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.
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.)
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.
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.
Ernie Bushmiller’s Nancy Classics for the 27th uses arithmetic as an economical way to demonstrate intelligence. At least, the ability to do arithmetic is used as proof of intelligence. Which shouldn’t surprise. The conventional appreciation for Ernie Bushmiller is of his skill at efficiently communicating the ideas needed for a joke. That said, it’s a bit surprising Sluggo asks the dog “six times six divided by two”; if it were just showing any ability at arithmetic “one plus one” or “two plus two” would do. But “six times six divided by two” has the advantage of being a bit complicated. That is, it’s reasonable Sluggo wouldn’t know it right away, and would see it as something only the brainiest would. But it’s not so complicated that Sluggo wouldn’t plausibly know the question.
Eric the Circle for the 28th, this one by AusAGirl, uses “Non-Euclidean” as a way to express weirdness in shape. My first impulse was to say that this wouldn’t really be a non-Euclidean circle. A non-Euclidean geometry has space that’s different from what we’re approximating with sheets of paper or with boxes put in a room. There are some that are familiar, or roughly familiar, such as the geometry of the surface of a planet. But you can draw circles on the surface of a globe. They don’t look like this mooshy T-circle. They look like … circles. Their weirdness comes in other ways, like how the circumference is not π times the diameter.
On reflection, I’m being too harsh. What makes a space non-Euclidean is … well, many things. One that’s easy to understand is to imagine that the space uses some novel definition for the distance between points. Distance is a great idea. It turns out to be useful, in geometry and in analysis, to use a flexible idea of of what distance is. We can define the distance between things in ways that look just like the Euclidean idea of distance. Or we can define it in other, weirder ways. We can, whatever the distance, define a “circle” as the set of points that are all exactly some distance from a chosen center point. And the appearance of those “circles” can differ.
There are literally infinitely many possible distance functions. But there is a family of them which we use all the time. And the “circles” in those look like … well, at the most extreme, they look like squares. Others will look like rounded squares, or like slightly diamond-shaped circles. I don’t know of any distance function that’s useful that would give us a circle like this picture of Eric. But there surely is one that exists and that’s enough for the joke to be certified factually correct. And that is what’s truly important in a comic strip.
Sandra Bell-Lundy’s Between Friends for the 29th is the Venn Diagram joke for the week. Formally, you have to read this diagram charitably for it to parse. If we take the “what” that Maeve says, or doesn’t say, to be particular sentences, then the intersection has to be empty. You can’t both say and not-say a sentence. But it seems to me that any conversation of importance has the things which we choose to say and the things which we choose not to say. And it is so difficult to get the blend of things said and things unsaid correct. And I realize that the last time Between Friends came up here I was similarly defending the comic’s Venn Diagram use. I’m a sympathetic reader, at least to most comic strips.
And that was the conclusion of comic strips through the 29th of June which mentioned mathematics enough for me to write much about. There were a couple other comics that brought up something or other, though. Wulff and Morgenthaler’s WuMo for the 27th of June has a Rubik’s Cube joke. The traditional Rubik’s Cube has three rows, columns, and layers of cubes. But there’s no reason there can’t be more rows and columns and layers. Back in the 80s there were enough four-by-four-by-four cubes sold that I even had one. Wikipedia tells me the officially licensed cubes have gotten only up to five-by-five-by-five. But that there was a 17-by-17-by-17 cube sold, with prototypes for 22-by-22-by-22 and 33-by-33-by-33 cubes. This seems to me like a great many stickers to peel off and reattach.
I concede I am late in wrapping up last week’s mathematically-themed comics. But please understand there were important reasons for my not having posted this earlier, like, I didn’t get it written in time. I hope you understand and agree with me about this.
Bill Griffith’s Zippy the Pinhead for the 9th brings up mathematics in a discussion about perfection. The debate of perfection versus “messiness” begs some important questions. What I’m marginally competent to discuss is the idea of mathematics as this perfect thing. Mathematics seems to have many traits that are easy to think of as perfect. That everything in it should follow from clearly stated axioms, precise definitions, and deductive logic, for example. This makes mathematics seem orderly and universal and fair in a way that the real world never is. If we allow that this is a kind of perfection then … does mathematics reach it?
Even the idea of a “precise definition” is perilous. If it weren’t there wouldn’t be so many pop mathematics articles about why 1 isn’t a prime number. It’s difficult to prove that any particular set of axioms that give us interesting results are also logically consistent. If they’re not consistent, then we can prove absolutely anything, including that the axioms are false. That seems imperfect. And few mathematicians even prepare fully complete, step-by-step proofs of anything. It takes ridiculously long to get anything done if you try. The proofs we present tend to show, instead, the reasoning in enough detail that we’re confident we could fill in the omitted parts if we really needed them for some reason. And that’s fine, nearly all the time, but it does leave the potential for mistakes present.
Zippy offers up a perfect parallelogram. Making it geometry is of good symbolic importance. Everyone knows geometric figures, and definitions of some basic ideas like a line or a circle or, maybe, a parallelogram. Nobody’s ever seen one, though. There’s never been a straight line, much less two parallel lines, and even less the pair of parallel lines we’d need for a parallellogram. There can be renderings good enough to fool the eye. But none of the lines are completely straight, not if we examine closely enough. None of the pairs of lines are truly parallel, not if we extend them far enough. The figure isn’t even two-dimensional, not if it’s rendered in three-dimensional things like atoms or waves of light or such. We know things about parallelograms, which don’t exist. They tell us some things about their shadows in the real world, at least.
Mark Litzler’s Joe Vanilla for the 9th is a play on the old joke about “a billion dollars here, a billion dollars there, soon you’re talking about real money”. As we hear more about larger numbers they seem familiar and accessible to us, to the point that they stop seeming so big. A trillion is still a massive number, at least for most purposes. If you aren’t doing combinatorics, anyway; just yesterday I was doing a little toy problem and realized it implied 470,184,984,576 configurations. Which still falls short of a trillion, but had I made one arbitrary choice differently I could’ve blasted well past a trillion.
Ruben Bolling’s Super-Fun-Pak Comix for the 9th is another monkeys-at-typewriters joke, that great thought experiment about probability and infinity. I should add it to my essay about the Infinite Monkey Theorem. Part of the joke is that the monkey is thinking about the content of the writing. This doesn’t destroy the prospect that a monkey given enough time would write any of the works of William Shakespeare. It makes the simple estimates of how unlikely that is, and how long it would take to do, invalid. But the event might yet happen. Suppose this monkey decided there was no credible way to delay Hamlet’s revenge to Act V, and tried to write accordingly. Mightn’t the monkey make a mistake? It’s easy to type a letter you don’t mean to. Or a word you don’t mean to. Why not a sentence you don’t mean to? Why not a whole act you don’t mean to? Impossible? No, just improbable. And the monkeys have enough time to let the improbable happen.
Eric the Circle for the 10th, this one by Kingsnake, declares itself set in “the 20th dimension, where shape has no meaning”. This plays on a pop-cultural idea of dimensions as a kind of fairyland, subject to strange and alternate rules. A mathematician wouldn’t think of dimensions that way. 20-dimensional spaces — and even higher-dimensional spaces — follow rules just as two- and three-dimensional spaces do. They’re harder to draw, certainly, and mathematicians are not selected for — or trained in — drawing, at least not in United States schools. So attempts at rendering a high-dimensional space tend to be sort of weird blobby lumps, maybe with a label “N-dimensional”.
And a projection of a high-dimensional shape into lower dimensions will be weird. I used to have around here a web site with a rotatable tesseract, which would draw a flat-screen rendition of what its projection in three-dimensional space would be. But I can’t find it now and probably it ran as a Java applet that you just can’t get to work anymore. Anyway, non-interactive videos of this sort of thing are common enough; here’s one that goes through some of the dimensions of a tesseract, one at a time. It’ll give some idea how something that “should” just be a set of cubes will not look so much like that.
Steve Kelly and Jeff Parker’s Dustin for the 11th is a variation on the “why do I have to learn this” protest. This one is about long division and the question of why one needs to know it when there’s cheap, easily-available tools that do the job better. It’s a fair question and Hayden’s answer is a hard one to refute. I think arithmetic’s worth knowing how to do, but I’ll also admit, if I need to divide something by 23 I’m probably letting the computer do it.
People reading my Reading the Comics post Sunday maybe noticed something. I mean besides my correct, reasonable complaining about the Comics Kingdom redesign. That is that all the comics were from before the 30th of March. That is, none were from the week before the 7th of April. The last full week of March had a lot of comic strips. The first week of April didn’t. So things got bumped a little. Here’s the results. It wasn’t a busy week, not when I filter out the strips that don’t offer much to write about. So now I’m stuck for what to post Thursday.
The strip explains things well enough. The Library holds every book that will ever be written. In the original story there are some constraints. Particularly, all the books are 410 pages. If you wanted, say, a 600-page book, though, you could find one book with the first 410 pages and another book with the remaining 190 pages and then some filler. The catch, as explained in the story and in the comic strip, is finding them. And there is the problem of finding a ‘correct’ text. Every possible text of the correct length should be in there. So every possible book that might be titled Mark Twain vs Frankenstein, including ones that include neither Mark Twain nor Frankenstein, is there. Which is the one you want to read?
Henry Scarpelli and Craig Boldman’s Archie for the 4th features an equal-divisions problem. In principle, it’s easy to divide a pizza (or anything else) equally; that’s what we have fractions for. Making them practical is a bit harder. I do like Jughead’s quick work, though. It’s got the slight-of-hand you expect from stage magic.
Scott Hilburn’s The Argyle Sweater for the 4th takes place in an algebra class. I’m not sure what algebraic principle demonstrates, but it probably came from somewhere. It’s 4,829,210. The exponentials on the blackboard do cue the reader to the real joke, of the sign reading “kick10 me”. I question whether this is really an exponential kicking situation. It seems more like a simple multiplication to me. But it would be harder to make that joke read clearly.
Tony Cochran’s Agnes for the 5th is part of a sequence investigating how magnets work. Agnes and Trout find just … magnet parts inside. This is fair. It’s even mathematics.
Thermodynamics classes teach one of the great mathematical physics models. This is about what makes magnets. Magnets are made of … smaller magnets. This seems like question-begging. Ultimately you get down to individual molecules, each of which is very slightly magnetic. When small magnets are lined up in the right way, they can become a strong magnet. When they’re lined up in another way, they can be a weak magnet. Or no magnet at all.
How do they line up? It depends on things, including how the big magnet is made, and how it’s treated. A bit of energy can free molecules to line up, making a stronger magnet out of a weak one. Or it can break up the alignments, turning a strong magnet into a weak one. I’ve had physics instructors explain that you could, in principle, take an iron rod and magnetize it just by hitting it hard enough on the desk. And then demagnetize it by hitting it again. I have never seen one do this, though.
This is more than just a physics model. The mathematics of it is … well, it can be easy enough. A one-dimensional, nearest-neighbor model, lets us describe how materials might turn into magnets or break apart, depending on their temperature. Two- or three-dimensional models, or models that have each small magnet affected by distant neighbors, are harder.
And we had another of those peculiar days where a lot of strips are on-topic enough for me to talk about.
Eric the Circle, this one by Kyle, for the 26th has a bit of mathematical physics in it. This is the kind of diagram you’ll see all the time, at least if you do the mathematics that tells you where things will be and when. The particular example is an easy problem, a thing rolling down an inclined plane. But the work done for it applies to more complicated problems. The question it’s for is, “what happens when this thing slides down the plane?” And that depends on the forces at work. There’s gravity, certainly . If there were something else it’d be labelled. Gravity’s represented with that arrow pointing straight down. That gives us the direction. The label (Eric)(g) gives us how strong this force is.
Where the diagram gets interesting, and useful, are those dashed lines ending in arrows. One of those lines is, or at least means to be, parallel to the incline. The other is perpendicular to it. These both reflect gravity. We can represent the force of gravity as a vector. That means, we can represent the force of gravity as the sum of vectors. This is like how we can can write “8” or we can write “3 + 5”, depending on what’s more useful for what we’re doing. (For example, if you wanted to work out “67 + 8”, you might be better off doing “67 + 3 + 5”.) The vector parallel to the plane and the one perpendicular to the plane add up to the original gravity vector.
The force that’s parallel to the plane is the only force that’ll actually accelerate Eric. The force perpendicular to the plane just … keeps it snug against the plane. (Well, it can produce friction. We try not to deal with that in introductory physics because it is so hard. At most we might look at whether there’s enough friction to keep Eric from starting to slide downhill.) The magnitude of the force parallel to the plane, and perpendicular to the plane, are easy enough to work out. These two forces and the original gravity can be put together into a little right triangle. It’s the same shape but different size to the right triangle made by the inclined plane plus a horizontal and a vertical axis. So that’s how the diagram knows the parallel force is the original gravity times the sine of x. And that the perpendicular force is the original gravity times the cosine of x.
The perpendicular force is often called the “normal” force. This because mathematical physicists noticed we had only 2,038 other, unrelated, things called “normal”.
Rick Detorie’s One Big Happy for the 26th sees Ruthie demand to know who this Venn person was. Fair question. Mathematics often gets presented as these things that just are. That someone first thought about these things gets forgotten.
John Venn, who lived from 1834 to 1923 — he died the 4th of April, it happens — was an English mathematician and philosopher and logician and (Anglican) priest. This is not a rare combination of professions. From 1862 he was a lecturer in Moral Science at Cambridge. This included work in logic, yes. But he also worked on probability questions. Wikipedia credits his 1866 Logic Of Chance with advancing the frequentist interpretation of probability. This is one of the major schools of thought about what the “probability of an event” is. It’s the one where you list all the things that could possibly happen, and consider how many of those are the thing you’re interested in. So, when you do a problem like “what’s the probability of rolling two six-sided dice and getting a total of four”? You’re doing a frequentist probability problem.
Venn Diagrams he presented to the world around 1880. These show the relationships between different sets. And the relationships of mathematical logic problems they represent. Venn, if my sources aren’t fibbing, didn’t take these diagrams to be a new invention of his own. He wrote of them as “Euler diagrams”. Venn diagrams, properly, need to show all the possible intersections of all the sets in play. You just mark in some way the intersections that happen to have nothing in them. Euler diagrams don’t require this overlapping. The name “Venn diagram” got attached to these pictures in the early 20th century. Euler here is Leonhard Euler, who created every symbol and notation mathematicians use for everything, and who has a different “Euler’s Theorem” that’s foundational to every field of mathematics, including the ones we don’t yet know exist. I exaggerate by 0.04 percent here.
Venn had his other projects, too. His biography at St Andrews writes of his composing The Biographical History of Gonville and Caius College (Cambridge). And then he had another history of the whole Cambridge University. It also mentions his skills in building machines, though only cites one, a device for bowling cricket balls. The St Andrews biography says that in 1909 “Venn’s machine clean bowled one of [the Australian Cricket Team’s] top stars four times”. I do not know precisely what it means but I infer it to be a pretty good showing for the machine. His Wikipedia biography calls him a “passionate gardener”. Apparently the Cambridgeshire Horticultural Society awarded him prizes for his roses in July 1885 and for white carrots in September that year. And that he was a supporter of votes for women.
Ashleigh Brilliant’s Pot-Shots for the 26th makes a cute and true claim about percentiles. That a person will usually be in the upper 99% of whatever’s being measured? Hard to dispute. But, measure enough things and eventually you’ll fall out of at least one of them. How many things? This is easy to calculate if we look at different things that are independent of each other. In that case we could look at 69 things before there we’d expect a 50% chance of at least one not being in the upper 99%.
It’s getting that independence that’s hard. There’s often links between things. For example, a person’s height does not tell us much about their weight. But it does tell us something. A person six foot, ten inches tall is almost certainly not also 35 pounds, even though a person could be that size or could be that weight. A person’s scores on a reading comprehension test and their income? But test-taking results and wealth are certainly tied together. Age and income? Most of us have a bigger income at 46 than at 6. This is part of what makes studying populations so hard.
T Shepherd’s Snow Sez for the 26th is finally a strip I can talk about briefly, for a change. Snow does a bit of arithmetic wordplay, toying with what an expression like “1 + 1” might represent.
Mark Anderson’s Andertoons for the 18th is the Mark Anderson’s Andertoons for the week. This features the kids learning some of the commonest terms in descriptive statistics. And, as Wavehead says, the similarity of names doesn’t help sorting them out. Each is a kind of average. “Mean” usually is the arithmetic mean, or the thing everyone including statisticians calls “average”. “Median” is the middle-most value, the one that half the data is less than and half the data is greater than. “Mode” is the most common value. In “normally distributed” data, these three quantities are all the same. In data gathered from real-world measurements, these are typically pretty close to one another. It’s very easy for real-world quantities to be normally distributed. The exceptions are usually when there are some weird disparities, like a cluster of abnormally high-valued (or low-valued) results. Or if there are very few data points.
The word “mean” derives from the Old French “meien”, that is, “middle, means”. And that itself traces to the Late Latin “medianus”, and the Latin “medius”. That traces back to the Proto-Indo-European “medhyo”, meaning “middle”. That’s probably what you might expect, especially considering that the mean of a set of data is, if the data is not doing anything weird, likely close to the middle of the set. The term appeared in English in the middle 15th century.
The word “median”, meanwhile, follows a completely different path. That one traces to the Middle French “médian”, which traces to the Late Latin “medianus” and Latin “medius” and Proto-Indo-European “medhyo”. This appeared as a mathematical term in the late 19th century; Etymology Online claims 1883, but doesn’t give a manuscript citation.
The word “mode”, meanwhile, follows a completely different path. This one traces to the Old French “mode”, itself from the Latin “modus”, meaning the measure or melody or style. We get from music to common values by way of the “style” meaning. Think of something being done “á la mode”, that is, “in the [ fashionable or popular ] style”. I haven’t dug up a citation about when this word entered the mathematical parlance.
So “mean” and “median” don’t have much chance to do anything but alliterate. “Mode” is coincidence here. I agree, it might be nice if we spread out the words a little more.
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th has Edison introduce a sequence to his grandfather. Doubling the number of things for each square of a checkerboard is an ancient thought experiment. The notion, with grains of wheat rather than cookies, seems to be first recorded in 1256 in a book by the scholar Ibn Khallikan. One story has it that the inventor of chess requested from the ruler that many grains of wheat as reward for inventing the game.
If we followed Edison Lee’s doubling through all 64 squares we’d have, in total, need for 263-1 or 18,446,744,073,709,551,615 cookies. You can see why the inventor of chess didn’t get that reward, however popular the game was. It stands as a good display of how exponential growth eventually gets to be just that intimidatingly big.
Edison, like many a young nerd, is trying to stagger his grandfather with the enormity of this. I don’t know that it would work. Grandpa ponders eating all that many cookies, since he’s a comical glutton. I’d estimate eating all that many cookies, at the rate of one a second, eight hours a day, to take something like eighteen billion centuries. If I’m wrong? It doesn’t matter. It’s a while. But is that any more staggering than imagining a task that takes a mere ten thousand centuries to finish?
Mathematics is, to an extent, about finding interesting true statements. What makes something interesting? That depends on the person surprised, certainly. A good guideline is probably “something not obvious before you’ve heard it, thatlooks inevitable after you have”. That is, a surprise. Learning mathematics probably has to be steadily surprising, and that’s good, because this kind of surprise is fun.
If it’s always a surprise there might be trouble. If you’re doing similar kinds of problems you should start to see them as pretty similar, and have a fair idea what the answers should be. So, from what Toby has said so far … I wouldn’t call him stupid. At most, just inexperienced.
Eric the Circle for the 19th, by Janka, is the Venn Diagram joke for the week. Properly any Venn Diagram with two properties has an overlap like this. We’re supposed to place items in both circles, and in the intersection, to reflect how much overlap there is. Using the sizes of each circle to reflect the sizes of both sets, and the size of the overlap to represent the size of the intersection, is probably inevitable. The shorthand calls on our geometric intuition to convey information, anyway.
Tony Murphy’s It’s All About You for the 19th has a bunch of things going on. The punch line calls “algebra” what’s really a statistics problem, calculating the arithmetic mean of four results. The work done is basic arithmetic. But making work seem like a more onerous task is a good bit of comic exaggeration, and algebra connotes something harder than arithmetic. But Murphy exaggerates with restraint: the characters don’t rate this as calculus.
Then there’s what they’re doing at all. Given four clocks, what’s the correct time? The couple tries averaging them. Why should anyone expect that to work?
There’s reason to suppose this might work. We can suppose all the clocks are close to the correct time. If they weren’t, they would get re-set, or not looked at anymore. A clock is probably more likely to be a little wrong than a lot wrong. You’d let a clock that was two minutes off go about its business, in a way you wouldn’t let a clock that was three hours and 42 minutes off. A clock is probably as likely to show a time two minutes too early as it is two minutes too late. This all suggests that the clock errors are normally distributed, or something like that. So the error of the arithmetic mean of a bunch of clock measurements we can expect to be zero. Or close to zero, anyway.
There’s reasons this might not work. For example, a clock might systematically run late. My mantle clock, for example, usually drifts about a minute slow over the course of the week it takes to wind. Or the clock might be deliberately set wrong: it’s not unusual to set an alarm clock to five or ten or fifteen minutes ahead of the true time, to encourage people to think it’s later than it really is and they should hurry up. Similarly with watches, if their times aren’t set by Internet-connected device. I don’t know whether it’s possible to set a smart watch to be deliberately five minutes fast, or something like that. I’d imagine it should be possible, but also that the people programming watches don’t see why someone might want to set their clock to the wrong time. From January to March 2018, famously, an electrical grid conflict caused certain European clocks to lose around six minutes. The reasons for this are complicated and technical, and anyway The Doctor sorted it out. But that sort of systematic problem, causing all the clocks to be wrong in the same way, will foil this take-the-average scheme.
Murphy’s not thinking of that, not least because this comic’s a rerun from 2009. He was making a joke, going for the funnier-sounding “it’s 8:03 and five-eights” instead of the time implied by the average, 8:04 and a half. That’s all right. It’s a comic strip. Being amusing is what counts.
It’s happened again: another slow week around here. My supposition is that Comic Strip Master Command was snowed in about a month ago, and I’m seeing the effects only now. There’s obviously no other reason that more comic strips didn’t address my particular narrow interest in one seven-day span.
Samson’s Dark Side of the Horse for the 18th is a numerals joke. The mathematics content is slight, I admit, but I’ve always had a fondness for Dark Side of the Horse. (I know it sounds like I have a fondness for every comic strip out there. I don’t quite, but I grant it’s close.) Conflating numerals and letters, and finding words represented by numerals, is an old tradition. It was more compelling in ancient days when letters were used as numerals so that it was impossible not to find neat coincidences. I suppose these days it’s largely confined to typefaces that make it easy to conflate a letter and a numeral. I mean moreso than the usual trouble telling apart 1 and l, 0 and O, or 5 and S. Or to special cases like hexadecimal numbers where, for ease of representation, we use the letters A through F as numerals.
Jef Mallett’s Frazz for the 18th is built on an ancient problem. I remember being frustrated with it. How is “questions 15 to 25” eleven questions when the difference between 15 and 25 is ten? The problem creeps into many fields. Most of the passion has gone out of the argument but around 1999 you could get a good fight going about whether the new millennium was to begin with January 2000 or 2001. The kind of problem is called a ‘fencepost error’. The name implies how often this has complicated someone’s work. Divide a line into ten segments. There are nine cuts on the interior of the line and the two original edges. I’m not sure I could explain to an elementary school student how the cuts and edges of a ten-unit-long strip match up to the questions in this assignment. I might ask how many birthdays someone’s had when they’re nine years old, though. And then flee the encounter.
Mark Parisi’s Off The Mark for the 19th is another numerals joke. This one’s also the major joke to make about an ice skater doing a figure eight: write the eight some other way. (I’d have sworn there was an M-G-M Droopy cartoon in which Spike demonstrates his ability to skate a figure 8, and then Droopy upstages him by skating ‘4 + 4’. I seem to be imagining it; the only cartoon where this seems to possibly fit is 1950’s The Chump Champ, and the joke isn’t in that one. If someone knows the cartoon I am thinking of, please let me know.) Here, the robot is supposed to be skating some binary numeral. It’s nothing close to an ‘8’, but perhaps the robot figures it needs to demonstrate some impressive number to stand out.
Bud Blake’s Tiger for the 21st has Tiger trying to teach his brother arithmetic. Working it out with fingers seems like a decent path to try, given Punkinhead’s age and background. And Punkinhead has a good point: why is the demonstration the easy problem and the homework the hard problem? I haven’t taught in a while, but do know I would do that sort of thing. My rationalization, I think, would be that a hard problem is usually hard because it involves several things. If I want to teach a thing, then I want to highlight just that thing. So I would focus on a problem in which that thing is the only tricky part, and everything else is something the students are so familiar with they don’t notice it. The result is usually an easy problem. There isn’t room for toughness. I’m not sure if that’s a thing I should change, though. Demonstrations of how to work harder problems are worth doing. But I usually think of those as teaching “how to use these several things we already know”. Using a tough problem to show one new thing, plus several already-existing tricky things, seems dangerous. It might be worth it, though.
Comic Strip Master Command decided this would be a light week, with about six comic strips worth discussing. I’ll go into four of them here, and in a day or two wrap up the remainder. There were several strips that didn’t quite rate discussion, and I’ll share those too. I never can be sure what strips will be best taped to someone’s office door.
But I got to another thought. We’re surprised to see lines in nature. We know what lines are, and understand properties of them pretty well. Even if we don’t specialize in geometry we can understand how we expect them to work. I don’t know how much of this is a cultural artifact: in the western mathematics tradition lines and polygons and circles are taught a lot, and from an early age. My impression is that enough different cultures have similar enough geometries, though. (Are there any societies that don’t seem aware of the Pythagorean Theorem?) So what is it that has got so many people making perfect lines and circles and triangles and squares out of crooked timbers?
Russell Myers’s Broom Hilda for the 13th is a lottery joke. Also, really, an accounting joke. Most of the players of a lottery will not win, of course. Nearly none of them will win more than they’ve paid into the lottery. If they didn’t, there would be an official inquiry. So, yes, nearly all people, even those who win money at the lottery, would have had more money if they skipped playing altogether.
Where it becomes an accounting question is how much did Broom Hilda expect to have when the week was through? If she planned to spend $20 on lottery tickets, and got exactly that? It seems snobbish to me to say that’s a dumber way to spend twenty bucks than, say, buying twenty bucks worth of magazines that you’ll throw away in a month would be. Or having dinner at a fast-casual place. Or anything else that you like doing even though it won’t leave you, in the long run, any better off. Has she come out ahead? That depends where she figures she should be.
Eric the Circle for the 13th, this one by Alabama_Al, is a plane- and solid-geometry joke. This gets it a bit more solidly on-topic than usual. But it’s still a strip focused on the connotations of mathematically-connected terms. There’s the metaphorical use of the ‘plane’ as in the thing people perceive as reality. There’s conflation between the idea of a ‘higher plane’ and ‘higher dimensions’. Also somewhere in here is the idea that ‘higher’ and ‘more’ dimensions of space are the same thing. ‘Transcendental’ here is used in the common English sense of surpassing something. ‘Transcendental’ has a mathematical definition too. That one relates to polynomials, because everything in mathematics is about polynomials. And, of course, one of the two numbers we know to be transcendental, and that people have any reason to care about, is π, which turns up all over circles.
Larry Wright’s Motley for the 13th riffs on the form of a story problem. Joey’s mother does ask something that seems like a plausible addition problem. I’m a bit surprised he hadn’t counted all the day’s cookies already, but perhaps he doesn’t dwell on past snacks.
Bill Holbrook’s On The Fastrack for the 18th is an anthropomorphic numerals joke. It’s part of Holbrook’s style to draw metaphors as literal happenings. It’s also a variation on a joke Holbrook used just last month, depicting then the phrase “accepting his numbers”. What I said about “accepting numbers” transfers over naturally to “trusting numbers”. It’s not that a number itself means anything. It’s that numbers are used to represent some narrative. If we can’t believe the narrative, we don’t believe the numbers. And the numbers used to represent something can give us reasons to trust, or reject, a narrative.
Eric the Circle for the 18th I can dub an anthropomorphic geometry joke for the week. At least it brings up one of the handful of geometry facts that people remember outside school. The relationship between the circumference and the diameter (or radius, if you rather) of a circle has been known just forever. It has the advantage of going through π, supporting and being supported by that celebrity number. … I’m not quite sure about the logic of this joke, though. My experience is that guys at least are fairly good about knowing their waist size (if you don’t know, it’s 38, although a 40 can feel so comfortable, and they’re sure they can wear a 36). Radius is a harder thing to keep in mind. But maybe it’s different for circles.
Russell Myers’s Broom Hilda for the 19th is a student-and-teacher problem. One thing is that Nerwin’s not wrong. It’s just that simply saying something true isn’t enough. We want to say things that are true and interesting.
But “you add two numbers and get a number” can be interesting. It depends on context. For example, in group theory, we will start by describing groups as a collection of things and an operation which works like addition. What does it mean to work like addition? Here, it means if you add two things from the collection, you get something from the collection. The collection of things is “closed” under your operation. And mathematical operations defined this abstractly — or defined this vaguely, if you don’t like the way it goes — can be great. We’re introduced to vectors, for example, as “ordered sets of numbers”. And that definition works all right. But when you start thinking of them instead as “things you can add to vectors and get other vectors out” you gain new power. You can use the mechanism developed for ordered sets of numbers to describe many things, including matrices and functions and shapes. But when we do that we’re saying things about how addition works, rather than what this particular addition is.
You know, on reflection, I’m not sure that Eric the Circle was more worthy of discussion than that Barney Google was. Hm.
There were just enough mathematically-themed comic strips last week for me to make two posts out of it. This current week? Is looking much slower, at least as of Wednesday night. But that’s a problem for me to worry about on Sunday.
Eric the Circle for the 20th, this one by Griffinetsabine, mentions a couple of shapes. That’s enough for me, at least on a slow comics week. There is a fictional tradition of X marking the spot. It can be particularly credited to Robert Louis Stevenson’s Treasure Island. Any symbol could be used to note a special place on maps, certainly. Many maps are loaded with a host of different symbols to convey different information. Circles and crosses have the advantage of being easy to draw and difficult to confuse for one another. Squares, triangles, and stars are good too.
Bill Whitehead’s Free Range for the 22nd spoofs Wheel of Fortune with “theoretical mathematics”. Making a game out of filling in parts of a mathematical expression isn’t ridiculous, although it is rather niche. I don’t see how the revealed string of mathematical expressions build to a coherent piece, but perhaps a few further pieces would help.
The parts shown are all legitimate enough expressions. Well, like is only true for some specific numbers ‘a’ and ‘b’, but you can find solutions. is just an expression, not picking out any particular values of ‘b’ or ‘x’ or ‘y’ as interesting. But in conjunction with or other expressions there might be something useful. On the second row is a graph, highlighting a region underneath a curve (and above the x-axis) between two vertical lines. This is often the sort of thing looked at in calculus. It also turns up in probability, as the area under a curve like this can show the chance that an experiment will turn up something in a range of values. And is a straightforward differential equation. Its solution is a family of similar-looking polynomials.
Mark Pett’s Lucky Cow for the 22nd has run before. I’ve even made it the title strip for a Reading the Comics post back in 2014. So it’s probably time to drop this from my regular Reading the Comics reporting. The physicists comes running in with the left half of the time-dependent Schrödinger Equation. This is all over quantum mechanics. In this form, quantum mechanics contains information about how a system behaves by putting it into a function named . Its value depends on space (‘x’). It can also depend on time (‘t’). The physicists pretends to not be able to complete this. Neil arranges to give the answer.
Schrödinger’s Equation looks very much like a diffusion problem. Normal diffusion problems don’t have that which appears in the part of Neil’s answer. But this form of equation turns up a lot. If you have something that acts like a fluid — and heat counts — then a diffusion problem is likely important in understanding it.
And, yes, the setup reminds me of a mathematical joke that I only encounter in lists of mathematics jokes. That one I told the last time this strip came up in the rotation. You might chuckle, or at least be convinced that it is a correctly formed joke.
I’ve settled to a pace of about four comics each essay. It makes for several Reading the Comics posts each week. But none of them are monsters that eat up whole evenings to prepare. Except that last week there were enough comics which made my initial cut that I either have to write a huge essay or I have to let last week’s strips spill over to Sunday. I choose that option. It’s the only way to square it with the demands of the A to Z posts, which keep creeping above a thousand words each however much I swear that this next topic is a nice quick one.
Roy Schneider’s The Humble Stumble for the 25th has some mathematics in a supporting part. It’s used to set up how strange Tommy is. Mathematics makes a good shorthand for this. It’s usually compact to write, important for word balloons. And it’s usually about things people find esoteric if not hilariously irrelevant to life. Tommy’s equation is an accurate description of what centripetal force would be needed to keep the Moon in a circular orbit at about the distance it really is. I’m not sure how to take Tommy’s doubts. If he’s just unclear about why this should be so, all right. Part of good mathematical learning can be working out the logic of some claim. If he’s not sure that Newtonian mechanics is correct — well, fair enough to wonder how we know it’s right. Spoiler: it is right. (For the problem of the Moon orbiting the Earth it’s right, at least to any reasonable precision.)
Stephan Pastis’s Pearls Before Swine for the 25th shows how we can use statistics to improve our lives. At least, it shows how tracking different things can let us find correlations. These correlations might give us information about how to do things better. It’s usually a shaky plan to act on a correlation before you have a working hypothesis about why the correlation should hold. But it can give you leads to pursue.
Eric the Circle for the 26th, this one by Vissoro, is a “two types of people in the world” joke. Given the artwork I believe it’s also riffing on the binary-arithmetic version of the joke. Which is, “there are 10 types of people in the world, those who understand binary and those who don’t”.
I figure to do something rare, and retire one of my comic strip tags after today. Which strip am I going to do my best to drop from Reading the Comics posts? Given how many of the ones I read are short-lived comics that have been rerun three or four times since I started tracking them? Read on and see!
Bill Holbrook’s On The Fastrack for the 29th of August continues the sequence of Fi talking with kids about mathematics. My understanding was that she tried to give talks about why mathematics could be fun. That there are different ways to express the same number seems like a pretty fine-grain detail to get into. But this might lead into some bigger point. That there are several ways to describe the same thing can be surprising and unsettling to discover. That you have, when calculating, the option to switch between these ways freely can be liberating. But you have to know the option is there, and where to look for it. And how to see it’ll make something simpler.
Bill Holbrook’s On The Fastrack for the 30th of August gets onto a thread about statistics. The point of statistics is to describe something complicated with something simple. So detail must be lost. That said, there are something like 2,038 different things called “average”. Each of them has a fair claim to the term, too. In Fi’s example here, 73 degrees (Fahrenheit) could be called the average as in the arithmetic mean, or average as in the median. The distribution reflects how far and how often the temperature is from 73. This would also be reflected in a quantity called the variance, or the standard deviation. Variance and standard deviation are different things, but they’re tied together; if you know one you know the other. It’s just sometimes one quantity is more convenient than the other to work with.
Bill Holbrook’s On The Fastrack for the 1st of September has Fi argue that apparent irrelevance makes mathematics boring. It’s a common diagnosis. I think I’ve advanced the claim myself. I remember a 1980s probability textbook asking the chance that two transistors out of five had broken simultaneously. Surely in the earlier edition of the textbook, it was two vacuum tubes out of five. Five would be a reasonable (indeed, common) number of vacuum tubes to have in a radio. And it would be plausible that two might be broken at the same time.
It seems obvious that wanting to know an answer makes it easier to do the work needed to find it. I’m curious whether that’s been demonstrated true. Like, it seems obvious that a reference to a thing someone doesn’t know anything about would make it harder to work on. But does it? Does it distract someone trying to work out the height of a ziggurat based on its distance and apparent angle, if all they know about a ziggurat is their surmise that it’s a thing whose height we might wish to know?
Tom Toles’s Randolph Itch, 2 am rerun for the 30th of August is an old friend that’s been here a couple times. I suppose I do have to retire the strip from my Reading the Comics posts, at least, although I’m still amused enough by it to keep reading it daily. Simon Garfield’s On The Map, a book about the history of maps, notes that the X-marks-the-spot thing is an invention of the media. Robert Louis Stevenson’s Treasure Island particularly. Stevenson’s treasure map, Garfield notes, had to be redrawn from the manuscript and the author’s notes. The original went missing in the mail to the publishers. I just mention because I think that adds a bit of wonder to the treasure map. And since, I guess, I won’t have the chance to mention this again.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 30th of August satisfies the need for a Venn Diagram joke this time around. It’s also the strange-geometry joke for the week. Klein bottles were originally described by Felix Klein. They exist in four (or more) dimensions, in much the way that M&oum;bius strips exist in three. And like the M&oum;bius strip the surface defies common sense. You can try to claim some spot on the surface is inside and some other spot outside. But you can get from your inside to your outside spot in a continuous path, one you might trace out on the surface without lifting your stylus.
If you were four-dimensional. Or more. If we were to see one in three dimensions we’d see a shape that intersects itself. As beings of only three spatial dimensions we have to pretend that doesn’t happen. It’s the same we we pretend a drawing of a cube shows six squares all of equal size and connected at right angles to one another, even though the drawing is nothing like that. The bottle-like shape Weinersmith draws is, I think, the most common representation of the Klein bottle. It looks like a fancy bottle, and you can buy one as a novelty gift for a mathematician. I don’t need one but do thank you for thinking of me. MathWorld shows another representation, a figure-eight-based one which looks to me like an advanced pasta noodle. But it doesn’t look anything like a bottle.
Eric the Circle for the 31st of August, this one by JohnG, is a spot of wordplay. The pun here is the sine of an angle in a (right) triangle. That would be the length of the leg opposite the angle divided by the length of the hypotenuse. This is still stuff relevant to circles, though. One common interpretation of the cosine and sine of an angle is to look at the unit circle. That is, a circle with radius 1 and centered on the origin. Draw a line segment opening up an angle θ from the positive x-axis. Draw it counterclockwise. That is, if your angle is a very small number, you’re drawing a line segment that’s a little bit above the positive x-axis. Draw the line segment long enough that it touches the unit circle. That point where the line segment and the circle intersect? Look at its Cartesian coordinates. The y-coordinate will be the sine of θ. The x-coordinate will be the cosine of θ. The triangle you’re looking at has vertices at the origin; at x-coordinate cosine θ, y-coordinate 0; and at x-coordinate cosine θ, y-coordinate sine θ.
Although the hyperbolic cosine is interesting and I could go on about it.
Eric the Circle for the 18th of June is a bit of geometric wordplay for the week. A secant is — well, many things. One of the important things is it’s a line that cuts across a circle. It intersects the circle in two points. This is as opposed to a tangent, which touch it in one. Or missing it altogether, which I think hasn’t got any special name. “Secant” also appears as one of the six common trig functions out there.
In value the secant of an angle is just the reciprocal of the cosine of that angle. Where the cosine is never smaller than -1 nor larger than 1, the secant is always either greater than 1 or smaller than -1. It’s a useful function to have by name. We can write “the secant of angle θ” as . The otherwise sensible-looking is unavailable, because we use that to mean “the angle whose cosine is θ”. We need to express that idea, the “arc-cosine” or “inverse cosine”, quite a bit too. And would look like we wanted the cosine of one divided by θ. Ultimately, we have a lot of ideas we’d like to write down, and only so many convenient quick shorthand ways to write them. And by using secant as its own function we can let the arc-cosine have a convenient shorthand symbol. These symbols are a point where you see the messy, human, evolutionary nature of mathematical symbols at work.
We can understand the cosine of an angle θ by imagining a right triangle with hypotenuse of length 1. Set that so the hypotenuse makes angle θ with respect to the x-axis. Then the opposite leg of that right triangle will be the cosine of θ away from the origin. The secant, now, that works differently. Again here imagine a right triangle, but this time one of the legs has length 1. And that leg is at an angle θ with respect to the x-axis. Then the far leg of that right triangle is going to cross the x-axis. And it’ll do that at a point that’s the secant of θ away from the origin.
Larry Wright’s Motley Classics for the 19th speaks of algebra as the way to explain any sufficiently complicated thing. Algebra’s probably not the right tool to analyze a soap opera, or any drama really. The interactions of characters are probably more a matter for graph theory. That’s the field that studies groups of things and the links between them. Occasionally you’ll see analyses of, say, which characters on some complicated science fiction show spend time with each other and which ones don’t. I’m not aware of any that were done on soap operas proper. I suspect most mathematics-oriented nerds view the soaps as beneath their study. But most soap operas do produce a lot of show to watch, and to summarize; I can’t blame them for taking a smaller, easier-to-summarize data set to study. (Also I’m not sure any of these graphs reveal anything more enlightening than, “Huh, really thought The Doctor met Winston Churchill more often than that”.)
Dave Whamond’s Reality Check for the 19th is our Venn Diagram strip for the week. I say the real punch line is the squirrel’s, though. Properly, yes, the Venn Diagram with the two having nothing in common should still have them overlap in space. There should be a signifier inside that there’s nothing in common, such as the null symbol or an x’d out intersection. But not overlapping at all is so commonly used that it might as well be standard.
Teresa Bullitt’s Frog Applause for the 21st uses a thought balloon full of mathematical symbols as icon for far too much deep thinking to understand. I would like to give my opinion about the meaningfulness of the expressions. But they’re too small for me to make out, and GoComics doesn’t allow for zooming in on their comics anymore. I looks like it’s drawn from some real problem, based on the orderliness of it all. But I have no good reason to believe that.
Hi, all. I apologize for being late in posting this, but my Friday and Saturday were eaten up by pinball competition. Pinball At The Zoo, particularly, in Kalamazoo, Michigan. There, Friday, I stepped up first thing and put in four games on the Classics, pre-1985, tournament bank and based on my entry scores was ranked the second-best player there. And then over the day my scores dwindled lower and lower on the list of what people had entered until, in the last five minutes of qualifying, they dropped off the roster altogether and I was knocked out. Meanwhile in the main tournament, I was never even close to making playoffs. But I did have a fantastic game of Bally/Midway’s World Cup Soccer, a game based on how much the United States went crazy for soccer that time we hosted the World Cup for some reason. The game was interrupted by one of the rubber straps around one of the kickers (the little triangular table just past the flippers that you would think would be called the bumpers) breaking, and then by the drain breaking in a way that later knocked the game entirely out of the competition. So anyway besides that glory I’ve been very busy trying to figure out what’s gone wrong and stepping outside to berate the fox squirrels out back, and that’s why I’m late with all this. I’m sure you relate.
Bill Holbrook’s Kevin and Kell rerun for the 15th is the anthropomorphic numerals strip for the week. Also the first of the anthropomorphic strips for the week. Calculating taxes has always been one of the compelling social needs for mathematics, arithmetic especially. If we consider the topic to be “accounting” then that might be the biggest use of mathematics in society. At least by humans; I’m not sure how to rate the arithmetic that computers do even for not explicitly mathematical tasks like sending messages back and forth. New comic strip tag for around here, too.
Bill Schorr’s The Grizzwells for the 17th sees Fauna not liking trigonometry class. I’m sympathetic. I remember it as seeming to be a lot of strange new definitions put to vague purposes. On the bright side, when you get into calculus trigonometry starts solving more problems than it creates. On the dim side, at least when I took it they tried to pass off “trigonometric substitution” as a thing we might need. (OK, it’s come in useful sometimes, but not as often as the presentation made it look.) Also a new comic strip tag.
Eric the Circle for the 18th, this one by sdhardie, is a joke in the Venn Diagram mode. The strip’s a little unusual for not having one of the circles be named Eric. Not a new comic strip tag.
Lord Birthday’s Dumbwitch Castle for the 19th is a small sketch and mostly a list of jokes. This is the normal format for this strip, which tests the idea of what makes something a comic strip. I grant it’s a marginal inclusion, but I am tickled by the idea of a math slap so here you go. This one’s another new comic strip tag.
And this should clear out last week’s mathematically-themed comic strips. I didn’t realize just how busy last week had been until I looked at what I thought was a backlog of just two days’ worth of strips and it turned out to be about two thousand comics. I exaggerate, but as ever, not by much. This current week seems to be a more relaxed pace. So I’ll have to think of something to write for the Tuesday and Thursday slots. Hm. (I’ll be all right. I’ve got one thing I need to stop bluffing about and write, and there’s usually a fair roundup of interesting tweets or articles I’ve seen that I can write. Those are often the most popular articles around here.)
Mark Tatulli’s Heart of the City rerun for the 1st finally has some specific mathematics mentioned in Heart’s efforts to avoid a mathematics tutor. The bit about the sum of adjacent angles forming a right line being 180 degrees is an important one. A great number of proofs rely on it. I can’t deny the bare fact seems dull, though. I know offhand, for example, that this bit about adjacent angles comes in handy in proving that the interior angles of a triangle add up to 180 degrees. At least for Euclidean geometry. And there are non-Euclidean geometries that are interesting and important and for which that’s not true. Which inspires the question: on a non-Euclidean surface, like say the surface of the Earth, is it that adjacent angles don’t add up to 180 degrees? Or does something else in the proof of a triangle’s interior angles adding up to 180 degrees go wrong?
Bill Whitehead’s Free Range for the 2nd features the classic page full of equations to demonstrate some hard mathematical work. And it is the sort of subject that is done mathematically. The equations don’t look to me anything like what you’d use for asteroid orbit projections. I’d expect forecasting just where an asteroid might hit the Earth to be done partly by analytic formulas that could be done on a blackboard. And then made precise by a numerical estimate. The advantage of the numerical estimate is that stuff like how air resistance affects the path of something in flight is hard to deal with analytically. Numerically, it’s tedious, but we can let the computer deal with the tedium. So there’d be just a boring old computer screen to show on-panel.
Bud Fisher’s Mutt and Jeff reprint for the 2nd is a little baffling. And not really mathematical. It’s just got a bizarre arithmetic error in it. Mutt’s fiancee Encee wants earrings that cost ten dollars (each?) and Mutt takes this to be fifty dollars in earring costs and I have no idea what happened there. Thomas K Dye, the web cartoonist who’s done artwork for various article series, has pointed out that the lettering on these strips have been redone with a computer font. (Look at the letters ‘S’; once you see it, you’ll also notice it in the slightly lumpy ‘O’ and the curly-arrow ‘G’ shapes.) So maybe in the transcription the earring cost got garbled? And then not a single person reading the finished product read it over and thought about what they were doing? I don’t know.
Zach Weinersmith’s Saturday Morning Breakfast Cereal reprint for the 2nd is based, as his efforts to get my attention often are, on a real mathematical physics postulate. As the woman postulates: given a deterministic universe, with known positions and momentums of every particle, and known forces for how all these interact, it seems like it should be possible to predict the future perfectly. It would also be possible to “retrodict” the past. All the laws of physics that we know are symmetric in time; there’s no reason you can’t predict the motion of something one second into the past just as well as you an one second into the future. This fascinating observation took a lot of battery in the 19th century. Many physical phenomena are better described by statistical laws, particularly in thermodynamics, the flow of heat. In these it’s often possible to predict the future well but retrodict the past not at all.
But that looks as though it’s a matter of computing power. We resort to a statistical understanding of, say, the rings of Saturn because it’s too hard to track the billions of positions and momentums we’d need to otherwise. A sufficiently powerful mathematician, for example God, would be able to do that. Fair enough. Then came the 1890s. Henri Poincaré discovered something terrifying about deterministic systems. It’s possible to have chaos. A mathematical representation of a system is a bit different from the original system. There’s some unavoidable error. That’s bound to make some, larger, error in any prediction of its future. For simple enough systems, this is okay. We can make a projection with an error as small as we need, at the cost of knowing the current state of affairs with enough detail. Poincaré found that some systems can be chaotic, though, ones in which any error between the current system and its representation will grow to make the projection useless. (At least for some starting conditions.) And so many interesting systems are chaotic. Incredibly simplified models of the weather are chaotic; surely the actual thing is. This implies that God’s projection of the universe would be an amusing but almost instantly meaningless toy. At least unless it were a duplicate of the universe. In which case we have to start asking our philosopher friends about the nature of identity and what a universe is, exactly.
Ruben Bolling’s Super-Fun-Pak Comix for the 2nd is an installment of Guy Walks Into A Bar featuring what looks like an arithmetic problem to start. It takes a turn into base-ten jokes. There are times I suspect Ruben Bolling to be a bit of a nerd.
Percy Crosby’s Skippy for the 3rd originally ran the 8th of December, 1930. It alludes to one of those classic probability questions: what’s the chance that in your lungs is one of the molecules exhaled by Julius Caesar in his dying gasp? Or whatever other event you want: the first breath you ever took, or something exhaled by Jesus during the Sermon on the Mount, or exhaled by Sue the T-Rex as she died. Whatever. The chance is always surprisingly high, which reflects the fact there’s a lot of molecules out there. This also reflects a confidence that we can say one molecule of air is “the same” as some molecule if air in a much earlier time. We have to make that supposition to have a problem we can treat mathematically. My understanding is chemists laugh at us if we try to suggest this seriously. Fair enough. But whether the air pumped out of a bicycle tire is ever the same as what’s pumped back in? That’s the same kind of problem. At least some of the molecules of air will be the same ones. Pretend “the same ones” makes sense. Please.
It was a slow week for mathematically-themed comic strips. What I have are meager examples. Small topics to discuss. The end of the week didn’t have anything even under loose standards of being on-topic. Which is fine, since I lost an afternoon of prep time to thunderstorms that rolled through town and knocked out power for hours. Who saw that coming? … If I had, I’d have written more the day before.
Mac King and Bill King’s Magic in a Minute for the 29th of October looks like a word problem. Well, it is a word problem. It looks like a problem about extrapolating a thing (price) from another thing (quantity). Well, it is an extrapolation problem. The fun is in figuring out what quantities are relevant. Now I’ve spoiled the puzzle by explaining it all so.
Olivia Walch’s Imogen Quest for the 30th doesn’t say it’s about a mathematics textbook. But it’s got to be. What other kind of textbook will have at least 28 questions in a section and only give answers to the odd-numbered problems in back? You never see that in your social studies text.
Eric the Circle for the 30th, this one by Dennill, tests how slow a week this was. I guess there’s a geometry joke in Jane Austen? I’ll trust my literate readers to tell me. My doing the world’s most casual search suggests there’s no mention of triangles in Pride and Prejudice. The previous might be the most ridiculously mathematics-nerdy thing I have written in a long while.
Tony Murphy’s It’s All About You for the 31st does some advanced-mathematics name-dropping. In so doing, it’s earned a spot taped to the door of two people in any mathematics department with more than 24 professors across the country. Or will, when they hear there was a gap unification theory joke in the comics. I’m not sure whether Murphy was thinking of anything particular in naming the subject “gap unification theory”. It sounds like a field of mathematical study. But as far as I can tell there’s just one (1) paper written that even says “gap unification theory”. It’s in partition theory. Partition theory is a rich and developed field, which seems surprising considering it’s about breaking up sets of the counting numbers into smaller sets. It seems like a time-waster game. But the game sneaks into everything, so the field turns out to be important. Gap unification, in the paper I can find, is about studying the gaps between these smaller sets.
There’s also a “band-gap unification” problem. I could accept this name being shortened to “gap unification” by people who have to say its name a lot. It’s about the physics of semiconductors, or the chemistry of semiconductors, as you like. The physics or chemistry of them is governed by the energies that electrons can have. Some of these energies are precise levels. Some of these energies are bands, continuums of possible values. When will bands converge? When will they not? Ask a materials science person. Going to say that’s not mathematics? Don’t go looking at the papers.
Whether partition theory or materials since it seems like a weird topic. Maybe Murphy just put together words that sounded mathematical. Maybe he has a friend in the field.
Bill Amend’s FoxTrot Classics for the 1st of November is aiming to be taped up to the high school teacher’s door. It’s easy to show how the square root of two is irrational. Takes a bit longer to show the square root of three is. Turns out all the counting numbers are either perfect squares — 1, 4, 9, 16, and so on — or else have irrational square roots. There’s no whole number with a square root of, like, something-and-three-quarters or something-and-85-117ths. You can show that, easily if tediously, for any particular whole number. What’s it look like to show for all the whole numbers that aren’t perfect squares already? (This strip originally ran the 8th of November, 2006.)
It’s another week where everything I have to talk about comes from GoComics.com. So, no pictures. The Comics Kingdom and the Creators.com strips are harder for non-subscribers to read so I feel better including those pictures. There’s not an overarching theme that I can fit to this week’s strips either, so I’m going to name it for the one that was most visually interesting to me.
Charlie Pondrebarac’s CowTown for the 22nd I just knew was a rerun. It turned up the 26th of August, 2015. Back then I described it as also “every graduate students’ thesis defense anxiety dream”. Now I wonder if I have the possessive apostrophe in the right place there. On reflection, if I have “every” there, then “graduate student” has to be singular. If I dropped the “every” then I could talk about “graduate students” in the plural and be sensible. I guess that’s all for a different blog to answer.
Mike Thompson’s Grand Avenue for the 22nd threatened to get me all cranky again, as Grandmom decided the kids needed to do arithmetic worksheets over the summer. The strip earned bad attention from me a few years ago when a week, maybe more, of the strip was focused on making sure the kids drudged their way through times tables. I grant it’s a true attitude that some people figure what kids need is to do a lot of arithmetic problems so they get better at arithmetic problems. But it’s hard enough to convince someone that arithmetic problems are worth doing, and to make them chores isn’t helping.
John Zakour and Scott Roberts’s Maria’s Day for the 22nd name-drops fractions as a worse challenge than dragon-slaying. I’m including it here for the cool partial picture of the fire-breathing dragon. Also I take a skeptical view of the value of slaying the dragons anyway. Have they given enough time for sanctions to work?
Eric the Circle for the 24th, this one by Dennill, gets in here by throwing some casual talk about arcs around. That and π. The given formula looks like nonsense to me. has parts that make sense. The first part will tell you what radian measure corresponds to 94 degrees, and that’s fine. Mathematicians will tend to look for radian measures rather than degrees for serious work. The sine of 94 degrees they might want to know. Subtracting the two? I don’t see the point. I dare to say this might be a bunch of silliness.
Cathy Law’s Claw for the 25th writes off another Powerball lottery loss as being bad at math and how it’s like algebra. Seeing algebra in lottery tickets is a kind of badness at mathematics, yes. It’s probability, after all. Merely playing can be defended mathematically, though, at least for the extremely large jackpots such as the Powerball had last week. If the payout is around 750 million dollars (as it was) and the chance of winning is about one in 250 million (close enough to true), then the expectation value of playing a ticket is about three dollars. If the ticket costs less than three dollars (and it does; I forget if it’s one or two dollars, but it’s certainly not three), then, on average you could expect to come out slightly ahead. Therefore it makes sense to play.
Except that, of course, it doesn’t make sense to play. On average you’ll lose the cost of the ticket. The on-average long-run you need to expect to come out ahead is millions of tickets deep. The chance of any ticket winning is about one in 250 million. You need to play a couple hundred million times to get a good enough chance of the jackpot for it to really be worth it. Therefore it makes no sense to play.
Mathematical logic therefore fails us: we can justify both playing and not playing. We must study lottery tickets as a different thing. They are (for the purposes of this) entertainment, something for a bit of disposable income. Are they worth the dollar or two per ticket? Did you have other plans for the money that would be more enjoyable? That’s not my ruling to make.
Grouping together three decimal digits as a block is as old, in the Western tradition, as decimal digits are. Leonardo of Pisa, in Liber Abbaci, groups the thousands and millions and thousands of millions and such together. By 1228 he had the idea to note this grouping with an arc above the set of digits, like a tie between notes on a sheet of music. This got cut down, part of the struggle in notation to write as little as possible. Johannes de Sacrobosco in 1256 proposed just putting a dot every third digit. In 1636 Thomas Blundeville put a | mark after every third digit. (I take all this, as ever, from Florian Cajori’s A History Of Mathematical Notations, because it’s got like everything in it.) We eventually settled on separating these stanzas of digits with a , or . mark. But that it should be three digits goes as far back as it could.