Here I’m just closing out last week’s mathematically-themed comics. The new week seems to be bringing some more in at a good pace, too. Should have stuff to talk about come Sunday.
Darrin Bell and Theron Heir’s Rudy Park for the 24th brings out the ancient question, why do people need to do mathematics when we have calculators? As befitting a comic strip (and Sadie’s character) the question goes unanswered. But it shows off the understandable confusion people have between mathematics and calculation. Calculation is a fine and necessary thing. And it’s fun to do, within limits. And someone who doesn’t like to calculate probably won’t be a good mathematician. (Or will become one of those master mathematicians who sees ways to avoid calculations in getting to an answer!) But put aside the obviou that we need mathematics to know what calculations to do, or to tell whether a calculation done makes sense. Much of what’s interesting about mathematics isn’t a calculation. Geometry, for an example that people in primary education will know, doesn’t need more than slight bits of calculation. Group theory swipes a few nice ideas from arithmetic and builds its own structure. Knot theory uses polynomials — everything does — but more as a way of naming structures. There aren’t things to do that a calculator would recognize.
John Atkinson’s Wrong Hands for the 25th is the awaited anthropomorphic-numerals and symbols joke for this past week. I enjoy the first commenter’s suggestion tha they should have stayed in unknown territory.
Rick Kirkman and Jerry Scott’s Baby Blues for the 26th does a little wordplay built on pre-algebra. I’m not sure that Zoe is quite old enough to take pre-algebra. But I also admit not being quite sure what pre-algebra is. The central idea of (primary school) algebra — that you can do calculations with a number without knowing what the number is — certainly can use some preparatory work. It’s a dazzling idea and needs plenty of introduction. But my dim recollection of taking it was that it was a bit of a subject heap, with some arithmetic, some number theory, some variables, some geometry. It’s all stuff you’ll need once algebra starts. But it is hard to say quickly what belongs in pre-algebra and what doesn’t.
Art Sansom and Chip Sansom’s The Born Loser for the 26th uses two ancient staples of jokes, probabilities and weather forecasting. It’s a hard joke not to make. The prediction for something is that it’s very unlikely, and it happens anyway? We all laugh at people being wrong, which might be our whistling past the graveyard of knowing we will be wrong ourselves. It’s hard to prove that a probability is wrong, though. A fairly tossed die may have only one chance in six of turning up a ‘4’. But there’s no reason to think it won’t, and nothing inherently suspicious in it turning up ‘4’ four times in a row.
We could do it, though. If the die turned up ‘4’ four hundred times in a row we would no longer call it fair. (This even if examination proved the die really was fair after all!) Or if it just turned up a ‘4’ significantly more often than it should; if it turned up two hundred times out of four hundred rolls, say. But one or two events won’t tell us much of anything. Even the unlikely happens sometimes.
Even the impossibly unlikely happens if given enough attempts. If we do not understand that instinctively, we realize it when we ponder that someone wins the lottery most weeks. Presumably the comic’s weather forecaster supposed the chance of snow was so small it could be safely rounded down to zero. But even something with literally zero percent chance of happening might.
Imagine tossing a fair coin. Imagine tossing it infinitely many times. Imagine it coming up tails every single one of those infinitely many times. Impossible: the chance that at least one toss of a fair coin will turn up heads, eventually, is 1. 100 percent. The chance heads never comes up is zero. But why could it not happen? What law of physics or logic would it defy? It challenges our understanding of ideas like “zero” and “probability” and “infinity”. But we’re well-served to test those ideas. They hold surprises for us.
Two things made repeat appearances in the mathematically-themed comics this week. They’re the comic strip Frazz and the idea of having infinitely many monkeys typing. Well, silly answers to word problems also turned up, but that’s hard to say many different things about. Here’s what I make the week in comics out to be.
Sandra Bell-Lundy’s Between Friends for the 6th introduces the infinite monkeys problem. I wonder sometimes why the monkeys-on-typewriters thing has so caught the public imagination. And then I remember it encourages us to stare directly into infinity and its intuition-destroying nature from the comfortable furniture of the mundane — typewriters, or keyboards, for goodness’ sake — with that childish comic dose of monkeys. Given that it’s a wonder we ever talk about anything else, really.
Monkeys writing Shakespeare has for over a century stood as a marker for what’s possible but incredibly improbable. I haven’t seen it compared to finding a four-digit PIN. It has got me wondering about the chance that four randomly picked letters will be a legitimate English word. I’m sure the chance is more than the one-in-a-thousand chance someone would guess a randomly drawn PIN correctly on one try. More than one in a hundred? I’m less sure. The easy-to-imagine thing to do is set a computer to try out all 456,976 possible sets of four letters and check them against a dictionary. The number of hits divided by the number of possibilities would be the chance of drawing a legitimate word. If I had a less capable computer, or were checking even longer words, I might instead draw some set number of words, never minding that I didn’t get every possibility. The fraction of successful words in my sample would be something close to the chance of drawing any legitimate word.
If I thought a little deeper about the problem, though, I’d just count how many four-letter words are already in my dictionary and divide that into 456,976. It’s always a mistake to start programming before you’ve thought the problem out. The trouble is not being able to tell when that thinking-out is done.
Richard Thompson’s Poor Richard’s Almanac for the 7th is the other comic strip to mention infinite monkeys. Well, chimpanzees in this case. But for the mathematical problem they’re not different. I’ve featured this particular strip before. But I’m a Thompson fan. And goodness but look at the face on the T S Eliot fan in the lower left corner there.
Jeff Mallet’s Frazz for the 6th gives Caulfield one of those flashes of insight that seems like it should be something but doesn’t mean much. He’s had several of these lately, as mentioned here last week. As before this is a fun discovery about Roman Numerals, but it doesn’t seem like it leads to much. Perhaps a discussion of how the subtractive principle — that you can write “four” as “IV” instead of “IIII” — evolved over time. But then there isn’t much point to learning Roman Numerals at all. It’s got some value in showing how much mathematics depends on culture. Not just that stuff can be expressed in different ways, but that those different expressions make different things easier or harder to do. But I suspect that isn’t the objective of lessons about Roman Numerals.
Frazz got my attention again the 12th. This time it just uses arithmetic, and a real bear of an arithmetic problem, as signifier for “a big pile of hard work”. This particular problem would be — well, I have to call it tedious, rather than hard. doing it is just a long string of adding together two numbers. But to do that over and over, by my count, at least 47 times for this one problem? Hardly any point to doing that much for one result.
Patrick Roberts’s Todd the Dinosaur for the 7th calls out fractions, and arithmetic generally, as the stuff that ruins a child’s dreams. (Well, a dinosaur child’s dreams.) Still, it’s nice to see someone reminding mathematicians that a lot of their field is mostly used by accountants. Actuaries we know about; mathematics departments like to point out that majors can get jobs as actuaries. I don’t know of anyone I went to school with who chose to become one or expressed a desire to be an actuary. But I admit not asking either.
Mike Thompson’s Grand Avenue started off a week of students-resisting-the-test-question jokes on the 7th. Most of them are hoary old word problem jokes. But, hey, I signed up to talk about it when a comic strip touches a mathematics topic and word problems do count.
Zach Weinersmith’s Saturday Morning Breakfast Cereal reprinted the 7th is a higher level of mathematical joke. It’s from the genre of nonsense calculation. This one starts off with what’s almost a cliche, at least for mathematics and physics majors. The equation it starts with, , is true. And famous. It should be. It links exponentiation, imaginary numbers, π, and negative numbers. Nobody would have seen it coming. And from there is the sort of typical gibberish reasoning, like writing “Pi” instead of π so that it can be thought of as “P times i”, to draw to the silly conclusion that P = 0. That much work is legitimate.
From there it sidelines into “P = NP”, which is another equation famous to mathematicians and computer scientists. It’s a shorthand expression of a problem about how long it takes to find solutions. That is, how many steps it takes. How much time it would take a computer to solve a problem. You can see why it’s important to have some study of how long it takes to do a problem. It would be poor form to tie up your computer on a problem that won’t be finished before the computer dies of old age. Or just take too long to be practical.
Most problems have some sense of size. You can look for a solution in a small problem or in a big one. You expect searching for the solution in a big problem to take longer. The question is how much longer? Some methods of solving problems take a length of time that grows only slowly as the size of the problem grows. Some take a length of time that grows crazy fast as the size of the problem grows. And there are different kinds of time growth. One kind is called Polynomial, because everything is polynomials. But there’s a polynomial in the problem’s size that describes how long it takes to solve. We call this kind of problem P. Another is called Non-Deterministic Polynomial, for problems that … can’t. We assume. We don’t know. But we know some problems that look like they should be NP (“NP Complete”, to be exact).
It’s an open question whether P and NP are the same thing. It’s possible that everything we think might be NP actually can be solved by a P-class algorithm we just haven’t thought of yet. It would be a revolution in our understanding of how to find solutions if it were. Most people who study algorithms think P is not NP. But that’s mostly (as I understand it) because it seems like if P were NP then we’d have some leads on proving that by now. You see how this falls short of being rigorous. But it is part of expertise to get a feel for what seems to make sense in light of everything else we know. We may be surprised. But it would be inhuman not to have any expectations of a problem like this.
Mark Anderson’s Andertoons for the 8th gives us the Andertoons content for the week. It’s a fair question why a right triangle might have three sides, three angles, three vertices, and just the one hypotenuse. The word’s origin, from Greek, meaning “stretching under” or “stretching between”. It’s unobjectionable that we might say this is the stretch from one leg of the right triangle to another. But that leaves unanswered why there’s just the one hypothenuse, since the other two legs also stretch from the end of one leg to another. Dr Sarah on The Math Forum suggests we need to think of circles. Draw a circle and a diameter line on it. Now pick any point on the circle other than where the diameter cuts it. Draw a line from one end of the diameter to your point. And from your point to the other end of the diameter. You have a right triangle! And the hypothenuse is the leg stretching under the other two. Yes, I’m assuming you picked a point above the diameter. You did, though, didn’t you? Humans do that sort of thing.
I don’t know if Dr Sarah’s explanation is right. It sounds plausible and sensible. But those are weak pins to hang an etymology on. But I have no reason to think she’s mistaken. And the explanation might help people accept there is the one hypothenuse and there’s something interesting about it.
The first (and as I write this only) commenter, Kristiaan, has a good if cheap joke there.
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.
John Kovaleski’s Bo Nanas rerun the 24th is about probability. There’s something wondrous and strange that happens when we talk about the probability of things like birth days. They are, if they’re in the past, determined and fixed things. The current day is also a known, determined, fixed thing. But we do mean something when we say there’s a 1-in-365 (or 366, or 365.25 if you like) chance of today being your birthday. It seems to me this is probability based on ignorance. If you don’t know when my birthday is then your best guess is to suppose there’s a one-in-365 (or so) chance that it’s today. But I know when my birthday is; to me, with this information, the chance today is my birthday is either 0 or 1. But what are the chances that today is a day when the chance it’s my birthday is 1? At this point I realize I need much more training in the philosophy of mathematics, and the philosophy of probability. If someone is aware of a good introductory book about it, or a web site or blog that goes into these problems in a way a lay reader will understand, I’d love to hear of it.
I’ve featured this installment of Poor Richard’s Almanac before. I’ll surely feature it again. I like Richard Thompson’s sense of humor. The first panel mentions non-Euclidean geometry, using the connotation that it does have. Non-Euclidean geometries are treated as these magic things — more, these sinister magic things — that defy all reason. They can’t defy reason, of course. And at least some of them are even sensible if we imagine we’re drawing things on the surface of the Earth, or at least the surface of a balloon. (There are non-Euclidean geometries that don’t look like surfaces of spheres.) They don’t work exactly like the geometry of stuff we draw on paper, or the way we fit things in rooms. But they’re not magic, not most of them.
Stephen Bentley’s Herb and Jamaal for the 25th I believe is a rerun. I admit I’m not certain, but it feels like one. (Bentley runs a lot of unannounced reruns.) Anyway I’m refreshed to see a teacher giving a student permission to count on fingers if that’s what she needs to work out the problem. Sometimes we have to fall back on the non-elegant ways to get comfortable with a method.
Berkeley Breathed’s Bloom County for the 28th is another rerun, from 1981. And it’s been featured here before too. As mentioned then, Milo is using calculus and logarithms correctly in his rather needless insult of Freida. 10,000 is a constant number, and as mentioned a few weeks back its derivative must be zero. Ten to the power of zero is 1. The log of 10, if we’re using logarithms base ten, is also 1. There are many kinds of logarithms but back in 1981, the default if someone said “log” would be the logarithm base ten. Today the default is more muddled; a normal person would mean the base-ten logarithm by “log”. A mathematician might mean the natural logarithm, base ‘e’, by “log”. But why would a normal person mention logarithms at all anymore?
Jef Mallett’s Frazz for the 28th is mostly a bit of wordplay on evens and odds. It’s marginal, but I do want to point out some comics that aren’t reruns in this batch.
Susan Camilleri Konar is a new cartoonist for the Six Chix collective. Her first strip to get mentioned around these parts is from the 5th. It’s a casual mention of the Fibonacci sequence, which is one of the few sequences that a normal audience would recognize as something going on forever. And yes, I noticed the spiral in the background. That’s one of the common visual representations of the Fibonacci sequence: it starts from the center. The rectangles inside have dimensions 1 by 2, then 2 by 3, then 3 by 5, then 5 by 8, and so on; the spiral connects vertices of these rectangles. It’s an attractive spiral and you can derive the overrated Golden Ratio from the dimensions of larger rectangles. This doesn’t make the Golden Ratio important or anything, but it is there.
Ryan North’s Dinosaur Comics for the 6th is part of a story about T-Rex looking for certain truth. Mathematics could hardly avoid coming up. And it does offer what look like universal truths: given the way deductive logic works, and some starting axioms, various things must follow. “1 + 1 = 2” is among them. But there are limits to how much that tells us. If we accept the rules of Monopoly, then owning four railroads means the rent for landing on one is a game-useful $200. But if nobody around you cares about Monopoly, so what? And so it is with mathematics. Utahraptor and Dromiceiomimus point out that the mathematics we know is built on premises we have selected because we find them interesting or useful. We can’t know that the mathematics we’ve deduced has any particular relevance to reality. Indeed, it’s worse than North points out: How do we know whether an argument is valid? Because we believe that its conclusions follow from its premises according to our rules of deduction. We rely on our possibly deceptive senses to tell us what the argument even was. We rely on a mind possibly upset by an undigested bit of beef, a crumb of cheese, or a fragment of an underdone potato to tell us the rules are satisfied. Mathematics seems to offer us absolute truths, but it’s hard to see how we can get there.
Mark Anderson’s Andertoons for the 7th is the long-awaited Andertoon for last week. It is hard getting education in through all the overhead.
Bill Watterson’s Calvin and Hobbes rerun for the 7th is a basic joke about Calvin’s lousy student work. Fun enough. Calvin does show off one of those important skills mathematicians learn, though. He does do a sanity check. He may not know what 12 + 7 and 3 + 4 are, but he does notice that 12 + 7 has to be something larger than 3 + 4. That’s a starting point. It’s often helpful before starting work on a problem to have some idea of what you think the answer should be.
As though to reinforce how nothing was basically wrong, Comic Strip Master Command sent a normal number of mathematically themed comics around this past week. They bunched the strips up in the first half of the week, but that will happen. It was a fun set of strips in any event.
Rob Harrell’s Adam @ Home for the 11th tells of a teacher explaining division through violent means. I’m all for visualization tools and if we are going to use them, the more dramatic the better. But I suspect Mrs Clark’s students will end up confused about what exactly they’ve learned. If a doll is torn into five parts, is that communicating that one divided by five is five? If the students were supposed to identify the mass of the parts of the torn-up dolls as the result of dividing one by five, was that made clear to them? Maybe it was. But there’s always the risk in a dramatic presentation that the audience will misunderstand the point. The showier the drama the greater the risk, it seems to me. But I did only get the demonstration secondhand; who knows how well it was done?
Greg Cravens’ The Buckets for the 11th has the kid, Toby, struggling to turn a shirt backwards and inside-out without taking it off. As the commenters note this is the sort of problem we get into all the time in topology. The field is about what can we say about shapes when we don’t worry about distance? If all we know about a shape is the ways it’s connected, the number of holes it has, whether we can distinguish one side from another, what else can we conclude? I believe Gocomics.com commenter Mike is right: take one hand out the bottom of the shirt and slide it into the other sleeve from the outside end, and proceed from there. But I have not tried it myself. I haven’t yet started wearing long-sleeve shirts for the season.
Bill Amend’s FoxTrot for the 11th — a new strip — does a story problem featuring pizzas cut into some improbable numbers of slices. I don’t say it’s unrealistic someone might get this homework problem. Just that the story writer should really ask whether they’ve ever seen a pizza cut into sevenths. I have a faint memory of being served a pizza cut into tenths by same daft pizza shop, which implies fifths is at least possible. Sevenths I refuse, though.
Mark Tatulli’s Heart of the City for the 12th plays on the show-your-work directive many mathematics assignments carry. I like Heart’s showiness. But the point of showing your work is because nobody cares what (say) 224 divided by 14 is. What’s worth teaching is the ability to recognize what approaches are likely to solve what problems. What’s tested is whether someone can identify a way to solve the problem that’s likely to succeed, and whether that can be carried out successfully. This is why it’s always a good idea, if you are stumped on a problem, to write out how you think this problem should be solved. Writing out what you mean to do can clarify the steps you should take. And it can guide your instructor to whether you’re misunderstanding something fundamental, or whether you just missed something small, or whether you just had a bad day.
Norm Feuti’s Gil for the 12th, another rerun, has another fanciful depiction of showing your work. The teacher’s got a fair complaint in the note. We moved away from tally marks as a way to denote numbers for reasons. Twelve depictions of apples are harder to read than the number 12. And they’re terrible if we need to depict numbers like one-half or one-third. Might be an interesting side lesson in that.
Brian Basset’s Red and Rover for the 14th is a rerun and one I’ve mentioned in these parts before. I understand Red getting fired up to be an animator by the movie. It’s been a while since I watched Donald Duck in Mathmagic Land but my recollection is that while it was breathtaking and visually inventive it didn’t really get at mathematics. I mean, not at noticing interesting little oddities and working out whether they might be true always, or sometimes, or almost never. There is a lot of play in mathematics, especially in the exciting early stages where one looks for a thing to prove. But it’s also in seeing how an ingenious method lets you get just what you wanted to know. I don’t know that the short demonstrates enough of that.
Bud Blake’s Tiger rerun for the 15th gives Punkinhead the chance to ask a question. And it’s a great question. I’m not sure what I’d say arithmetic is, not if I’m going to be careful. Offhand I’d say arithmetic is a set of rules we apply to a set of things we call numbers. The rules are mostly about how we can take two numbers and a rule and replace them with a single number. And these turn out to correspond uncannily well with the sorts of things we do with counting, combining, separating, and doing some other stuff with real-world objects. That it’s so useful is why, I believe, arithmetic and geometry were the first mathematics humans learned. But much of geometry we can see. We can look at objects and see how they fit together. Arithmetic we have to infer from the way the stuff we like to count works. And that’s probably why it’s harder to do when we start school.
What’s not good about that as an answer is that it actually applies to a lot of mathematical constructs, including those crazy exotic ones you sometimes see in science press. You know, the ones where there’s this impossibly complicated tangle with ribbons of every color and a headline about “It’s Revolutionary. It’s 46-Dimensional. It’s Breaking The Rules Of Geometry. Is It The Shape That Finally Quantizes Gravity?” or something like that. Well, describe a thing vaguely and it’ll match a lot of other things. But also when we look to new mathematical structures, we tend to look for things that resemble arithmetic. Group theory, for example, is one of the cornerstones of modern mathematical thought. It’s built around having a set of things on which we can do something that looks like addition. So it shouldn’t be a surprise that many groups have a passing resemblance to arithmetic. Mathematics may produce universal truths. But the ones we see are also ones we are readied to see by our common experience. Arithmetic is part of that common experience.
Here in the United States schools are just lurching back into the mode where they have students come in and do stuff all day. Perhaps this is why it was a routine week. Comic Strip Master Command wants to save up a bunch of story problems for us. But here’s what the last seven days sent into my attention.
Jeff Harris’s Shortcuts educational feature for the 21st is about algebra. It’s got a fair enough blend of historical trivia and definitions and examples and jokes. I don’t remember running across the “number cruncher” joke before.
Mark Anderson’s Andertoons for the 23rd is your typical student-in-lecture joke. But I do sympathize with students not understanding when a symbol gets used for different meanings. It throws everyone. But sometimes the things important to note clearly in one section are different from the needs in another section. No amount of warning will clear things up for everybody, but we try anyway.
Tom Thaves’s Frank and Ernest for the 23rd tells a joke about collapsing wave functions, which is why you never see this comic in a newspaper but always see it on a physics teacher’s door. This is properly physics, specifically quantum mechanics. But it has mathematical import. The most practical model of quantum mechanics describes what state a system is in by something called a wave function. And we can turn this wave function into a probability distribution, which describes how likely the system is to be in each of its possible states. “Collapsing” the wave function is a somewhat mysterious and controversial practice. It comes about because if we know nothing about a system then it may have one of many possible values. If we observe, say, the position of something though, then we have one possible value. The wave functions before and after the observation are different. We call it collapsing, reflecting how a universe of possibilities collapsed into a mere fact. But it’s hard to find an explanation for what that is that’s philosophically and physically satisfying. This problem leads us to Schrödinger’s Cat, and to other challenges to our sense of how the world could make sense. So, if you want to make your mark here’s a good problem for you. It’s not going to be easy.
The biggest thing I learned from my Theorem Thursdays project was: don’t do this for Thursdays. The appeal is obvious. If things were a little different I’d have no problem with Thursdays. But besides being a slightly-read pop-mathematics blogger I’m also a slightly-read humor blogger. And I try to have a major piece, about seven hundred words that are more than simply commentary on how a comic strip’s gone wrong, ready for Thursday evenings my time.
That’s all my doing. It’s a relic of my thinking that the humor blog should run at least a bit like a professional syndicated columnist’s, with a fixed deadline for bigger pieces. While I should be writing more ahead of deadline than this, what I would do is get to Wednesday realizing I have two major things to write in a day. I’d have an idea for one of them, the mathematics thing, since I would pick a topic the previous Thursday. And once I’ve picked an idea the rest is easy. (Part of the process of picking is realizing whether there’s any way to make seven hundred words about something.) But that’s a lot of work for something that’s supposed to be recreational. Plus Wednesdays are, two weeks a month, a pinball league night.
So Thursday is right out, unless I get better about having first drafts of stuff done Monday night. So Thursday is right out. This has problems for future appearances of the gimmick. The alliterative pull is strong. The only remotely compelling alternative is Theorems on the Threes, maybe one the 3rd, 13th, and 23rd of the month. That leaves the 30th and 31st unaccounted for, and room for a good squabble about whether they count in an “on the threes” scheme.
There’s a lot of good stuff to say about the project otherwise. The biggest is that I had fun with it. The Theorem Thursday pieces sprawled into for-me extreme lengths, two to three thousand words. I had space to be chatty and silly and autobiographic in ways that even the A To Z projects don’t allow. Somehow those essays didn’t get nearly as long, possibly because I was writing three of them a week. I didn’t actually write fewer things in July than I did in, say, May. But it was fewer kinds of things; postings were mostly Theorem Thursdays and Reading the Comics posts. Still, overall readership didn’t drop and people seemed to quite like what I did write. It may be fewer but longer-form essays are the way I should go.
Also I found that people like stranger stuff. There’s an understandable temptation in doing pop-mathematics to look for topics that are automatically more accessible. People are afraid enough of mathematics. They have good reason to be terrified of some topic even mathematics majors don’t encounter until their fourth year. So there’s a drive to simpler topics, or topics that have fewer prerequisites, and that’s why every mathematics blogger has an essay about how the square root of two is irrational and how there’s different sizes to infinitely large sets. And that’s produced some excellent writing about topics like those, which are great topics. They have got the power to inspire awe without requiring any warming up. That’s special.
But it also means they’re hard to write anything new or compelling about if you’re like me, and in somewhere like the second hundred billion of mathematics bloggers. I can’t write anything better than what’s already gone about that. Liouville’s Theorem? That’s something I can be a good writer about. With that, I can have a blog personality. It’s like having a real personality but less work.
As I did with the Leap Day 2016 A To Z project, I threw the topics open to requests. I didn’t get many. Possibly the form gave too much freedom. Picking something to match a letter, as in the A to Z, gives a useful structure for choosing something specific. Pick a theorem from anywhere in mathematics? Something from algebra class? Something mentioned in a news report about a major breakthrough the reporter doesn’t understand but had an interesting picture? Something that you overheard the name of once without any context? How should people know what the scope of it is, before they’ve even seen a sample? And possibly people don’t actually remember the names of theorems unless they stay in mathematics or mathematics-related fields. Those folks hardly need explained theorems with names they remember. This is a hard problem to imagine people having, but it’s something I must consider.
So this is what I take away from the two-month project. There’s a lot of fun digging into the higher-level mathematics stuff. There’s an interest in it, even if it means I write longer and therefore fewer pieces. Take requests, but have a structure for taking them that makes it easy to tell what requests should look like. Definitely don’t commit to doing big things for Thursday, not without a better scheme for getting the humor blog pieces done. Free up some time Wednesday and don’t put up an awful score on Demolition Man like I did last time again. Seriously, I had a better score on The Simpsons Pinball Party than I did on Demolition Man and while you personally might not find this amusing there’s at least two people really into pinball who know how hilarious that is. (The games have wildly different point scorings. This like having a basketball score be lower than a hockey score.) That isn’t so important to mathematics blogging but it’s a good lesson to remember anyway.
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.
The last day of July and first day of August saw enough mathematically-themed comic strips to fill a standard-issue entry. The rest of the week wasn’t so well-stocked. But I’ll cover those comics on Tuesday if all goes well. This may be a silly plan, but it is a plan, and I will stick to that.
Johnny Hart’s Back To BC reprints the venerable and groundbreaking comic strip from its origins. On the 31st of July it reprinted a strip from February 1959 in which Peter discovers mathematics. The work’s elaborate, much more than we would use to solve the problem today. But it’s always like that. Newly-discovered mathematics is much like any new invention or innovation, a rickety set of things that just barely work. With time we learn better how the idea should be developed. And we become comfortable with the cultural assumptions going into the work. So we get more streamlined, faster, easier-to-use mathematics in time.
Mac King and Bill King’s Magic in a Minute for the 31st maybe isn’t really mathematics. I guess there’s something in the modular-arithmetic implied by it. But it depends on a neat coincidence. Follow the directions in the comic about picking a number from one to twelve and counting out the letters in the word for that number. And then the letters in the word for the number you’re pointing to, and then once again. It turns out this leads to the same number. I’d never seen this before and it’s neat that it does.
Rick Detorie’s One Big Happy rerun for the 31st features Ruthie teaching, as she will. She mentions offhand the “friendlier numbers”. By this she undoubtedly means the numbers that are attractive in some way, like being nice to draw. There are “friendly numbers”, though, as number theorists see things. These are sets of numbers. For each number in this set you get the same index if you add together all its divisors (including 1 and the original number) and divide it by the original number. For example, the divisors of six are 1, 2, 3, and 6. Add that together and you get 12; divide that by the original 6 and you get 2. The divisors of 28 are 1, 2, 4, 7, 14, and 28. Add that pile of numbers together and you get 56; divide that by the original 28 and you get 2. So 6 and 28 are friendly numbers, each the friend of the other.
As often happens with number theory there’s a lot of obvious things we don’t know. For example, we know that 1, 2, 3, 4, and 5 have no friends. But we do not know whether 10 has. Nor 14 nor 20. I do not know if it is proved whether there are infinitely many sets of friendly numbers. Nor do I know if it is proved whether there are infinitely many numbers without friends. Those last two sentences are about my ignorance, though, and don’t reflect what number theory people know. I’m open to hearing from people who know better.
There are also things called “amicable numbers”, which are easier to explain and to understand than “friendly numbers”. A pair of numbers are amicable if the sum of one number’s divisors is the other number. 220 and 284 are the smallest pair of amicable numbers. Fermat found that 17,296 and 18,416 were an amicable pair; Descartes found that 9,363,584 and 9,437,056 were. Both pairs were known to Arab mathematicians already. Amicable pairs are easy enough to produce. From the tenth century we’ve had Thâbit ibn Kurrah’s rule, which lets you generate sets of numbers. Ruthie wasn’t thinking of any of this, though, and was more thinking how much fun it is to write a 7.
Terry Border’s Bent Objects for the 1st just missed the anniversary of John Venn’s birthday and all the joke Venn Diagrams that were going around at least if your social media universe looks anything like mine.
Jon Rosenberg’s Scenes from a Multiverse for the 1st is set in “Mathpinion City”, in the “Numerically Flexible Zones”. And I appreciate it’s a joke about the politicization of science. But science and mathematics are human activities. They are culturally dependent. And especially at the dawn of a new field of study there will be long and bitter disputes about what basic terms should mean. It’s absurd for us to think that the question of whether 1 + 1 should equal 2 or 3 could even arise.
But we think that because we have absorbed ideas about what we mean by ‘1’, ‘2’, ‘3’, ‘plus’, and ‘equals’ that settle the question. There was, if I understand my mathematics history right — and I’m not happy with my reading on this — a period in which it was debated whether negative numbers should be considered as less than or greater than the positive numbers. Absurd? Thermodynamics allows for the existence of negative temperatures, and those represent extremely high-energy states, things that are hotter than positive temperatures. A thing may get hotter, from 1 Kelvin to 4 Kelvin to a million Kelvin to infinitely many Kelvin to -1000 Kelvin to -6 Kelvin. If there are intuition-defying things to consider about “negative six” then we should at least be open to the proposition that the universal truths of mathematics are understood by subjective processes.
I thank Comic Strip Master Command for the steady pace of mathematically-themed comics this past week. It fit quite nicely with my schedule, which you might get hints about in weeks to come. Depends what I remember to write about. I did have to search a while for any unifying motif of this set. The idea of stuff you use to help learn turned up several times over, and that will do.
Steve Breen and Mike Thompson’s Grand Avenue threatened on the 24th to resume my least-liked part of reading comics for mathematics themes. This would be Grandma’s habit of forcing the kids to spend their last month of summer vacation doing arithmetic drills. I won’t say that computing numbers isn’t fun because I know what it’s like to work out how many seconds are in 50 years in your head. But that’s never what this sort of drill is about. The strip’s diverted from that subject, but it might come back to spoil the end of summer vacation. (I’m not positive what my least-liked part of the comics overall is. I suspect it might be the weird anti-participation-trophy bias comic strip writers have.)
Ryan North’s Dinosaur Comics reprint for the 25th is about the end of the universe. We’ve got several competing theories about how the universe is likely to turn out, several trillion years down the road. The difference between them is in the shape of space and how that shape is changing. I’ve mentioned sometimes the wonder of being able to tell something about a whole shape from local information, things we can tell without being far from a single point. The fate of the universe must be the greatest example of this. Considering how large the universe is and how little of it we will ever be able to send an instrument to, we measure the shape of space from a single point. And we can realistically project what will happen in unimaginably distant times. Admittedly, if we get it wrong, we’ll never know, which takes off some of the edge.
Dinosaur Comics reappears the 28th with some talk about number bases. It’s all fine and accurate enough, except for the suggestion that anyone would use base five for something other than explaining how bases work. I like learning about bases. When I was a kid this concept explained much to me about how our symbols for numbers work. It also helped appreciate that symbols are not these fixed or universal things. They’re our creations and ours to adapt for whatever reason we find convenient. In the past we’ve found bases as high as sixty to be convenient. (The division of angles into 360 degrees each of 60 minutes, each of those of 60 seconds, is an echo of that.) But when I was a kid doing alternate-base problems nobody knew what I was doing or why, except the mathematics teacher who said I might like the optional sections in the book. We only really need base ten, base two, and base sixteen, which might as well be base two written more compactly. The rest are toys, good for instruction and for fun. Sorry, base seven.
Scott Meyer’s Basic Instructions rerun for the 27th is about everyone’s favorite bit of intransitivity. Rock-Paper-Scissors and its related games are all about systems in which any two results can be decisive but any three might not be. This prospect turns up whenever there are three or more possible outcomes. And it doesn’t require a system to be irrational or random. Chaos and counterintuitive results just happen when there’s three of a thing.
I remember, and possibly you remember too, learning of a computer system that can consistently beat humans at Rock-Paper-Scissors. It manages to do that by the oldest of game theory exploits, cheating. Its sensors look for the twitches suggesting what a person is going to throw and then it changes its throw to beat that. I don’t know what that’s supposed to prove since anyone who’s played a Sid Meier’s Civilization game knows that computers already know how to cheat.
Bill Schorr’s The Grizzwells for the 28th is a resisted word problem joke. It doesn’t use the classic railroad or airplane forms, but it’s the same joke anyway.
Benita Epstein’s Six Chix for the 29th is probably familiar to the folks taking electronics. The chart is a compact map used as a mnemonic for the different relationships between the current (I), the voltage (V), the resistance (R), and the power (P) in a circuit. When I was a student we got this as two separate circles, one for current-voltage-resistance and one for power-current-voltage. Each was laid out like the T-and-O maps which pre-Renaissance Western Europe used to diagram the world. While I now see that as a convenient and useful tool, as a student, I was skeptical that it was any easier to use the mnemonic aid than it was to just remember “voltage equals current times resistance” and “power equals voltage times current”. I’ve always had an irrational suspicion of mnemonic devices. I’m trying to do better.
Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 30th is a return of the whiteboard full of symbols to represent deep thinking. The symbols don’t mean anything as equations, though that might be my limited perspective. And that also might represent the sketchy, shorthand way serious work is done. As an idea is sketched out weird bundles of symbols that don’t literally parse do appear. In a publishable paper this is all turned into neatly formatted and standard stuff. Or we introduce symbols with clear explanations of what they mean so that others can learn to read what we write. But for ourselves, in the heat of work, we’ll produce what looks like gibberish to others and that’s all right as long as we don’t forget what the gibberish means. Sometimes we do, but the gibberish typically helps us recapture a lost idea. (I offer the tale of a mathematician with pages of notes for a brilliant insight which she has to reconstruct from a lost memory to would-be short story writers looking for a Romantic hook.)
I know, it’s impolitic for me to say something like my title. But I noticed a particular rerun in this set of mathematically-themed comics. And it left me wondering if I should drop that from my daily routine. There are strips I read more out of a fear of missing out than anything else. Most of them are in perpetual reruns, though some of them are so delightful I wouldn’t dare drop them. (Here I mean Cul de Sac and Peanuts.) An individual comic takes typically little time to read, but add that up and it does take a while, especially on vacation or the like. I won’t actually change anything; I’m too stubborn in lazy ways for that. But it crosses my mind.
Tim Lachowski’s Get A Life for the 14th is what set me off. Lachowski’s rerun this before, and I’ve mentioned it before, back in March of 2015 and back in November 2012. Given this I wonder if there’s a late-2013 or early-2014 reuse of the strip I failed to note around here. Or just missed, possibly because I was on vacation.
Nicholas Gurewitch’s Perry Bible Fellowship reprint for the 14th gives me the title for this edition. It uses symbols and diagrams of mathematics for their graphical artistry, the sort of thing I’m surprised doesn’t get done more. Back in college the creative-writing-and-arts editor for the unread leftist weekly asked me to do a page of physics calculations as an aesthetic composition and I was glad to do it. Good notation has a beauty to it; I wonder if people would like mathematics more if they got to spend time at play with its shapes.
Morrie Turner’s Wee Pals rerun for the 14th name-checks the New Math. The New Math was this attempt to reform mathematics in the 1970s. It was great for me, and my love remembers only liking or understanding mathematics while in New Math-guided classes. But it was an attempt at educational reform that didn’t promise that people at the cash registers would make change fast enough, and so was doomed to failure. (I am being reductive here. Much about the development of New Math went wrong, and it’s unfair to blame it all on the resistance of parents to new teaching methods. But educational reform always crashes hard against parents’ reasonable question, “Why should my child be your test case?”)
Many of the New Math ideas grew out of the work of Nicholas Bourbaki, and the attempt to explain mathematics on completely rigorous logical foundations, as free from intuition as possible to get. That sounds like an odd thing to do; intuition is a guide to useful ways to spend one’s time and energy. But that supposes the intuition is good.
Much of late 19th and early 20th century mathematics was spent discovering cases in which intuitive understandings of things were wrong. Deterministic systems can be unpredictable. A curve can be continuous at a single point and nowhere else in space. Infinitely large sets can be bigger or smaller than other sets. A line can wriggle around so much that it has a volume, it fills space. In that context wanting to ditch intuition a a once-useful but now-unreliable guide is not a bad idea.
I like the New Math. I suppose we always like the way we first learned things. But I still think it’s got a healthy focus. The idea that mathematics is built on rules we agree to use, and that we are free to change if we find they’re not doing things we need, is true. It’s one easy to forget considering mathematics’ primary job, which has always been making trade, accounting, and record-keeping go smoothly. Changing those systems are perilous. But we should know something about how to pick tools to use.
Zoe Piel’s At The Zoo for the 15th uses the blackboard-full-of-mathematics image to suggest deep thinking. (Toby the lion’s infatuated with the vet, which is why he’s thinking how to get her to visit again.) Really there’s a bunch of iconic cartoon images of deep thinking, including a mid-century-esque big-tin-box computer with reel-to-reel memory tape. Modern computers are vastly more powerful than that sort of 50s/60s contraption, but they’re worthless artistically if you want to suggest any deep thinking going on. You need stuff with moving parts for that, even in a still image.
Scott Adams’s Dilbert Classics for the 16th originally ran the 21st of May, 1993. And it comes back to a practical use for mathematics and the sort of thing we do need to know how to calculate. It also uses the image of mathematics as obscurant nonsense.
Great 12th-century English historian William of Malmesbury was no fan of maths. He called it 'dangerous Saracen magic'. @holland_tom
That tweet’s interesting in itself, although one of the respondents wonders if William meant astrology, often called “mathematics” at the time. That would be a fairer thing to call magic. But it would be only a century after William of Malmesbury’s death that Arabic numerals would become familiar in Europe. They would bring suspicions that merchants and moneylenders were trying to cheat their customers, by using these exotic specialist notations with unrecognizable rules, instead of the traditional and easy-to-follow Roman numerals. If this particular set of mathematics comics were mostly reruns, that’s all right; sometimes life is like that.
I confess I spent the last week on vacation, away from home and without the time to write about the comics. And it was another of those curiously busy weeks that happens when it’s inconvenient. I’ll try to get caught up ahead of the weekend. No promises.
Art and Chip Samson’s The Born Loser for the 10th talks about the statistics of body measurements. Measuring bodies is one of the foundations of modern statistics. Adolphe Quetelet, in the mid-19th century, found a rough relationship between body mass and the square of a person’s height, used today as the base for the body mass index.Francis Galton spent much of the late 19th century developing the tools of statistics and how they might be used to understand human populations with work I will describe as “problematic” because I don’t have the time to get into how much trouble the right mind at the right idea can be.
No attempt to measure people’s health with a few simple measurements and derived quantities can be fully successful. Health is too complicated a thing for one or two or even ten quantities to describe. Measures like height-to-waist ratios and body mass indices and the like should be understood as filters, the way temperature and blood pressure are. If one or more of these measurements are in dangerous ranges there’s reason to think there’s a health problem worth investigating here. It doesn’t mean there is; it means there’s reason to think it’s worth spending resources on tests that are more expensive in time and money and energy. And similarly just because all the simple numbers are fine doesn’t mean someone is perfectly healthy. But it suggests that the person is more likely all right than not. They’re guides to setting priorities, easy to understand and requiring no training to use. They’re not a replacement for thought; no guides are.
Jeff Harris’s Shortcuts educational panel for the 10th is about zero. It’s got a mix of facts and trivia and puzzles with a few jokes on the side.
John Hambrock’s The Brilliant Mind of Edison Lee for the 13th of July riffs on the world’s leading exporter of statistics, baseball. Organized baseball has always been a statistics-keeping game. The Olympic Ball Club of Philadelphia’s 1837 rules set out what statistics to keep. I’m not sure why the game is so statistics-friendly. It must be in part that the game lends itself to representation as a series of identical events — pitcher throws ball at batter, while runners wait on up to three bases — with so many different outcomes.
Alan Schwarz’s book The Numbers Game: Baseball’s Lifelong Fascination With Statistics describes much of the sport’s statistics and record-keeping history. The things recorded have varied over time, with the list of things mostly growing. The number of statistics kept have also tended to grow. Sometimes they get dropped. Runs Batted In were first calculated in 1880, then dropped as an inherently unfair statistic to keep; leadoff hitters were necessarily cheated of chances to get someone else home. How people’s idea of what is worth measuring changes is interesting. It speaks to how we change the ways we look at the same event.
Dana Summers’s Bound And Gagged for the 13th uses the old joke about computers being abacuses and the like. I suppose it’s properly true that anything you could do on a real computer could be done on the abacus, just, with a lot ore time and manual labor involved. At some point it’s not worth it, though.
Nate Fakes’s Break of Day for the 13th uses the whiteboard full of mathematics to denote intelligence. Cute birds, though. But any animal in eyeglasses looks good. Lab coats are almost as good as eyeglasses.
David L Hoyt and Jeff Knurek’s Jumble for the 13th is about one of geometry’s great applications, measuring how large the Earth is. It’s something that can be worked out through ingenuity and a bit of luck. Once you have that, some clever argument lets you work out the distance to the Moon, and its size. And that will let you work out the distance to the Sun, and its size. The Ancient Greeks had worked out all of this reasoning. But they had to make observations with the unaided eye, without good timekeeping — time and position are conjoined ideas — and without photographs or other instantly-made permanent records. So their numbers are, to our eyes, lousy. No matter. The reasoning is brilliant and deserves respect.
I had thought we were barely entering Final Exam season. But I hear reports many (United States) colleges and universities have already got them started. And I see what people are searching for around my writing here. So let me help folks out here.
Good luck. Read the syllabus and any test preparation sheets the instructor gives. Get a full night’s sleep before and eat well the day of the exam. Don’t pester with e-mails asking for extra credit. Only bother your professor with requests to correct errors of fact, which would be recorded grades or an error in calculation. Have your returned assignment to show, and understand how weighted grades work, before you do.
In today’s installment of Reading The Comics, mathematics gets name-dropped a bunch in strips that aren’t really about my favorite subject other than my love. Also, I reveal the big lie we’ve been fed about who drew the Henry comic strip attributed to Carl Anderson. Finally, I get a question from Queen Victoria. I feel like this should be the start of a podcast.
Patrick Roberts’ Todd the Dinosaur for the 6th of April just name-drops mathematics. The flash cards suggest it. They’re almost iconic for learning arithmetic. I’ve seen flash cards for other subjects. But apart from learning the words of other languages I’ve never been able to make myself believe they’d work. On the other hand, I haven’t used flash cards to learn (or teach) things myself.
Joe Martin’s Boffo for the 7th of April is a solid giggle. (I have a pretty watery giggle myself.) There are unknowable, or at least unprovable, things in mathematics. Any logic system with enough rules to be interesting has ideas which would make sense, and which might be true, but which can’t be proven. Arithmetic is such a system. But just fractions and long division by itself? No, I think we need something more abstract for that.
Carl Anderson’s Henry for the 7th of April is, of course, a rerun. It’s also a rerun that gives away that the “Carl Anderson” credit is a lie. Anderson turned over drawing the comic strip in 1942 to John Liney, for weekday strips, and Don Trachte for Sundays. There is no possible way the phrase “New Math” appeared on the cover of a textbook Carl Anderson drew. Liney retired in 1979, and Jack Tippit took over until 1983. Then Dick Hodgins, Jr, drew the strip until 1990. So depending on how quickly word of the New Math penetrated Comic Strip Master Command, this was drawn by either Liney, Tippit, or possibly Hodgins. (Peanuts made New Math jokes in the 60s, but it does seem the older the comic strip the longer it takes to mention new stuff.) I don’t know when these reruns date from. I also don’t know why Comics Kingdom is fibbing about the artist. But then they went and cancelled The Katzenjammer Kids without telling anyone either.
Eric the Circle for the 8th, this one by “lolz”, declares that Eric doesn’t like being graphed. This is your traditional sort of graph, one in which points with coordinates x and y are on the plot if their values make some equation true. For a circle, that equation’s something like (x – a)2 + (y – b)2 = r2. Here (a, b) are the coordinates for the point that’s the center of the circle, and r is the radius of the circle. This looks a lot like Eric is centered on the origin, the point with coordinates (0, 0). It’s a popular choice. Any center is as good. Another would just have equations that take longer to work with.
Richard Thompson’s Cul de Sac rerun for the 10th is so much fun to look at that I’m including it even though it just name-drops mathematics. The joke would be the same if it were something besides fractions. Although see Boffo.
Norm Feuti’s Gil rerun for the 10th takes on mathematics’ favorite group theory application, the Rubik’s Cube. It’s the way I solved them best. This approach falls outside the bounds of normal group theory, though.
Mac King and Bill King’s Magic in a Minute for the 10th shows off a magic trick. It’s also a non-Rubik’s-cube problem in group theory. One of the groups that a mathematics major learns, after integers-mod-four and the like, is the permutation group. In this, the act of swapping two (or more) things is a thing. This puzzle restricts the allowed permutations down to swapping one item with the thing next to it. And thanks to that, an astounding result emerges. It’s worth figuring out why the trick would work. If you can figure out the reason the first set of switches have to leave a penny on the far right then you’ve got the gimmick solved.
Comic Strip Master Command slowed down the pace at which the newspaper comics were to talk mathematical subjects. All right, that’s their prerogative. But it leaves me here, at Thursday, with slightly too few comics for my tastes. On the other hand, if I don’t run with what I have, I might not have anything to post for the 31st of March, and it would be a shame to go this whole month with something posted every day only to spoil it on the 31st. This is a pretty juvenile reason to do a thing, so here we are. Enjoy, please.
Tom Thaves’s Frank and Ernest for the 25th of March is a students-grumbling joke. I’m not sure what to make of the argument “arithmetic might be education, but that algebra stuff is indoctrination”. I imagine it reflects the feeling that the rules of arithmetic are all these nice straightforward things, and then algebra’s rules seem a bewildering set of near-gibberish. I can understand people looking at the quadratic formula, being told it has something to do with parabolas and an axis, throwing up their hands, and declaring it all this crazy game they’ll never play.
What people are forgetting in this is that everything sounds like this crazy gibberish game at first. The confusion you felt when first trying to factor a quadratic polynomial? It’s the same confusion you felt when first doing long division. And when you first multiplied a three-digit by a two-digit number. And when you had to subtract with borrowing. It’s also the same confusion you have when you first hear the first European settlement of Manhattan was driven by the Netherlands’ war for independence from Spain. Learning is changing the baffling confusion of life into an understandable pattern.
Which is not to deny that we could do a better job motivating stuff. You have no idea how many drafts of the Dedekind Domain essay I threw out because there were just too many words describing conditions and not why any of them mattered. I’m lazy; I don’t like scrapping that much text. And I’m still not quite happy with Normal Groups.
Jeff Mallet’s Frazz for the 27th is an easier joke to explain. It’s also one whose appeal I really understand. There is a compelling beauty to the notation and the symbols of higher mathematics. I remember when a kid I peered at one of my parents’ calculus textbooks. The reference page of common integrals was enchanting. It wasn’t the only thing that drove me towards mathematics. But the aesthetic beauty is there.
And it’s not just mathematicians and mathematics-based fields that see it. The arts editor for my undergraduate school’s unread leftist weekly newspaper asked me to work out a problem, any problem, to include as graphic arts. I was happy to. (I was the managing editor for the paper at the time.) I even had a great problem, from the final exam in my freshman Classical Mechanics course. The problem was to derive the equivalent of Kepler’s Laws of Motion with a different force law. Instead of the inverse-square attraction of gravity we used the exponential-decay-style interactions of the weak force. It was a brilliant exam question, frankly, and made for a page of symbols that maybe nobody understood but that I’ll bet everyone thought pretty.
John Forgetta and L A Rose’s The Meaning of Lila for the 27th is probably a rerun. The strip mostly is, although a few new or updated comics are fit into the rotation. It’s an example of a census joke, in which you classify away the whole population of the world. I remember first seeing it, as a kid, in a church bulletin. That one worked out how the entire working population of the United States was actually only two people and that’s why you’re always so tired. You could probably use the logic of this sort of joke to teach Venn diagrams. The logic that produces a funny low count relies on counting people several times, once for each of many categories they might fit in.
Mark Anderson’s Andertoons for the 30th made me giggle. I suppose there’s an essay to be written about whether we need mathematics, and what we need it for. But wouldn’t that just take away from the fun of it?
Terri Libenson’s The Pajama Diaries for the 20th of March mentions, among “reasons for ice cream”, the stress of having “helped with New Math”. It’s a curious reference, to me. I expect it refers to the stress of how they teach arithmetic differently from how it was when you grew up. I expect that feeds any adult’s natural anxiety about having forgot, or never really been good at, arithmetic. Add to that the anxiety of not being able to help your child when you’re called on. And add to that the ever-present fear of looking like a fool. There’s plenty of reason to be anxious.
Still, the reference to “New Math” is curious since, at least in the United States, that refers to a specific era. In the 1960s and 70s mathematics education saw a major revision, called the “New Math”. This revision tried many different approaches, but built around the theory that students should know why mathematics looks like it does. The hope was that in this way students wouldn’t just know what eight times seven was, but would agree that it made sense for this to be 56. The movement is, generally, regarded as a well-meant failure. The reasons are diverse, but many of them amount to it being very hard to explain why mathematics looks like it does. And it’s even harder to explain it to parents, who haven’t gone to school for years and aren’t going to go back to learn eight times seven. And it’s hard for many teachers, who often aren’t specialists in mathematics, to learn eight times seven in a new way either.
Still, the New Math was dead and buried in the United States by the 1980s. And more, Libenson is Canadian. I don’t know what educational fashions, and reform fashions, are like in Canada. I’m curious if Canadian parents or teachers could let me know, what is going on in reforming Canadian mathematics education? Is “New Math” a term of art in Canada now? Or did Libenson pick a term that would communicate efficiently “mathematics but not like I learned it”?
Rudolph Dirk’s The Katzenjammer Kids on the 20th reprinted the strip from the 5th of September, 1943. I mention it here because it contains an example of mathematics talk being used as signifier of great intelligence. The kids expound: “Now, der t’eory uf der twerpsicosis iss dot er sum uf circumvegetatium und der horizontal triggernometry iss equal to der … ” and that’s as far as it needs to go. It isn’t quite mathematics, but it is certainly using a painting of mathematics to make one look bright.
Mark Anderson’s Andertoons got its appearance in here the 20th. It’s got a student resisting the equivalent fractions idea. he kid notes that 1/2 might equal 2/4 and 4/8 and 8/16, but “the ones on the right feel like more bang for your buck”. The kid has a point. These are all the same number. It’s usually easiest to work with the smallest representation that means what you need. But they might convey their meanings differently. I get a different picture, at least, in speaking of “half the class not being done with the assignment” versus “16 of the 32 students aren’t done with the assignment”.
Charlie Podrebarac’s CowTown for the 20th of March claims Charlie could “literally paper the Earth” with losing NCAA brackets. As I make it out, he’s right. There are 263 possible NCAA brackets, because there are 63 matches in the college basketball tournament. All but one of these are losing. If each bracket fits on one sheet of paper — well, how big is a sheet of paper? If each bracket is on a sheet of A4-size paper, then, each page is 1/16th of a square meter. This is easy to work with. Unfortunately, if Charlie cares about the NCAA college basketball tournament, he’s probably in the United States. So he would print out on paper that’s 8 ½ inches by 11 inches. That’s not quite 1/16th of a square meter or any other convenient-to-work-with size. It’s 93.5 square inches but what good is that?
Well, I will pretend that the 8 ½ by 11 inch paper is close enough to A4. It’s going to turn out not to matter. 263 is 9,223,372,036,854,775,808. Subtract one and we have 9,223,372,036,854,775,807. Big difference. Multiply this by one-sixteenth of a square meter and we have about 576,460,752,000,000,000 square meters of paper. I’m rounding off because it is beyond ridiculous that I didn’t before. The surface area of the Earth is about 510,000,000,000,000 square meters. So if Bob picked every possible losing bracket he could indeed literally paper the Earth a thousand times over and have some paper to spare.
Ruben Bolling’s Super-Fun-Pak Comix for the 21st of March is a Guy Walks Into A Bar that depends on non-base-ten arithmetic for its punch line. I’m amused. I learned about different bases as a kid, in the warm glow of the New Math. The different bases and how they changed what arithmetic looked like enchanted me. Today I know there’s not much need for bases besides ten (normal mathematics), two (used by computers), and sixteen (used by people dealing with computers). (Base sixteen converts easily to base two, so people can understand what the computer is actually doing, while being much more compact, so people don’t have to write out prodigiously long sequences of digits.) But for a while there you can play around with base five or base twelve or, as a horse might, base four. It can help you better appreciate how much thought there is behind something as straightforward as “10”.
It always feels odd to toss folks from my mathematics to my humor blog. I suppose it only sometimes seems on-point. Last week, though, I ran a series of essays about the old-time radio series Vic and Sade. One of them, happening to star neither Vic nor Sade, was all about Uncle Fletcher trying to explain algebra, or arithmetic, or something or other. The radio program won’t be to everyone’s tastes. It had a very dry style, closer in tone to a modern one-camera sitcom than anything where there’s a studio audience and easy-to-quote patter. And it does start with an interminable advertisement for sponsor Crisco. (It also includes a contest that adds to the announcement’s length. I assume we’ve missed the contest deadline.) But past the first three minutes and twenty seconds you get some fine mathematics exposition. I hope you enjoy.
My love and I saw Only Yesterday recently. It’s a 1991 Studio Ghibli film, directed by Isao Takahata. It hasn’t had a United States release before, which is a pity; it’s quite good. The movie is about a woman, Taeko, reflecting on her childhood as she considers changing her life. One of the many wonderfully-realized scenes is about ten-year-old Taeko’s struggles with arithmetic. You probably guessed that, as otherwise the movie would seem outside the remit of this blog.
In the scene Taeko has had a disastrous arithmetic test. Her older sister is trying to coach her through how to divide fractions. It goes lousy. Her older sister insists it’s just a matter of inverting and multiplying. This is a useful tip if you understand how to divide fractions and need to keep straight what you’re doing. If you don’t understand, then it’s whatever the modern equivalent is for instructions on how to set a VCR.
Taeko tries to understand one problem. . She pictures it as an apple and draws a circle, blacking out a third of it. She cuts the rest into four equally-sized pieces and concludes that you could fit six slices into the original apple. Her sister stammers over this and fumes. She declares “that’s multiplication!”. She complains her sister isn’t doing the right thing, she’s not inverting and multiplying. I recognize her sister’s panic. It’s the bluster of someone trying to explain something not actually understood, on watching someone going far off the script.
The scene’s filled with irony. Taeko has a better understanding of what she’s doing than her sister has, but never knows it. Her sister understands a procedure but not what fractions dividing signifies. She can’t say why one wants to invert anything or multiply something. Taeko knows what the question she’s asked means, but not how to relate that to what she’s asked to do.
I don’t want to undervalue learning procedures. They’re worth knowing. They are, once you master them, efficient ways to compute. But there are many ways to master a procedure. I can’t believe there is one way to learn anything that works for everyone. One of many challenges teachers face is exploring the different ways their students best learn. Another is getting close enough to how they best learn that most of the students can understand something. It’s a pity when real people akin to Taeko can’t get that little bridge to connect their drawings of an apple to the page of fractions to be worked out.
Elzie Segar’s Thimble Theater is a comic strip you maybe vaguely remember hearing about for some reason. The reason is that, ten years into its run, Segar discovered a charismatic sailor named Popeye. People who read my humor blog know I’m a bit Popeye-mad, even still. Comics Kingdom has in its Vintage comics run the strips from the first story where Popeye appeared. This isn’t it. That story resolved, and the comic tried to carry on with the old cast. It didn’t last. After a few dull weeks Segar started making excuses to put Popeye back on-screen. It’s quite like Dickens’s Pickwick Papers and the discovery of Sam Weller, right down to this being the character that made the author famous.
As part of Segar’s excuses to keep Popeye on panel, nominal lead Castor Oyl has hired a tutor. It’s not going well. I blame the tutor, who’s berating Popeye for being wrong and giving no hint what to do right. But in this installment, originally run the 14th of September, 1929, we get around to arithmetic. Popeye is either a natural, has experience we don’t know about, or is quite lucky. It wouldn’t be absurd for Popeye to be good at some kinds of arithmetic. If he’s trained in navigation he’d probably pick up a good bit of practice calculating. I don’t know anything but the most trivial points of how to calculate one’s position at sea. So I can’t say if it’s plausible Popeye would have practiced calculations like “six and a half times 656”. He may just be lucky.
Mark Tatulli’s Lio for the 26th features soap bubbles made into geometry diagrams. I like that; it’s cute. Coincidentally, Guy Gilchrist’s Nancy for the 29th turns the pieces of a geometry puzzle into pizza. I think that’s a lesser version of the joke. It’s less absurd.
Nick Seluk’s The Awkard Yeti for the 2nd of March is a Schrödinger’s Cat reference alongside a butterfly reference. It seems Comic Strip Master Command challenges my “I’ve said all I can say, for now, about Schrödinger’s Cat and Chaos Butterflies” policy.
Missy Meyer’s Holiday Dodles mentions the 2nd of March was World Maths Day. I hadn’t heard about this; had you? Wikipedia indicates it’s a worldwide mathematics competition event sponsored by 3P Learning. Also that the first one was held on “Pi Day”, the 14th of March, which would make sense. I didn’t know it was Dr Seuss’s birthday either until I ran across a third comic strip doing some Dr Seuss jokes. Comic strips sometimes line up by accident. But I’m always impressed when they spontaneously (I assume) line up for some minor event like that.
Charles Schulz’s Peanuts for the 3rd of March originally ran the 6th of March, 1969. It’s part of a storyline in which Linus’s favorite teacher, Miss Othmar, is replaced following a teacher’s strike. This is why he complains to the new teacher about how Miss Othmar never did things that way.
It gets to appear here because Linus suggests that for some problem or other “we could divide instead of subtract”. I’m a little curious what the problem might have been. Division is often presented as a sort of hurried-up subtraction, or at least it was when I was Linus’s age. But they don’t quite address the same sorts of questions. I suppose something like “how many times eight goes into thirty-two”. But I wouldn’t do that by subtraction except to point out how division answers that question so much better. Still, there is a good point in showing how there can be several ways to do a problem. There almost always are. Sometimes a particular approach is faster than another. Sometimes it’s less confusing than another. Sometimes it gives better insight into other problems than another. If all you are interested in is the right answer, then you can use whatever method works, including letting Popeye guess for you. But, except on the frontier of research where we don’t quite know what we’re studying, there are always choices in how to find an answer.
Tom Toles’s Randolph Itch, 2 am for the 3rd I feel confident I’ve shown before. The strip didn’t run long originally and it’s in its third or fourth rerun cycle on Gocomics.com. It’s still an amusing bit of figure drawing, drawn by figures, being figured out. I make it out to 111,193.