Reading the Comics, March 9, 2019: In Which I Explain Eleven Edition


I thought I had a flood of mathematically-themed comic strips last week. On reflection, many of them were slight enough not to need further context. You’ll see in the paragraph of not-discussed strips at the end of this. What did rate discussion turned out to get more interesting to me the more I wrote about them.

Stephen Beals’s Adult Children for the 6th uses mathematics as icon of things that are indisputably true. Two plus two equals four is a good example of such. If we take the ordinary meanings of ‘two’ and ‘plus’ and ‘equals’ and ‘four’ there’s no disputing it. The result follows from some uncontroversial-seeming axioms and a lot of deduction. By the rules of logic, the conclusion has to be true, whoever makes it. Even, for that matter, if nobody makes it. It’s difficult to imagine a universe in which nobody ever notices two plus two equals four. But we can imagine that there are mathematical truths that will never be noticed by anyone. (Here’s one. There is some largest finite whole number that any human-created project will ever use in any context. Consider the equation represented by “that number plus two equals (even bigger number)”.)

Harvey: 'Everyone ignores facts! Two plus two equals four, you know what I mean?' Friend: 'Yes. In your opinion, two plus two equals four.' Harvey: 'Noooo! Facts aren't opinions! There are no true facts, fake facts, iffy facts ... just facts! Let's judge things based on the facts!' Friend: 'And how do these facts make you feel?' Harvey, clutching his chest. 'Like you're giving me a fact attack.'
Stephen Beals’s Adult Children for the 6th of March, 2019. Essays inspired by something mentioned in Adult Children appear at this link.

But you see cards palmed there. What do we mean by ‘two’? Have we got a good definition? Might there be a different definition that’s more useful? Probably not, for ‘two’ anyway. But a part of mathematics, especially as a field develops, is working out what are the important concepts, and what their definitions should be. What a ‘function’ is, for example, went through a lot of debate and change over the 19th century. There is an elusiveness to facts, even in mathematics, where you’d think epistemology would be simpler.

Lauren's problem: '(x^2 y - 3y^2 + 5xy^2) - (-x^2 y + 3xy^2 - 3y^2). Which of the following is equivalent to the expression above? a. 4x^2 y^2. b. 8xy^2 - 6y^2. c. 2x^2 + 2xy^2. d. 2x^2 y + 8xy^2 - 6y^2.' Next problem: 'If a/b = 2 what's the value of 4b/a? a. 0. b. 1. c. 2. d. 4.' Bob, holding up empty ice trays: 'If a and b are empty because Lauren is selfish and not thinking of Bob, what are the chances he gets to have an iced drink? a. slim, b. none, c. all of the above?'
Frank Page’s Bob the Squirrel for the 6th of March, 2019. When I’m moved to write something based on Bob the Squirrel the essays should be tagged to appear at this link.

Frank Page’s Bob the Squirrel for the 6th continues the SAT prep questions from earlier in the week. There’s two more problems in shuffling around algebraic expressions here. The first one, problem 5, is probably easiest to do by eliminating wrong answers. (x^2 y - 3y^2 + 5xy^2) - (-x^2 y + 3xy^2 - 3y^2) is a tedious mess. But look at just the x^2 y terms: they have to add up to 2x^2 y , so, the answer has to be either c or d. So next look at the 3y^2 terms and oh, that’s nice. They add up to zero. The answer has to be c. If you feel like checking the 5xy^2 terms, go ahead; that’ll offer some reassurance, if you do the addition correctly.

The second one, problem 8, is probably easier to just think out. If \frac{a}{b} = 2 then there’s a lot of places to go. What stands out to me is that 4\frac{b}{a} has the reciprocal of \frac{a}{b} in it. So, the reciprocal of \frac{a}{b} has to equal the reciprocal of 2 . So \frac{a}{b} = \frac{1}{2} . And 4\frac{b}{a} is, well, four times \frac{b}{a} , so, four times one-half, or two. There’s other ways to go about this. In honestly, what I did when I looked at the problem was multiply both sides of \frac{a}{b} = 2 by \frac{b}{a} . But it’s harder to explain why that struck me as an obviously right thing to do. It’s got shortcuts I grew into from being comfortable with the more methodical approach. Someone who does a lot of problems like these will discover shortcuts.

Ruthie on the phone: 'Hello, homework hotline? I have an arithmetic question. Why isn't eleven called oneteen, and twelve called twoteen? ... You don't know? ... May I speak to your supervisor, please?'
Rick Detorie’s One Big Happy for the 6th of March, 2019. This particular strip is several years old, but I can’t pin down its original run more precisely than that. Essays featuring One Big Happy should be at this link.

Rick Detorie’s One Big Happy for the 6th asks one of those questions you need to be a genius or a child to ponder. Why don’t the numbers eleven and twelve follow the pattern of the other teens, or for that matter of twenty-one and thirty-two, and the like? And the short answer is that they kind of do. At least, “eleven” and “twelve”, etymologists agree, derive from the Proto-Germanic “ainlif” and “twalif”. If you squint your mouth you can get from “ain” to “one” (it’s probably easier if you go through the German “ein” along the way). Getting from “twa” to “two” is less hard. If my understanding is correct, etymologists aren’t fully agreed on the “lif” part. But they are settled on it means the part above ten. Like, “ainlif” would be “one left above ten”. So it parses as one-and-ten, putting it in form with the old London-English preference for one-and-twenty or two-and-thirty as word constructions.

It’s not hard to figure how “twalif” might over centuries mutate to “twelve”. We could ask why “thirteen” didn’t stay something more Old Germanic. My suspicion is that it amounts to just, well, it worked out like that. It worked out the same way in German, which switches to “-zehn” endings from 13 on. Lithuanian has all the teens end with “-lika”; Polish, similarly, but with “-ście”. Spanish — not a Germanic language — has “custom” words for the numbers up to 15, and then switches to “diecis-” as a prefix to the numbers 6 through 9. French doesn’t switch to a systematic pattern until 17. (And no I am not going to talk about France’s 80s and 90s.) My supposition is that different peoples came to different conclusions about whether they needed ten, or twelve, or fifteen, or sixteen, unique names for numbers before they had to resort to systemic names.

Here’s some more discussion of the teens, though, including some exploration of the controversy and links to other explanations.

Caption: '4 out of 5 Doctors agree ... ' Four, of five, chickens dressed as doctors: 'We are 80% of the doctors!'
Doug Savage’s Savage Chickens for the 6th of March, 2019. And the occasional essay based on Savage Chickens should be gathered at this link.

Doug Savage’s Savage Chickens for the 6th is a percentages comic. It makes reference to an old series of (American, at least) advertisements in which four out of five dentists would agree that chewing sugarless gum is a good thing. Shifting the four-out-of-five into 80% riffs is not just fun with tautologies. Percentages have this connotation of technical precision; 80% sounds like a more rigorously known number than “four out of five”. It doesn’t sound as scientific as “0.80”, quite. But when applied to populations a percentage seems less bizarre than a decimal.


Oh, now, and what about comic strips I can’t think of anything much to write about?
Ruben Bolling’s Super-Fun-Pak Comix for the 4th featured divisibility, in a panel titled “Fun Facts for the Obsessive-Compulsive”. Olivia James’s Nancy on the 6th was avoiding mathematics homework. Jonathan Mahood’s Bleeker: The Rechargeable Dog for the 7th has Skip avoiding studying for his mathematics test. Bob Scott’s Bear With Me for the 7th has Molly mourning a bad result on her mathematics test. (The comic strip was formerly known as Molly And The Bear, if this seems familiar but the name seems wrong.) These are all different comic strips, I swear. Bill Holbrook’s Kevin and Kell for the 8th has Rudy and Fiona in mathematics class. (The strip originally ran in 2013; Comics Kingdom has started running Holbrook’s web comic, but at several years’ remove.) And, finally, Alex Hallatt’s Human Cull for the 8th talks about “110%” as a phrase. I don’t mind the phrase, but the comic strip has a harder premise.


And that finishes the comic strips from last week. But Pi Day is coming. I’ll be ready for it. Shall see you there.

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Reading the Comics, December 19, 2018: Andertoons Is Back Edition


I had not wanted to mention, for fear of setting off a panic. But Mark Anderson’s Andertoons, which I think of as being in every Reading the Comics post, hasn’t been around lately. If I’m not missing something, it hasn’t made an appearance in three months now. I don’t know why, and I’ve been trying not to look too worried by it. Mostly I’ve been forgetting to mention the strange absence. This even though I would think any given Tuesday or Friday that I should talk about the strip not having anything for me to write about. Fretting about it would make a great running theme. But I have never spotted a running theme before it’s finished. In any event the good news is that the long drought has ended, and Andertoons reappears this week. Yes, I’m hoping that it won’t be going to long between appearances this time.

Mrs Olsen: 'How do you know I haven't got my flu shot?' Caulfield: 'Just playing the odds.' Mrs Olsen: 'Maybe I was playing some odds myself. Maybe I got to the pharmacy and remembered that this year's vaccine is 30-40% effective.' Caulfield: 'I'd take those odds.' Mrs Olsen: 'They're not my kind of odds.' Caulfield: 'And what are the odds you bought a lottery ticket on your way out?' (Pause.) Mrs Olsen: 'You are getting under my skin.' Caulfield: 'That's good news. Now there's a 30-40% chance you'll develop a resistance.'
Jef Mallett’s Frazz for the 16th of December, 2018. Other essays discussing topics raised by Frazz are at this link.

Jef Mallett’s Frazz for the 16th talks about probabilities. This in the context of assessing risks. People are really bad at estimating probabilities. We’re notoriously worse at assessing risks, especially when it’s a matter of balancing a present cost like “fifteen minutes waiting while the pharmacy figures out whether insurance will pay for the flu shot” versus a nebulous benefit like “lessened chance of getting influenza, or at least having a less severe influenza”. And it’s asymmetric, too. We view improbable but potentially enormous losses differently from the way we view improbable but potentially enormous gains. And it’s hard to make the rationally-correct choice reliably, not when there are so many choices of this kind every day.

Guard, to new prisoner: 'Never mind Professor Phillip. He's always preoccupied with some theory of escape probability.' The cell walls are covered with mathematical scrawls.
Tak Bui’s PC and Pixel for the 16th of December, 2018. This and other essays, when they’re written, inspired by PC and Pixel should be at this link. It’s a new tag, which surprises me.

Tak Bui’s PC and Pixel for the 16th features a wall full of mathematical symbols, used to represent deep thought about a topic. The symbols are gibberish, yes. I’m not sure that an actual “escape probability” could be done in a legible way, though. Or even what precisely Professor Phillip might be calculating. I imagine it would be an estimate of the various ways he might try to escape, and what things might affect that. This might be for the purpose of figuring out what he might do to maximize his chances of a successful escape. Although I wouldn’t put it past the professor to just be quite curious what the odds are. There’s a thrill in having a problem solved, even if you don’t use the answer for anything.

Amazing Yet Tautological strip: 'Each year America consumes enough EGG SALAD ... ' (Picture of a woman holding up a lumpy pile that context indicates is egg salad.) ' ... to give EACH AMERICAN an annualized national-average serving of the tasty concoction!'
Ruben Bolling’s Super-Fun-Pak Comix for the 18th of December, 2018. Essays based on Super-Fun-Pak Comix are at this link. (Amazing Yet Tautological is one of the features that turns up in Super-Fun-Pak Comix, which is why it doesn’t rate a tag on its own)

Ruben Bolling’s Super-Fun-Pak Comix for the 18th has a trivia-panel-spoof dubbed Amazing Yet Tautological. One could make an argument that most mathematics trivia fits into this category. At least anything about something that’s been proven. Anyway, whether this is a tautological strip depends on what the strip means by “average” in the phrase “average serving”. There’s about four jillion things dubbed “average” and each of them has a context in which they make sense. The thing intended here, and the thing meant if nobody says anything otherwise, is the “arithmetic mean”. That’s what you get from adding up everything in a sample (here, the amount of egg salad each person in America eats per year) and dividing it by the size of the sample (the number of people in America that year). Another “average” which would make sense, but would break this strip, would be the median. That would be the amount of egg salad that half of all Americans eat more than, and half eat less than. But whether every American could have that big a serving really depends on what that median is. The “mode”, the most common serving, would also be a reasonable “average” to expect someone to talk about.

Teacher showing solid geometry to the class. Wavehead: 'I saw a movie where the robot monster came right at me. If you want me to get excited about 3D shapes, you're going to have to do better than that.'
Mark Anderson’s Andertoons for the 19th of December, 2018. The many essays which discuss Andertoons are at this link.

Mark Anderson’s Andertoons for the 19th is that strip’s much-awaited return to my column here. It features solid geometry, which is both an important part of geometry and also a part that doesn’t get nearly as much attention as plane geometry. It’s reductive to suppose the problem is that it’s harder to draw solids than planar figures. I suspect that’s a fair part of the problem, though. Mathematicians don’t get much art training, not anymore. And while geometry is supposed to be able to rely on pure reasoning, a good picture still helps. And a bad picture will lead us into trouble.


Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

Reading the Comics, February 17, 2018: Continuing Deluge Month


February’s been a flooding month. Literally (we’re about two blocks away from the Voluntary Evacuation Zone after the rains earlier this week) and figuratively, in Comic Strip Master Command’s suggestions about what I might write. I have started thinking about making a little list of the comics that just say mathematics in some capacity but don’t give me much to talk about. (For example, Bob the Squirrel having a sequence, as it does this week, with a geometry tutor.) But I also know, this is unusually busy this month. The problem will recede without my having to fix anything. One of life’s secrets is learning how to tell when a problem’s that kind.

Patrick Roberts’s Todd the Dinosaur for the 12th just shows off an arithmetic problem — fractions — as the thing that can be put on the board and left for students to do.

Todd: *Sniff sniff* 'Hey! What's that on the floor?' (He follows a trail of beef jerky, eating, until he's at the chalkboard.) Teacher: 'Well, hello, Todd! Say, while you're up there, why don't you do that fractions problem on the board?' Todd: 'Darn you, tasty Slim jims!'
Patrick Roberts’s Todd the Dinosaur for the 12th of February, 2018. I’ll risk infecting you with one of my problems: I look at this particular comic and wonder what happened right before the first panel to lead to this happening.

Ham’s Life on Earth for the 12th has a science-y type giving a formula as “something you should know”. The formula’s gibberish, so don’t worry about it. I got a vibe of it intending to be some formula from statistics, but there’s no good reason for that. I’ve had some statistical distribution problems on my mind lately.

Eric Teitelbaum and Bill Teitelbaum’s Bottomliners for the 12th maybe influenced my thinking. It has a person claiming to be a former statistician, and his estimate of how changing his job’s affected his happiness. Could really be any job that encourages people to measure and quantify things. But “statistician” is a job with strong connotations of being able to quantify happiness. To have that quantity feature a decimal point, too, makes him sound more mathematical and thus, more surely correct. I’d be surprised if “two and a half times” weren’t a more justifiable estimate, given the margin for error on happiness-measurement I have to imagine would be there. (This seems to be the first time I’ve featured Bottomliners at least since I started tagging the comic strips named. Neat.)

Ruben Bolling’s Super-Fun-Pak Comix for the 12th reprinted a panel called The Uncertainty Principal that baffled commenters there. It’s a pun on “Uncertainty Principle”, the surprising quantum mechanics result that there are some kinds of measurements that can’t be taken together with perfect precision. To know precisely where something is destroys one’s ability to measure its momentum. To know the angular momentum along one axis destroys one’s ability to measure it along another. This is a physics result (note that the panel’s signed “Heisenberg”, for the name famously attached to the Uncertainty Principle). But the effect has a mathematical side. The operations that describe finding these incompatible pairs of things are noncommutative; it depends what order you do them in.

We’re familiar enough with noncommutative operations in the real world: to cut a piece of paper and then fold it usually gives something different to folding a piece of paper and then cutting it. To pour batter in a bowl and then put it in the oven has a different outcome than putting batter in the oven and then trying to pour it into the bowl. Nice ordinary familiar mathematics that people learn, like addition and multiplication, do commute. These come with partners that don’t commute, subtraction and division. But I get the sense we don’t think of subtraction and division like that. It’s plain enough that ‘a’ divided by ‘b’ and ‘b’ divided by ‘a’ are such different things that we don’t consider what’s neat about that.

In the ordinary world the Uncertainty Principle’s almost impossible to detect; I’m not sure there’s any macroscopic phenomena that show it off. I mean, that atoms don’t collapse into electrically neutral points within nanoseconds, sure, but that isn’t as compelling as, like, something with a sodium lamp and a diffraction grating and an interference pattern on the wall. The limits of describing certain pairs of properties is about how precisely both quantities can be known, together. For everyday purposes there’s enough uncertainty about, say, the principal’s weight (and thus momentum) that uncertainty in his position won’t be noticeable. There’s reasons it took so long for anyone to suspect this thing existed.

Samson’s Dark Side of the Horse for the 13th uses a spot of arithmetic as the sort of problem coffee helps Horace solve. The answer’s 1.

Mike Baldwin’s Cornered for the 14th is a blackboard-full-of-symbols panel. Well, a whiteboard. It’s another in the line of mathematical proofs of love.

Dana Simpson’s Ozy and Millie rerun for the 14th has the title characters playing “logical fallacy tag”. Ozy is, as Millie says, making an induction argument. In a proper induction argument, you characterize something with some measure of size. Often this is literally a number. You then show that if it’s true that the thing is true for smaller problems than you’re interested in, then it has to also be true for the problem you are interested in. Add to that a proof that it’s true for some small enough problem and you’re done. In this case, Ozy’s specific fallacy is an appeal to probability: all but one of the people playing tag are not it, and therefore, any particular person playing the game isn’t it. That it’s fallacious really stands out when there’s only two people playing.

Ed: 'Only recently, scientists discovered pigeons understand space and time.' Pigeon: 'They never questioned us before. We're waiting for them to ask us about the Grand Unified Theory of Physics next.'
Alex Hallatt’s Arctic Circle for the 16th of February, 2018. As ever, I learn things from doing this! Specifically the names of the penguins which I’d somehow not thought about before. Ed’s the one with a pair of antenna-like feathers on his head. Oscar has the smooth head. Gordo has the set of bumps.

Alex Hallatt’s Arctic Circle for the 16th riffs on the mathematics abilities of birds. Pigeons, in this case. The strip starts from their abilities understanding space and time (which are amazing) and proposes pigeons have some insight into the Grand Unified Theory. Animals have got astounding mathematical abilities, should point out. Don’t underestimate them. (This also seems to be the first time I’ve tagged Arctic Circle which doesn’t seem like it could be right. But I didn’t remember naming the penguins before so maybe I haven’t? Huh. Mind, I only started tagging the comic strip titles a couple months ago.)

Tony Cochrane’s Agnes for the 17th has the title character try bluffing her way out of mathematics homework. Could there be a fundamental flaw in mathematics as we know it? Possibly. It’s hard to prove that any field complicated enough to be interesting is also self-consistent. And there’s a lot of mathematics out there. And mathematics subjects often develop with an explosion of new ideas and then a later generation that cleans them up and fills in logical gaps. Symplectic geometry is, if I’m following the news right, going into one of those cleaning-up phases now. Is it likely to be uncovered by a girl in elementary school? I’m skeptical, and also skeptical that she’d have a replacement system that would be any better. I admire Agnes’s ambition, though.

Mike Baldwin’s Cornered for the 17th plays on the reputation for quantum mechanics as a bunch of mathematically weird, counter-intuitive results. In fairness to the TV program, I’ve had series run longer than I originally planned too.

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.

The Geometry. A pair of parallel lines, one with a rectangular lump. 'Not true --- parallel lines *do* meet. In fact, Peter and I are expected.' ('We met at a crossroads in both our lives.')
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st of February, 2018. All right, but they’re line segments, but I suppose you can’t reasonably draw infinitely vast things in a daily newspaper strip’s space. The lean of that triangle makes it look way more skeptical, even afraid, than I think Price and Piccolo intended, but I’m not sure there’s a better way to get these two in frame without making the composition weird.

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, 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.

'So, I finally used my Algebra 2 ... helping my kid with his Algebra 2.'
Hilary Price (w/KG)’s Rhymes with Orange for the 17th of December, 2017. I’m not sure who KG is. Daily strips lately have been co-signed by Rina Piccolo, formerly of Tina’s Groove.

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 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, November 25, 2017: Shapes and Probability Edition


This week was another average-grade week of mathematically-themed comic strips. I wonder if I should track them and see what spurious correlations between events and strips turn up. That seems like too much work and there’s better things I could do with my time, so it’s probably just a few weeks before I start doing that.

Ruben Bolling’s Super-Fun-Pax Comics for the 19th is an installment of A Voice From Another Dimension. It’s in that long line of mathematics jokes that are riffs on Flatland, and how we might try to imagine spaces other than ours. They’re taxing things. We can understand some of the rules of them perfectly well. Does that mean we can visualize them? Understand them? I’m not sure, and I don’t know a way to prove whether someone does or does not. This wasn’t one of the strips I was thinking of when I tossed “shapes” into the edition title, but you know what? It’s close enough to matching.

Olivia Walch’s Imogen Quest for the 20th — and I haven’t looked, but it feels to me like I’m always featuring Imogen Quest lately — riffs on the Monty Hall Problem. The problem is based on a game never actually played on Monty Hall’s Let’s Make A Deal, but very like ones they do. There’s many kinds of games there, but most of them amount to the contestant making a choice, and then being asked to second-guess the choice. In this case, pick a door and then second-guess whether to switch to another door. The Monty Hall Problem is a great one for Internet commenters to argue about while the rest of us do something productive. The trouble — well, one trouble — is that whether switching improves your chance to win the car is that whether it does depends on the rules of the game. It’s not stated, for example, whether the host must open a door showing a goat behind it. It’s not stated that the host certainly knows which doors have goats and so chooses one of those. It’s not certain the contestant even wants a car when, hey, goats. What assumptions you make about these issues affects the outcome.

If you take the assumptions that I would, given the problem — the host knows which door the car’s behind, and always offers the choice to switch, and the contestant would rather have a car, and such — then Walch’s analysis is spot on.

Jonathan Mahood’s Bleeker: The Rechargeable Dog for the 20th features a pretend virtual reality arithmetic game. The strip is of incredibly low mathematical value, but it’s one of those comics I like that I never hear anyone talking about, so, here.

Richard Thompson’s Cul de Sac rerun for the 20th talks about shapes. And the names for shapes. It does seem like mathematicians have a lot of names for slightly different quadrilaterals. In our defense, if you’re talking about these a lot, it helps to have more specific names than just “quadrilateral”. Rhomboids are those parallelograms which have all four sides the same length. A parallelogram has to have two pairs of equal-sized legs, but the two pairs’ sizes can be different. Not so a rhombus. Mathworld says a rhombus with a narrow angle that’s 45 degrees is sometimes called a lozenge, but I say they’re fibbing. They make even more preposterous claims on the “lozenge” page.

Todd Clark’s Lola for the 20th does the old “when do I need to know algebra” question and I admit getting grumpy like this when people ask. Do French teachers have to put up with this stuff?

Brian Fies’s Mom’s Cancer rerun for the 23rd is from one of the delicate moments in her story. Fies’s mother just learned the average survival rate for her cancer treatment is about five percent and, after months of things getting haltingly better, is shaken. But as with most real-world probability questions context matters. The five-percent chance is, as described, the chance someone who’d just been diagnosed in the state she’d been diagnosed in would survive. The information that she’s already survived months of radiation and chemical treatment and physical therapy means they’re now looking at a different question. What is the chance she will survive, given that she has survived this far with this care?

Mark Anderson’s Andertoons for the 24th is the Mark Anderson’s Andertoons for the week. It’s a protesting-student kind of joke. For the student’s question, I’m not sure how many sides a polygon has before we can stop memorizing them. I’d say probably eight. Maybe ten. Of the shapes whose names people actually care about, mm. Circle, triangle, a bunch of quadrilaterals, pentagons, hexagons, octagons, maybe decagon and dodecagon. No, I’ve never met anyone who cared about nonagons. I think we could drop heptagons without anyone noticing either. Among quadrilaterals, ugh, let’s see. Square, rectangle, rhombus, parallelogram, trapezoid (or trapezium), and I guess diamond although I’m not sure what that gets you that rhombus doesn’t already. Toss in circles, ellipses, and ovals, and I think that’s all the shapes whose names you use.

Stephan Pastis’s Pearls Before Swine for the 25th does the rounding-up joke that’s been going around this year. It’s got a new context, though.

Reading the Comics, April 24, 2017: Reruns Edition


I went a little wild explaining the first of last week’s mathematically-themed comic strips. So let me split the week between the strips that I know to have been reruns and the ones I’m not so sure were.

Bill Amend’s FoxTrot for the 23rd — not a rerun; the strip is still new on Sundays — is a probability question. And a joke about story problems with relevance. Anyway, the question uses the binomial distribution. I know that because the question is about doing a bunch of things, homework questions, each of which can turn out one of two ways, right or wrong. It’s supposed to be equally likely to get the question right or wrong. It’s a little tedious but not hard to work out the chance of getting exactly six problems right, or exactly seven, or exactly eight, or so on. To work out the chance of getting six or more questions right — the problem given — there’s two ways to go about it.

One is the conceptually easy but tedious way. Work out the chance of getting exactly six questions right. Work out the chance of getting exactly seven questions right. Exactly eight questions. Exactly nine. All ten. Add these chances up. You’ll get to a number slightly below 0.377. That is, Mary Lou would have just under a 37.7 percent chance of passing. The answer’s right and it’s easy to understand how it’s right. The only drawback is it’s a lot of calculating to get there.

So here’s the conceptually harder but faster way. It works because the problem says Mary Lou is as likely to get a problem wrong as right. So she’s as likely to get exactly ten questions right as exactly ten wrong. And as likely to get at least nine questions right as at least nine wrong. To get at least eight questions right as at least eight wrong. You see where this is going: she’s as likely to get at least six right as to get at least six wrong.

There’s exactly three possibilities for a ten-question assignment like this. She can get four or fewer questions right (six or more wrong). She can get exactly five questions right. She can get six or more questions right. The chance of the first case and the chance of the last have to be the same.

So, take 1 — the chance that one of the three possibilities will happen — and subtract the chance she gets exactly five problems right, which is a touch over 24.6 percent. So there’s just under a 75.4 percent chance she does not get exactly five questions right. It’s equally likely to be four or fewer, or six or more. Just-under-75.4 divided by two is just under 37.7 percent, which is the chance she’ll pass as the problem’s given. It’s trickier to see why that’s right, but it’s a lot less calculating to do. That’s a common trade-off.

Ruben Bolling’s Super-Fun-Pax Comix rerun for the 23rd is an aptly titled installment of A Million Monkeys At A Million Typewriters. It reminds me that I don’t remember if I’d retired the monkeys-at-typewriters motif from Reading the Comics collections. If I haven’t I probably should, at least after making a proper essay explaining what the monkeys-at-typewriters thing is all about.

'This new math teacher keeps shakin' us down every morning, man ... what's she looking for, anyway?' 'Pocket calculators.'
Ted Shearer’s Quincy from the 28th of February, 1978. So, that FoxTrot problem I did? The conceptually-easy-but-tedious way is not too hard to do if you have a calculator. It’s a buch of typing but nothing more. If you don’t have a calculator, though, the desire not to do a whole bunch of calculating could drive you to the conceptually-harder-but-less-work answer. Is that a good thing? I suppose; insight is a good thing to bring. But the less-work answer only works because of a quirk in the problem, that Mary Lou is supposed to have a 50 percent chance of getting a question right. The low-insight-but-tedious problem will aways work. Why skip on having something to do the tedious part?

Ted Shearer’s Quincy from the 28th of February, 1978 reveals to me that pocket calculators were a thing much earlier than I realized. Well, I was too young to be allowed near stuff like that in 1978. I don’t think my parents got their first credit-card-sized, solar-powered calculator that kind of worked for another couple years after that. Kids, ask about them. They looked like good ideas, but you could use them for maybe five minutes before the things came apart. Your cell phone is so much better.

Bil Watterson’s Calvin and Hobbes rerun for the 24th can be classed as a resisting-the-word-problem joke. It’s so not about that, but who am I to slow you down from reading a Calvin and Hobbes story?

Garry Trudeau’s Doonesbury rerun for the 24th started a story about high school kids and their bad geography skills. I rate it as qualifying for inclusion here because it’s a mathematics teacher deciding to include more geography in his course. I was amused by the week’s jokes anyway. There’s no hint given what mathematics Gil teaches, but given the links between geometry, navigation, and geography there is surely something that could be relevant. It might not help with geographic points like which states are in New England and where they are, though.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is built on a plot point from Carl Sagan’s science fiction novel Contact. In it, a particular “message” is found in the digits of π. (By “message” I mean a string of digits that are interesting to us. I’m not sure that you can properly call something a message if it hasn’t got any sender and if there’s not obviously some intended receiver.) In the book this is an astounding thing because the message can’t be; any reasonable explanation for how it should be there is impossible. But short “messages” are going to turn up in π also, as per the comic strips.

I assume the peer review would correct the cartoon mathematicians’ unfortunate spelling of understanding.

Reading the Comics, April 15, 2017: Extended Week Edition


It turns out last Saturday only had the one comic strip that was even remotely on point for me. And it wasn’t very on point either, but since it’s one of the Creators.com strips I’ve got the strip to show. That’s enough for me.

Henry Scarpelli and Craig Boldman’s Archie for the 8th is just about how algebra hurts. Some days I agree.

'Ugh! Achey head! All blocked up! Throbbing! Completely stuffed!' 'Sounds like sinuses!' 'No. Too much algebra!'
Henry Scarpelli and Craig Boldman’s Archie for the 8th of April, 2017. Do you suppose Archie knew that Dilton was listening there, or was he just emoting his fatigue to himself?

Ruben Bolling’s Super-Fun-Pak Comix for the 8th is an installation of They Came From The Third Dimension. “Dimension” is one of those oft-used words that’s come loose of any technical definition. We use it in mathematics all the time, at least once we get into Introduction to Linear Algebra. That’s the course that talks about how blocks of space can be stretched and squashed and twisted into each other. You’d expect this to be a warmup act to geometry, and I guess it’s relevant. But where it really pays off is in studying differential equations and how systems of stuff changes over time. When you get introduced to dimensions in linear algebra they describe degrees of freedom, or how much information you need about a problem to pin down exactly one solution.

It does give mathematicians cause to talk about “dimensions of space”, though, and these are intuitively at least like the two- and three-dimensional spaces that, you know, stuff moves in. That there could be more dimensions of space, ordinarily inaccessible, is an old enough idea we don’t really notice it. Perhaps it’s hidden somewhere too.

Amanda El-Dweek’s Amanda the Great of the 9th started a story with the adult Becky needing to take a mathematics qualification exam. It seems to be prerequisite to enrolling in some new classes. It’s a typical set of mathematics anxiety jokes in the service of a story comic. One might tsk Becky for going through university without ever having a proper mathematics class, but then, I got through university without ever taking a philosophy class that really challenged me. Not that I didn’t take the classes seriously, but that I took stuff like Intro to Logic that I was already conversant in. We all cut corners. It’s a shame not to use chances like that, but there’s always so much to do.

Mark Anderson’s Andertoons for the 10th relieves the worry that Mark Anderson’s Andertoons might not have got in an appearance this week. It’s your common kid at the chalkboard sort of problem, this one a kid with no idea where to put the decimal. As always happens I’m sympathetic. The rules about where to move decimals in this kind of multiplication come out really weird if the last digit, or worse, digits in the product are zeroes.

Mel Henze’s Gentle Creatures is in reruns. The strip from the 10th is part of a story I’m so sure I’ve featured here before that I’m not even going to look up when it aired. But it uses your standard story problem to stand in for science-fiction gadget mathematics calculation.

Dave Blazek’s Loose Parts for the 12th is the natural extension of sleep numbers. Yes, I’m relieved to see Dave Blazek’s Loose Parts around here again too. Feels weird when it’s not.

Bill Watterson’s Calvin and Hobbes rerun for the 13th is a resisting-the-story-problem joke. But Calvin resists so very well.

John Deering’s Strange Brew for the 13th is a “math club” joke featuring horses. Oh, it’s a big silly one, but who doesn’t like those too?

Dan Thompson’s Brevity for the 14th is one of the small set of punning jokes you can make using mathematician names. Good for the wall of a mathematics teacher’s classroom.

Shaenon K Garrity and Jefferey C Wells’s Skin Horse for the 14th is set inside a virtual reality game. (This is why there’s talk about duplicating objects.) Within the game, the characters are playing that game where you start with a set number (in this case 20) tokens and take turn removing a couple of them. The “rigged” part of it is that the house can, by perfect play, force a win every time. It’s a bit of game theory that creeps into recreational mathematics books and that I imagine is imprinted in the minds of people who grow up to design games.

Reading the Comics, February 15, 2017: SMBC Does Not Cut In Line Edition


On reflection, that Saturday Morning Breakfast Cereal I was thinking about was not mathematically-inclined enough to be worth including here. Helping make my mind up on that was that I had enough other comic strips to discuss here that I didn’t need to pad my essay. Yes, on a slow week I let even more marginal stuff in. Here’s the comic I don’t figure to talk about. Enjoy!

Jack Pullan’s Boomerangs rerun for the 16th is another strip built around the “algebra is useless in real life” notion. I’m too busy noticing Mom in the first panel saying “what are you doing play [sic] video games?” to respond.

Ruben Bolling’s Super-Fun-Pak Comix excerpt for the 16th is marginal, yeah, but fun. Numeric coincidence and numerology can sneak into compulsions with terrible ease. I can believe easily the need to make the number of steps divisible by some favored number.

Rich Powell’s Wide Open for the 16th is a caveman science joke, and it does rely on a chalkboard full of algebra for flavor. The symbols come tantalizingly close to meaningful. The amount of kinetic energy, K or KE, of a particle of mass m moving at speed v is indeed K = \frac{1}{2} m v^2 . Both 16 and 32 turn up often in the physics of falling bodies, at least if we’re using feet to measure. a = -\frac{k}{m} x turns up in physics too. It comes from the acceleration of a mass on a spring. But an equation of the same shape turns up whenever you describe things that go through tiny wobbles around the normal value. So the blackboard is gibberish, but it’s a higher grade of gibberish than usual.

Rick Detorie’s One Big Happy rerun for the 17th is a resisting-the-word-problem joke, made fresher by setting it in little Ruthie’s playing at school.

T Lewis and Michael Fry’s Over The Hedge for the 18th mentions the three-body problem. As Verne the turtle says, it’s a problem from physics. The way two objects — sun and planet, planet and moon, pair of planets, whatever — orbit each other if they’re the only things in the universe is easy. You can describe it all perfectly and without using more than freshman physics majors know. Introduce a third body, though, and we don’t know anymore. Chaos can happen.

Emphasis on can. There’s no good way to solve the “general” three-body problem, the one where the star and planets can have any sizes and any starting positions and any starting speeds. We can do well for special cases, though. If you have a sun, a planet, and a satellite — each body negligible compared to the other — we can predict orbits perfectly well. If the bodies have to stay in one plane of motion, instead of moving in three-dimensional space, we can do pretty well. If we know two of the bodies orbit each other tightly and the third is way off in the middle of nowhere we can do pretty well.

But there’s still so many interesting cases for which we just can’t be sure chaos will not break out. Three interacting bodies just offer so much more chance for things to happen. (To mention something surely coincidental, it does seem to be a lot easier to write good comedy, or drama, with three important characters rather than two. Any pair of characters can gang up on the third, after all. I notice how much more energetic Over The Hedge became when Hammy the Squirrel joined RJ and Verne as the core cast.)

Dave Whamond’s Reality Check for the 18th is your basic mathematics-illiteracy joke, done well enough.