## Reading the Comics, February 11, 2018: February 11, 2018 Edition

And it’s not always fair to say that the gods mock any plans made by humans, but Comic Strip Master Command has been doing its best to break me of reading and commenting on any comic strip with a mathematical theme. I grant that I could make things a little easier if I demanded more from a comic strip before including it here. But even if I think a theme is slight that doesn’t mean the reader does, and it’s easy to let the eye drop to the next paragraph if the reader does think it’s too slight. The anthology nature of these posts is part of what works for them. And then sometimes Comic Strip Master Command sends me a day like last Sunday when everybody was putting in some bit of mathematics. There’ll be another essay on the past week’s strips, never fear. But today’s is just for the single day.

Susan Camilleri Konar’s Six Chix for the 11th illustrates the Lemniscate Family. The lemniscate is a shape well known as the curve made by a bit of water inside a narrow tube by people who’ve confused it with a meniscus. An actual lemniscate is, as the chain of pointing fingers suggests, a figure-eight shape. You get — well, I got — introduced to them in prealgebra. They’re shapes really easy to describe in polar coordinates but a pain to describe in Cartesian coordinates. There are several different kinds of lemniscates, each satisfying slightly different conditions while looking roughly like a figure eight. If you’re open to the two lobes of the shape not being the same size there’s even a kind of famous-ish lemniscate called the analemma. This is the figure traced out by the sun if you look at its position from a set point on the surface of the Earth at the same clock time each day over the course of the year. That the sun moves north and south from the horizon is easy to spot. That it is sometimes east or west of some reference spot is a surprise. It shows the difference between the movement of the mean sun, the sun as we’d see it if the Earth had a perfectly circular orbit, and the messy actual thing. Dr Helmer Aslasken has a fine piece about this, and how it affects when the sun rises earliest and latest in the year.

There’s also a thing called the “polynomial lemniscate”. This is a level curve of a polynomial. That is, what are all the possible values of the independent variable which cause the polynomial to evaluate to some particular number? This is going to be a polynomial in a complex-valued variable, in order to get one or more closed and (often) wriggly loops. A polynomial of a real-valued variable would typically give you a boring shape. There’s a bunch of these polynomial lemniscates that approximate the boundary of the Mandelbrot Set, that fractal that you know from your mathematics friend’s wall in 1992.

Mark Anderson’s Andertoons took care of being Mark Anderson’s Andertoons early in the week. It’s a bit of optimistic blackboard work.

Lincoln Pierce’s Big Nate features the formula for calculating the wind chill factor. Francis reads out what is legitimately the formula for estimating the wind chill temperature. I’m not going to get into whether the wind chill formula makes sense as a concept because I’m not crazy. The thinking behind it is that a windless temperature feels about the same as a different temperature with a particular wind. How one evaluates those equivalences offers a lot of room for debate. The formula as the National Weather Service, and Francis, offer looks frightening, but isn’t really hard. It’s not a polynomial, in terms of temperature and wind speed, but it’s close to that in form. The strip is rerun from the 15th of February, 2009, as Lincoln Pierce has had some not-publicly-revealed problem taking him away from the comic for about a month and a half now.

Jim Scancarelli’s Gasoline Alley included a couple of mathematics formulas, including the famous E = mc2 and the slightly less famous πr2, as part of Walt Wallet’s fantasy of advising scientists and inventors. (Scientists have already heard both.) There’s a curious stray bit in the corner, writing out 6.626 x 102 x 3 that I wonder about. 6.626 is the first couple digits of Planck’s Constant, as measured in Joule-seconds. (This is h, not h-bar, I say for the person about to complain.) It’d be reasonable for Scancarelli to have drawn that out of a physics book or reference page. But the exponent is all wrong, even if you suppose he mis-wrote 1023. It should be 6.626 x 10-34. So I don’t know whether Scancarelli got things very garbled, or if he just picked a nice sciencey-looking number and happened to hit on a significant one. (There’s enough significant science numbers that he’d have a fair chance of finding something.) The strip is a reprint from the 4th of February, 2007, as Jim Scancarelli has been absent for no publicly announced reason for four months now.

Greg Evans and Karen Evans’s Luann is not perfectly clear. But I think it’s presenting Gunther doing mathematics work to support his mother’s contention that he’s smart. There’s no working out what work he’s doing. But then we might ask how smart his mother is to have made that much food for just the two of them. Also that I think he’s eating a potato by hand? … Well, there are a lot of kinds of food that are hard to draw.

Greg Evans’s Luann Againn reprints the strip from the 11th of February (again), 1990. It mentions as one of those fascinating things of arithmetic an easy test to see if a number’s a multiple of nine. There are several tricks like this, although the only ones anybody can remember are finding multiples of 3 and finding multiples of 9. Well, they know the rules for something being a multiple of 2, 5, or 10, but those hardly look like rules, and there’s no addition needed. Similarly with multiples of 4.

Modular arithmetic underlies all these rules. Once you know the trick you can use it to work out your own add-up-the-numbers rules to find what numbers are multiples of small numbers. Here’s an example. Think of a three-digit number. Suppose its first digit is ‘a’, its second digit ‘b’, and its third digit ‘c’. So we’d write the number as ‘abc’, or, 100a + 10b + 1c. What’s this number equal to, modulo 9? Well, 100a modulo 9 has to be equal to whatever a modulo 9 is: (100 a) modulo 9 is (100) modulo 9 — that is, 1 — times (a) modulo 9. 10b modulo 9 is (10) modulo 9 — again, 1 — times (b) modulo 9. 1c modulo 9 is … well, (c) modulo 9. Add that all together and you have a + b + c modulo 9. If a + b + c is some multiple of 9, so must be 100a + 10b + 1c.

The rules about whether something’s divisible by 2 or 5 or 10 are easy to work with since 10 is a multiple of 2, and of 5, and for that matter of 10, so that 100a + 10b + 1c modulo 10 is just c modulo 10. You might want to let this settle. Then, if you like, practice by working out what an add-the-digits rule for multiples of 11 would be. (This is made a lot easier if you remember that 10 is equal to 11 – 1.) And if you want to show off some serious arithmetic skills, try working out an add-the-digits rule for finding whether something’s a multiple of 7. Then you’ll know why nobody has ever used that for any real work.

J C Duffy’s Lug Nuts plays on the equivalence people draw between intelligence and arithmetic ability. Also on the idea that brain size should have something particularly strong link to intelligence. Really anyone having trouble figuring out 15% of $10 is psyching themselves out. They’re too much overwhelmed by the idea of percents being complicated to realize that it’s, well, ten times 15 cents. Advertisements ## Reading the Comics, February 10, 2018: I Meant To Post This Thursday Edition Ah, yes, so, in the midst of feeling all proud that I’d gotten my Reading the Comics workflow improved, I went out to do my afternoon chores without posting the essay. I’m embarrassed. But it really only affects me looking at the WordPress Insights page. It publishes this neat little calendar-style grid that highlights the days when someone’s posted and this breaks up the columns. This can only unnerve me. I deserve it. Tom Thaves’s Frank and Ernest for the 8th of February is about the struggle to understand zero. As often happens, the joke has a lot of truth to it. Zero bundles together several ideas, overlapping but not precisely equal. And part of that is the idea of “nothing”. Which is a subtly elusive concept: to talk about the properties of a thing that does not exist is hard. As adults it’s easy to not notice this anymore. Part’s likely because mastering a concept makes one forget what it took to understand. Part is likely because if you don’t have to ponder whether the “zero” that’s “one less than one” is the same as the “zero” that denotes “what separates the count of thousands from the count of tens in the numeral 2,038” you might not, and just assume you could explain the difference or similarity to someone who has no idea. John Zakour and Scott Roberts’s Maria’s Day for the 8th has maria and another girl bonding over their hatred of mathematics. Well, at least they’re getting something out of it. The date in the strip leads me to realize this is probably a rerun. I’m not sure just when it’s from. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th proposes a prank based on mathematical use of the word “arbitrarily”. This is a word that appears a lot in analysis, and the strip makes me realize I’m not sure I can give a precise definition. An “arbitrarily large number”, for example, would be any number that’s large enough. But this also makes me realize I’m not sure precisely what joke Weinersmith is going for. I suppose that if someone were to select an arbitrarily large number they might pick 53, or a hundred, or million billion trillion. I suppose Weinersmith’s point is that in ordinary speech an arbitrarily made choice is one selection from all the possible alternatives. In mathematical speech an arbitrarily made choice reflects every possible choice. To speak of an arbitrarily large number is to say that whatever selection is made, we can go on to show this interesting stuff is true. We’d typically like to prove the most generically true thing possible. But picking a single example can be easier to prove. It can certainly be easier to visualize. 53 is probably easier to imagine than “every number 52 or larger”, for example. Ted Shearer’s Quincy for the 16th of December, 1978 was rerun the 9th of February. It just shows Quincy at work on his mathematics homework, and considering dedicating it to his grandmother. Mathematics books have dedications, just as any other book does. I’m not aware of dedications of proofs or other shorter mathematics works, but there’s likely some. There’s often a note of thanks, usually given to people who’ve made the paper’s writers think harder about the subjects. But I don’t think there’s any reason a paper wouldn’t thank someone who provided “mere” emotional support. I just don’t have examples offhand. Jef Mallet’s Frazz for the 9th looks like one of those creative-teaching exercises I sometimes see in Mathematics Education Twitter: the teacher gives answers and the students come up with story problems to match. That’s not a bad project. I’m not sure how to grade it, but I haven’t done anything that creative when I’ve taught. I’m sorry I haven’t got more to say about it since the idea seems fun. Gordon Bess’s Redeye for the 30th of September, 1971 was rerun the 10th. It’s a bit of extremely long division and I don’t blame Pokey for giving up on that problem. Starting from 5,967,342 divided by 973 I’d say, well, that’s about six million divided by a thousand, so the answer should be near six thousand. I don’t think the last digits of 2 and 3 suggest anything about what the final digit should be, if this divides evenly. So the only guidance I have is that my answer ought to be around six thousand and then we have to go into actually working. It turns out that 973 doesn’t go into 5,967,342 a whole number of times, so I sympathize more with Pokey. The answer is a little more than 6,132.9311. ## Reading the Comics, February 7, 2018: Not Taking Algebra Too Seriously Edition There were nearly a dozen mathematically-themed comic strips among what I’d read, and they almost but not quite split mid-week. Better, they include one of my favorite ever mathematics strips from Charles Schulz’s Peanuts. Jimmy Halto’s Little Iodine for the 4th of December, 1956 was rerun the 2nd of February. Little Iodine seeks out help with what seems to be story problems. The rate problem — “if it takes one man two hours to plow seven acros, how long will it take five men and a horse to … ” — is a kind I remember being particularly baffling. I think it’s the presence of three numbers at once. It seems easy to go from, say, “if you go two miles in ten minutes, how long will it take to go six miles?” to an answer. To go from “if one person working two hours plows seven acres then how long will five men take to clear fourteen acres” to an answer seems like a different kind of problem altogether. It’s a kind of problem for which it’s even wiser than usual to carefully list everything you need. Kieran Meehan’s Pros and Cons for the 5th uses a bit of arithmetic. It looks as if it’s meant to be a reminder about following the conclusions of one’s deductive logic. It’s more common to use 1 + 1 equalling 2, or 2 + 2 equalling 4. Maybe 2 times 2 being 4. But then it takes a little turn into numerology, trying to read more meaning into numbers than is wise. (I understand why people should use numerological reasoning, especially given how much mathematicians like to talk up mathematics as descriptions of reality and how older numeral systems used letters to represent words. And that before you consider how many numbers have connotations.) Charles Schulz’s Peanuts for the 5th of February reruns the strip from the 8th of February, 1971. And it is some of the best advice about finding the values of x and y, and about approaching algebra, that I have ever encountered. Mort Walker and Dik Browne’s Hi and Lois for the 10th of August, 1960 was rerun the 6th of February. It’s a counting joke. Babies do have some number sense. At least babies as old as Trixie do, I believe, in that they’re able to detect that something weird is going on when they’re shown, eg, two balls put into a box and four balls coming out. (Also it turns out that stage magicians get called in to help psychologists study just how infants and toddlers understand the world, which is neat.) John Zakour and Scott Roberts’s Maria’s Day for the 7th is Ms Payne’s disappointed attempt at motivating mathematics. I imagine she’d try going on if it weren’t a comic strip limited to two panels. ## Reading the Comics, February 3, 2018: Overworked Edition And this should clear out last week’s mathematically-themed comic strips. I didn’t realize just how busy last week had been until I looked at what I thought was a backlog of just two days’ worth of strips and it turned out to be about two thousand comics. I exaggerate, but as ever, not by much. This current week seems to be a more relaxed pace. So I’ll have to think of something to write for the Tuesday and Thursday slots. Hm. (I’ll be all right. I’ve got one thing I need to stop bluffing about and write, and there’s usually a fair roundup of interesting tweets or articles I’ve seen that I can write. Those are often the most popular articles around here.) Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st of February, 2018 gives us an anthropomorphic geometric figures joke for the week. Also a side of these figures that I don’t think I’ve seen in the newspaper comics before. It kind of raises further questions. Jason Chatfield’s Ginger Meggs for the 1st just mentions that it’s a mathematics test. Ginger isn’t ready for it. Mark Tatulli’s Heart of the City rerun for the 1st finally has some specific mathematics mentioned in Heart’s efforts to avoid a mathematics tutor. The bit about the sum of adjacent angles forming a right line being 180 degrees is an important one. A great number of proofs rely on it. I can’t deny the bare fact seems dull, though. I know offhand, for example, that this bit about adjacent angles comes in handy in proving that the interior angles of a triangle add up to 180 degrees. At least for Euclidean geometry. And there are non-Euclidean geometries that are interesting and important and for which that’s not true. Which inspires the question: on a non-Euclidean surface, like say the surface of the Earth, is it that adjacent angles don’t add up to 180 degrees? Or does something else in the proof of a triangle’s interior angles adding up to 180 degrees go wrong? The Eric the Circle rerun for the 2nd, by JohnG, is one of the occasional Erics that talk about π and so get to be considered on-topic here. Bill Whitehead’s Free Range for the 2nd features the classic page full of equations to demonstrate some hard mathematical work. And it is the sort of subject that is done mathematically. The equations don’t look to me anything like what you’d use for asteroid orbit projections. I’d expect forecasting just where an asteroid might hit the Earth to be done partly by analytic formulas that could be done on a blackboard. And then made precise by a numerical estimate. The advantage of the numerical estimate is that stuff like how air resistance affects the path of something in flight is hard to deal with analytically. Numerically, it’s tedious, but we can let the computer deal with the tedium. So there’d be just a boring old computer screen to show on-panel. Bud Fisher’s Mutt and Jeff reprint for the 2nd is a little baffling. And not really mathematical. It’s just got a bizarre arithmetic error in it. Mutt’s fiancee Encee wants earrings that cost ten dollars (each?) and Mutt takes this to be fifty dollars in earring costs and I have no idea what happened there. Thomas K Dye, the web cartoonist who’s done artwork for various article series, has pointed out that the lettering on these strips have been redone with a computer font. (Look at the letters ‘S’; once you see it, you’ll also notice it in the slightly lumpy ‘O’ and the curly-arrow ‘G’ shapes.) So maybe in the transcription the earring cost got garbled? And then not a single person reading the finished product read it over and thought about what they were doing? I don’t know. Zach Weinersmith’s Saturday Morning Breakfast Cereal reprint for the 2nd is based, as his efforts to get my attention often are, on a real mathematical physics postulate. As the woman postulates: given a deterministic universe, with known positions and momentums of every particle, and known forces for how all these interact, it seems like it should be possible to predict the future perfectly. It would also be possible to “retrodict” the past. All the laws of physics that we know are symmetric in time; there’s no reason you can’t predict the motion of something one second into the past just as well as you an one second into the future. This fascinating observation took a lot of battery in the 19th century. Many physical phenomena are better described by statistical laws, particularly in thermodynamics, the flow of heat. In these it’s often possible to predict the future well but retrodict the past not at all. But that looks as though it’s a matter of computing power. We resort to a statistical understanding of, say, the rings of Saturn because it’s too hard to track the billions of positions and momentums we’d need to otherwise. A sufficiently powerful mathematician, for example God, would be able to do that. Fair enough. Then came the 1890s. Henri Poincaré discovered something terrifying about deterministic systems. It’s possible to have chaos. A mathematical representation of a system is a bit different from the original system. There’s some unavoidable error. That’s bound to make some, larger, error in any prediction of its future. For simple enough systems, this is okay. We can make a projection with an error as small as we need, at the cost of knowing the current state of affairs with enough detail. Poincaré found that some systems can be chaotic, though, ones in which any error between the current system and its representation will grow to make the projection useless. (At least for some starting conditions.) And so many interesting systems are chaotic. Incredibly simplified models of the weather are chaotic; surely the actual thing is. This implies that God’s projection of the universe would be an amusing but almost instantly meaningless toy. At least unless it were a duplicate of the universe. In which case we have to start asking our philosopher friends about the nature of identity and what a universe is, exactly. Ruben Bolling’s Super-Fun-Pak Comix for the 2nd is an installment of Guy Walks Into A Bar featuring what looks like an arithmetic problem to start. It takes a turn into base-ten jokes. There are times I suspect Ruben Bolling to be a bit of a nerd. Nate Fakes’s Break of Day for the 3rd looks like it’s trying to be an anthropomorphic-numerals joke. At least it’s an anthropomorphic something joke. Percy Crosby’s Skippy for the 3rd originally ran the 8th of December, 1930. It alludes to one of those classic probability questions: what’s the chance that in your lungs is one of the molecules exhaled by Julius Caesar in his dying gasp? Or whatever other event you want: the first breath you ever took, or something exhaled by Jesus during the Sermon on the Mount, or exhaled by Sue the T-Rex as she died. Whatever. The chance is always surprisingly high, which reflects the fact there’s a lot of molecules out there. This also reflects a confidence that we can say one molecule of air is “the same” as some molecule if air in a much earlier time. We have to make that supposition to have a problem we can treat mathematically. My understanding is chemists laugh at us if we try to suggest this seriously. Fair enough. But whether the air pumped out of a bicycle tire is ever the same as what’s pumped back in? That’s the same kind of problem. At least some of the molecules of air will be the same ones. Pretend “the same ones” makes sense. Please. ## Reading the Comics, January 31, 2018: Workload Edition I thought my new workflow of writing my paragraph or two about each comic was going to help me keep up and keep fresher with the daily comics. And then Comic Strip Master Command decided that everybody had to do comics that at least touched on some mathematical subject. I don’t know. I’m trying to keep up but will admit, I didn’t get to writing anything about Friday’s or Saturday’s strips yet. They’ll keep a couple days. Bill Amend’s FoxTrot Classicsfor the 29th of January reprints the strip from the 5th of February, 1996. (The Classics reprints finally reached the point where Amend retired from daily strips, and jumped back a dozen years to continue printing.) It just mentions mathematics exams, and high performances on both is all. Josh Shalek’s Kid Shay Comics reprint for the 29th tosses off a mention of Uncle Brian attempting a great mathematical feat. In this case it’s the Grand Unification Theory, some logically coherent set of equations that describe the fundamental forces of the universe. I think anyone with a love for mathematics makes a couple quixotic attempts on enormously vast problems like this. Or the Riemann Hypothesis, or Goldbach’s Conjecture, or Fermat’s Last Theorem. Yes, Fermat’s Last Theorem has been proven, but there’s no reason there couldn’t be an easier proof. Similarly there’s no reason there couldn’t be a better proof of the Four Color Map theorem. Most of these attempts end up the way Brian’s did. But there’s value in attempting this anyway. Even when you fail, you can have fun and learn fascinating things in the attempt. Carol Lay’s Lay Lines for the 29th is a vignette about a statistician. And one of those statisticians with the job of finding surprising correlations between things. I think it’s also a riff on the hypothesis that free markets are necessarily perfect: if there’s any advantage to doing something one way, it’ll quickly be found and copied until that is the normal performance of the market. Anyone doing better than average is either taking advantage of concealed information, or else is lucky. Matt Lubchansky’s Please Listen To Me for the 29th depicts a person doing statistical work for his own purposes. In this case he’s trying to find what factors might be screwing up the world. The expressions in the second panel don’t have an obvious meaning to me. The start of the expression $\int exp\left(\frac{1}{N_0}\right)$ at the top line suggests statistical mechanics to me, for what that’s worth, and the H and Ψ underneath suggest thermodynamics or quantum mechanics. So if Lubchansky was just making up stuff, he was doing it with a good eye for mathematics that might underly everything. Rick Stromoski’s Soup to Nutz for the 29th circles around the anthropomorphic numerals idea. It’s not there exactly, but Andrew is spending some time giving personality to numerals. I can’t say I give numbers this much character. But there are numbers that seem nicer than others. Usually this relates to what I can do with the numbers. 10, for example, is so easy to multiply or divide by. If I need to multiply a number by, say, something near thirty, it’s a delight to triple it and then multiply by ten. Twelve and 24 and 60 are fun because they’re so relatively easy to find parts of. Even numbers often do seem easier to work with, just because splitting an even number in half saves us from dealing with decimals or fractions. Royboy sees all this as silliness, which seems out of character for him, really. I’d expect him to be up for assigning traits to numbers like that. Bill Griffith’s Zippy the Pinhead for the 30th mentions Albert Einstein and relativity. And Zippy ruminates on the idea that there’s duplicates of everything, in the vastness of the universe. It’s an unsettling idea that isn’t obviously ruled out by mathematics alone. There’s, presumably, some chance that a bunch of carbon and hydrogen and oxygen and other atoms happened to come together in such a way as to make our world as we know it today. If there’s a vast enough universe, isn’t there a chance that a bunch of carbon and hydrogen and oxygen and other atoms happened to come together that same way twice? Three times? If the universe is infinitely large, might it not happen infinitely many times? In any number of variations? It’s hard to see why not, but even if it is possible, that’s no reason to think it must happen either. And whether those duplicates are us is a question for philosophers studying the problem of identity and what it means to be one person rather than some other person. (It turns out to be a very difficult problem and I’m glad I’m not expected to offer answers.) Tony Cochrane’s Agnes attempts to use mathematics to reason her way to a better bedtime the 31st. She’s not doing well. Also this seems like it’s more of an optimization problem than a simple arithmetic one. What’s the latest bedtime she can get that still allows for everything that has to be done, likely including getting up in time and getting enough sleep? Also, just my experience but I didn’t think Agnes was old enough to stay up until 10 in the first place. ## Reading the Comics, January 27, 2018: Working Through The Week Edition And today I bring the last couple mathematically-themed comic strips sent my way last week. GoComics has had my comics page working intermittently this week. And I was able to get a response from them, by e-mailing their international sales office, the only non-form contact I could find. Anyway, this flood of comics does take up the publishing spot I’d figured for figuring how I messed up Wronski’s formula. But that’s all right, as I wanted to spend more time thinking about that. Here’s hoping spending more time thinking works out for me. Nate Fakes’s Break of Day for the 24th was the big anthropomorphic numerals joke for the week. And it’s even dubbed the numbers game. Mark Tatulli’s Heart of the City from the 24th got into a storyline about Heart needing a mathematics tutor. It’s a rerun sequence, although if you remember a particular comic storyline from 2009 you’re doing pretty well. Nothing significantly mathematical has turned up in the story so far, past the mention of fractions as things that exist and torment students. But the stories are usually pretty good for this sort of strip. Mikael Wulff and Anders Morganthaler’s WuMo for the 24th includes a story problems freak out. I’m not sure what’s particularly implausible about buying nine apples. I’d agree a person is probably more likely to buy an even number of things, since we seem to like numbers like “ten” and “eight” so well, but it’s hardly ridiculous. Tim Rickard’s Brewster Rockit for the 25th is an arithmetic class on the Snowman Planet. So there’s some finger-counting involved. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a reminder that most of my days are spent seeing how Zach Weinersmith wants my attention. It also includes what I suppose is a legitimate attempt to offer a definition for what all mathematics is. It’s hard to come up with something that does cover all the stuff mathematicians do. Bear in mind, this includes counting, calculating how far the Sun is based on the appearance of a lunar eclipse, removing static from a recording, and telling how many queens it’s possible to place eight queens on a chess board that’s wrapped around a torus without any being able to capture another, among other problems. My instinct is to dismiss the proposed “anything you can think deeply about that has no reference to the real world”. That seems over-broad, and to cover a lot of areas that are really philosophy’s beat. And I think there’s something unseemly in mathematicians gloating about their work having no “practical” use. I grant I come from an applied school, and I came to there through an interest in physics. But to build up “inapplicability to the real word” as if it were some ideal, as opposed to just how something has turned out to be right now, strikes me as silly. Applicability is so dependent on context, on culture, and accidents of fate that there’s no way it can be important to characterizing mathematics. And it would imply that once we found a use for something it would stop being mathematically interesting. I don’t see evidence of that in mathematical history. Mikael Wulff and Anders Morganthaler’s WuMo pops back in on the 27th with an appearance of sudoku, presenting the logic puzzle as one of the many things beyond the future Disgraced Former President’s abilities. ## Reading the Comics, January 23, 2018: Adult Content Edition I was all set to say how complaining about GoComics.com’s pages not loading had gotten them fixed. But they only worked for Monday alone; today they’re broken again. Right. I haven’t tried sending an error report again; we’ll see if that works. Meanwhile, I’m still not through last week’s comic strips and I had just enough for one day to nearly enough justify an installment for the one day. Should finish off the rest of the week next essay, probably in time for next week. Mark Leiknes’s Cow and Boy rerun for the 23rd circles around some of Zeno’s Paradoxes. At the heart of some of them is the question of whether a thing can be divided infinitely many times, or whether there must be some smallest amount of a thing. Zeno wonders about space and time, but you can do as well with substance, with matter. Mathematics majors like to say the problem is easy; Zeno just didn’t realize that a sum of infinitely many things could be a finite and nonzero number. This misses the good question of how the sum of infinitely many things, none of which are zero, can be anything but infinitely large? Or, put another way, what’s different in adding $\frac11 + \frac12 + \frac13 + \frac14 + \cdots$ and adding $\frac11 + \frac14 + \frac19 + \frac{1}{16} + \cdots$ that the one is infinitely large and the other not? Or how about this. Pick your favorite string of digits. 23. 314. 271828. Whatever. Add together the series $\frac11 + \frac12 + \frac13 + \frac14 + \cdots$except that you omit any terms that have your favorite string there. So, if you picked 23, don’t add $\frac{1}{23}$, or $\frac{1}{123}$, or $\frac{1}{802301}$ or such. That depleted series does converge. The heck is happening there? (Here’s why it’s true for a single digit being thrown out. Showing it’s true for longer strings of digits takes more work but not really different work.) J C Duffy’s Lug Nuts for the 23rd is, I think, the first time I have to give a content warning for one of these. It’s a porn-movie advertisement spoof. But it mentions Einstein and Pi and has the tagline “she didn’t go for eggheads … until he showed her a new equation!”. So, you know, it’s using mathematics skill as a signifier of intelligence and riffing on the idea that nerds like sex too. John Graziano’s Ripley’s Believe It or Not for the 23rd has a trivia that made me initially think “not”. It notes Vince Parker, Senior and Junior, of Alabama were both born on Leap Day, the 29th of February. I’ll accept this without further proof because of the very slight harm that would befall me were I to accept this wrongly. But it also asserted this was a 1-in-2.1-million chance. That sounded wrong. Whether it is depends on what you think the chance is of. Because what’s the remarkable thing here? That a father and son have the same birthday? Surely the chance of that is 1 in 365. The father could be born any day of the year; the son, also any day. Trusting there’s no influence of the father’s birthday on the son’s, then, 1 in 365 it is. Or, well, 1 in about 365.25, since there are leap days. There’s approximately one leap day every four years, so, surely that, right? And not quite. In four years there’ll be 1,461 days. Four of them will be the 29th of January and four the 29th of September and four the 29th of August and so on. So if the father was born any day but leap day (a “non-bissextile day”, if you want to use a word that starts a good fight in a Scrabble match), the chance the son’s birth is the same is 4 chances in 1,461. 1 in 365.25. If the father was born on Leap Day, then the chance the son was born the same day is only 1 chance in 1,461. Still way short of 1-in-2.1-million. So, Graziano’s Ripley’s is wrong if that’s the chance we’re looking at. Ah, but what if we’re looking at a different chance? What if we’re looking for the chance that the father is born the 29th of February and the son is also born the 29th of February? There’s a 1-in-1,461 chance the father’s born on Leap Day. And a 1-in-1,461 chance the son’s born on Leap Day. And if those events are independent, the father’s birth date not influencing the son’s, then the chance of both those together is indeed 1 in 2,134,521. So Graziano’s Ripley’s is right if that’s the chance we’re looking at. Which is a good reminder: if you want to work out the probability of some event, work out precisely what the event is. Ordinary language is ambiguous. This is usually a good thing. But it’s fatal to discussing probability questions sensibly. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 23rd presents his mathematician discovering a new set of numbers. This will happen. Mathematics has had great success, historically, finding new sets of things that look only a bit like numbers were understood. And showing that if they follow rules that are, as much as possible, like the old numbers, we get useful stuff out of them. The mathematician claims to be a formalist, in the punch line. This is a philosophy that considers mathematical results to be the things you get by starting with some symbols and some rules for manipulating them. What this stuff means, and whether it reflects anything of interest in the real world, isn’t of interest. We can know the results are good because they follow the rules. This sort of approach can be fruitful. It can force you to accept results that are true but intuition-defying. And it can give results impressive confidence. You can even, at least in principle, automate the creating and the checking of logical proofs. The disadvantages are that it takes forever to get anything done. And it’s hard to shake the idea that we ought to have some idea what any of this stuff means. ## Reading the Comics, January 22, 2018: Breaking Workflow Edition So I was travelling last week, and this threw nearly all my plans out of whack. We stayed at one of those hotels that’s good enough that its free Internet is garbage and they charge you by day for decent Internet. So naturally Comic Strip Master Command sent a flood of posts. I’m trying to keep up and we’ll see if I wrap up this past week in under three essays. And I am not helped, by the way, by GoComics.com rejiggering something on their server so that My Comics Page won’t load, and breaking their “Contact Us” page so that that won’t submit error reports. If someone around there can break in and turn one of their servers off and on again, I’d appreciate the help. Hy Eisman’s Katzenjammer Kids for the 21st of January is a curiously-timed Tax Day joke. (Well, the Katzenjammer Kids lapsed into reruns a dozen years ago and there’s probably not much effort being put into selecting seasonally appropriate ones.) But it is about one of the oldest and still most important uses of mathematics, and one that never gets respect. Morrie Turner’s Wee Pals rerun for the 21st gets Oliver the reputation for being a little computer because he’s good at arithmetic. There is something that amazes in a person who’s able to calculate like this without writing anything down or using a device to help. Steve Kelley and Jeff Parker’s Dustin for the 22nd seems to be starting off with a story problem. It might be a logic problem rather than arithmetic. It’s hard to say from what’s given. Mark Anderson’s Andertoons for the 22nd is the Mark Anderson’s Andertoons for the week. Well, for Monday, as I write this. It’s got your classic blackboard full of equations for the people in over their head. The equations look to me like gibberish. There’s a couple diagrams of aromatic organic compounds, which suggests some quantum-mechanics chemistry problem, if you want to suppose this could be narrowed down. Greg Evans’s Luann Againn for the 22nd has Luann despair about ever understanding algebra without starting over from scratch and putting in excessively many hours of work. Sometimes it feels like that. My experience when lost in a subject has been that going back to the start often helps. It can be easier to see why a term or a concept or a process is introduced when you’ve seen it used some, and often getting one idea straight will cause others to fall into place. When that doesn’t work, trying a different book on the same topic — even one as well-worn as high school algebra — sometimes helps. Just a different writer, or a different perspective on what’s key, can be what’s needed. And sometimes it just does take time working at it all. Richard Thompson’s Richard’s Poor Almanac rerun for the 22nd includes as part of a kit of William Shakespeare paper dolls the Typing Monkey. It’s that lovely, whimsical figure that might, in time, produce any written work you could imagine. I think I’d retired monkeys-at-typewriters as a thing to talk about, but I’m easily swayed by Thompson’s art and comic stylings so here it is. Darrin Bell and Theron Heir’s Rudy Park for the 18th throws around a lot of percentages. It’s circling around the sabermetric-style idea that everything can be quantified, and measured, and that its changes can be tracked. In this case it’s comments on Star Trek: Discovery, but it could be anything. I’m inclined to believe that yeah, there’s an astounding variety of things that can be quantified and measured and tracked. But it’s also easy, especially when you haven’t got a good track record of knowing what is important to measure, to start tracking what amounts to random noise. (See any of my monthly statistics reviews, when I go looking into things like views-per-visitor-per-post-made or some other dubiously meaningful quantity.) So I’m inclined to side with Randy and his doubts that the Math Gods sanction this much data-mining. ## Reading the Comics, January 20, 2018: Increased Workload Edition It wasn’t much of an increased workload, really. I mean, none of the comics required that much explanation. But Comic Strip Master Command donated enough topics to me last week that I have a second essay for the week. And here it is; sorry there’s no pictures. Mark Anderson’s Andertoons for the 17th is the Mark Anderson’s Andertoons we’ve been waiting for. It returns to fractions and their frustrations for its comic point. Jef Mallet’s Frazz for the 17th talks about story problems, although not to the extent of actually giving one as an example. It’s more about motivating word-problem work. Mike Thompson’s Grand Avenue for the 17th is an algebra joke. I’d call it a cousin to the joke about mathematics’s ‘x’ not coming back and we can’t say ‘y’. On the 18th was one mentioning mathematics, although in a joke structure that could have been any subject. Lorrie Ransom’s The Daily Drawing for the 18th is another name-drop of mathematics. I guess it’s easier to use mathematics as the frame for saying something’s just a “problem”. I don’t think of, say, identifying the themes of a story as a problem in the way that finding the roots of a quadratic is. Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 18th is an anthropomorphic-geometric-figures joke that I’m all but sure is a rerun I’ve shared here before. I’ll try to remember to check before posting this. Mikael Wulff and Anders Morgenthaler’s WuMo for the 20th gives us a return of the pie chart joke that seems like it’s been absent a while. Worth including? Eh, why not. ## Reading the Comics, January 16, 2017: Better Workflow Edition So one little secret of my Reading the Comics posts is I haven’t been writing them in a way that makes sense to me. To me, I should take each day’s sufficiently relevant comics, describe them in a paragraph or two, and then have a nice pile of text all ready for the posting Sunday and, if need be, later. I haven’t been doing that. I’ve let links pile up until Friday or Saturday, and then try to process them all, and if you’ve ever wondered why the first comic of the week gets 400 words about some subtlety while the last gets “this is a comic that exists”, there you go. This time around, let me try doing each day’s strips per day and see how that messes things up. Jef Mallett’s Frazz for the 14th of January is another iteration of the “when will we ever use mathematics” complaint. The answer of “you’ll use it on the test” is unsatisfactory. But somehow, the answer of “you’ll use it to think deeply about something you had never considered before” also doesn’t satisfy. Anyway I’d like to see the idea that education is job-training abolished; I think it should be about making a person conversant with the history of human thought. That can’t be done perfectly, and we might ask whether factoring 32 is that important a piece, but it should certainly be striven for. Ham’s Life on Earth for the 14th is a Gary Larsonesque riff on that great moment of calculus and physics history, Newton’s supposition that gravity has to follow a universally true law. I’m not sure this would have made my cut if I reviewed a week’s worth of strips at a time. Hm. Mason Mastroianni’s B.C. for the 15th is a joke about story problem construction, and how the numbers in a story problem might be obvious nonsense. It’s also a cheap shot at animal hoarders, I suppose, but that falls outside my territory here. Anthony Blades’s Bewley rerun for the 15th riffs on the natural number sense we all have. And we do have a number sense, remarkably. We might not be able to work out 9 times 6 instantly. But asked to pick from a list of possible values, we’re more likely to think that 58 is credible than that 78 or 38 are. It’s quite imprecise, but isn’t it amazing that it’s there at all? Bill Amend’s FoxTrot Classics for the 15th is a story problem joke, in this case, creating one with a strong motivation for its solution to be found. The strip originally ran the 22nd of January, 1996. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 16th is maybe marginal to include, too. It’s about the kinds of logic puzzles that mathematicians grow up reading and like to pass around. And the way you can fake out someone by presenting a problem with too obvious a solution. It’s not just professors who’ll be stymied by having the answer look too obvious, by the way. Everyone’s similarly vulnerable. To see anything, including an abstract thing like the answer to a puzzle, you need some idea of what you are looking at. If you don’t think the answer could be something that simple, you won’t see it there. Gordon Bess’s Redeye for the 6th of September, 1971, was reprinted the 17th. It’s about the fun of teaching a subject you aren’t all that good on yourself. The mathematics is a name-drop here, but the joke wouldn’t make sense if it were about social studies. Elzie Segar’s Thimble Theatre for the 10th of August, 1931, was also reprinted the 17th. It’s an old gag, even back when it was first run. But I suppose there’s some numerical-conversion mathematics to wring out of it. Given the rate of exchange, a pezozee would seem to be 24 pazimees. I’m not sure we need so many units in-between the pazimee and the pezozee, but perhaps King Blozo’s land set its units in a time when fractions were less familiar to the public. The punch line depends on the pazimee being worth nothing and, taken literally, that has sad implications for the pezozee too. If you take the King as speaking roughly, though, sixteen times a small amount is … at least a less small amount. It wouldn’t take many doublings to go from an infinitesimally tiny sum to a respectable one. And it turns out there were enough comic strips I need to split this into two segments. So I should schedule that to appear. It’s already written and everything. ## Reading the Comics, January 13, 2018: Barney Google Is Messing With My Head For Some Reason Edition I do not know what’s possessed John Rose, cartoonist for Barney Google and Snuffy Smith — possibly the oldest syndicated comic strip not in perpetual reruns — to decide he needs to mess with my head. So far as I’m aware we haven’t ever even had any interactions. While I’ll own up to snarking about the comic strip here and there, I mean, the guy draws Barney Google and Snuffy Smith. He won’t attract the snark community of, say, Marmaduke, but he knew the job was dangerous when he took it. There’s lots of people who’ve said worse things about the comic than I ever have. He can’t be messing with them all. There’s no mathematical content to it, but here, continuing the curious thread of Elviney and Miss Prunelly looking the same, and Elviney turning out to have a twin sister, is the revelation that Elviney’s husband also has a twin. This means something and I don’t know what. To mathematics: Zach Weinersmith’s Saturday Morning Breakfast Cereal gets my attention again for the 10th. There is this famous quotation from Leopold Kronecker, one of the many 19th century German mathematicians who challenged, and set, our ideas of what mathematics is. In debates about what should count as a proof Kronecker said something translated in English to, “God created the integers, all else is the work of man”. He favored proofs that only used finite numbers, and only finitely many operations, and was skeptical of existence proofs. Those are ones that show something with desired properties must exist, without necessarily showing how to find it. Most mathematicians accept existence proofs. If you can show how to find that thing, that’s a constructive proof. Usually mathematicians like those better. Mark Tatulli’s Heart of the City for the 11th uses a bunch of arithmetic and word problems to represent all of Dean’s homework. All looks like reasonable homework for my best guess about his age. Jon Rosenberg’s Scenes From A Multiverse for the 11th is a fun, simple joke with some complex stuff behind it. It’s riffing on the kind of atheist who wants moral values to come from something in the STEM fields. So here’s a mathematical basis for some moral principles. There are, yes, ethical theories that have, or at least imply having, mathematics behind them. Utilitarianism at least supposes that ethical behavior can be described as measurable and computable quantities. Nobody actually does that except maybe to make video games more exciting. But it’s left with the idea that one could, and hope that this would lead to guidance that doesn’t go horribly wrong. Don Asmussen’s Bad Reporter for the 12th uses knowledge of arithmetic as a signifier of intelligence. Common enough joke style. Thom Bluemel’s Birdbrains for the 13th starts Pi Day observances early, or maybe supposed the joke would be too out of season were it to come in March. Greg Evans and Karen Evans’s Luann for the 13th uses mathematics to try building up the villainy of one of the strip’s designated villains. Ann Eiffel, there, uses a heap of arithmetic to make her lingerie sale sound better. This isn’t simply a riff on people not wanting to do arithmetic, although I understand people not wanding to work out what five percent of a purchase of over$200 is. There’s a good deal of weird psychology in getting people to buy things. Merely naming a number, for example, gets people to “anchor” their expectations to it. To speak of a free gift worth $75 makes any purchase below$75 seem more economical. To speak of a chance to win $1,000 prepares people to think they’ve got a thousand dollars coming in, and that they can safely spend under that. It’s amazing stuff to learn about, and it isn’t all built on people being too lazy to figure out what five percent off of$220 would be.

T Lewis and Michael Fry’s Over the Hedge for the 13th uses &infty; along the way to making nonsense out of ice-skating judging. It’s a good way to make a hash of a rating system. Most anything done with infinitely large numbers or infinitely large sets challenges one’s intuition at least. This is part of what Leopold Kronecker was talking about.

## Reading the Comics, January 9, 2018: Be Squared Edition

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.

Mark O’Hare’s Citizen Dog rerun for the 7th continues last essay’s storyline about Fergus taking Maggie’s place at school. He’s having trouble understanding the story within a story problem. I sympathize.

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.

Tim Lachowski’s Get A Life for the 9th is at least the fourth time I’ve seen this panel since I started doing Reading the Comics posts regularly. (Previous times: the 5th of November, 2012 and the 10th of March, 2015 and the 14th of July, 2016.) I’m like this close to concluding the strip’s in perpetual rerun and I can drop it from my daily reading.

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.

Charles Schulz’s Peanuts for the 8th of October, 1953, is about practical uses of mathematics. It got rerun on the 9th of January.

## Reading the Comics, January 6, 2018: Terms Edition

The last couple days of last week saw a rush of comics, although most of them were simpler things to describe. Bits of play on words, if you like.

Samson’s Dark Side of the Horse for the 4th of January, 2018, is one that plays on various meanings of “average”. The mean, alluded to in the first panel, is the average most people think of first. Where you have a bunch of values representing instances of something, add up the values, and divide by the number of instances. (Properly that’s the arithmetic mean. There’s some others, such as the geometric mean, but if someone’s going to use one of those they give you clear warning.) The median, in the second, is the midpoint, the number that half of all instances are less than. So you see the joke. If the distribution of intelligence is normal — which is a technical term, although it does mean “not freakish” — then the median and the mean should be equal. If you had infinitely many instances, and they were normally distributed, the two would be equal. With finitely many instances, the mean and the median won’t be exactly in line, for the same reason if you fairly toss a coin two million times it won’t turn up heads exactly one million times.

Dark Side of the Horse for the 5th delivers the Roman numerals joke of the year. And I did have to think about whether ‘D’ is a legitimate Roman numeral. This would be easier to remember before 1900.

Mike Lester’s Mike du Jour for the 4th is geometry wordplay. I’m not sure the joke stands up to scrutiny, but it lands well enough initially.

Johnny Hart’s Back to BC for the 5th goes to the desire to quantify and count things. And to double-check what other people tell you about this counting. It’s easy, today, to think of the desire to quantify things as natural to humans. I’m not confident that it is. The history of statistics shows this gradual increase in the number and variety of things getting tracked. This strip originally ran the 11th of July, 1960.

Bill Watterson’s Calvin and Hobbes for the 5th talks about averages again. And what a population average means for individuals. It doesn’t mean much. The glory of statistics is that groups are predictable in a way that individuals are not.

John Graziano’s Ripley’s Believe It Or Not for the 5th features a little arithmetic coincidence, that multiplying 21,978 by four reverses its digits. It made me think of Ray Kassinger’s question the other day about parasitic numbers. But this isn’t a parasitic number. A parasitic number is one with a value, multiplied by a particular number, that’s the same as you get by moving its last digit to the front. Flipping the order of digits seems like it should be something and I don’t know what.

Mark Anderson’s Andertoons for the 6th is a confident reassurance that 2018 is a normal, healthy year after all. Or can be. Prime numbers.

Mark O’Hare’s Citizen Dog rerun for the 6th is part of a sequence in which Fergus takes a (human) child’s place in school. Mathematics gets used as a subject that’s just a big pile of unfamiliar terms if you just jump right in. Most subjects are like this if you take them seriously, of course. But mathematics has got an economy of technical terms to stuff into people’s heads, and that have to be understood to make any progress. In grad school my functional analysis professor took great mercy on us, and started each class with re-writing the definitions of all the technical terms introduced the previous class. Also of terms that might be a bit older, but that are important to get right, which is why I got through it confident I knew what a Sobolev Space was. (It’s a collection of functions that have enough derivatives to do your differential equations problem.) Numerator and denominator, we’re experts on by now.

## Reading the Comics, January 3, 2018: Explaining Things Edition

There were a good number of mathematically-themed comic strips in the syndicated comics last week. Those from the first part of the week gave me topics I could really sink my rhetorical teeth into, too. So I’m going to lop those off into the first essay for last week and circle around to the other comics later on.

Jef Mallett’s Frazz started a week of calendar talk on the 31st of December. I’ve usually counted that as mathematical enough to mention here. The 1st of January as we know it derives, as best I can figure, from the 1st of January as Julius Caesar established for 45 BCE. This was the first Roman calendar to run basically automatically. Its length was quite close to the solar year’s length. It had leap days added according to a rule that should have been easy enough to understand (one day every fourth year). Before then the Roman calendar year was far enough off the solar year that they had to be kept in synch by interventions. Mostly, by that time, adding a short extra month to put things more nearly right. This had gotten all confusingly messed up and Caesar took the chance to set things right, running 46 BCE to 445 days long.

But why 445 and not, say, 443 or 457? And I find on research that my recollection might not be right. That is, I recall that the plan was to set the 1st of January, Reformed, to the first new moon after the winter solstice. A choice that makes sense only for that one year, but, where to set the 1st is literally arbitrary. While that apparently passes astronomical muster (the new moon as seen from Rome then would be just after midnight the 2nd of January, but hitting the night of 1/2 January is good enough), there’s apparently dispute about whether that was the objective. It might have been to set the winter solstice to the 25th of December. Or it might have been that the extra days matched neatly the length of two intercalated months that by rights should have gone into earlier years. It’s a good reminder of the difficulty of reading motivation.

Brian Fies’s The Last Mechanical Monster for the 1st of January, 2018, continues his story about the mad scientist from the Fleischer studios’ first Superman cartoon, back in 1941. In this panel he’s describing how he realized, over the course of his long prison sentence, that his intelligence was fading with age. He uses the ability to do arithmetic in his head as proof of that. These types never try naming, like, rulers of the Byzantine Empire. Anyway, to calculate the cube root of 50,653 in his head? As he used to be able to do? … guh. It’s not the sort of mental arithmetic that I find fun.

But I could think of a couple ways to do it. The one I’d use is based on a technique called Newton-Raphson iteration that can often be used to find where a function’s value is zero. Raphson here is Joseph Raphson, a late 17th century English mathematician known for the Newton-Raphson method. Newton is that falling-apples fellow. It’s an iterative scheme because you start with a guess about what the answer would be, and do calculations to make the answer better. I don’t say this is the best method, but it’s the one that demands me remember the least stuff to re-generate the algorithm. And it’ll work for any positive number ‘A’ and any root, to the ‘n’-th power.

So you want the n-th root of ‘A’. Start with your current guess about what this root is. (If you have no idea, try ‘1’ or ‘A’.) Call that guess ‘x’. Then work out this number:

$\frac{1}{n}\left( (n - 1) \cdot x + \frac{A}{x^{n - 1}} \right)$

Ta-da! You have, probably, now a better guess of the n-th root of ‘A’. If you want a better guess yet, take the result you just got and call that ‘x’, and go back calculating that again. Stop when you feel like your answer is good enough. This is going to be tedious but, hey, if you’re serving a prison term of the length of US copyright you’ve got time. (It’s possible with this sort of iterator to get a worse approximation, although I don’t think that happens with n-th root process. Most of the time, a couple more iterations will get you back on track.)

But that’s work. Can we think instead? Now, most n-th roots of whole numbers aren’t going to be whole numbers. Most integers aren’t perfect powers of some other integer. If you think 50,653 is a perfect cube of something, though, you can say some things about it. For one, it’s going to have to be a two-digit number. 103 is 1,000; 1003 is 1,000,000. The second digit has to be a 7. 73 is 343. The cube of any number ending in 7 has to end in 3. There’s not another number from 1 to 9 with a cube that ends in 3. That’s one of those things you learn from playing with arithmetic. (A number ending in 1 cubes to something ending in 1. A number ending in 2 cubes to something ending in 8. And so on.)

So the cube root has to be one of 17, 27, 37, 47, 57, 67, 77, 87, or 97. Again, if 50,653 is a perfect cube. And we can do better than saying it’s merely one of those nine possibilities. 40 times 40 times 40 is 64,000. This means, first, that 47 and up are definitely too large. But it also means that 40 is just a little more than the cube root of 50,653. So, if 50,653 is a perfect cube, then it’s most likely going to be the cube of 37.

Bill Watterson’s Calvin and Hobbes rerun for the 2nd is a great sequence of Hobbes explaining arithmetic to Calvin. There is nothing which could be added to Hobbes’s explanation of 3 + 8 which would make it better. I will modify Hobbes’s explanation of what the numerator. It’s ridiculous to think it’s Latin for “number eighter”. The reality is possibly more ridiculous, as it means “a numberer”. Apparently it derives from “numeratus”, meaning, “to number”. The “denominator” comes from “de nomen”, as in “name”. So, you know, “the thing that’s named”. Which does show the terms mean something. A poet could turn “numerator over denominator” into “the number of parts of the thing we name”, or something near enough that.

Hobbes continues the next day, introducing Calvin to imaginary numbers. The term “imaginary numbers” tells us their history: they looked, when first noticed in formulas for finding roots of third- and fourth-degree polynomials, like obvious nonsense. But if you carry on, following the rules as best you can, that nonsense would often shake out and you’d get back to normal numbers again. And as generations of mathematicians grew up realizing these acted like numbers we started to ask: well, how is an imaginary number any less real than, oh, the square root of six?

Hobbes’s particular examples of imaginary numbers — “eleventenn” and “thirty-twelve” — are great-sounding compositions. They put me in mind, as many of Watterson’s best words do, of a 1960s Peanuts in which Charlie Brown is trying to help Sally practice arithmetic. (I can’t find it online, as that meme with edited text about Sally Brown and the sixty grapefruits confounds my web searches.) She offers suggestions like “eleventy-Q” and asks if she’s close, which Charlie Brown admits is hard to say.

And finally, James Allen’s Mark Trail for the 3rd just mentions mathematics as the subject that Rusty Trail is going to have to do some work on instead of allowing the experience of a family trip to Mexico to count. This is of extremely marginal relevance, but it lets me include a picture of a comic strip, and I always like getting to do that.

Are … are [ the Smiths’ next-door neighbor Elviney and Jughaid’s teacher Miss Prunelly ] the same character, just wearing different glasses? I’ve been reading this comic strip for like forty years and I’ve never noticed this before.

Really. Apart from their accessories the characters are the same.

And then, published by John Rose today, the 3rd of January, 2018:

So. Per the US Navy’s Julian Date converter today is 2458121.5. The 17th of September was Julian Date 2458013.5. (Never try to work out the difference between two dates by yourself. Use a Julian Date converter.) So that’s 108 days, or just over 15 weeks. Good to know.

## Reading the Comics, December 30, 2017: Looking To 2018 Edition

The last full week of 2017 was also a slow one for mathematically-themed comic strips. You can tell by how many bits of marginally relevant stuff I include. In this case, it also includes a couple that just mention the current or the upcoming year. So you’ve been warned.

Mac King and Bill King’s Magic in a Minute activity for the 24th is a logic puzzle. I’m not sure there’s deep mathematics to it, but it’s some fun to reason out.

John Graziano’s Ripley’s Believe It Or Not for the 24th mentions the bit of recreational group theory that normal people know, the Rubik’s Cube. The group theory comes in from rotations: you can take rows or columns on the cube and turn them, a quarter or a half or a three-quarters turn. Which rows you turn, and which ways you turn them, form a group. So it’s a toy that inspires deep questions. Who wouldn’t like to know in how few moves a cube could be solved? We know there are at least some puzzles that take 18 moves to solve. (You can calculate the number of different cube arrangements there are, and how many arrangements you could make by shuffling a cube around with 17 moves. There’s more possible arrangements than there are ones you can get to in 17 moves; therefore, there must be at least one arrangement that takes 18 moves to solve.) A 2010 computer-assisted proof by Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge showed that at most 20 face turns are needed for every possible cube to be solved. I don’t know if there’s been any success figuring out whether 19 or even 18 is necessarily enough.

Bill Griffith’s Zippy the Pinhead for the 26th just mentions algebra as a thing that Griffith can’t really remember, even in one of his frequent nostalgic fugues. I don’t know that Zippy’s line about the fifth dimension is meant to refer to geometry. It might refer to the band, but that would be a bit odd. Yes, I know, Zippy the Pinhead always speaks oddly, but in these nostalgic fugue strips he usually provides some narrative counterpoint.

Larry Wright’s Motley Classics for the 26th originally ran in 1986. I mention this because it makes the odd dialogue of getting “a new math program” a touch less odd. I confess I’m not sure what the kid even got. An educational game? Something for numerical computing? The coal-fired, gear-driven version of Mathematica that existed in the 1980s? It’s a mystery, it is.

Ryan Pagelow’s Buni for the 27th is really a calendar joke. It seems to qualify as an anthropomorphic numerals joke, though. It’s not a rare sentiment either.

Jef Mallett’s Frazz for the 29th is similarly a calendar joke. It does play on 2017 being a prime number, a fact that doesn’t really mean much besides reassuring us that it’s not a leap year. I’m not sure just what’s meant by saying it won’t repeat for another 2017 years, at least that wouldn’t be just as true for (say) 2015 or 2019. But as Frazz points out, we do cling to anything that floats in times like these.

## Reading the Comics, December 23, 2017: Slow Week Edition

Comic Strip Master Command apparently wants everybody to have a quiet time ahead of Christmas. How quiet? Quiet enough that I’m including a strip I skipped last week and probably shouldn’t have. Here goes.

Ruben Bolling’s Super-Fun-Pak Comix for the 15th was an installment of Uncle Cap’n’s Puzzle Pontoon, an activity puzzle that’s always about Uncle Cap’n running some low-competence scam. In this case the scam is bitcoins, which makes me wonder how old this particular panel rerun is. (I thought I saw a bitcoin joke in Barney Google, mind, although I can’t find the reference to prove it.)

I don’t feel confident that I understand the full mathematics behind the scheme, so I’ll pass on that. I can talk about the SHA-256 Hash Function and what it’s for, though. To be part of the bitcoin process your computer needs to do two things: it has to do some computing work, and it has to convince other computers that it’s done that. The trick is to prove it was done without giving the original work away. The answer is one that humans have known for centuries. Probably millennia. Possibly since the invention of secrets. To show you’re in on a secret, publicize something that makes no sense except to other people who know the secret. A hash is one way to do it.

It’s a function which matches a string of numbers that represent your original message to the real numbers. It should be easy to make the hash from the original string. But it should be hard to go from the hash back to the original string. So then you can publicize the hash of whatever your secret is. And someone else can know that they have the same secret by checking whether it hashes to the same number. (I’m reminded of how Galileo secured his priority of the discovery that Venus shows phases by writing a short sentence describing the phenomenon, and then publicizing an anagram of it. The anagram made no sense, but if you knew his original message you verify that yes, indeed, he did publicize that string of letters. I suppose that’s not properly a hash, but it serves much the same role.) It’s an easy enough way to add some authentication to a message, and to make it more tamper-proof. Hash functions for this kind of security are believed to be reasonably collision-proof. It might be possible to find two original messages with the same hash. But we believe it would take so long to do that it would be more effective to just break into your target’s house and steal their computer instead of counterfeiting the message.

Hilary Price’s Rhymes with Orange for the 17th is a joke about the uselessness of Algebra 2. It’s a joke of a kind with jokes about philosophy professors having jobs training students to be philosophy professors (a joke mathematicians get too, come to think of it). I’m a bit more sympathetic to joking about Algebra 2, rather than Algebra at all. There are some classes with a purpose that doesn’t seem quite clear. I’m more likely to name pre-algebra as a course whose purpose I can’t quite pin down. Algebra 2 I would, generically, expect to cover stuff like functions of several variables that you’re prepared for the first time you take Algebra, and you should be comfortable with before you start Calculus (or Pre-Calculus), but that aren’t essential to knowing algebra in the first place.

Sam Hurt’s Eyebeam for the 18th is the anthropomorphic numerals segment for this slow week and makes literal an ancient joke. Incidentally, has anyone else been seeing the follow-up joke on their social media feeds? I don’t remember seeing it before about two months ago. (The follow up is, why was it that seven ate nine? … Because one should eat three-square meals a day.)

Brant Parker and Johnny Hart’s Wizard of Id Classics for the 21st mentions mathematicians, engineers, and wizards as the epitome of intelligence and ability. Flattering thought. My love’s father just yesterday proclaimed his confidence that as a mathematics PhD I could surely figure out how to do something mechanical. Related note: in three decades of being in an adult-like state I have never once successfully changed my car’s tire without outside aid. The strip originally ran the 25th of December, 1967.

There’s no Andertoons this week. I told you it was slow.

## Reading the Comics, December 16, 2017: Andertoons Drought Ended Edition

And now, finally, we get what we’ve been waiting so long for: my having enough energy and time to finish up last week’s comics. And I make excuses to go all fanboy over Elzie Segar’s great Thimble Theatre. Also more attention to Zach Weinersmith. You’ve been warned.

Mark Anderson’s Andertoons for the 13th is finally a breath of Mark Anderson’s Andertoons around here. Been far too long. Anyway it’s an algebra joke about x’s search for identity. And as often happens I’m sympathetic here. It’s not all that weird to think of ‘x’ as a label for some number. Knowing whether it means “a number whose value we haven’t found yet” or “a number whose value we don’t care about” is one trick, though. It’s not something you get used to from learning about, like, ‘6’. And knowing whether we can expect ‘x’ to have held whatever value it represented before, or whether we can expect it to be something different, is another trick.

Doug Bratton’s Pop Culture Shock Therapy for the 13th I feel almost sure has come up here before. Have I got the energy to find where? Oh, yes. It ran the 5th of September, 2015.

David Gilbert’s Buckles for the 14th is a joke on animals’ number sense. In fairness, after that start I wouldn’t know whether to go for four or five barks myself.

Bud Blake’s Tiger for the 15th is a bit of kid logic about how to make a long column of numbers easier to add. I endorse the plan of making the column shorter, although I’d do that by trying to pair up numbers that, say, add to 10 or 20 or something else easy to work with. Partial sums can make the overall work so much easier. And probably avoid mistakes.

Elzie Segar’s Thimble Theatre for the 8th of July, 1931, is my most marginal inclusion yet. It was either that strip or the previous day’s worth including. I’m throwing it in here because Segar’s Thimble Theatre keeps being surprisingly good. And, heck, slowing a count by going into fractions is viable way to do it. As the clobbered General Bunzo points out, you can drag this out longer by going into hundredths. Or smaller units. There is no largest real number less than ten; if it weren’t incredibly against the rules, boxers could make use of that.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th is about those mathematics problems with clear and easy-to-understand statements whose answers defy intuition. Weinersmith is completely correct about all of this. I’m surprised he doesn’t mention the one about how you could divide an orange into five pieces, reassemble the pieces, and get back two spheres each the size of a sun.

## Reading the Comics, December 11, 2017: Vamping For Andertoons Edition

So Mark Anderson’s Andertoons has been missing from the list of mathematically-themed the last couple weeks. Don’t think I haven’t been worried about that. But it’s finally given another on-topic-enough strip and I’m not going to include it here. I’ve had a terrible week and I’m going to use the comics we got in last week slowly.

Hector D Cantu and Carlos Castellanos’s Baldo for the 10th of December uses algebra as the type for homework you’d need help with. It reads plausibly enough to me, at least so far as I remember learning algebra.

Greg Evans’s Luann Againn for the 10th reprints the strip of the 10th of December, 1989. And as often happens, mathematics is put up as the stuff that’s too hard to really do. The expressions put up don’t quite parse; there’s nothing to solve. But that’s fair enough for a panicked brain. To not recognize what the problem even is makes it rather hard to solve.

Ruben Bolling’s Super-Fun-Pak Comix for the 10th is an installation of Quantum Mechanic, playing on the most fun example of non-commutative processes I know. That’s the uncertainty principle, which expresses itself as pairs of quantities that can’t be precisely measured simultaneously. There are less esoteric kinds of non-commutative processes. Like, rotating something 90 degrees along a horizontal and then along a vertical axis will turn stuff different from 90 degrees vertical and then horizontal. But that’s too easy to understand to capture the imagination, at least until you’re as smart as an adult and as thoughtful as a child.

Maria Scrivan’s Half Full for the 11th features Albert Einstein and one of the few equations that everybody knows. So that’s something.

Jeff Stahler’s Moderately Confused for the 11th features the classic blackboard full of equations, this time to explain why Christmas lights wouldn’t work. There is proper mathematics in lights not working. It’s that electrical-engineering work about the flow of electricity. The problem is, typically, a broken or loose bulb. Maybe a burnt-out fuse, although I have never fixed a Christmas lights problem by replacing the fuse. It’s just something to do so you can feel like you’ve taken action before screaming in rage and throwing the lights out onto the front porch. More interesting to me is the mathematics of strands getting tangled. The idea — a foldable thread, marked at regular intervals by points that can hook together — seems trivially simple. But it can give insight into how long molecules, particularly proteins, will fold together. It may help someone frustrated to ponder that their light strands are knotted for the same reasons life can exist. But I’m not sure it ever does.

## Reading the Comics, December 9, 2017: Zach Weinersmith Wants My Attention Edition

If anything dominated the week in mathematically-themed comic strips it was Zach Weinersmith’s Saturday Morning Breakfast Cereal. I don’t know how GoComics selects the strips to (re?)print on their site. But there were at least four that seemed on-point enough for me to mention. So, okay. He’s got my attention. What’s he do with it?

On the 3rd of December is a strip I can say is about conditional probability. The mathematician might be right that the chance someone will be murdered by a serial killer are less than one in ten million. But that is the chance of someone drawn from the whole universe of human experiences. There are people who will never be near a serial killer, for example, or who never come to his attention or who evade his interest. But if we know someone is near a serial killer, or does attract his interest? The information changes the probability. And this is where you get all those counter-intuitive and somewhat annoying logic puzzles about, like, the chance someone’s other child is a girl if the one who just walked in was, and how that changes if you’re told whether the girl who just entered was the elder.

On the 5th is a strip about sequences. And built on the famous example of exponential growth from doubling a reward enough times. Well, you know these things never work out for the wise guy. The “Fibonacci Spiral” spoken of in the next-to-last panel is a spiral, like you figure. The dimensions of the spiral are based on those of golden-ratio rectangles. It looks a great deal like a logarithmic spiral to the untrained eye. Also to the trained eye, but you knew that. I think it’s supposed to be humiliating that someone would call such a spiral “random”. But I admit I don’t get that part.

The strip for the 6th has a more implicit mathematical content. It hypothesizes that mathematicians, given the chance, will be more interested in doing recreational puzzles than even in eating and drinking. It’s amusing, but I’ll admit I’ve found very few puzzles all that compelling. This isn’t to say there aren’t problems I keep coming back to because I’m curious about them, just that they don’t overwhelm my common sense. Don’t ask me when I last received actual pay for doing something mathematical.

And then on the 9th is one more strip, about logicians. And logic puzzles, such as you might get in a Martin Gardner collection. The problem is written out on the chalkboard with some shorthand logical symbols. And they’re symbols both philosophers and mathematicians use. The letter that looks like a V with a crossbar means “for all”. (The mnemonic I got was “it’s an A-for-all, upside-down”. This paired with the other common symbol, which looks like a backwards E and means there exists: “E-for-exists, backwards”. Later I noticed upside-down A and backwards E could both be just 180-degree-rotated A and E. But try saying “180-degree-rotated” in a quick way.) The curvy E between the letters ‘x’ and ‘S’ means “belongs to the set”. So that first line says “for all x that belong to the set S this follows”. Writing out “isLiar(x)” instead of, say, “L(x)”, is more a philosopher’s thing than a mathematician’s. But it wouldn’t throw anyway. And the T just means emphasizing that this is true.

And that is as much about Saturday Morning Breakfast Cereal as I have to say this week.

Sam Hurt’s Eyebeam for the 4th tells a cute story about twins trying to explain infinity to one another. I’m not sure I can agree with the older twin’s assertion that infinity means there’s no biggest number. But that’s just because I worry there’s something imprecise going on there. I’m looking forward to the kids learning about negative numbers, though, and getting to wonder what’s the biggest negative real number.

Percy Crosby’s Skippy for the 4th starts with Skippy explaining a story problem. One about buying potatoes, in this case. I’m tickled by how cranky Skippy is about boring old story problems. Motivation is always a challenge. The strip originally ran the 7th of October, 1930.

Dave Whamond’s Reality Check for the 6th uses a panel of (gibberish) mathematics as an example of an algorithm. Algorithms are mathematical, in origin at least. The word comes to us from the 9th century Persian mathematician Al-Khwarizmi’s text about how to calculate. The modern sense of the word comes from trying to describe the methods by which a problem can be solved. So, legitimate use of mathematics to show off the idea. The symbols still don’t mean anything.

Rick Detorie’s One Big Happy for the 7th has Joe trying to get his mathematics homework done at the last minute. … And it’s caused me to reflect on how twenty multiplication problems seems like a reasonable number to do. But there’s only fifty multiplications to even do, at least if you’re doing the times tables up to the 10s. No wonder students get so bored seeing the same problems over and over. It’s a little less dire if you’re learning times tables up to the 12s, but not that much better. Yow.

Olivia Walch’s Imogen Quest for the 8th looks pretty legitimate to me. It’s going to read as gibberish to people who haven’t done parametric functions, though. Start with the plane and the familiar old idea of ‘x’ and ‘y’ representing how far one is along a horizontal and a vertical direction. Here, we’re given a dummy variable ‘t’, and functions to describe a value for ‘x’ and ‘y’ matching each value of ‘t’. The plot then shows all the points that ever match a pair of ‘x’ and ‘y’ coordinates for some ‘t’. The top drawing is a shape known as the cardioid, because it kind of looks like a Valentine-heart. The lower figure is a much more complicated parametric equation. It looks more anatomically accurate,

Still no sign of Mark Anderson’s Andertoons and the drought is worrying me, yes.

But they’re still going on the cartoonist’s web site, so there’s that.