives us the identity element. So if we were talking addition and were 5, then would be -5. This might not sound like very much, but we can make it complicated.
Also distinctive to me: that first line. I’m not perfectly sure I’m transcribing this right. But it looks a good deal to me like the binomial distribution. This is the probability of seeing something happen k times, if you give it n chances to happen, and every chance has the same probability p of it happening. The formula isn’t quite right. It’s missing a power on the (1 – p) term at the end. But it’s wrong in ways that make sense for the need to draw something legible.
Just under Dumbo’s pencil, too, is a line that I had to look up how to render in WordPress’s LaTeX. It’s the one about . The union symbol, the U there, speaks of set theory. It means to form a new set, one that has all the elements in the set called X or the set called Y or both. The straight vertical lines flanking these set names or descriptions are how we describe taking the norm, finding the size, of a set. This is ordinarily how many things are inside the set. If the sets X and Y have no elements in common, then the size of the union of X and Y will be the size of the set X plus the size of the set Y.
There’s other lines that come near making sense. The line about has the form of the “mapping” way to define a function. I just don’t understand what the rule here means. The final line, , first … well, this sort of e-raised-to-the-minus-something-squared form turns up all the time. But second, to end a bit of work with an exclamation point really captures the surprise and joy of having reached a goal. Mathematicians take delight in their work, like you’d expect.
Maria Scrivan’s Half Full for the 29th is a Rubik’s Cube joke. A variation of it ran back in June 2018. I hate that this time I noticed that on the right, the cubelet — with white on top, red on the lower left, and green on the lower right — is inconsistent with the ordered cube. The corresponding cubelet there has blue on top, red on the lower left, and green on the lower right. Well, maybe the cube on the right had its color stickers applied differently. This is a little thing. But it’s close to a problem that turns up all the time in representing geometry. It’s easy to say you have, say, axes going in the x, y, and z directions. But which direction is x? Which is y? Which is z? You can lay all three out so every pair makes a right angle. Whatever way you lay them out will turn out to be, up to a rotation, one of two patterns. Let’s say the x axis points east, and the y axis points north. Then the z axis can point up. Or it can point down. You can pick which one makes sense for your problem. The two choices are mirror images of the other. You get primed to notice this when you do mathematical physics. The Rubik’s Cube on the left is just this kind of representation, with (let’s say) the red face pointing in the x direction, the green face pointing in the y direction, and the blue pointing in the z direction. Which is a lot of thought to put into what was an arbitrary choice, as I’m sure the cartoonist (or whoever did the coloring) just wanted a cube that looked attractive.
As teased with the Andertoons I featured Tuesday, there’s some mathematics comics slight enough I can’t write paragraphs about them. But people like seeing comics that at least say “mathematics”, so here’s your heads-up to them.
Mark Parisi’s Off The Mark for the 18th is an anthropomorphic numerals joke. The numerals in a paint-by-numbers kit are really serving the role of indices, rather than anything numerical. The instructions would be the same if, say, a letter ‘p’ or a small square represented purple.
Gene Mora’s Graffiti for the 23rd is also a spot of wordplay mentioning geometry. And it comes back to the joke about one shape being a kind of another that New Adventures of Queen Victoria was on about.
While there were a good number of comic strips to mention mathematics this past week, there were only a few that seemed substantial to me. This works well enough. This probably is going to be the last time I keep the Reading the Comics post until after Sunday, at least until the Fall 2019 A To Z is finished.
Gordon Bess’s Redeye rerun for the 18th is a joke building on animals’ number sense. And, yeah, about dumb parents too. Horses doing arithmetic have a noteworthy history. But more in the field of understanding how animals learn, than in how they do arithmetic. In particular in how animals learn to respond to human cues, and how slight a cue has to be to be recognized and acted on. I imagine this reflects horses being unwieldy experimental animals. Birds — pigeons and ravens, particularly — make better test animals.
Art Sansom and Chip Sansom’s The Born Loser for the 18th gives a mental arithmetic problem. It’s a trick question, yes. But Brutus gives up too soon on what the problem is supposed to be. Now there’s no calculating, in your head, exactly how many seconds are in a year; that’s just too much work. But an estimate? That’s easy.
At least it’s easy if you remember one thing: a million seconds is about eleven and a half days. I find this easy to remember because it’s one of the ideas used all the time to express how big a million, a billion, and a trillion are. A million seconds are about eleven and a half days. A billion seconds are a little under 32 years. A trillion seconds are about 32,000 years, which is about how long it’s been since the oldest known domesticated dog skulls were fossilized. I’m sure that gives everyone a clear idea of how big a trillion is. The important thing, though, is that a million seconds is about eleven and a half days.
So. Think of the year. There are — as the punch line to Hattie’s riddle puts it — twelve 2nd’s in the year. So there are something like a million seconds spent each year on days that are the 2nd of the month. There about a million seconds spent each year on days that are the 1st of the month, too. There are about a million seconds spent each year on days that are the 3rd of the month. And so on. So, there’s something like 31 million seconds in the year.
You protest. There aren’t a million seconds in twelve days; there’s a million seconds in eleven and a half days. True. Also there aren’t 31 days in every month; there’s 31 days in seven months of the year. There’s 30 days in four months, and 28 or 29 in the remainder. That’s fine. This is mental arithmetic. I’m undercounting the number of seconds by supposing that a million seconds makes twelve days. I’m overcounting the number of seconds by supposing that there are twelve months of 31 days each. I’m willing to bet this undercount and this overcount roughly balance out. How close do I get?
There are 31,536,000 seconds in a common year. That is, a non-leap-year. So “31 million” is a bit low. But it’s not bad for working without a calculator.
Ryan North’s Dinosaur Comics for the 19th lays on us the Eubulides Paradox. It’s traced back to the fourth century BCE. Eubulides was a Greek philosopher, student of “Not That” Euclid of Megara. We know Eubulides for a set of paradoxes, including the Sorites paradox. As T-Rex’s friends point out, we’ve all heard this paradox. We’ve all gone on with our lives, knowing that the person who said it wanted us to say they were very clever. Fine.
But if we take this seriously we find … this keeps not being simple. We can avoid the problem by declaring self-referential statements exist outside of truth or falsity. This forces us to declare the sentence “this sentence is true” can’t be true. This seems goofy. We can avoid the problem by supposing there are things that are neither true nor false. That solves our problem here at the mere cost of ruining our ability to prove stuff by contradiction. There’s a lot of stuff we prove by contradiction. It’s hard to give that all up for this (Although, so far as I’m aware, anything that can be proved by contradiction can also be proven by a direct line of reasoning. The direct line may just be tedious.) We can solve this problem by saying that our words are fuzzy imprecise things. This is true enough, as see any time my love and I debate how many things are in “a couple of things”. But declaring that we just can’t express the problem well enough to answer it seems like running away from the question. We can resolve things by accepting there are limits to what can be proved by logic. Gödel’s Incompleteness Theorem shows that any interesting enough logic system has statements that are true but unprovable. A version of this paradox helps us get to this interesting conclusion.
So this is one of those things it should be easy to laugh off, but why it should be easy is hard.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st is about the other great logic problem of the 20th century. The Halting Problem here refers to Turing Machines. This is the algorithmic model for computing devices. It’s rather abstract, so the model won’t help you with your C++ homework, but nothing will. But it turns out we can represent a computer running a program as a string of cells. Each cell holds one of a couple possible values. The program is a series of steps. Each step starts at one cell. The program resets the value of that cell to something dictated by the algorithm. Then, the program moves focus to another cell, again as the algorithm dictates. Do enough of this and you get SimCity 2000. I don’t know all the steps in-between.
So. The Halting Program is this: take a program. Run it. What happens in the long run? Well, it does something or other, yes. But there’s three kinds of things it can do. It can run for a while and then finish, that is, ‘halt’. It can run for a while and then get into a repeating loop, after which it repeats things forever. It can run forever without repeating itself. (Yeah, I see the structural resemblance to terminating decimals, repeating decimals, and irrational numbers too, but I don’t know of any link there.) The Halting Problem asks, if all we know is the algorithm, can we know what happens? Can we say for sure the program will always end, regardless of what the data it works on are? Can we say for sure the program won’t end if we feed it the right data to start?
If the program is simple enough — and it has to be extremely simple — we can say. But, basically, if the program is complicated enough to be even the least bit interesting, it’s impossible to say. Even just running the program isn’t enough: how do you know the difference between a program that takes a trillion seconds to finish and one that never finishes?
For human needs, yes, a program that needs a trillion seconds might as well be one that never finishes. Which is not precisely the joke Weinersmith makes here, but is circling around similar territory.
Last week was another light week of work from Comic Strip Master Command. One could fairly argue that nothing is worth my attention. Except … one comic strip got onto the calendar. And that, my friends, is demanding I pay attention. Because the comic strip got multiple things wrong. And then the comments on GoComics got it more wrong. Got things wrong to the point that I could not be sure people weren’t trolling each other. I know how nerds work. They do this. It’s not pretty. So since I have the responsibility to correct strangers online I’ll focus a bit on that.
Robb Armstrong’s JumpStart for the 13th starts off all right. The early Roman calendar had ten months, December the tenth of them. This was a calendar that didn’t try to cover the whole year. It just started in spring and ran into early winter and that was it. This may seem baffling to us moderns, but it is, I promise you, the least confusing aspect of the Roman calendar. This may seem less strange if you think of the Roman calendar as like a sports team’s calendar, or a playhouse’s schedule of shows, or a timeline for a particular complicated event. There are just some fallow months that don’t need mention.
Things go wrong with Rob’s claim that December will have five Saturdays, five Sundays, and five Mondays. December 2019 will have no such thing. It has four Saturdays. There are five Sundays, Mondays, and Tuesdays. From Crunchy’s response it sounds like Joe’s run across some Internet Dubious Science Folklore. You know, where you see a claim that (like) Saturn will be larger in the sky than anytime since the glaciers receded or something. And as you’d expect, it’s gotten a bit out of date. December 2018 had five Saturdays, Sundays, and Mondays. So did December 2012. And December 2007.
And as this shows, that’s not a rare thing. Any month with 31 days will have five of some three days in the week. August 2019, for example, has five Thursdays, Fridays, and Saturdays. October 2019 will have five Tuesdays, Wednesdays, and Thursdays. This we can show by the pigeonhole principle. And there are seven months each with 31 days in every year.
It’s not every year that has some month with five Saturdays, Sundays, and Mondays in it. 2024 will not, for example. But a lot of years do. I’m not sure why December gets singled out for attention here. From the setup about December having long ago been the tenth month, I guess it’s some attempt to link the fives of the weekend days to the ten of the month number. But we get this kind of December about every five or six years.
This 823 years stuff, now that’s just gibberish. The Gregorian calendar has its wonders and mysteries yes. None of them have anything to do with 823 years. Here, people in the comments got really bad at explaining what was going on.
So. There are fourteen different … let me call them year plans, available to the Gregorian calendar. January can start on a Sunday when it is a leap year. Or January can start on a Sunday when it is not a leap year. January can start on a Monday when it is a leap year. January can start on a Monday when it is not a leap year. And so on. So there are fourteen possible arrangements of the twelve months of the year, what days of the week the twentieth of January and the thirtieth of December can occur on. The incautious might think this means there’s a period of fourteen years in the calendar. This comes from misapplying the pigeonhole principle.
Here’s the trouble. January 2019 started on a Tuesday. This implies that January 2020 starts on a Wednesday. January 2025 also starts on a Wednesday. But January 2024 starts on a Monday. You start to see the pattern. If this is not a leap year, the next year starts one day of the week later than this one. If this is a leap year, the next year starts two days of the week later. This is all a slightly annoying pattern, but it means that, typically, it takes 28 years to get back where you started. January 2019 started on Tuesday; January 2020 on Wednesday, and January 2021 on Friday. the same will hold for January 2047 and 2048 and 2049. There are other successive years that will start on Tuesday and Wednesday and Friday before that.
The important difference between the Julian and the Gregorian calendars is century years. 1900. 2000. 2100. These are all leap years by the Julian calendar reckoning. Most of them are not, by the Gregorian. Only century years divisible by 400 are. 2000 was a leap year; 2400 will be. 1900 was not; 2100 will not be, by the Gregorian scheme.
These exceptions to the leap-year-every-four-years pattern mess things up. The 28-year-period does not work if it stretches across a non-leap-year century year. By the way, if you have a friend who’s a programmer who has to deal with calendars? That friend hates being a programmer who has to deal with calendars.
There is still a period. It’s just a longer period. Happily the Gregorian calendar has a period of 400 years. The whole sequence of year patterns from 2000 through 2019 will reappear, 2400 through 2419. 2800 through 2819. 3200 through 3219.
(Whether they were also the year patterns for 1600 through 1619 depends on where you are. Countries which adopted the Gregorian calendar promptly? Yes. Countries which held out against it, such as Turkey or the United Kingdom? No. Other places? Other, possibly quite complicated, stories. If you ask your computer for the 1619 calendar it may well look nothing like 2019’s, and that’s because it is showing the Julian rather than Gregorian calendar.)
This is all in reference to the days of the week. The date of Easter, and all of the movable holidays tied to Easter, is on a completely different cycle. Easter is set by … oh, dear. Well, it’s supposed to be a simple enough idea: the Sunday after the first spring full moon. It uses a notional moon that’s less difficult to predict than the real one. It’s still a bit of a mess. The date of Easter is periodic again, yes. But the period is crazy long. It would take 5,700,000 years to complete its cycle on the Gregorian calendar. It never will. Never try to predict Easter. It won’t go well. Don’t believe anything amazing you read about Easter online.
Michael Jantze’s The Norm (Classics) for the 15th is much less trouble. It uses some mathematics to represent things being easy and things being hard. Easy’s represented with arithmetic. Hard is represented with the calculations of quantum mechanics. Which, oddly, look very much like arithmetic. even has fewer symbols than has. But the symbols mean different abstract things. In a quantum mechanics context, ‘A’ and ‘B’ represent — well, possibly matrices. More likely operators. Operators work a lot like functions and I’m going to skip discussing the ways they don’t. Multiplying operators together — B times A, here — works by using the range of one function as the domain of the other. Like, imagine ‘B’ means ‘take the square of’ and ‘A’ means ‘take the sine of’. Then ‘BA’ would mean ‘take the square of the sine of’ (something). The fun part is the ‘AB’ would mean ‘take the sine of the square of’ (something). Which is fun because most of the time, those won’t have the same value. We accept that, mathematically. It turns out to work well for some quantum mechanics properties, even though it doesn’t work like regular arithmetic. So holds complexity, or at least strangeness, in its few symbols.
There were some more comic strips which just mentioned mathematics in passing.
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 11th has a blackboard of mathematics used to represent deep thinking. Also, it I think, the colorist didn’t realize that they were standing in front of a blackboard. You can see mathematicians doing work in several colors, either to convey information in shorthand or because they had several colors of chalk. Not this way, though.
Mark Leiknes’s Cow and Boy rerun for the 16th mentions “being good at math” as something to respect cows for. The comic’s just this past week started over from its beginning. If you’re interested in deeply weird and long-since cancelled comics this is as good a chance to jump on as you can get.
That’s the mathematically-themed comic strips for last week. All my Reading the Comics essays should be at this link. I’ve traditionally run at least one essay a week on Sunday. But recently that’s moved to Tuesday for no truly compelling reason. That seems like it’s working for me, though. I may stick with it. If you do have an opinion about Sunday versus Tuesday please let me know.
Don’t let me know on Twitter. I continue to have this problem where Twitter won’t load on Safari. I don’t know why. I’m this close to trying it out on a different web browser.
And, again, I’m planning a fresh A To Z sequence. It’s never to early to think of mathematics topics that I might explain. I should probably have already started writing some. But you’ll know the official announcement when it comes. It’ll have art and everything.