This installment took longer to write than you’d figure, because it’s the time of year we’re watching a lot of mostly Rankin/Bass Christmas specials around here. So I have to squeeze words out in-between baffling moments of animation and, like, arguing whether there’s any possibility that Jack Frost was not meant to be a Groundhog Day special that got rewritten to Christmas because the networks weren’t having it otherwise.
Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 3rd is the anthropomorphic numerals joke for the week. … You know, I’ve always wondered in this sort of setting, what are two-digit numbers like? I mean, what’s the difference between a twelve and a one-and-two just standing near one another? How do people recognize a solitary number? This is a darned silly thing to wonder so there’s probably a good web comic about it.
John Hambrock’s The Brilliant Mind of Edison Lee for the 4th has Edison forecast the outcome of a basketball game. I can’t imagine anyone really believing in forecasting the outcome, though. The elements of forecasting a sporting event are plausible enough. We can suppose a game to be a string of events. Each of them has possible outcomes. Some of them score points. Some block the other team’s score. Some cause control of the ball (or whatever makes scoring possible) to change teams. Some take a player out, for a while or for the rest of the game. So it’s possible to run through a simulated game. If you know well enough how the people playing do various things? How they’re likely to respond to different states of things? You could certainly simulate that.
But all sorts of crazy things will happen, one game or another. Run the same simulation again, with different random numbers. The final score will likely be different. The course of action certainly will. Run the same simulation many times over. Vary it a little; what happens if the best player is a little worse than average? A little better? What if the referees make a lot of mistakes? What if the weather affects the outcome? What if the weather is a little different? So each possible outcome of the sporting event has some chance. We have a distribution of the possible results. We can judge an expected value, and what the range of likely outcomes is. This demands a lot of data about the players, though. Edison Lee can have it, I suppose. The premise of the strip is that he’s a genius of unlimited competence. It would be more likely to expect for college and professional teams.
Brian Basset’s Red and Rover for the 4th uses arithmetic as the homework to get torn up. I’m not sure it’s just a cameo appearance. It makes a difference to the joke as told that there’s division and long division, after all. But it could really be any subject.
Last week Comic Strip Master Command sent out just enough on-theme comics for two essays, the way I do them these days. The first half has some multiplication in two of the strips. So that’s enough to count as a theme for me.
Aaron Neathery’s Endtown for the 26th depicts a dreary, boring school day by using arithmetic. A lot of times tables. There is some credible in-universe reason to be drilling on multiplication like this. The setting is one where the characters can’t expect to have computers available. That granted, I’m not sure there’s a point to going up to memorizing four times 27. Going up to twelve-times seems like enough for common uses. For multiplying two- and longer-digit numbers together we usually break the problem up into a string of single-digit multiplications.
There are a handful of bigger multiplications that can make your life easier to know, like how four times 25 is 100. Or three times 33 is pretty near 100. But otherwise? … Of course, the story needs the class to do something dull and seemingly pointless. Going deep into multiplication tables communicates that to the reader quickly.
Thaves’s Frank and Ernest for the 26th is a spot of wordplay. Also a shout-out to my friends who record mathematics videos for YouTube. It is built on the conflation between the ideas of something multiplying and the amount of something growing. It’s easy to see where the idea comes from; just keep hitting ‘x 2’ on a calculator and the numbers grow excitingly fast. You get even more exciting results with ‘x 3’ or ‘x π’. But multiplying by 1 is still multiplication. As is multiplying by a number smaller than 1. Including negative numbers. That doesn’t hurt the joke any. That multiplying two things together doesn’t necessarily give you something larger is a consideration when you’re thinking rigorously about what multiplication can do. It doesn’t have to be part of normal speech.
Nate Frakes’s Break of Day for the 27th is the anthropomorphic numerals joke for the week. I don’t know that there’s anything in the other numerals being odds rather than evens, or a mixture of odds and evens. It might just be that they needed to be anything but 1.
So I’m going to have a third Reading the Comics essay for last week’s strips. This happens sometimes. Two of the four strips for this essay mention percentages. But one of the others is so important to me that it gets naming rights for the essay. You’ll understand when I’m done. I hope.
Angie Bailey’s Texts From Mittens for the 2nd talks about percentages. That’s a corner of arithmetic that many people find frightening and unwelcoming. I’m tickled that Mittens doesn’t understand how easy it is to work out a percentage of 100. It’s a good, reasonable bit of characterization for a cat.
John Graziano’s Ripley’s Believe It Or Not for the 2nd is about a subject close to my heart. At least a third of it is. The mention of negative Kelvin temperatures set off a … heated … debate on the comments thread at GoComics.com. Quite a few people remember learning in school that the Kelvin temperature scale. It starts with the coldest possible temperature, which is zero. And that’s that. They have taken this to denounce Graziano as writing obvious nonsense. Well.
Something you should know about anything you learned in school: the reality is more complicated than that. This is true for thermodynamics. This is true for mathematics. This is true for anything interesting enough for humans to study. This also applies to stuff you learned as an undergraduate. Also to grad school.
So what are negative temperatures? At least on an absolute temperature scale, where the answer isn’t an obvious and boring “cold”? One clue is in the word “absolute” there. It means a way of measuring temperature that’s in some way independent of how we do the measurement. In ordinary life we measure temperatures with physical phenomena. Fluids that expand or contract as their temperature changes. Metals that expand or contract as their temperatures change. For special cases like blast furnaces, sample slugs of clays that harden or don’t at temperature. Observing the radiation of light off a thing. And these are all fine, useful in their domains. They’re also bound in particular physical experiments, though. Is there a definition of temperature that … you know … we can do mathematically?
Of course, or I wouldn’t be writing this. There are two mathematical-physics components to give us temperature. One is the internal energy of your system. This is the energy of whatever your thing is, less the gravitational or potential energy that reflects where it happens to be sitting. Also minus the kinetic energy that comes of the whole system moving in whatever way you like. That is, the energy you’d see if that thing were in an otherwise empty universe. The second part is — OK, this will confuse people. It’s the entropy. Which is not a word for “stuff gets broken”. Not in this context. The entropy of a system describes how many distinct ways there are for a system to arrange its energy. Low-entropy systems have only a few ways to put things. High-entropy systems have a lot of ways to put things. This does harmonize with the pop-culture idea of entropy. There are many ways for a room to be messy. There are few ways for it to be clean. And it’s so easy to make a room messier and hard to make it tidier. We say entropy tends to increase.
So. A mathematical physicist bases “temperature” on the internal energy and the entropy. Imagine giving a system a tiny bit more energy. How many more ways would the system be able to arrange itself with that extra energy? That gives us the temperature. (To be precise, it gives us the reciprocal of the temperature. We could set this up as how a small change in entropy affects the internal energy, and get temperature right away. But I have an easier time thinking of going from change-in-energy to change-in-entropy than the other way around. And this is my blog so I get to choose how I set things up.)
This definition sounds bizarre. But it works brilliantly. It’s all nice clean mathematics. It matches perfectly nice easy-to-work-out cases, too. Like, you may kind of remember from high school physics how the temperature of a gas is something something average kinetic energy something. Work out the entropy and the internal energy of an ideal gas. Guess what this change-in-entropy/change-in-internal-energy thing gives you? Exactly something something average kinetic energy something. It’s brilliant.
In ordinary stuff, adding a little more internal energy to a system opens up new ways to arrange that energy. It always increases the entropy. So the absolute temperature, from this definition, is always positive. Good stuff. Matches our intuition well.
So in 1956 Dr Norman Ramsey and Dr Martin Klein published some interesting papers in the Physical Review. (Here’s a link to Ramsey’s paper and here’s Klein’s, if you can get someone else to pay for your access.) Their insightful question: what happens if a physical system has a maximum internal energy? If there’s some way of arranging the things in your system so that no more energy can come in? What if you’re close to but not at that maximum?
It depends on details, yes. But consider this setup: there’s one, or only a handful, of ways to arrange the maximum possible internal energy. There’s some more ways to arrange nearly-the-maximum-possible internal energy. There’s even more ways to arrange not-quite-nearly-the-maximum-possible internal energy.
Look at what that implies, though. If you’re near the maximum-possible internal energy, then adding a tiny bit of energy reduces the entropy. There’s fewer ways to arrange that greater bit of energy. Greater internal energy, reduced entropy. This implies the temperature is negative.
So we have to allow the idea of negative temperatures. Or we have to throw out this statistical-mechanics-based definition of temperature. And the definition works so well otherwise. Nobody’s got an idea nearly as good for it. So mathematical physicists shrugged, and noted this as a possibility, but mostly ignored it for decades. If it got mentioned, it was because the instructor was showing off a neat weird thing. This is how I encountered it, as a young physics major full of confidence and not at all good on wedge products. But it was sitting right there, in my textbook, Kittel and Kroemer’s Thermal Physics. Appendix E, four brisk pages before the index. Still, it was an enchanting piece.
And a useful one, possibly the most useful four-page aside I encountered as an undergraduate. My thesis research simulated a fluid-equilibrium problem run at different temperatures. There was a natural way that this fluid would have a maximum possible internal energy. So, a good part — the most fascinating part — of my research was in the world of negative temperatures. It’s a strange one, one where entropy seems to work in reverse. Things build, spontaneously. More heat, more energy, makes them build faster. In simulation, a shell of viscosity-free gas turned into what looked for all the world like a solid shell.
All right, but you can simulate anything on a computer, or in equations, as I did. Would this ever happen in reality? … And yes, in some ways. Internal energy and entropy are ideas that have natural, irresistible fits in information theory. This is the study of … information. I mean, how you send a signal and how you receive a signal. It turns out a lot of laser physics has, in information theory terms, behavior that’s negative-temperature. And, all right, but that’s not what anybody thinks of as temperature.
Well, these ideas happen still. They usually need some kind of special constraint on the things. Atoms held in a magnetic field so that their motions are constrained. Vortices locked into place on a two-dimensional surface (a prerequisite to my little fluids problems). Atoms bound into a lattice that keeps them from being able to fly free. All weird stuff, yes. But all exactly as the statistical-mechanics temperature idea calls on.
And notice. These negative temperatures happen only when the energy is extremely high. This is the grounds for saying that they’re hotter than positive temperatures. And good reason, too. Getting into what heat is, as opposed to temperature, is an even longer discussion. But it seems fair to say something with a huge internal energy has more heat than something with slight internal energy. So Graziano’s Ripley’s claim is right.
(GoComics.com commenters, struggling valiantly, have tried to talk about quantum mechanics stuff and made a hash of it. As a general rule, skip any pop-physics explanation of something being quantum mechanics.)
If you’re interested in more about this, I recommend Stephen J Blundell and Katherine M Blundell’s Concepts in Thermal Physics. Even if you’re not comfortable enough in calculus to follow the derivations, the textbook prose is insightful.
John Hambrock’s The Brilliant Mind of Edison Lee for the 3rd is a probability joke. And it’s built on how impossible putting together a particular huge complicated structure can be. I admit I’m not sure how I’d go about calculating the chance of a heap of Legos producing a giraffe shape. Imagine working out the number of ways Legos might fall together. Imagine working out how many of those could be called giraffe shapes. It seems too great a workload. And figuring it by experiment, shuffling Legos until a giraffe pops out, doesn’t seem much better.
This approaches an argument sometimes raised about the origins of life. Grant there’s no chance that a pile of Legos could be dropped together to make a giraffe shape. How can the much bigger pile of chemical elements have been stirred together to make an actual giraffe? Or, the same problem in another guise. If a monkey could go at a typewriter forever without typing any of Shakespeare’s plays, how did a chain of monkeys get to writing all of them?
And there’s a couple of explanations. At least partial explanations. There is much we don’t understand about the origins of life. But one is that the universe is huge. There’s lots of stars. It looks like most stars have planets. There’s lots of chances for chemicals to mix together and form a biochemistry. Even an impossibly unlikely thing will happen, given enough chances.
And another part is selection. A pile of Legos thrown into a pile can do pretty much anything. Any piece will fit into any other piece in a variety of ways. A pile of chemicals are more constrained in what they can do. Hydrogen, oxygen, and a bit of activation energy can make hydrogen-plus-hydroxide ions, water, or hydrogen peroxide, and that’s it. There can be a lot of ways to arrange things. Proteins are chains of amino acids. These chains can be about as long as you like. (It seems.) (I suppose there must be some limit.) And they curl over and fold up in some of the most complicated mathematical problems anyone can even imagine doing. How hard is it to find a set of chemicals that are a biochemistry? … That’s hard to say. There are about twenty amino acids used for proteins in our life. It seems like there could be a plausible life with eighteen amino acids, or 24, including a couple we don’t use here. It seems plausible, though, that my father could have had two brothers growing up; if there were, would I exist?
Jason Chatfield’s Ginger Meggs for the 3rd is a story-problem joke. Familiar old form to one. The question seems to be a bit mangled in the asking, though. Thirty percent of Jonson’s twelve apples is a nasty fractional number of apples. Surely the question should have given Jonson ten and Fitzclown twelve apples. Then thirty percent of Jonson’s apples would be a nice whole number.
It wasn’t just another busy week from Comic Strip Master Command. And a week busy enough for me to split the mathematics comics into two essays. It was one where I recognized one of the panels as one I’d featured before. Multiple times. Some of the comics I feature are in perpetual reruns and don’t have your classic, deep, Peanuts-style decades of archives to draw from. I don’t usually go checking my archives to see if I’ve mentioned a comic before, not unless something about it stands out. So for me to notice I’ve seen this strip repeatedly can mean only one thing: there was something a little bit annoying about it. Recognize it yet? You will.
Hy Eisman’s Popeye for the 7th of January, 2018 is an odd place for mathematics to come in. J Wellington Wimpy regales Popeye with all the intellectual topics he tried to impress his first love with, and “Euclidean postulates in the original Greek” made the cut. And, fair enough. Euclid’s books are that rare thing that’s of important mathematics (or scientific) merit and that a lay person can just pick up and read, even for pleasure. These days we’re more likely to see a division between mathematics writing that’s accessible but unimportant (you know, like, me) or that’s important but takes years of training to understand. Doing it in the original Greek is some arrogant showing-off, though. Can’t blame Carolyn for bailing on someone pulling that stunt.
John Hambrock’s The Brilliant Mind of Edison Lee for the 8th is set in mathematics class. And Edison tries to use a pile of mathematically-tinged words to explain why it’s okay to read a Star Wars book instead of paying attention. Or at least to provide a response the teacher won’t answer. Maybe we can make something out of this by allowing the monetary value of something to be related to its relevance. But if we allow that then Edison’s messed up. I don’t know what quantity is measured by multiplying “every Star Wars book ever written” by “all the movies and merchandise”. But dividing that by the value of the franchise gets … some modest number in peculiar units divided by a large number of dollars. The number value is going to be small. And the dimensions are obviously crazy. Edison needs to pay better attention to the mathematics.
Johnny Hart’s B.C. for the 14th of July, 1960 shows off the famous equation of the 20th century. All part of the comic’s anachronism-comedy chic. The strip reran the 9th of January. “E = mc2” is, correctly, associated with Albert Einstein and some of his important publications of 1905. But the expression does have some curious precursors, people who had worked out the relationship (or something close to it) before Einstein and who didn’t quite know what they had. A short piece from Scientific American a couple years back describes pre-Einstein expressions of the equation from Oliver Heaviside, Henri Poincaré, and Fritz Hasenöhrl. I’m not surprised Poincaré had something close to this; it seems like he spent twenty years almost discovering Relativity. That’s all right; he did enough in dynamical systems that mathematicians aren’t going to forget him.
Jason Chatfield’s Ginger Meggs for the 9th draws my eye just because the blackboard lists “Prime Numbers”. Fair enough place setting, although what’s listed are 1, 3, 5, and 7. These days mathematicians don’t tend to list 1 as a prime number; it’s inconvenient. (A lot of proofs depend on their being exactly one way to factorize a number. But you can always multiply a number by ‘1’ a couple more times without changing its value. So ‘6’ is 3 times 2, but it’s also 3 times 2 times 1, or 3 times 2 times 1 times 1, or 3 times 2 times 1145,388,434,247. You can write around that, but it’s easier to define ‘1’ as not a prime.) But it could be defended. I can’t think any reason to leave ‘2’ off a list of prime numbers, though. I think Chatfield conflated odd and prime numbers. If he’d had a bit more blackboard space we could’ve seen whether the next item was 9 or 11 and that would prove the matter.
Paul Trap’s Thatababy for the 9th uses arithmetic — square roots — as the kind of thing to test whether a computer’s working. Everyone has their little tests like this. My love’s father likes to test whether the computer knows of the band Walk The Moon or of Christine Korsgaard (a prominent philosopher in my love’s specialty). I’ve got a couple words I like to check dictionaries for. Of course the test is only any good if you know what the answer should be, and what’s the actual square root of 3,278? Goodness knows. It’s got to be between 50 (50 squared is 25 hundred) and 60 (60 squared is 36 hundred). Since 3,278 is so much closer 3,600 than 2,500 its square root should be closer to 60 than to 50. So 57-point-something is plausible. Unfortunately square roots don’t lend themselves to the same sorts of tricks from reading the last digit that cube roots do. And 3,278 isn’t a perfect square anyway. Alexa is right on this one. Also about the specific gravity of cobalt, at least if Wikipedia is right and not conspiring with the artificial intelligences on this one. Catch you in 2021.
I have my reasons for this installment’s title. They involve my deductions from a comic strip. Give me a few paragraphs.
Mark Anderson’s Andertoons for the 16th asks for attention from whatever optician-written blog reads the comics for the eye jokes. And meets both the Venn Diagram and the Mark Anderson’s Andertoons content requirements for this week. Good job! Starts the week off strong.
Lincoln Pierce’s Big Nate: First Class for the 16th, rerunning the strip from 1993, is about impossibly low-probability events. We can read the comic as a joke about extrapolating a sequence from a couple examples. Properly speaking we can’t; any couple of terms can be extended in absolutely any way. But we often suppose a sequence follows some simple pattern, as many real-world things do. I’m going to pretend we can read Jenny’s estimates of the chance she’ll go out with him as at all meaningful. If Jenny’s estimate of the chance she’d go out with Nate rose from one in a trillion to one in a billion over the course of a week, this could be a good thing. If she’s a thousand times more likely each week to date him — if her interest is rising geometrically — this suggests good things for Nate’s ego in three weeks. If she’s only getting 999 trillionths more likely each week — if her interest is rising arithmetically — then Nate has a touch longer to wait before a date becomes likely.
(I forget whether she has agreed to a date in the 24 years since this strip first appeared. He has had some dates with kids in his class, anyway, and some from the next grade too.)
Is the point of the first problem, Farmer Joe’s apples, to see whether a student can do a not-quite-long division? Or is it to see whether the student can extract a price-per-quantity for something, and apply that to find the quantity to fit a given price? If it’s the latter then the numbers don’t make a difference. One would want to avoid marking down a student who knows what to do, and could divide 15 cents by three, but would freeze up if a more plausible price of, say, $2.25 per pound had to be divided by three.
But then the second problem, Mr Schad driving from Belmont to Cadillac, got me wondering. It is about 84 miles between the two Michigan cities (and there is a Reed City along the way). The time it takes to get from one city to another is a fair enough problem. But these numbers don’t make sense. At 55 miles per hour the trip takes an awful 1.5273 hours. Who asks elementary school kids to divide 84 by 55? On purpose? But at the state highway speed limit (for cars) of 70 miles per hour, the travel time is 1.2 hours. 84 divided by 70 is a quite reasonable thing to ask elementary school kids to do.
And then I thought of this: you could say Belmont and Cadillac are about 88 miles apart. Google Maps puts the distance as 86.8 miles, along US 131; but there’s surely some point in the one town that’s exactly 88 miles from some point in the other, just as there’s surely some point exactly 84 miles from some point in the other town. 88 divided by 55 would be another reasonable problem for an elementary school student; 1.6 hours is a reasonable answer. The (let’s call it) 1980s version of the question ought to see the car travel 88 miles at 55 miles per hour. The contemporary version ought to see the car travel 84 miles at 70 miles per hour. No reasonable version would make it 84 miles at 55 miles per hour.
So did Mallett take a story problem that could actually have been on an era-appropriate test and ancient it up?
Before anyone reports me to Comic Strip Master Command let me clarify what I’m wondering about. I don’t care if the details of the joke don’t make perfect sense. They’re jokes, not instruction. All the story problem needs to set up the joke is the obsolete speed limit; everything else is fluff. And I enjoyed working out variation of the problem that did make sense, so I’m happy Mallett gave me that to ponder.
Here’s what I do wonder about. I’m curious if story problems are getting an unfair reputation. I’m not an elementary school teacher, or parent of a kid in school. I would like to know what the story problems look like. Do you, the reader, have recent experience with the stuff farmers, drivers, and people weighing things are doing in these little stories? Are they measuring things that people would plausibly care about today, and using values that make sense for the present day? I’d like to know what the state of story problems is.
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th uses mental arithmetic as the gauge of intelligence. Pretty harsly, too. I wouldn’t have known the square root of 8649 off the top of my head either, although it’s easy to tell that 92 can’t be right: the last digit of 92 squared has to be 4. It’s also easy to tell that 92 has to be about right, though, as 90 times 90 will be about 8100. Given this information, if you knew that 8,649 was a perfect square, you’d be hard-pressed to think of a better guess for its value than 93. But since most whole numbers are not perfect squares, “a little over 90” is the best I’d expect to do.