I have been reading Mapping In Michigan and the Great Lakes Region, edited by David I Macleod, because — look, I understand that I have a problem. I just live with it. The book is about exactly what you might imagine from the title. And it features lots of those charming old maps where, you know, there wasn’t so very much hard data available and everyone did the best with what they had. So you get these maps with spot-on perfect Lake Eries and the eastern shore of Lake Huron looking like you pulled it off of Open Street Maps. And then Michigan looks like a kid’s drawing of a Thanksgiving turkey. Also sometimes they drop a mountain range in the middle of the state because I guess it seemed a little empty without.
The first chapter, by Mary Sponberg Pedley, is a biography and work-history of Louis Charles Karpinski, 1878-1956. Karpinski did a lot to bring scholastic attention to maps of the Great Lakes area. He was a professor of mathematics for the University of Michigan. And he commented a good bit about the problems of teaching mathematics. Pedley quoted this bit that I thought was too good not to share. It’s from Arithmetic For The Farm. It’s about the failure of textbooks to provide examples that actually reflected anything anyone might want to know. I quote here Pedley’s endnote:
Karpinski disparaged the typical “story problems” found in contemporary textbooks, such as the following: “How many sacks, holding 2 bushels, 3 pecks and 2 quarts each can be filled from a bin containing 366 bushels, 3 pecks, 4 quarts of what?” Karpinski comments: “How carefully would you have to fill a sack to make it hold 3 pecks 2 quarts of anything? And who filled the bin so marvelously that the capacity is known with an accuracy of one-25th of 1% of the total?” He recommended an easier, more practical means of doing such problems, noting that a bushel is about 1 & 1/4 or 5/4 cubic feet. Therefore the number of bushels in the bin is the length times width times 4/5; the easiest way to get 4/5 of anything is to take away one-fifth of it.
This does read to me like Pedley jumped a track somewhere. It seems to go from the demolition of the plausibility of one problem’s setup to demolishing the plausibility of how to answer a problem. Still, the core complaint is with us yet. It’s hard to frame problems that might actually come up in ways that clearly test specific mathematical skills.
And on another note. This is the 1,000th mathematical piece that I’ve published since I started in September of 2011. If I’m not misunderstanding this authorship statistic on WordPress, which is never a safe bet. I’m surprised that it has taken as long as this to get to a thousand posts. Also I’m surprised that I should be surprised. I know roughly how many days there are in a year. And I know I need special circumstances to post something more often than every other day. Still, I’m glad to reach this milestone, and gratified that there’s anyone interested in what I have to say. In my next thousand posts I hope to say something.
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.
It’s too many comics to call this a famine edition, after last week’s feast. But there’s not a lot of theme to last week’s mathematically-themed comic strips. There’s a couple that include vintage comic strips from before 1940, though, so let’s run with that as a title.
Bill Hinds’s Tank McNamara for the 4th is built on mathematics’ successful invasion and colonization of sports management. Analytics, sabermetrics, Moneyball, whatever you want to call it, is built on ideas not far removed from the quality control techniques that changed corporate management so. Look for patterns; look for correlations; look for the things that seem to predict other things. It seems bizarre, almost inhuman, that we might be able to think of football players as being all of a kind, that what we know about (say) one running back will tell us something about another. But if we put roughly similarly capable people through roughly similar training and set them to work in roughly similar conditions, then we start to see why they might perform similarly. Models can help us make better, more rational, choices.
Morrie Turner’s Wee Pals rerun for the 4th is another word-problem resistance joke. I suppose it’s also a reminder about the unspoken assumptions in a problem. It also points out why mathematicians end up speaking in an annoyingly precise manner. It’s an attempt to avoid being shown up like Oliver is.
Which wouldn’t help with Percy Crosby’s Skippy for the 7th of April, 1930, and rerun the 5th. Skippy’s got a smooth line of patter to get out of his mother’s tutoring. You can see where Percy Crosby has the weird trait of drawing comics in 1930 that would make sense today still; few pre-World-War-II comics do.
Niklas Eriksson’s Carpe Diem for the 7th is a joke about mathematics anxiety. I don’t know that it actually explains anything, but, eh. I’m not sure there is a rational explanation for mathematics anxiety; if there were, I suppose it wouldn’t be anxiety.
George Herriman’s Krazy Kat for the 15th of July, 1939, and rerun the 8th, extends that odd little faintly word-problem-setup of the strips I mentioned the other day. I suppose identifying when two things moving at different speeds will intersect will always sound vaguely like a story problem.
You know we’re getting near the end of the (United States) school year when Comic Strip Master Command orders everyone to clear out their mathematics jokes. I’m assuming that’s what happened here. Or else a lot of cartoonists had word problems on their minds eight weeks ago. Also eight weeks ago plus whenever they originally drew the comics, for those that are deep in reruns. It was busy enough to split this week’s load into two pieces and might have been worth splitting into three, if I thought I had publishing dates free for all that.
Larry Wright’s Motley Classics for the 28th of May, a rerun from 1989, is a joke about using algebra. Occasionally mathematicians try to use the the ability of people to catch things in midair as evidence of the sorts of differential equations solution that we all can do, if imperfectly, in our heads. But I’m not aware of evidence that anyone does anything that sophisticated. I would be stunned if we didn’t really work by a process of making a guess of where the thing should be and refining it as time allows, with experience helping us make better guesses. There’s good stuff to learn in modeling how to catch stuff, though.
Michael Jantze’s The Norm Classics rerun for the 28th opines about why in algebra you had to not just have an answer but explain why that was the answer. I suppose mathematicians get trained to stop thinking about individual problems and instead look to classes of problems. Is it possible to work out a scheme that works for many cases instead of one? If it isn’t, can we at least say something interesting about why it’s not? And perhaps that’s part of what makes algebra classes hard. To think about a collection of things is usually harder than to think about one, and maybe instructors aren’t always clear about how to turn the specific into the general.
Also I want to say some very good words about Jantze’s graphical design. The mock textbook cover for the title panel on the left is so spot-on for a particular era in mathematics textbooks it’s uncanny. The all-caps Helvetica, the use of two slightly different tans, the minimalist cover art … I know shelves stuffed full in the university mathematics library where every book looks like that. Plus, “[Mathematics Thing] And Their Applications” is one of the roughly four standard approved mathematics book titles. He paid good attention to his references.
Gary Wise and Lance Aldrich’s Real Life Adventures for the 28th deploys a big old whiteboard full of equations for the “secret” of the universe. This makes a neat change from finding the “meaning” of the universe, or of life. The equations themselves look mostly like gibberish to me, but Wise and Aldrich make good uses of their symbols. The symbol , a vector-valued quantity named B, turns up a lot. This symbol we often use to represent magnetic flux. The B without a little arrow above it would represent the intensity of the magnetic field. Similarly an turns up. This we often use for magnetic field strength. While I didn’t spot a — electric field — which would be the natural partner to all this, there are plenty of bare E symbols. Those would represent electric potential. And many of the other symbols are what would naturally turn up if you were trying to model how something is tossed around by a magnetic field. Q, for example, is often the electric charge. ω is a common symbol for how fast an electromagnetic wave oscillates. (It’s not the frequency, but it’s related to the frequency.) The uses of symbols is consistent enough, in fact, I wonder if Wise and Aldrich did use a legitimate sprawl of equations and I’m missing the referenced problem.
Bill Amend’s FoxTrot Classicfor the 31st, a rerun from the 7th of June, 2006, shows the conflation of “genius” and “good at mathematics” in everyday use. Amend has picked a quixotic but in-character thing for Jason Fox to try doing. Euclid’s Fifth Postulate is one of the classic obsessions of mathematicians throughout history. Euclid admitted the thing — a confusing-reading mess of propositions — as a postulate because … well, there’s interesting geometry you can’t do without it, and there doesn’t seem any way to prove it from the rest of his geometric postulates. So it must be assumed to be true.
There isn’t a way to prove it from the rest of the geometric postulates, but it took mathematicians over two thousand years of work at that to be convinced of the fact. But I know I went through a time of wanting to try finding a proof myself. It was a mercifully short-lived time that ended in my humbly understanding that as smart as I figured I was, I wasn’t that smart. We can suppose Euclid’s Fifth Postulate to be false and get interesting geometries out of that, particularly the geometries of the surface of the sphere, and the geometry of general relativity. Jason will surely sometime learn.
I admit to padding this week’s collection of mathematically-themed comic strips. There’s just barely enough to justify my splitting this into a Sunday and a Tuesday installment. I’m including a follow-the-bouncing-ball cartoon to make up for that though. Enjoy!
Jimmy Hatlo’s Little Iodine from the 20th originally ran the 18th of September, 1955. It’s a cute enough bit riffing on realistic word problems. If the problems do reflect stuff ordinary people want to know, after all, then they’re going to be questions people in the relevant fields know how to solve. A limitation is that word problems will tend to pick numbers that make for reasonable calculations, which may be implausible for actual problems. None of the examples Iodine gives seem implausible to me, but what do I know about horses? But I do sometimes encounter problems which have the form but not content of a reasonable question, like an early 80s probability book asking about the chances of one or more defective transistors in a five-transistor radio set. (The problem surely began as one about burned-out vacuum tubes in a radio.)
Daniel Beyer’s Long Story Short for the 21st is another use of Albert Einstein as iconic for superlative first-rate genius. I’m curious how long it did take for people to casually refer to genius as Einstein. The 1930 song Kitty From Kansas City (and its 1931 Screen Songs adaptation, starring Betty Boop) mention Einstein as one of those names any non-stupid person should know. But that isn’t quite the same as being the name for a genius.
My love asked if I’d include Stephen Pastis’s Pearls Before Swine of the 22nd. It has one of the impossibly stupid crocodiles say, poorly, that he was a mathematics major. I admitted it depended how busy the week was. On a slow week I’ll include more marginal stuff.
Is it plausible that the Croc is, for all his stupidity, a mathematics major? Well, sure. Perseverance makes it possible to get any degree. And given Croc’s spent twenty years trying to eat Zebra without getting close clearly perseverance is one of his traits. But are mathematics majors bad at communication?
Certainly we get the reputation for it. Part of that must be that any specialized field — whether mathematics, rocket science, music, or pasta-making — has its own vocabulary and grammar for that vocabulary that outsiders just don’t know. If it were easy to follow it wouldn’t be something people need to be trained in. And a lay audience starts scared of mathematics in a way they’re not afraid of pasta technology; you can’t communicate with people who’ve decided they can’t hear you. And many mathematical constructs just can’t be explained in a few sentences, the way vacuum extrusion of spaghetti noodles could be. And, must be said, it’s often the case a mathematics major (or a major in a similar science or engineering-related field) has English as a second (or third) language. Even a slight accent can make someone hard to follow, and build an undeserved reputation.
The Pearls crocodiles are idiots, though. The main ones, anyway; their wives and children are normal.
Ernie Bushmiller’s Nancy Classics for the 23rd originally appeared the 23rd of November, 1949. It’s just a name-drop of mathematics, though, using it as the sort of problem that can be put on the blackboard easily. And it’s not the most important thing going on here, but I do notice Bushmiller drawing the blackboard as … er … not black. It makes the composition of the last panel easier to read, certainly. And makes the visual link between the paper in the second panel and the blackboard in the last stronger. It seems more common these days to draw a blackboard that’s black. I wonder if that’s so, or if it reflects modern technology making white-on-black-text easier to render. A Photoshop select-and-invert is instantaneous compared to what Bushmiller had to do.
Comic Strip Master Command gave me a light load this week, which suit me fine. I’ve been trying to get the End 2016 Mathematics A To Z comfortably under way instead. It does strike me that there were fewer Halloween-themed jokes than I’d have expected. For all the jokes there are to make about Halloween I’d imagine some with some mathematical relevance would come up. But they didn’t and, huh. So it goes. The one big exception is the one I’d have guessed would be the exception.
Bill Amend’s FoxTrot for the 30th — a new strip — plays with the scariness of mathematics. Trigonometry specifically. Trig is probably second only to algebra for the scariest mathematics normal people encounter. And that’s probably more because people get to algebra before they might get to trigonometry. Which is madness, in its way. Trigonometry is about how we can relate angles, arcs, and linear distances. It’s about stuff anyone would like to know, like how to go from an easy-to-make observation of the angle spanned by a thing to how big the thing must be. But the field does require a bunch of exotic new functions like sine and tangent and novelty acts like “arc-cosecant”. And the numbers involved can be terrible things. The sine of an angle, for example, is almost always going to be some irrational number. For common angles we use a lot it’ll be an irrational number with an easy-to-understand form. For example the sine of 45 degrees, mentioned here, is “one-half the square root of two”. Anyone not trying to be intimidating will use that instead. But the sine of, say, 50 degrees? I don’t know what that is either except that it’s some never-ending sequence of digits. People love to have digits, but when they’re asked to do something with them, they get afraid and I don’t blame them.
Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 30th uses sudoku as shorthand for “genius thinking”. I am aware some complain sudoku isn’t mathematics. It’s certainly logic, though, and if we’re going to rule out logic puzzles from mathematics we’re going to lose a lot of fun fields. One of the commenters provided what I suppose the solution to be. (I haven’t checked.) If wish to do the puzzle be careful about scrolling.
In Jef Mallet’s Frazz for the 2nd Caulfield notices something cute about 100. A perfect square is a familiar enough idea; it’s a whole number that’s the square of another whole number. The “roundest of round numbers” is a value judgement I’m not sure I can get behind. It’s a good round number, anyway, at least for stuff that’s sensibly between about 50 and 150. Or maybe between 50 and 500 if you’re just interested in about how big something might be. An irrational number, well, you know where that joke’s going.
Mrs Olsen doesn’t seem impressed by Caulfield’s discovery, although in fairness we don’t see the actual aftermath. Sometimes you notice stuff like that and it is only good for a “huh”. But sometimes you get into some good recreational mathematics. It’s the sort of thinking that leads to discovering magic squares and amicable numbers and palindromic prime numbers and the like. Do they lead to important mathematics? Some of them do. Or at least into interesting mathematics. Sometimes they’re just passingly amusing.
Greg Curfman’s Meg rerun for the 12th quotes Einstein’s famous equation as the sort of thing you could just expect would be asked in school. I’m not sure I ever had a class where knowing E = mc2 was the right answer to a question, though. Maybe as I got into physics since we did spend a bit of time on special relativity and E = mc2 turns up naturally there. Maybe I’ve been out of elementary school too long to remember.
Mark Tatulli’s Heart of the City for the 4th has Heart and Dean talking about postapocalyptic society. Heart doubts that postapocalyptic society would need people like him, “with long-division experience”. Ah, but, grant the loss of computing devices. People will still need to compute. Before the days of electrical, and practical mechanical, computing people who could compute accurately were in demand. The example mathematicians learn to remember is Zacharias Dase, a German mental calculator. He was able to do astounding work and in his head. But he didn’t earn so much money as pro-mental-arithmetic propaganda would like us to believe. And why work entirely in your head if you don’t need to?
Larry Wright’s Motley Classics rerun for the 5th is a word problem joke. And it’s mixed with labor relations humor for the sake of … I’m not quite sure, actually. Anyway I would have sworn I’d featured this strip in a long-ago Reading The Comics post, but I don’t see it on a casual search. So, go figure.
I have no idea how many or how few comic strips on Saturday included some mathematical content. I was away most of the day. We made a quick trip to the Michigan’s Adventure amusement park and then to play pinball in a kind-of competitive league. The park turned out to have every person in the world there. If I didn’t wave to you from the queue on Shivering Timbers I apologize but it hasn’t got the greatest lines of sight. The pinball stuff took longer than I expected too and, long story short, we got back home about 4:15 am. So I’m behind on my comics and here’s what I did get to.
Tak Bui’s PC and Pixel for the 8th depicts the classic horror of the cleaning people wiping away an enormous amount of hard work. It’s a primal fear among mathematicians at least. Boards with a space blocked off with the “DO NOT ERASE” warning are common. At this point, though, at least, the work is probably savable. You can almost always reconstruct work, and a few smeared lines like this are not bad at all.
The work appears to be quantum mechanics work. The tell is in the upper right corner. There’s a line defining E (energy) as equal to something including . This appears in the time-dependent Schrödinger Equation. It describes how probability waveforms look when the potential energies involved may change in time. These equations are interesting and impossible to solve exactly. We have to resort to approximations, including numerical approximations, all the time. So that’s why the computer lab would be working on this.
Mark Anderson’s Andertoons! Where would I be without them? Besides short on content. The strip for the 10th depicts a pollster saying to “put the margin of error at 50%”, guaranteeing the results are right. If you follow elections polls you do see the results come with a margin of error, usually of about three percent. But every sampling technique carries with it a margin of error. The point of a sample is to learn something about the whole without testing everything in it, after all. And probability describes how likely it is the quantity measured by a sample will be far from the quantity the whole would have. The logic behind this is independent of the thing being sampled. It depends on what the whole is like. It depends on how the sampling is done. It doesn’t matter whether you’re sampling voter preferences or whether there are the right number of peanuts in a bag of squirrel food.
So a sample’s measurement will almost never be exactly the same as the whole population’s. That’s just requesting too much of luck. The margin of error represents how far it is likely we’re off. If we’ve sampled the voting population fairly — the hardest part — then it’s quite reasonable the actual vote tally would be, say, one percent different from our poll. It’s implausible that the actual votes would be ninety percent different. The margin of error is roughly the biggest plausible difference we would expect to see.
Except. Sometimes we do, even with the best sampling methods possible, get a freak case. Rarely noticed beside the margin of error is the confidence level. This is what the probability is that the actual population value is within the sampling error of the sample’s value. We don’t pay much attention to this because we don’t do statistical-sampling on a daily basis. The most normal people do is read election polling results. And most election polls settle for a confidence level of about 95 percent. That is, 95 percent of the time the actual voting preference will be within the three or so percentage points of the survey. The 95 percent confidence level is popular maybe because it feels like a nice round number. It’ll be off only about one time out of twenty. It also makes a nice balance between a margin of error that doesn’t seem too large and that doesn’t need too many people to be surveyed. As often with statistics the common standard is an imperfectly-logical blend of good work and ease of use.
Anthony Blades’s Bewley for the 12th is a rerun. It’s at least the third time this strip has turned up since I started writing these Reading The Comics posts. For the record it ran also the 27th of April, 2015 and on the 24th of May, 2013. It also suggests mathematicians have a particular tell. Try this out next time you do word problem poker and let me know how it works for you.
Bill Amend’s FoxTrot Classics for the 12th is a bunch of gags about a mathematics fighting game. I think Amend might be on to something here. I assume mathematics-education contest games have evolved from what I went to elementary school on. That was a Commodore PET with a game where every time you got a multiplication problem right your rocket got closer to the ASCII Moon. But the game would probably quickly turn into people figuring how to multiply the other person’s function by zero. I know a game exploit when I see it.
The most obscure reference is in the third panel one. Jason speaks of “a z = 0 transform”. This would seem to be some kind of z-transform, a thing from digital signals processing. You can represent the amplification, or noise-removal, or averaging, or other processing of a string of digits as a polynomial. Of course you can. Everything is polynomials. (OK, sometimes you must use something that looks like a polynomial but includes stuff like the variable z raised to a negative power. Don’t let that throw you. You treat it like a polynomial still.) So I get what Jason is going for here; he’s processing Peter’s function down to zero.
That said, let me warn you that I don’t do digital signal processing. I just taught a course in it. (It’s a great way to learn a subject.) But I don’t think a “z = 0 transform” is anything. Maybe Amend encountered it as an instructor’s or friend’s idiosyncratic usage. (Amend was a physics student in college, and shows his comfort with mathematics-major talk often. He by the way isn’t even the only syndicated cartoonist with a physics degree. Bud Grace of The Piranha Club was also a physics major.) I suppose he figured “z = 0 transform” would read clearly to the non-mathematician and be interpretable to the mathematician. He’s right about that.
And now to close out the rest of last week’s comics, those from between the 1st and the 6th of the month. It’s a smaller set. Take it up with the traffic division of Comic Strip Master Command.
Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 2nd is mostly a word problem joke. It’s boosted some by melting into it a teacher complaining about her pay. It does make me think some about what the point of a story problem is. That is, why is the story interesting? Often it isn’t. The story is just an attempt to make a computation problem look like the sort of thing someone might wonder in the real world. This is probably why so many word problems are awful as stories and as incentive to do a calculation. There’s a natural interest that one might have in, say, the total distance travelled by a rubber ball dropped and bouncing until it finally comes to a rest. But that’s only really good for testing how one understands a geometric series. It takes more storytelling to work out why you might want to find a cube root of x2 minus eight.
Dave Whamond’s Reality Check for the 3rd uses mathematics on the blackboard as symbolic for all the problems one might have. Also a solution, if you call it that. It wouldn’t read so clearly if Ms Haversham had an English problem on the board.
Mark Anderson’s Andertoons for the 5th keeps getting funnier to me. At first reading I didn’t connect the failed mathematics problem of 2 x 0 with the caption. Once I did, I realized how snugly fit the comic is.
Greg Curfman’s Meg Classics for the 5th ran originally the 23rd of May, 1998. The application of mathematics to everyday sports was a much less developed thing back then. It’s often worthwhile to methodically study what you do, though, to see what affects the results. Here Mike has found the team apparently makes twelve missed shots for each goal. This might not seem like much of a formula, but these are kids. We shouldn’t expect formulas with a lot of variables under consideration. Since Meg suggests Mike needed to account for “the whiff factor” I have to suppose she doesn’t understand the meaning of the formula. Or perhaps she wonders why missed kicks before getting to the goal don’t matter. Well, every successful model starts out as a very simple thing to which we add complexity, and realism, as we’re able to handle them. If lucky we end up with a good balance between a model that describes what we want to know and yet is simple enough to understand.
For this week’s roundup of mathematically themed comic strips I have a picture again! After a month or so. It’s great to see again. Also there’s several comics I could swear I’ve shown and featured before. But it’s really quite hot here and I don’t feel like going to the effort of looking. If I repeat myself, so I do. I bet you’ve forgotten the last time I did this Robbie and Bobby too.
Carol Lay’s Lay Lines for the 6th implicitly uses mathematics as an example of perfection. The idea of the straight line is in that territory shared by both mathematics and Platonic ideals. We can imagine a straight line and understand many properties of it even though it can’t be manifest in our real world. The Gods, allegedly, would be able to overcome that and offer perfect circles around imperfect lines. I suppose that’s one way to tell there’s a god involved. The strip also take a moment to riff on the ontological problem, although I don’t know if that’s part of Lay’s intent.
Jonathan Lemon’s Rabbits Against Magic for the 6th uses a bit of mathematics to represent having a theory. It’s true enough that mathematics serves this role in many sciences. We can often put a good explanation for phenomena in a set of equations. But that’s so if you have a good idea what quantities to measure, and how they affect one another. Lettuce’s equation just describes how long an arc within a circle is. It’s true, although I don’t think it rates the status of a theory; it just describes one thing we’d like to know in terms of another thing. And it’s all a setup for a π joke anyway.
Bud Blake’s Tiger for the 8th, as a King Features comic, broke my drought of having images to include with Reading the Comics posts! Celebrate! It’s also the one that made me think I was getting reruns in. But it’s more mysterious than that. The Tiger rerun (Blake died several years ago and all Tiger strips are reruns) for the 20th of April, 2015, is the same joke. I featured it in a Reading The Comics post back then. But it’s not the same strip. The art’s completely redrawn. I can’t fault Blake for having reused a setup-and-punchline. Every comic strip creator does this. Sometimes the cartoonist has improved the joke. (Berkeley Breathed did this several times over.) Sometimes the cartoonist probably just forgot it was done before. (There’s several Peanuts strips suggesting this.) I’m just delighted to catch someone at it.
Ryan Pagelow’s Buni for the 8th uses a blackboard full of mathematics as signifier for explaining the Big Questions of life. And features the traditional little error spotted by someone else. The scribbles are gibberish altogether, but they don’t need to be (and in truth couldn’t be) meaningful. I will defend the backwards-capital-sigma in the upper left of the first panel, though. Sigmas are some of those letters that get pretty sloppy treatment. You get swept up in inspiration and penmanship just collapses. Other Greek letters take some shabby treatment too. And there was a stretch of about three years when I would’ve sworn there was a letter ‘ksee’, a sort of topheavy squiggle. It doesn’t exist, but it’s pretty convenient when you need one more easy Greek letter to use.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th features the “scariest equation” in the universe. The board gives a good description of the quantities in the equation and the relationship makes superficial sense. But it does depend on an assumption I’m not sure about, but I will go with. Weinersmith’s argument supposes that a mis-sent text is equally likely to go to any of your contacts. I am not an experienced texter. But it seems to me that a mis-sent text is more likely to go to a contact you’d recently messaged, or one that’s close to the person you meant to contact. Suppose parents are among the people you text often, or whose contact information is stored where it’s easy to pick by accident. Then you likely send them more messages by accident than this expects. On the other hand, suppose you don’t text parents often or you store their information well away from your significant’s. Then the number of mis-sent messages given to them is lower. Without information about how you organize your contacts, we can’t say what’s a better estimate. So in ignorance we may suppose you mis-send texts to every one of your contacts equally often.
Mell Lazarus’s Momma rerun for the 10th uses a word problem to try to quantify love. Marylou decries the result as “differential calculus”, although it’s really just high school algebra. “Differential calculus” is the funnier term, must admit. Differential calculus refers, generally, to the study of how much one quantity depends on another. On average you can expect something to change if one or more of the variables that describe it change. For example, if you make a rectangle a little larger, its area gets larger. What’s the ratio between how much the area changes and how much the lengths of the rectangle change? If you make an angled corridor wider, then a longer straight object can be fit through the corner. How much longer an object can you bring through the corridor if you make the width a tiny bit bigger? And this also tells us where maximums and minimums are. At a maximum or minimum, a quantity doesn’t change appreciably as the variables that describe it change a little bit. So we can find maximums and minimums by the differential calculus.
I haven’t got Sunday’s comics under review yet. But the past seven days were slow ones for mathematically-themed comics. Maybe Comic Strip Master Command is under the impression that it’s the (United States) summer break already. It’s not, although Funky Winkerbean did a goofy sequence graduating its non-player-character students. And Zits has been doing a summer reading storyline that only makes sense if Jeremy Duncan is well into summer. Maybe Comic Strip Master Command thinks it’s a month later than it actually is?
Tony Cochrane’s Agnes for the 29th of May looks at first like a bit of nonsense wordplay. But whether a book with the subject “All About Books” would discuss itself, and how it would discuss itself, is a logic problem. And not just a logic problem. Start from pondering how the book All About Books would describe the content of itself. You can go from that to an argument that it’s impossible to compress every possible message. Imagine an All About Books which contained shorthand descriptions of every book. And the descriptions have enough detail to exactly reconstruct each original book. But then what would the book list for the description of All About Books?
And self-referential things can lead to logic paradoxes swiftly. You’d have some fine ones if Agnes were to describe a book All About Not-Described Books. Is the book described in itself? The question again sounds silly. But thinking seriously about it leads us to the decidability problem. Any interesting-enough logical system will always have statements that are meaningful and true that no one can prove.
Furthermore, the suggestion of an “All About `All About Books’ Book” suggests to me power sets. That’s the set of all the ways you can collect the elements of a set. Power sets are always bigger than the original set. They lead to the staggering idea that there are many sizes of infinitely large sets, a never-ending stack of bigness.
Robb Armstrong’s Jump Start for the 31st of May is part of a sequence about getting a tutor for a struggling kid. That it’s mathematics is incidental to the storyline, must be said. (It’s an interesting storyline, partly about the Jojo’s father, a police officer, coming to trust Ray, an ex-convict. Jump Start tells many interesting and often deeply weird storylines. And it never loses its camouflage of being an ordinary family comic strip.) It uses the familiar gimmick of motivating a word problem by making it about something tangible.
Ken Cursoe’s Tiny Sepuku for the 2nd of June uses the motif of non-Euclidean geometry as some supernatural magic. It’s a small reference, you might miss it. I suppose it is true that a high-dimensional analogue to conic sections would focus things from many dimensions. If those dimensions match time and space, maybe it would focus something from all humanity into the brain. I would try studying instead, though.
Russell Myers’s Broom Hilda for the 3rd is a resisting-the-word-problems joke. It’s funny to figure on missing big if you have to be wrong at all. But something you learn in numerical mathematics, particularly, is that it’s all right to start from a guess. Often you can take a wrong answer and improve it. If you can’t get the exact right answer, you can usually get a better answer. And often you can get as good as you need. So in practice, sorry to say, I can’t recommend going for the ridiculous answer. You can do better.
It’s been a normal cluster of mathematically-themed jokes this past week. But one of them lets me show off my ability to do introductory calculus.
Norm Feuti’s Gil for the 15th of March is a resisting-the-word-problems joke. It’s also a rerun, sad to say. King Features syndicated Feuti’s strip for a couple of years, but couldn’t make a go of it. GoComics.com is reprinting what ran and that’s something, at least.
Justin Boyd’s Invisible Bread for the 16th plays on alarm clocks that make you solve problems. I’ve heard of these things, and I suppose they exist or something. The idea is that making you do a bit of arithmetic proves you’ve gotten up enough to not fall right back asleep. The clockmakers are underestimating my ability to get back to sleep. Anyway, I like the escalation of this.
The integral that has to be solved, , is a good problem for people taking their first calculus course. Let me spoil it as a homework problem by saying how I’d solve it. If you haven’t got the first idea what calculus is about and don’t wish to know, go ahead and skip to the bit about Rudy Park. Or just enjoy the parts of the sentences below that aren’t mathematics.
The first thing I notice is the integrand, the thing inside the integral. That’s , which is the same as . Distributive law, as if you didn’t know. That strikes me as worth doing because, if the integral converges, the integral of the sum of two things is the same as the sum of the integral of two things. I’m willing to suppose it converges until given evidence otherwise. So this integral is the same as .
I think that’s worth doing because that first integral is incredibly easy. It’ll be a number equal to whatever -e-x is, when x is infinitely large, minus what -e-x is when x is zero. When x is infinitely large, -e-x is zero. When x is 0, -e-x is -1. So 0 minus -1 is … 1.
is harder. But it suggests how to evaluate it. The integrand is the quantity 2x times the quantity e-x. 2x is easy to take the derivative of. e-x is easy to integrate. (It’s also easy to take the derivative of, but it’s easier to integrate.) This suggests trying out integration by parts.
When you integrate by parts, you notice the original integral is the product of a part that’s easy to differentiate and a part that’s easy to integrate. My Intro Calculus textbooks generically label the easy-to-differentiate part u, and the easy-to-integrate part dv. Then the derivatie of the easy-to-differentiate part is du, and the integral of the easy-to-integrate part is v. When you integrate by parts, the integral of u times dv turns out to be equal to u times v (no integral signs there) minus the integral of v du. This may sound like we’ve just turned one integral into another. So we have. But we’ve often made it into an easier integral to evaluate. This is why we ever try it.
So if u equals 2x, then its derivative du is equal to 2 dx. If dv is equal to e-xdx (we want to carry those little d’s along), then v is equal to -e-x. And this means we have this:
That middle part, , is not an integral. It’s been integrated. The notation there means to evaluate the thing when x is infinitely large, and evaluate the thing when x is zero. Then subtract the x-is-zero value from the x-is-infinitely-large value. The x-is-zero value of this expression turns out to be zero, as you realize when you start writing “2 times 0 times oh wait we’re done here”. The x-is-infinitely-large value of this expression takes longer to get done. If you want to do it right you have to invoke l’Hôpital’s Rule. But it’s also zero.
The right-hand part, , is equal to , and that’s equal to . Which will be 0 minus a -2. Or 2 altogether.
So the integral is 1 plus 2, or in total, 3. The strip got its integration right.
Darrin Bell and Theron Heir’s Rudy Park for the 16th speaks of some architect who said the job didn’t demand being good at mathematics. I hadn’t heard the original claim and didn’t feel my constitution up to finding it. It was hard enough reading the comments at GoComics.com.
Comic Strip Master Command had the regular pace of mathematically-themed comic strips the last few days. But it remembered what the 14th would be. You’ll see that when we get there.
Ray Billingsley’s Curtis for the 11th of March is a student-resists-the-word-problem joke. But it’s a more interesting word problem than usual. It’s your classic problem of two trains meeting, but rather than ask when they’ll meet it asks where. It’s just an extra little step once the time of meeting is made, but that’s all right by me. Anything to freshen the scenario up.
Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 12th of March name-drops statisticians. Statisticians are almost expected to produce interesting pictures of their results. It is the field that gave us bar charts, pie charts, scatter plots, and many more. Statistics is, in part, about understanding a complicated set of data with a few numbers. It’s also about turning those numbers into recognizable pictures, all in the hope of finding meaning in a confusing world (ours).
Brian Anderson’s Dog Eat Doug for the 13th of March uses walls full of mathematical scrawl as signifier for “stuff thought deeply about’. I don’t recognize any of the symbols specifically, although some of them look plausibly like calculus. I would not be surprised if Anderson had copied equations from a book on string theory. I’d do it to tell this joke.
And then came the 14th of March. That gave us a bounty of Pi Day comics. Among them:
Edited To Add: And I forgot to mention, after noting to myself that I ought to mention it. The Price Is Right (the United States edition) hopped onto the Pi Day fuss. It used the day as a thematic link for its Showcase prize packages, noting how you could work out π from the circumference of your new bicycles, or how π was a letter from your vacation destination of Greece, and if you think there weren’t brand-new cars in both Showcases you don’t know the game show well. Did anyone learn anything mathematical from this? I am skeptical. Do people come away thinking mathematics is more fun after this? … Conceivably. At least it was a day fairly free of people declaring they Hate Math and Can Never Do It.
I haven’t been skipping the comics, even with the effort of keeping up on the Leap Day 2016 A To Z Glossary. I just try to keep to the pace which Comic Strip Master Command sets.
The kids-information feature Short Cuts, by Jeff Harris, got ahead of “Pi Day” last Sunday. I imagine the feature gets run mid-week in some features, so that it’s better to run a full week before March 14th. But here’s a bundle of trivia, some jokes, some activities, that sort of thing. I am curious about one of Harris’s trivias, that Pi “plays an important role in some of the equations used in Einstein’s famous general theory of relativity”. That’s true, but it’s not as if general relativity is a rare appearance for pi in physics. Maybe Harris chose it on aesthetic grounds. General relativity has a familiar name and exotic concepts. And it allowed him to put in an equation that’s mysterious yet attractive-looking.
Samson’s Dark Side Of The Horse for the 7th of March made me wonder how many sudoku puzzles there are. The answer is — well, you have to start thinking carefully about what you mean by “how many”. For example: start with one puzzle. Swap out every appearance of a 1 with a 2, and a 2 with a 1. Is this new one actually a different puzzle? You can make a case for yes or for no. And that’s before we get into the question of how many clues to give to solve the puzzle. If I’m not misreading Wikipedia’s “Mathematics of Sudoku” page, the number of different nine-by-nine combinations of digits that can be legitimate sudoku puzzle solutions is 6,670,903,752,021,072,936,960. This was worked out in 2005 by Bertram Felgenhauer and Frazer Jarvis. They worked it out partly by logic, partly by brute force. Brute force is trying all the possibilities to see what works. It’s a method that rewards endurance. We like that we can turn it over to computers now. Or cartoon horses, whichever. They’re good at endurance.
Jef Mallett’s Frazz started a sequence about problem-writing on the 7th of March. Caulfield’s setup, complaining about trains and apple bushels, suggests he was annoyed by mathematics problems. I understand. Much of real mathematics starts with curiosity about something (how many sudoku puzzles are there?). Then it’s working out what computation might answer that question. Then it’s doing that calculation. And then it’s verifying that the calculation is right. Mathematics educators have to teach ways to do a calculation, and test that. And to teach how to know what calculation to do, and test that. That’s challenging enough. Add to that working out something to be curious about and you understand the appeal of stock setups. Maybe mathematics should include some courses in creative writing and short-short fiction. (Verification is, in my experience, the part nobody cares about. This is a shame. The hardest part of doing numerical mathematics is making sure your computation makes any sense.)
Richard Thompson’s Richard’s Poor Almanac rerun the 7th of March features the Non-Euclidean Creeper. It’s a plant perhaps related to the Cubist Fir Christmas tree and to the Otterloops’ troublesome non-Euclidean tree. Non-Euclidean geometry will probably always sound more intimidating and exotic. Euclidean geometry describes the way objects on the human scale behave. Shapes that fit on the table, or in your garden, follow Euclidean rules. But non-Euclidean isn’t magic; it’s the way that shapes on the surface of a globe work, for example. And the idea of drawing a thing like a square on the surface of the Earth isn’t so bizarre.
My love and I were talking the other day about Jim Toomey’s Sherman’s Lagoon. It’s a bit odd as comic strips go. It’s been around forever, for one, but nobody talks about it. It’s stayed reliably funny. Comic strips that’ve been around forever tend to … you know … not be. The strip’s done as a work-and-home strip except the cast is all sea life. And the thing is, Toomey keeps paying attention to new discoveries in sea life, and other animal research. And this is a fantastic era for discoveries in sea life, aside from how humans have now eaten all of it and we don’t have any left. I am not joking when I say the comic strip is an effortless way to keep up with new discoveries about the oceans.
I missed it when in December the discovery was announced to the world. But the setup, about the common name being given by a group of kids, is apparently quite correct. So we should expect from Toomey. (The scientific name is Etmopterus benchleyi. The last name refers to Peter Benchley, repentant Jaws novelist.) LiveScience.com’s article says lead author Dr Vicky Vásquez had to “scale them back” from their starting point, the “super ninja”. This differs from Hawthorne’s claim that the kids started from the “math stinks” shark, but it’s still a delight anyway.
Is there a unifying theme between many of the syndicated comic strips with mathematical themes the last few days? Of course there is. It’s students giving snarky answers to their teachers’ questions. That’s the theme every week. But other stuff comes up.
Joe Martin’s Boffo for the 12th of depicts “the early days before all the bugs were worked out” of mathematics. And the early figure got a whole string of operations which don’t actually respect the equals sign, before getting finally to the end. Were I to do this, I would use an arrow, =>, and I suspect many mathematicians would too. It’s a way of indicating the flow of one’s thoughts without trying to assert that 2+2 is actually the same number as 1 + 1 + 1 + 1 + 6.
And this comic is funny, in part, because it’s true. New mathematical discoveries tend to be somewhat complicated, sloppy messes to start. Over time, if the thing is of any use, the mathematical construct gets better. By better I mean the logic behind it gets better explained. You’d expect that, of course, just because time to reflect gives time to improve exposition. But the logic also tends to get better. We tend to find arguments that are, if not shorter, then better-constructed. We get to see how something gets used, and how to relate it to other things we’d like to do, and how to generalize the pieces of argument that go into it. If we think of a mathematical argument as a narrative, then, we learn how to write the better narrative.
Then, too, we get better at notation, at isolating what concepts we want to describe and how to describe them. For example, to write the fourth power of a number such as ‘x’, mathematicians used to write ‘xxxx’ — fair enough, but cumbersome. Or then xqq — the ‘q’ standing for quadratic, that is, square, of the thing before. That’s better. At least it’s less stuff to write. How about “xiiii” (as in the Roman numeral IV)? Getting to “x4” took time, and thought, and practice with what we wanted to raise numbers to powers to do. In short, we had to get the bugs worked out.
Randy Glasbergen’s Glasbergen Cartoons for the 12th (a rerun; Galsbergen died last year) is a similar student-resisting-problems joke. Arithmetic gets an appearance no doubt because it’s the easiest kind of problem to put on the board and not distract from the actual joke.
Mark Pett’s Lucky Cow for the 14th (a rerun from the early 2000s) mentions the chaos butterfly. I am considering retiring chaos butterfly mentions from these roundups because I seem to say the same thing each time. But I haven’t yet, so I’ll say it. Part of what makes a system chaotic is that it’s deterministic and unpredictable. Most different outcomes result from starting points so similar they can’t be told apart. There’s no guessing whether any action makes things better or worse, and whether that’s in the short or the long term.
Historically, being the sole party that understands the financial calculations has not brought money lenders appreciation.
Tony Cochran’s Agnes for the 17th also can’t be a response to that Pearls Before Swine. The lead times just don’t work that way. But it gives another great reason to learn mathematics. I encourage anyone who wants to be Lord and Queen of Mathdom; it’s worth a try.
Tom Thaves’s Frank and Ernest for the 17th tells one of the obvious jokes about infinite sets. Fortunately mathematicians aren’t expected to list everything that goes into an infinitely large set. It would put a terrible strain on our wrists. Usually it’s enough to describe the things that go in it. Some descriptions are easy, especially if there’s a way to match the set with something already familiar, like counting numbers or real numbers. And sometimes a description has to be complicated.
There are urban legends among grad students. Many of them are thesis nightmares. One is about such sets. The story goes of the student who had worked for years on a set whose elements all had some interesting collection of properties. At the defense her advisor — the person who’s supposed to have guided her through finding and addressing an interesting problem — actually looks at the student’s work for the first time in ages, or ever. And starts drawing conclusions from it. And proves that the only set whose elements all have these properties is the null set, which hasn’t got anything in it. The whole thesis is a bust. Thaves probably didn’t have that legend in mind. But you could read the comic that way.
Percy Crosby’s Skippy for the 17th gives a hint how long kids in comic strips have been giving smart answers to teachers. This installment’s from 1928 sometime. Skippy’s pretty confident in himself, it must be said.
I’d hoped that running a slightly-too-soon edition of Reading the Comics would let me have a better-sized edition for later in this week. Then everybody did comics for the 11th of December. I can have a series of awkward-sized essays or just run what I have. I wonder which I’ll do.
Aaron McGruder’s The Boondocks for the 6th of December is a student-resisting-the-problem joke. It ran originally the 24th of September, 2000, if the copyright information is right. The original problem — “what is 24 divided by 4 minus 2” — is a reasonable one for at least some level of elementary school. (I’m vague on just what grade Caesar is supposed to be in. It’s a problem for any strip with wise-beyond-their-years children. Peanuts plays with this by having the kids give book reports on Peter Rabbit and Tess of the d’Urbervilles.) What makes it a challenge is that you know to know the order of operations. Should you divide 24 by 4 first, and subtract 2 from that, or should you take 4 minus 2 and then divide 24 by whatever that number is?
Absent any confounding information, you should always do multiplication and division before you do addition and subtraction. So this suggests 24 divided by 4, giving us 6, and then subtract 2, giving us 4. The only relevant confounding information, though, would be the direction to do something else first. That’s indicated by putting something in parentheses. (Or brackets, if you have so many parentheses the symbols are getting confusing.) A thing in parentheses has higher priority and should be calculated first. But there’s no way to tell parentheses in dialogue. The best the teacher could do is say something like “24 divided by the quantity four minus two”, or even, “24 divided by parenthesis four minus two close parenthesis”. That’s awkward but it is what we resort to even in the mathematics department.
Eric the Circle for the 6th of December, this one by “Scooterpiggy”, is the anthropomorphic-numerals joke this essay. You might fuss that there’s a difference between a circle and zero. The earliest examples of zero seem to have been simple dots. But the circle, or at least elliptical, shape of zero grew pretty fast. Maybe in a couple of centuries. Maybe there’s something in the empty loop that suggests what it stands for.
Tom Thaves’s Frank and Ernest for the 6th of December tosses in a statistics pun for the final panel. The statistics use of “median” is the number that half the data is less than and half the data is greater than. It’s one of several quantities that get called an “average”. In this case it’s average because if you picked a data point at random you’d be as likely to be above as below the median. In data sets that aren’t too weird, that will usually be pretty close to the arithmetic mean. The arithmetic mean is the thing normal people mean by “average”. It’ll also typically be near the most common value. That most common value mathematicians and statisticians call the “mode”.
I don’t know if the use of “median” for the middle strip of a divided road shares an etymology with the statistics use of the word. It might be one use might have inspired the other, perhaps as metaphor. But the similarity between “being in the middle of the data” and “being in the middle of the street” is straightforward for English.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th of December pinpoints a common failure mode of experts. (The strip almost surely ran before, sometime. The only method I have to find out when, though, is to post an incorrect date and make someone correct me. So let me say it originally ran on Singapore National Day, 2009.) Mathematics is especially prone to it. It’s so seductive to teach something the way an expert sees it. This is usually in a rigorously thought-out, open-ended, flexible method. After all, why would you ever teach something that wasn’t exactly right, with “right” being “the ways experts see things”? A teacher knows the answer: the expert understanding of a thing is hard to get to. That’s why having it takes expertise. The comic strip’s explanation of fractions is correct and reasonable. But it brings up why Bertrand Russell and Alfred North Whitehead needed over four hundred pages to establish 1 + 1 equals 2. That’s a lot of intellectual scaffolding for the quality of paint job required. Sometimes it’s easier to start with a quick and dirty explanation, and then go back later and rebuild the understanding if a student needs it.
Rick Stromoski’s Soup to Nutz for the 7th of December puts forth a kind of Zeno’s paradox problem in the guise of compound interest. If doing something increases life expectancy by a certain percentage, then, how much of the extra time one gets do you need to be immortal? I’m amused by this although I can’t imagine modest alcohol consumption increasing lifespan by 20 percent. (I assume 20 percent of the average expected lifespan.) If the effect were anything near that big the actuaries would have noticed and ordered people to drink long ago.
On looking at all this, I think I’ll save the December 11th strips for later. This is enough text for this early in the morning.
I confess I’m dissatisfied with this batch of Reading the Comics posts. I like having something like six to eight comics for one of these roundups. But there was this small flood of mathematically-themed comics on the 6th of December. I could either make do with a slightly short edition, or have an overstuffed edition. I suppose it’s possible to split one day’s comics across two Reading the Comics posts, but that’s crazy talk. So, a short edition today.
Jef Mallett’s Frazz for the 4th of December was part of a series in which Caulfield resists learning about reciprocals. The 4th offers a fair example of the story. At heart the joke is just the student-resisting-class, or student-resisting-story-problems. It certainly reflects a lack of motivation to learn what they are.
We use reciprocals most often to write division problems as multiplication. “a ÷ b” is the same as “a times the reciprocal of b”. But where do we get the reciprocal of b from? … Well, we can say it’s the multiplicative inverse of b. That is, it’s whatever number you have to multiply ‘b’ by in order to get ‘1’. But we’re almost surely going to find that taking 1 and dividing it by b. So we’ve swapped out one division problem for a slightly different one. This doesn’t seem to be getting us anywhere.
But we have gotten a new idea. If we can define the multiplication of things, we might be able to get division for almost free. Could we divide one matrix by another? We can certainly multiply a matrix by the inverse of another. (There are complications at work here. We’ll save them for another time.) A lot of sets allow us to define things that make sense as addition and multiplication. And if we can define a complicated operation in terms of addition and multiplication … If we follow this path, we get to do things like define the cosine of a matrix. Then we just have to figure out why we’d want have a cosine of a matrix.
There’s a simpler practical use of reciprocals. This relates to numerical mathematics, computer work. Computer chips do addition (and subtraction) really fast. They do multiplication a little slower. They do division a lot slower. Division is harder than multiplication, as anyone who’s done both knows. However, dividing by (say) 4 is the same thing as multiplying by 0.25. So if you know you need to divide by a number a lot, then it might make for a faster program to change division into multiplication by a reciprocal. You have to work out the reciprocal, but if you only have to do that once instead of many times over, this might make for faster code. Reciprocals are one of the tools we can use to change a mathematical process into something faster.
(In practice, you should never do this. You have a compiler that does this, and you should let it do its work. But it’s enlightening to know these are the sorts of things your compiler is looking for when it turns your code into something the computer does. And looking for ways to do the same work in less time is a noble side of mathematics.)
Charles Schulz’s Peanuts for the 4th of December (originally from 1968, on the same day) sees Peppermint Patty’s education crash against a word problem. It’s another problem in motivating a student to do a word problem. I admit when I was a kid I’d have been enchanted by this puzzle. But I was a weird one.
Dave Coverly’s Speed Bump for the 4th of December is a mathematics-symbols joke as applied to toast. I think you could probably actually sell those. At least the greater-than and the less-than signs. The approximately-equal-to signs would be hard to use. And people would think they were for bacon anyway.
Ruben Bolling’s Super-Fun-Pak Comix for the 4th of December showcases Young Albert Einstein. That counts as mathematical content, doesn’t it? The strip does make me wonder if they’re still selling music CDs and other stuff for infant or even prenatal development. I’m skeptical that they ever did any good, but it isn’t a field I’ve studied.
Bill Whitehead’s Free Range for the 5th of December uses a blackboard full of mathematical and semi-mathematical symbols to denote “stuff too complicated to understand”. The symbols don’t parse as anything. It is authentic to mathematical work to sometimes skip writing all the details of a thing and write in instead a few words describing it. Or to put in an abbreviation for the thing. That often gets circled or boxed or in some way marked off. That keeps us from later on mistaking, say, “MUB” as the product of M and U and B, whatever that would mean. Then we just have to remember we meant “minimum upper bound” by that.
Bill Amend’s FoxTrot Classics for the 28th of November (originally run in 2004) depicts a “Christmas Card For Smart People”. It uses the familiar motif of “ability to do arithmetic” as denoting smartness. The key to the first word is remembering that mathematicians use the symbol ‘e’ to represent a number that’s just a little over 2.71828. We call the number ‘e’, or something ‘the base of the natural logarithm’. It turns up all over the place. If you have almost any quantity that grows or that shrinks at a speed proportional to how much there is, and describe how much of stuff there is over time, you’ll find an ‘e’. Leonhard Euler, who’s renowned for major advances in every field of mathematics, is also renowned for major advances in notation in physics, and he gave us ‘e’ for that number.
The key to the second word there is remembering from physics that force equals mass times acceleration. Therefore the force divided by the acceleration is …
And so that inspires this essay’s edition title. There are several comics in this selection that are about the symbols or the representations of mathematics, and that touch on the subject as a visual art.
Matt Janz’s Out of the Gene Pool for the 28th of November first ran the 26th of October, 2002. It would make for a good word problem, too, with a couple of levels: given the constraints of (a slightly looser) budget, how do they get the greatest number of cookies? Or if some cookies are better than others, how do they get the most enjoyment from their cookie purchase? Working out the greatest amount of enjoyment within a given cookie budget, with different qualities of cookies, can be a good introduction to optimization problems and how subtle they can be.
Bill Holbrook’s On The Fastrack for the 29th of November speaks in support of accounting. It’s a worthwhile message. It doesn’t get much respect, not from the general public, and not from typical mathematics department. The general public maybe thinks of accounting as not much more than a way companies nickel-and-dime them. If the mathematics departments I’ve associated with are fair representatives, accounting isn’t even thought of except by the assistant professor doing a seminar on financial mathematics. (And I’m not sure accounting gets mentioned there, since there’s exciting stuff about the Black-Scholes Equation and options markets to think about instead.) This despite that accounting is probably, by volume, the most used part of mathematics. Anyway, Holbrook’s strip probably won’t get the field a better reputation. But it has got some great illustrations of doing things with numbers. The folks in mathematics departments certainly have had days feeling like they’ve done each of these things.
Dave Coverly’s Speed Bump for the 30th of November is a compound interest joke. I admit I’ve told this sort of joke myself, proposing that the hour cut out of the day in spring when Daylight Saving Time starts comes back as a healthy hour and three minutes in autumn when it’s taken out of saving. If I can get the delivery right I might have someone going for that three minutes.
Mikael Wulff and Anders Morgenthaler’s Truth Facts for the 30th of November is a Venn diagram joke for breakfast. I would bet they’re kicking themselves for not making the intersection be the holes in the center.
Mark Anderson’s Andertoons for this week interests me. It uses a figure to try explaining how to relate gallon and quart an pint and other units relate to each other. I like it, but I’m embarrassed to say how long it took in my life to work out the relations between pints, quarts, gallons, and particularly whether the quart or the pint was the larger unit. I blame part of that on my never really having to mix a pint of something with a quart of something else, which ought to have sorted that out. Anyway, let’s always cherish good representations of information. Good representations organize information and relationships in ways that are easy to remember, or easy to reconstruct or extend.
John Graziano’s Ripley’s Believe It or Not for the 2nd of December tries to visualize how many ways there are to arrange a Rubik’s Cube. Counting off permutations of things by how many seconds it’d take to get through them all is a common game. The key to producing a staggering length of time is that it one billion seconds are nearly 32 years, and the number of combinations of things adds up really really fast. There’s over eight billion ways to draw seven letters in a row, after all, if every letter is equally likely and if you don’t limit yourself to real or even imaginable words. Rubik’s Cubes have a lot of potential arrangements. Graziano misspells Rubik, but I have to double-check and make sure I’ve got it right every time myself. I didn’t know that about the pigeons.
Charles Schulz’s Peanuts for the 2nd of December (originally run in 1968) has Peppermint Patty reflecting on the beauty of numbers. I don’t think it’s unusual to find some numbers particularly pleasant and others not. Some numbers are easy to work with; if I’m trying to add up a set of numbers and I have a 3, I look instinctively for a 7 because of how nice 10 is. If I’m trying to multiply numbers, I’d so like to multiply by a 5 or a 25 than by a 7 or an 18. Typically, people find they do better on addition and multiplication with lower numbers like two and three, and get shaky with sevens and eights and such. It may be quirky. My love is a wizard with 7’s, but can’t do a thing with 8. But it’s no more irrational than the way a person might a pyramid attractive but a sphere boring and a stellated icosahedron ugly.
I’ve seen some comments suggesting that Peppermint Patty is talking about numerals, that is, the way we represent numbers. That she might find the shape of the 2 gentle, while 5 looks hostile. (I can imagine turning a 5 into a drawing of a shouting person with a few pencil strokes.) But she doesn’t seem to say one way or another. She might see a page of numbers as visual art; she might see them as wonderful things with which to play.
Yes, I feel a bit bad that I didn’t have anything posted yesterday. I’d had a nice every-other-day streak going for a couple weeks there. But I had honestly expected more mathematically themed comic strips, and there just weren’t enough in my box by the end of the 17th. So I didn’t have anything to schedule for a post the 18th. The 18th came through, though, and now I’ve got enough to talk about. And that before I get to reading today’s comics. So, please, enjoy.
Scott Adams’s Dilbert Classics for the 16th of November (originally published the 21st of September, 1992) features Dilbert discovering Bell’s Theorem. Bell’s Theorem is an important piece of our understanding of quantum mechanics. It’s a theorem that excites people who first hear about it. It implies quantum mechanics can’t explain reality unless it can allow information to be transmitted between interacting particles faster than light. And quantum mechanics does explain reality. The thing is, and the thing that casual readers don’t understand, is that there’s no way to use this to send a signal. Imagine that I took two cards, one an ace and one an eight, seal them in envelopes, and gave them to astronauts. The astronauts each travel to ten light-years away from me in opposite directions. (They took extreme offense at something I said and didn’t like one another anyway.) Then one of them opens her envelope, finding that she’s got the eight. Then instantly, even though they’re twenty light-years apart, she knows the other astronaut has an ace in her envelope. But there is no way the astronauts can use this to send information to one another, which is what people want Bell’s Theorem to tell us. (My example is not legitimate quantum mechanics and do not try to use it to pass your thesis defense. It just shows why Bell’s Theorem does not give us a way to send information we care about faster than light.) The next day Dilbert’s Garbageman, the Smartest Man in the World, mentions Dilbert’s added something to Bell’s Theorem. It’s the same thing everybody figuring they can use quantum entanglement to communicate adds to the idea.
Tom Thaves’ Frank and Ernest for the 16th of November riffs on the idea of a lottery as a “tax on people who are bad at math”. Longtime readers here know that I have mixed feelings about that, and not just because I’m wary of cliché. If the jackpot is high enough, you can reach the point where the expectation value of the prize is positive. That is, you would expect to make money if you played the game under the same conditions often enough. But that chance is still vanishingly small. Even playing a million times would not make it likely you would more earn money than you spent. I’m not dogmatic enough to say what your decision should be, at least if the prize is big enough. (And that’s not considering the value placed on the fun of playing. One may complain that it shouldn’t be any fun to buy a soon-to-be-worthless ticket. But many people do enjoy it and I can’t bring myself to say they’re all wrong about feeling enjoyment.)
And it happens that on the 18th Brant Parker and Johnny Hart’s Wizard of Id Classics (originally run the 20th of November, 1965) did a lottery joke. That one is about a lottery one shouldn’t play, except that the King keeps track of who refuses to buy a ticket. I know when we’re in a genre.
Peter Mann’s The Quixote Syndrome for the 16th of November explores something I had never known but that at least the web seems to think is true. Apparently in 1958 Samuel Beckett knew the 12-year-old André Roussimoff. People of my age cohort have any idea who that is when they hear Roussimoff became pro wrestling star André the Giant. And Beckett drove the kid to school. Mann — taking, I think, a break from his usual adaptations of classic literature — speculates on what they might have talked about. His guess: Beckett attempting to ease one of his fears through careful study and mathematical treatment. The problem is goofily funny. But the treatment is the sort of mathematics everyone understands needing and understands using.
Tony Cochrane’s Agnes for the 18th of November tells a resisting-the-word-problem joke. I admit expecting better from Cochrane. But casting arithmetic problems into word problems is fraught with peril. It isn’t enough to avoid obsolete references. (If we accept trains as obsolete. I’m from the United States Northeast, where subways and even commuter trains are viable things.) The problem also has to ask something the problem-solver can imagine wanting to know. It may not matter whether the question asks how far apart two trains, two cars, or two airplanes are, if the student can’t see their distance as anything but trivia. We may need better practice in writing stories if we’re to write story problems.
The good news is I’ve got a couple of comic strips I feel responsible including the pictures for. (While I’m confident I could include all the comics I talk about as fair use — I make comments which expand on the strips’ content and which don’t make sense without the original — Gocomics.com links seem reasonably stable and likely to be there in the future. Comics Kingdom links generally expire after a month except to subscribers and I don’t know how long Creators.com links last.) And a couple of them talk about rock bands, so, that’s why I picked that titel.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 17th of September is a subverted-fairy-tale-moral strip, naturally enough. It’s also a legitimate point, though. Unlikely events do happen sometimes, and it’s a mistake to draw too-strong conclusions from them. This is why it’s important to reproduce interesting results. It’s also why, generally, we like larger sample sizes. It’s not likely that twenty fair coins flipped will all come up tails at once. But it’s far more likely that will happen than that two hundred fair coins flipped will all come up tails. And that’s far more likely than that two thousand fair coins will. For that matter, it’s more likely that three-quarters of twenty fair coins flipped will come up tails than that three-quarters of two hundred fair coins will. And the chance that three-quarters of two thousand fair coins will come up tails is ignorable. If that happens, then something interesting has been found.
In Juba’s Viivi and Wagner for the 17th of September, Wagner announces his decision to be a wandering mathematician. I applaud his ambition. If I had any idea where to find someone who needed mathematics done I’d be doing that myself. If you hear something give me a call. I’ll be down at the City Market, in front of my love’s guitar case, multiplying things by seven. I may get it wrong, but nobody will know how to correct me.
Daniel Beyer’s Long Story Short for the 18th of September uses a page full of calculations to predict when prog-rock band Tool will release their next album. (Wikipedia indicates they’re hoping for sometime before the end of 2015, but they’ve been working on it since 2008.) Some of the symbols make a bit of sense as resembling those of quantum physics. An expression like (in the lower left of the board) resembles a probability distribution calculation. (There should be a ^ above the H there, but that’s a little beyond what WordPress can render in the simple mathematical LaTeX tools it has available. It’s in the panel, though.) The letter ψ stands for a probability wave, containing somehow all the information about a system. The composition of symbols literally means to calculate how an operator — a function that has a domain of functions and a range of functions — changes that probability distribution. In quantum mechanics every interesting physical property has a matching operator, and calculating this set of symbols tells us the distribution of whatever that property is. H generally suggests the total energy of the system, so the implication is this measures, somehow, what energies are more and are less probable. I’d be interested to know if Beyer took the symbols from a textbook or paper and what the original context was.
Dave Whamond’s Reality Check for the 19th of September brings in another band to this review. It uses a more basic level of mathematics, though.
Percy Crosby’s Skippy from the 19th of September — rerun from sometime in 1928 — is a clever way to get a word problem calculated. It also shows off what’s probably been the most important use of arithmetic, which is keeping track of money. Accountants and shopkeepers get little attention in histories of mathematics, but a lot of what we do has been shaped by their needs for speed, efficiency, and accuracy. And one of Gocomics’s commenters pointed out that the shopkeeper didn’t give the right answer. Possibly the shopkeeper suspected what was up.
Paul Trap’s Thatababy for the 20th of September uses a basic geometry fact as an example of being “very educated”. I don’t think the area of the circle rises to the level of “very” — the word means “truly”, after all — but I would include it as part of the general all-around awareness of the world people should have. Also it fits in the truly confined space available. I like the dad’s eyes in the concluding panel. Also, there’s people who put eggplant on pizza? Really? Also, bacon? Really?
Alex Hallatt’s Arctic Circle for the 21st of September is about making your own luck. I find it interesting in that it rationalizes magic as a thing which manipulates probability. As ways to explain magic for stories go that isn’t a bad one. We can at least imagine the rigging of card decks and weighting of dice. And its plot happens in the real world, too: people faking things — deceptive experimental results, rigged gambling devices, financial fraud — can often be found because the available results are too improbable. For example, a property called Benford’s Law tells us that in many kinds of data the first digit is more likely to be a 1 than a 2, a 2 than a 3, a 3 than a 4, et cetera. This fact serves to uncover fraud surprisingly often: people will try to steal money close to but not at some limit, like the $10,000 (United States) limit before money transactions get reported to the federal government. But that means they work with checks worth nine thousand and something dollars much more often than they do checks worth one thousand and something dollars, which is suspicious. Randomness can be a tool for honesty.
Peter Maresca’s Origins of the Sunday Comics feature for the 21st of September ran a Rube Goldberg comic strip from the 19th of November, 1913. That strip, Mike and Ike, precedes its surprisingly grim storyline with a kids-resisting-the-word-problem joke. The joke interests me because it shows a century-old example of the joke about word problems being strings of non sequiturs stuffed with unpleasant numbers. I enjoyed Mike and Ike’s answer, and the subversion of even that answer.
Mark Anderson’s Andertoons for the 22nd of September tries to optimize its targeting toward me by being an anthropomorphized-mathematical-objects joke and a Venn diagram joke. Also being Mark Anderson’s Andertoons today. If I didn’t identify this as my favorite strip of this set Anderson would just come back with this, but featuring monkeys at typewriters too.
I admit I’ve been a little unnerved lately. Between the A To Z project and the flood of mathematics-themed jokes from Comic Strip Master Command — and miscellaneous follies like my WordPress statistics-reading issues — I’ve had a post a day for several weeks now. The streak has to end sometime, surely, right? So it must, but not today. I admit the bunch of comics mentioning mathematical topics the past couple days was more one of continuing well-explored jokes rather than breaking new territory. But every comic strip is somebody’s first, isn’t it? (That’s an intimidating thought.)
Disney’s Mickey Mouse (June 6, rerun from who knows when) is another example of the word problem that even adults can’t do. I think it’s an interesting one for being also a tongue-twister. I tend to think of this sort of problem as a calculus question, but that’s surely just that I spend more time with calculus than with algebra or simpler arithmetic.
Justin Thompson’s Mythtickle (June 6, again a rerun) is about the curious way that objects are mostly empty space. The first panel shows on the alien’s chalkboard legitimate equations from quantum mechanics. The first line describes (in part) a function called psi that describes where a particle is likely to be found over time. The second and third lines describe how the probability distribution — where a particle is likely to be found — changes over time.
Doug Bratton’s Pop Culture Shock Therapy (July 7) just name-drops mathematics as something a kid will do badly in. In this case the kid is Calvin, from Calvin and Hobbes. While it’s true he did badly in mathematics I suspect that’s because it’s so easy to fit an elementary-school arithmetic question and a wrong answer in a single panel.
Comic Strip Master Command was pretty kind to me this week, and didn’t overload me with too many comics when my computer problems were the most time-demanding. You’ve seen how bad that is by how long it’s taken me to get to answering people’s comments. But they have kept publishing mathematical comic strips, and so I’m ready for another review. This time around a couple of the strips talk about the symbols of mathematics, so that’s enough of a hook for my titling needs.
Henry Scarpelli and Craig Boldman’s Archie (June 30, rerun) is about living with long odds. People react to very improbable events in strange ways. Moose is being maybe more consistent than normal for folks in figuring that if he’s going to be lucky enough to win a contest then he’s just lucky enough to be hit by a meteor too. (It feels like a lottery to me, although I guess Moose has to be too young to enter a lottery.) And I’m amused by the logic of someone’s behavior becoming funny because it is logically consistent.
The past several days produced a good number of comic strips mentioning mathematical topics. Strangely, they seem to be carefully targeted to appeal to me. Here’s how.
Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (June 12) is your classic resisting-the-world-problems joke. I admit I haven’t done anything at this level of mathematics in a long while. I’m curious if actual teachers, or students, could say whether problems with ridiculous numbers of fruits actually appear in word problems, or if this is one of those motifs that’s popular despite a nearly imaginary base in the real world.
Dan Thompson’s Brevity (June 13) is aimed very precisely at the professional knot theorist. Also, mathematics includes a thing called knot theory which is almost exactly what you imagine. For a while it looked like I might get into knot theory, although ultimately I wasn’t able to find a problem interesting enough to work on that I was able to prove anything interesting about. I’m delighted a field that so many people wouldn’t imagine existed got a comic strip in this manner; I wonder if this is what dinosaur researchers felt when The Far Side was still in production.
Steve Sicula’s Home and Away (June 14) name-drops the New Math, though the term’s taken literally. The joke feels anachronistic to me. Would a kid that age have even heard of a previous generation’s effort to make mathematics about understanding what you’re doing and why? New Math (admittedly, on the way out) was my elementary school thing.
Mark Litzler’s Joe Vanilla (June 15) tickles me with the caption, “the clarity of the equation befuddles”. It’s a funny idea. Ideally, the point of an equation is to provide clarity and insight, maybe by solving it, maybe by forming it. A befuddling equation is usually a signal the problem needs to be thought out some more.
Lincoln Pierce’s Big Nate: First Class (June 16, originally run June 11, 1991) is aimed at the Mathletes out there. It throws in a slide rule mention for good measure. Given Nate’s Dad’s age in the 1991 setting it’s plausible he’d have had a slide rule. (He’s still the same age in the comic strip being produced today, so he wouldn’t have had one if the strip were redrawn.) I don’t remember being on a competitive mathematics team in high school, although I did participate in some physics contests. My recollection is that I was an inconsistent performer, though. I don’t think I had the slightly obsessive competitive urge needed to really excel in high school academic competition.
We’re drawing upon the end of the school year, on the United States calendar. Comic Strip Master Command may have ordered fewer mathematically-themed comic strips to be run. That’s all right. I have plans. I also may need to stop paying attention to the Disney comic strips, reruns of Mickey Mouse and Donald Duck. I explain why within.
Niklas Eriksson’s Carpe Diem made its first appearance in my pages here on the 16th of May. (It’s been newly introduced to United States comics. I’m sorry, I just can’t read all the syndicated newspaper comic strips in the world for mathematical content. If someone wants to franchise the Reading The Comics idea for a country they like, let’s talk. We can negotiate reasonable terms.) Anyway, it uses the usual string of mathematical symbols to express the idea of a lot of hard mathematical work. The big down-arrow just before superstar equation E = mc2 is authentic enough. Trying to show the chain of thought, or to point out the conclusions one hopes follow from the work done, is a common part of discovering or inventing mathematics.
Mike Peters’s Mother Goose and Grimm (May 17) riffs on that ancient and transparently stupid bit of folklore about the chance of an older woman having a better chance of dying in some improbable manner than of marrying successfully. It’s always been obvious nonsense and people passing along the claim uncritically should be ashamed of themselves. I’ll give Peters a pass since the point is to set up a joke, and joke-setup can get away with a lot. Still.
And now I get caught up again, if briefly, to the mathematically-themed comic strips I can find. I’ve dubbed this one the trapezoid edition because one happens to mention the post that will outlive me.
Todd Clark’s Lola (May 4) is a straightforward joke. Monty’s given his chance of passing mathematics and doesn’t understand the prospect is grim.
Joe Martin’s Willy and Ethel (May 4) shows an astounding feat of mind-reading, or of luck. How amazing it is to draw a number at random from a range depends on many things. It’s less impressive to pick the right number if there are only three possible answers than it is to pick the right number out of ten million possibilities. When we ask someone to pick a number we usually mean a range of the counting numbers. My experience suggests it’s “one to ten” unless some other range is specified. But the other thing affecting how amazing it is is the distribution. There might be ten million possible responses, but if only a few of them are likely then the feat is much less impressive.
The distribution of a random number is the interesting thing about it. The number has some value, yes, and we may not know what it is, but we know how likely it is to be any of the possible values. And good mathematics can be done knowing the distribution of a value of something. The whole field of statistical mechanics is an example of that. James Clerk Maxwell, famous for the equations which describe electromagnetism, used such random variables to explain how the rings of Saturn could exist. It isn’t easy to start solving problems with distributions instead of particular values — I’m not sure I’ve seen a good introduction, and I’d be glad to pass one on if someone can suggest it — but the power it offers is amazing.
I have to make two confessions for this round of mathematics comic strips. First is that I was busy for like two days and missed about a jillion comic strips. So this is the first part of some catching-up to do. The second is that I don’t have a favorite of this bunch. The most interesting, I suppose, is the Mr Boffo, because it lets me get into a little trivia about Albert Einstein. But there’s not any in this bunch that made me smile much or that gave me a juicy topic to discuss. Maybe tomorrow.
Steve Breen and Mike Thompson’s Grand Avenue ran a week of snarky-answers-to-word-problems strips. April 28th, April 30th, and May 2nd featured mathematics questions. This must reflect how easy it is to undermine the logic of a mathematics question. The April 27th strip is about using Roman numerals, which I suppose is arithmetic. I’m not sure there’s much point to learning Roman numerals. We don’t do any calculations using the Roman numeral scheme except to show why Arabic numerals are better. All you get from Roman numerals is an ability to read building cornerstones and movie copyright dates. At least learning cursive handwriting provides the learner with a way to make illegible notes.
I try to avoid doing Reading The Comics entries back-to-back since I know they can get a bit repetitive. How many ways can I say something is a student-resisting-the-word-problem joke? But if Comic Strip Master Command is going to send a half-dozen strips at least mentioning mathematical topics in a single day, how can I resist the challenge? Worse, what might they have waiting for me tomorrow? So here’s a bunch of comic strips from the 21st of April, 2015:
Mark Anderson’s Andertoons plays on the idea of a number being used up. I’m most tickled by this one. I have heard that the New York Yankees may be running short on uniform numbers after having so many retired. It appears they’ve only retired 17 numbers, but they do need numbers for a 40-player roster as well as managers and coaches and other participants. Also, and this delights me, two numbers are retired for two people each. (Number 8, for Yogi Berra and Bill Dickey, and Number 42, for Jackie Robinson and Mariano Rivera.)
This is a bit of a broad claim, but it seems Comic Strip Master Command was thinking of all mathematics one lead-time ago. There’s a comic about the original invention of mathematics, and another showing off 20th century physics equations. This seems as much of the history of mathematics as one could reasonably expect from the comics page.
Mark Anderson’s Andertoons gets its traditional appearance around here with the April 17th strip. It features a bit of arithmetic that is indeed lovely but wrong.
I know it’s been like forever, or four days, since the last time I had a half-dozen or so mathematically themed comic strips to write about, but if Comic Strip Master Command is going to order cartoonists to give me stuff to write about I’m not going to turn them away. Several seemed to me about the struggle to get someone to buy into a story — the thing being asked after in a word problem, perhaps, or about the ways mathematics is worth knowing, or just how the mathematics in a joke’s setup are presented — and how skepticism about these things can turn up. So I’ll declare that the theme of this collection.
Steve Sicula’s Home And Away started a sequence on April 7th about “is math really important?”, with the father trying to argue that it’s so very useful. I’m not sure anyone’s ever really been convinced by the argument that “this is useful, therefore it’s important, therefore it’s interesting”. Lots of things are useful or important while staying fantastically dull to all but a select few souls. I would like to think a better argument for learning mathematics is that it’s beautiful, and astounding, and it allows you to discover new ways of studying the world; it can offer all the joy of any art, even as it has a practical side. Anyway, the sequence goes on for several days, and while I can’t say the arguments get very convincing on any side, they do allow for a little play with the fourth wall that I usually find amusing in comics which don’t do that much.
It’s been another week of Comic Strip Master Command supporting my most popular regular feature around here. As sometimes happens there were so many comics in a row that I can’t catch them all up in a single post. Actually, there were enough just on the 29th of March to justify another Reading The Comics post, but I didn’t want to overload what was already a pretty busy month with more postings. This is a Gocomics.com-heavy entry, so I’m afraid folks have to click the links to see images. I hope you’ll be all right.
Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (March 29) is a bit of a geography joke built around the idea that a circle hasn’t got a side. Whether it does or not — besides “inside” and “outside”, source for another joke — requires thinking carefully what you mean by a shape’s side: does it have to be straight? If it can be curved, can it curve so sharply that it looks like it’s a corner? For that matter, can you tell a circle apart from, for example, the chiliagon, a regular polygon with a thousand equal sides? (If you can, then, how about a regular polygon with a million, or a billion, or more equal sides, to the point that you can’t tell the difference?) If you can’t, then how do you know a circle was in the story at all?
I’m sorry to have fallen silent the last few days; it’s been a bit busy and I’ve been working on follow-ups to a couple of threads. Fortunately Comic Strip Master Command is still around and working to make sure I don’t disappear altogether, and I have a selection of comic strips which at least include a Jumble world puzzle, which should be a fun little diversion.
Tony Rubino and Gary Markstein’s Daddy’s Home (March 23) asks what seems like a confused question to me, “if you believe in infinity, does that mean anything is possible?” As I say, I’m not sure I understand how belief in infinity comes into play, but that might just reflect my background: I’ve been thoroughly convinced that one can describe collections of things that have infinitely many elements — the counting numbers, rectangles, continuous functions — as well as that one can subdivide things — like segments of a number line — infinitely many times — as well as of quantities that are larger than any finite number and so must be infinitely large; so, what’s to not believe in? (I’m aware that there are philosophical and theological questions that get into things termed “potential” and “actual” infinities, but I don’t understand the questions those terms are meant to address.) The phrasing of “anything is possible” seems obviously flawed to me. But if we take it to mean instead “anything not logically inconsistent or physically prohibited is possible” then we seem to have a reasonable question, if that hasn’t just reduced to “anything not impossible is possible”. I guess ultimately I just wonder if the kid is actually trying to understand anything or if he’s just procrastinating.
After the flurry of comic strips that did Pi Day jokes last time around, and that one had worked in a March Madness joke, I’d expected there to be at least a couple of mathematically-mind college basketball tournament strips coming up this week. If they did, they didn’t appear on the comics sites I normally read, though. This time around turned out to be much more about word problems and the problem-answerer resisting the actual answering of the word problems. It’s possible that Comic Strip Master Command didn’t notice that this would be the weekend that United States readers would spend the most of their time complaining about how their bracket picks weren’t working right.
Phil Frank and Joe Troise’s The Elderberries (March 17, rerun) mentions sudoku, and how to play it, and also shows off how explaining things really is a pleasure, at least as long as you have someone who wants to know listening to the explanation. The strip’s also made me realize I don’t remember what the Professor’s background was. Certainly anyone of any background might enjoy sudoku puzzles, or at least know them well enough to explain how to do them, though I wonder if there’s not a use of the motif here that “professors are smart people, mathematics-or-logic puzzles require smartness, so professors are skilled at mathematics-or-logic puzzles”. (For what it’s worth, I’m not much on this sort of puzzle, though I believe that just reflects that I don’t care to do them very much, so I don’t have the experience needed to do them impressively well.)
Dan Thompson’s Rip Haywire (March 17) features a word problem as part of an aptitude test. Interesting to me is that the test is a multiple-choice, which means one should be able to pick the right answer without doing the whole multiplication of “3.29 times 6.5”: 3.29 is pretty near 3.30, so the answer will be about 3 times 6.5 plus a tenth of 3 times 6.5. And 3 times 6.5 is going to be 3 times 6 plus 3 times a half, or 18 plus 1.5. So, look for the answer that’s about 19.5 plus 1.95, which will be around 21.45. In particular, look for an answer a little bit less than that (to be exact, 0.01 times 6.5 less than that.) Of course, if the exam-writer was clever, 21.45 was included as a plausible yet incorrect answer, but at least the problem can be worked out in one’s head.
I haven’t had the chance to read today’s comics, what with it having snowed just enough last night that we have to deal with it instead of waiting for the sun to melt it, so, let me go with what I have. There’s a sad lack of strips I feel justified including the images of, since they’re all Gocomics.com representatives and I’m used to those being reasonably stable links. Too bad.
Eric the Circle has a pair of strips by Griffinetsabine, the first on the 7th of February, and the next on February 13, both returning to “the Shape Single’s Bar” and both working on “complementary angles” for a pun. That all may help folks remember the difference between complementary angles — those add up to a right angle — and supplementary angles — those add up to two right angles, a straight line — although what it makes me wonder is the organization behind the Eric the Circle art collective. It hasn’t got any nominal author, after all, and there’s what appear to be different people writing and often drawing it, so, who does the scheduling so that the same joke doesn’t get repeated too frequently? I suppose there’s some way of finding that out for myself, but this is the Internet, so it’s easier to admit my ignorance and let the answer come up to me.
Mark Anderson’s Andertoons (February 10) surprised me with a joke about the Dewey decimal system that I hadn’t encountered before. I don’t know how that happened; it just did. This is, obviously, a use of decimal that’s distinct from the number system, but it’s so relatively rare to think of decimals as apart from representations of numbers that pointing it out has the power to surprise me at least.
I do occasionally worry that my little blog is going to become nothing but a review of mathematics-themed comic strips, especially when Comic Strip Master Command sends out abundant crops like it has the past few weeks. This week’s offerings bring out the return of a lot of familiar motifs, like fighting with word problems and anthropomorphized numbers; and there’s one strip that suggests a pair of articles I wrote a while back might be useful yet.
I have no idea why Comic Strip Master Command decided this week should see everybody do some mathematics-themed comic strips, but, so they did, and here’s my collection of the, I estimate, six hundred comic strips that touched on something recently. Good luck reading it all.
Scott Adams’s Dilbert Classics (November 10) started a sequence in which Dilbert gets told the big boss was a geometry major, so, what can he say about rectangles? Further rumors indicate he’s more a geography fan, shifting Dilbert’s topic to the “many” rectangular states of the United States. Of course, there’s only two literally rectangular states, but — and Mark Stein’s How The States Got Their Shapes contains a lot of good explanations of this — many of the states are approximately rectangular. After all, when many of the state boundaries were laid out, the federal government had only vague if any idea what the landscapes looked like in detail, and there weren’t many existing indigenous boundaries the white governments cared about. So setting a proposed territory’s bounds to be within particular lines of latitude and longitude, with some modification for rivers or shorelines or mountain ranges known to exist, is easy, and can be done with rather little of the ambiguity or contradictory nonsense that plagued the eastern states (where, say, a colony’s boundary might be defined as where a river intersects a line of latitude that in fact it never touches). And while perfect rectangularity may be achieved only by Colorado and Wyoming, quite a few states — the Dakotas, Washington, Oregon, Missisippi, Alabama, Iowa — are rectangular enough.
Darrin Bell’s Candorville (November 12) starts talking about Zeno’s paradox — not the first time this month that a comic strip’s gotten to the apparent problem of covering any distance when distance is infinitely divisible. On November 13th it’s extended to covering stretches of time, which has exactly the same problem. Now it’s worth reminding people, because a stunning number of them don’t seem to understand this, that Zeno was not suggesting that there’s no such thing as motion (or that he couldn’t imagine an infinite convergent sequence; it’s easy to think of a geometric construction that would satisfy any ancient geometer); he was pointing out that there’s things that don’t make perfect sense about it. Either distance (and time) are infinitely divisible into indistinguishable units, or they are not; and either way has implications that seem contrary to the way motion works. Perhaps they can be rationalized; perhaps they can’t; but when you can find a question that’s easy to pose and hard to answer, you’re probably looking at something really worth thinking hard about.
Bill Amend’s FoxTrot Classics (November 12, a rerun) puns on the various meanings of “irrational”. A fun little fact you might want to try proving sometime, though I wouldn’t fault you if you only tried it out for a couple specific numbers and decided the general case too much to do: any whole number — like 2, 3, 4, or so on — has a square root that’s either another whole number, or else has a square root that’s irrational. There’s not a case where, say, the square root is exactly 45.144 or something like that, though it might be close.
Sandra Bell-Lundy’sBetween Friends (November 13) shows one of those cases where mental arithmetic really is useful, as Susan tries to work out — actually, staring at it, I’m not precisely sure what she is trying to work out. Her and her coffee partner’s ages in Grade Ten, probably, or else just when Grade Ten was. That’s most likely her real problem: if you don’t know what you’re looking for it’s very difficult to find it. Don’t start calculating before you know what you’re trying to work out.
If I wanted to work out what year was 35 years ago I’d probably just use a hack: 35 years before 2014 is one year before “35 years before 2015”, which is a much easier problem to do. 35 years before 2015 is also 20 years before 2000, which is 1980, so subtract one and you get 1979. (Alternatively, I might remember it was 35 years ago that the Buggles’ “Video Killed The Radio Star” first appeared, which I admit is not a method that would work for everyone, or for all years.) If I wanted to work out my (and my partner’s) age in Grade Ten … well, I’d use a slightly different hack: I remember very well that I was ten years old in Grade Five (seriously, the fact that twice my grade was my age overwhelmed my thinking on my tenth birthday, which is probably why I had to stay in mathematics), so, add five to that and I’d be 15 in Grade Ten.
Rick Detorie’s One Big Happy (November 13), this one a rerun from a couple years ago because that’s how his strip works on Gocomics, goes to one of its regular bits of the kid Ruthie teaching anyone she can get in range, and while there’s a bit more to arithmetic than just adding two numbers to get a bigger number, she is showing off an understanding of a useful sanity check: if you add together two (positive) numbers, you have to get a result that’s bigger than either of the ones you started with. As for the 14th, and counting higher, well, there’s not much she could do about that.
Steve McGarry’s Badlands (November 14) talks about the kind of problem people wish to have: how to win a lottery where nobody else picks the same numbers, so that the prize goes undivided? The answer, of course, is to have a set of numbers that nobody else picked, but is there any way to guarantee that? And this gets into the curious psychology of random numbers: there is absolutely no reason that 1-2-3-4-5-6, or for that matter 7-8-9-10-11-12, would not come up just as often as, say, 11-37-39-51-52-55, but the latter set looks more random. But we see some strings of numbers as obviously a pattern, while others we don’t see, and we tend to confuse “we don’t know the pattern” with “there is no pattern”. I have heard the lore that actually a disproportionate number of people pick such obvious patterns like 1-2-3-4-5-6, or numbers that form neat pictures on a lottery card, no doubt cackling at how much more clever they are than the average person, and guaranteeing that if such a string ever does come out there’ll a large number of very surprised lottery winners. All silliness, really; the thing to do, obviously, is buy two tickets with the exact same set of numbers, so that if you do win, you get twice the share of anyone else, unless they’ve figured out the same trick.
I’m not precisely sure how to embed this, so, let me just point over to the Five Triangles blog (on blogspot) where there’s a neat little puzzle. It starts with Pythagorean triplets, the sets of numbers (a, b, c) so that a2 plus b2 equals c2. Pretty much anyone who knows the term “Pythagorean triplet” knows the set (3, 4, 5), and knows the set (5, 12, 13), and after that knows that there’s more if you really have to dig them up but who can be bothered?
Anyway, the problem at Five Triangle’s “Inscribed Circle” here draws that second Pythagorean triplet triangle, the one with sides of length 5, 12, and 13, and inscribes a circle within it. The problem: find the radius of the circle?
I’m embarrassed to say how much time I took to work it out, but that’s because I was looking for purely geometric approaches, when casting it over to algebra turns this into a pretty quick problem. I do feel like there should be an obvious geometric solution, though, and I’m sure I’ll wake in the middle of the night feeling like an idiot for not having that before I talked about this.