Reading the Comics, April 5, 2019: The Slow Week Edition


People reading my Reading the Comics post Sunday maybe noticed something. I mean besides my correct, reasonable complaining about the Comics Kingdom redesign. That is that all the comics were from before the 30th of March. That is, none were from the week before the 7th of April. The last full week of March had a lot of comic strips. The first week of April didn’t. So things got bumped a little. Here’s the results. It wasn’t a busy week, not when I filter out the strips that don’t offer much to write about. So now I’m stuck for what to post Thursday.

Jason Poland’s Robbie and Bobby for the 3rd is a Library of Babel comic strip. This is mathematical enough for me. Jorge Luis Borges’s Library is a magnificent representation of some ideas about infinity and probability. I’m surprised to realize I haven’t written an essay specifically about it. I have touched on it, in writing about normal numbers, and about the infinite monkey theorem.

At a tower. Bobby: 'The library of Babel!' Robbie: 'Inside is every book that will ever be written! It may take the rest of our lives to search, but it'll be worth it!' Bobby: 'What? No index?' Robbie: 'The search for meaning has no index.' Bobby (on the phone): 'I just downloaded one.' Robbie: 'It can't have everything. ... Mark Twain vs Frankenstein? Dante in Space? Harry Potter Infinity?' Bobby: 'Yep. All available as e-books too! Wow, Jeff Goldblum does the audio books.' Robbie: 'pfff. Well, forget this place!' (They leave a 'BORING' sign across the library's door.)
Jason Poland’s Robbie and Bobby for the 3rd of April, 2019. I would have sworn that I write more about this strip. But this seems to be the first time I’ve mentioned it since 2017. Well, that and other Robbie and Bobby-based essays are at this link.

The strip explains things well enough. The Library holds every book that will ever be written. In the original story there are some constraints. Particularly, all the books are 410 pages. If you wanted, say, a 600-page book, though, you could find one book with the first 410 pages and another book with the remaining 190 pages and then some filler. The catch, as explained in the story and in the comic strip, is finding them. And there is the problem of finding a ‘correct’ text. Every possible text of the correct length should be in there. So every possible book that might be titled Mark Twain vs Frankenstein, including ones that include neither Mark Twain nor Frankenstein, is there. Which is the one you want to read?

Over a pizza. Reggie: 'Don't let Jughead near the pizza! He always ends up eating half of it!' Jughead, with the cutter: 'Relax! I've divided it into four equal slices! Check it yourself!' Reggie: 'OK, I guess they do look equal.' Archie: 'Except for one thing! There are only three of us!' (Reggie and Archie each have one slice; Jughead has two.)
Henry Scarpelli and Craig Boldman’s Archie for the 4th of April, 2019. Now this strip I’ve written about as recently as October. That appearance, and other Archie strips, are discussed at this link.

Henry Scarpelli and Craig Boldman’s Archie for the 4th features an equal-divisions problem. In principle, it’s easy to divide a pizza (or anything else) equally; that’s what we have fractions for. Making them practical is a bit harder. I do like Jughead’s quick work, though. It’s got the slight-of-hand you expect from stage magic.

Caterpillars in an algebra classroom. On the back of one caterpillar student is a sign, 'Kick^{10} me'.
Scott Hilburn’s The Argyle Sweater for the 4th of April, 2019. And this strip I’ve written about … wait, can I really have gone since early March without mentioning? Huh. Well, so it appears. Essays discussing The Argyle Sweater appear at this link.

Scott Hilburn’s The Argyle Sweater for the 4th takes place in an algebra class. I’m not sure what algebraic principle 7^4 \times 13^6 demonstrates, but it probably came from somewhere. It’s 4,829,210. The exponentials on the blackboard do cue the reader to the real joke, of the sign reading “kick10 me”. I question whether this is really an exponential kicking situation. It seems more like a simple multiplication to me. But it would be harder to make that joke read clearly.

Tony Cochran’s Agnes for the 5th is part of a sequence investigating how magnets work. Agnes and Trout find just … magnet parts inside. This is fair. It’s even mathematics.

Looking over a pile of debris and a hammer on the table. Agnes: 'OK, we smashed a magnet. What do we see?' Trout: 'Uh. Magnet crumbs.' Agnes: 'Me too. I see magnet crumbs.' Trout: 'No gizmos, no gears, no wires. Just dirty black magnet crumbs.' Agnes: 'So what does this tell us about magnet function?' Trout: 'That it's one of God's many mysteries. Let's go eat.'
Tony Cochran’s Agnes for the 5th of April, 2019. And this strip I quite like, but don’t get to discuss enough. My essays featuring Agnes appears at this link.

Thermodynamics classes teach one of the great mathematical physics models. This is about what makes magnets. Magnets are made of … smaller magnets. This seems like question-begging. Ultimately you get down to individual molecules, each of which is very slightly magnetic. When small magnets are lined up in the right way, they can become a strong magnet. When they’re lined up in another way, they can be a weak magnet. Or no magnet at all.

How do they line up? It depends on things, including how the big magnet is made, and how it’s treated. A bit of energy can free molecules to line up, making a stronger magnet out of a weak one. Or it can break up the alignments, turning a strong magnet into a weak one. I’ve had physics instructors explain that you could, in principle, take an iron rod and magnetize it just by hitting it hard enough on the desk. And then demagnetize it by hitting it again. I have never seen one do this, though.

This is more than just a physics model. The mathematics of it is … well, it can be easy enough. A one-dimensional, nearest-neighbor model, lets us describe how materials might turn into magnets or break apart, depending on their temperature. Two- or three-dimensional models, or models that have each small magnet affected by distant neighbors, are harder.


And then there’s the comic strips that didn’t offer much to write about.
Brian Basset’s Red and Rover for the 3rd,
Liniers’s Macanudo for the 5th, Stephen Bentley’s Herb and Jamaal rerun for the 5th, and Gordon Bess’s Redeye rerun for the 5th all idly mention mathematics class, or things brought up in class.

Doug Savage’s Savage Chickens for the 2nd is another more-than-100-percent strip. Richard Thompson’s Richard’s Poor Almanac for the 3rd is a reprint of his Christmas Tree guide including a fir that “no longer inhabits Euclidean space”.

Mike Baldwin’s Cornered for the 31st depicts a common idiom about numbers. Eric the Circle for the 5th, by Rafoliveira, plays on the ∞ symbol.


And that covers the mathematically-themed comic strips from last week. There are more coming, though. I’ll show them on Sunday. Thanks for reading.

Advertisements

Reading the Comics, March 26, 2019: March 26, 2019 Edition


And we had another of those peculiar days where a lot of strips are on-topic enough for me to talk about.

Eric the Circle, this one by Kyle, for the 26th has a bit of mathematical physics in it. This is the kind of diagram you’ll see all the time, at least if you do the mathematics that tells you where things will be and when. The particular example is an easy problem, a thing rolling down an inclined plane. But the work done for it applies to more complicated problems. The question it’s for is, “what happens when this thing slides down the plane?” And that depends on the forces at work. There’s gravity, certainly . If there were something else it’d be labelled. Gravity’s represented with that arrow pointing straight down. That gives us the direction. The label (Eric)(g) gives us how strong this force is.

Caption: Eric on an inclined plane. It shows a circle on a right triangle, with the incline of the angle labelled 'x'. The force of gravity is pointing vertically down, labelled (Eric)(g). The force parallel to the incline is labelled (Eric)(g)sin(x); the force perpendicular to the incline is labelled (Eric)(g)cos(x).
Eric the Circle, by Kyle, for the 26th of March, 2019. Essays inspired at all by Eric the Circle are at this link.

Where the diagram gets interesting, and useful, are those dashed lines ending in arrows. One of those lines is, or at least means to be, parallel to the incline. The other is perpendicular to it. These both reflect gravity. We can represent the force of gravity as a vector. That means, we can represent the force of gravity as the sum of vectors. This is like how we can can write “8” or we can write “3 + 5”, depending on what’s more useful for what we’re doing. (For example, if you wanted to work out “67 + 8”, you might be better off doing “67 + 3 + 5”.) The vector parallel to the plane and the one perpendicular to the plane add up to the original gravity vector.

The force that’s parallel to the plane is the only force that’ll actually accelerate Eric. The force perpendicular to the plane just … keeps it snug against the plane. (Well, it can produce friction. We try not to deal with that in introductory physics because it is so hard. At most we might look at whether there’s enough friction to keep Eric from starting to slide downhill.) The magnitude of the force parallel to the plane, and perpendicular to the plane, are easy enough to work out. These two forces and the original gravity can be put together into a little right triangle. It’s the same shape but different size to the right triangle made by the inclined plane plus a horizontal and a vertical axis. So that’s how the diagram knows the parallel force is the original gravity times the sine of x. And that the perpendicular force is the original gravity times the cosine of x.

The perpendicular force is often called the “normal” force. This because mathematical physicists noticed we had only 2,038 other, unrelated, things called “normal”.

Rick Detorie’s One Big Happy for the 26th sees Ruthie demand to know who this Venn person was. Fair question. Mathematics often gets presented as these things that just are. That someone first thought about these things gets forgotten.

Ruthie, on the phone: 'Homework hot line? On the Same/Different page of our workbook there are two circles like this. They're called Venn diagrams and I wanna know who this Venn person is. And if I put two squares together, can we call it the Ruthie diagram, and how much money do I get for that? ... Huh? Well, I'll wait here 'til you find somebody who DOES know!'
Rick Detorie’s One Big Happy for the 26th of March, 2019. This is a rerun from … 2007, I want to say? There are two separate feeds, one of current and one of several-years-old, strips on the web. Essays including One Big Happy, current or years-old reruns, should be at this link.

John Venn, who lived from 1834 to 1923 — he died the 4th of April, it happens — was an English mathematician and philosopher and logician and (Anglican) priest. This is not a rare combination of professions. From 1862 he was a lecturer in Moral Science at Cambridge. This included work in logic, yes. But he also worked on probability questions. Wikipedia credits his 1866 Logic Of Chance with advancing the frequentist interpretation of probability. This is one of the major schools of thought about what the “probability of an event” is. It’s the one where you list all the things that could possibly happen, and consider how many of those are the thing you’re interested in. So, when you do a problem like “what’s the probability of rolling two six-sided dice and getting a total of four”? You’re doing a frequentist probability problem.

Venn Diagrams he presented to the world around 1880. These show the relationships between different sets. And the relationships of mathematical logic problems they represent. Venn, if my sources aren’t fibbing, didn’t take these diagrams to be a new invention of his own. He wrote of them as “Euler diagrams”. Venn diagrams, properly, need to show all the possible intersections of all the sets in play. You just mark in some way the intersections that happen to have nothing in them. Euler diagrams don’t require this overlapping. The name “Venn diagram” got attached to these pictures in the early 20th century. Euler here is Leonhard Euler, who created every symbol and notation mathematicians use for everything, and who has a different “Euler’s Theorem” that’s foundational to every field of mathematics, including the ones we don’t yet know exist. I exaggerate by 0.04 percent here.

Although we always start Venn diagrams off with circles, they don’t have to be. Circles are good shapes if you have two or three sets. It gets hard to represent all the possible intersections with four circles, though. This is when you start seeing weirder shapes. Wikipedia offers some pictures of Venn diagrams for four, five, and six sets. Meanwhile Mathworld has illustrations for seven- and eleven-set Venn diagrams. At this point, the diagrams are more for aesthetic value than to clarify anything, though. You could draw them with squares. Some people already do. Euler diagrams, particularly, are often squares, sometimes with rounded corners.

Venn had his other projects, too. His biography at St Andrews writes of his composing The Biographical History of Gonville and Caius College (Cambridge). And then he had another history of the whole Cambridge University. It also mentions his skills in building machines, though only cites one, a device for bowling cricket balls. The St Andrews biography says that in 1909 “Venn’s machine clean bowled one of [the Australian Cricket Team’s] top stars four times”. I do not know precisely what it means but I infer it to be a pretty good showing for the machine. His Wikipedia biography calls him a “passionate gardener”. Apparently the Cambridgeshire Horticultural Society awarded him prizes for his roses in July 1885 and for white carrots in September that year. And that he was a supporter of votes for women.

An illustration of an abacus. Caption: 'No matter what the category, you'll usually find me in the upper 99%.'
Ashleigh Brilliant’s Pot-Shots for the 26th of March, 2019. The strip originally appeared sometime in 1979. Essays discussing anything from Pot-Shots should appear at this link.

Ashleigh Brilliant’s Pot-Shots for the 26th makes a cute and true claim about percentiles. That a person will usually be in the upper 99% of whatever’s being measured? Hard to dispute. But, measure enough things and eventually you’ll fall out of at least one of them. How many things? This is easy to calculate if we look at different things that are independent of each other. In that case we could look at 69 things before there we’d expect a 50% chance of at least one not being in the upper 99%.

It’s getting that independence that’s hard. There’s often links between things. For example, a person’s height does not tell us much about their weight. But it does tell us something. A person six foot, ten inches tall is almost certainly not also 35 pounds, even though a person could be that size or could be that weight. A person’s scores on a reading comprehension test and their income? But test-taking results and wealth are certainly tied together. Age and income? Most of us have a bigger income at 46 than at 6. This is part of what makes studying populations so hard.

Snow, cat, to a kitten: '1 + 1 = 2 ... unless it's spring.' (Looking at a bird's nest with five eggs.) 'Then 1 + 1 = 5.'
T Shepherd’s Snow Sez for the 26th of March, 2019. Essays including an appearance of Essays inspired at all by Snow Sez should be gathered at this link. They will be, anyway; this is a new tag.

T Shepherd’s Snow Sez for the 26th is finally a strip I can talk about briefly, for a change. Snow does a bit of arithmetic wordplay, toying with what an expression like “1 + 1” might represent.


There were a lot of mathematically-themed comic strips last week. There’ll be another essay soon, and it should appear at this link. And then there’s always Sunday, as long as I stay ahead of deadline. I am never ahead of deadline.

Reading the Comics, March 19, 2019: Average Edition


This time around, averages seem important.

Mark Anderson’s Andertoons for the 18th is the Mark Anderson’s Andertoons for the week. This features the kids learning some of the commonest terms in descriptive statistics. And, as Wavehead says, the similarity of names doesn’t help sorting them out. Each is a kind of average. “Mean” usually is the arithmetic mean, or the thing everyone including statisticians calls “average”. “Median” is the middle-most value, the one that half the data is less than and half the data is greater than. “Mode” is the most common value. In “normally distributed” data, these three quantities are all the same. In data gathered from real-world measurements, these are typically pretty close to one another. It’s very easy for real-world quantities to be normally distributed. The exceptions are usually when there are some weird disparities, like a cluster of abnormally high-valued (or low-valued) results. Or if there are very few data points.

On the blackboard the teacher's written Median, Mode, and Mean, with a bunch of numbers from 3 through 15. Wavehead: 'I know they're all subtly different, but I have to say, the alliteration doesn't help.'
Mark Anderson’s Andertoons for the 18th of March, 2019. Essays which discuss topics raised by Andertoons can be found at this link. Also at this link, nearly enough.

The word “mean” derives from the Old French “meien”, that is, “middle, means”. And that itself traces to the Late Latin “medianus”, and the Latin “medius”. That traces back to the Proto-Indo-European “medhyo”, meaning “middle”. That’s probably what you might expect, especially considering that the mean of a set of data is, if the data is not doing anything weird, likely close to the middle of the set. The term appeared in English in the middle 15th century.

The word “median”, meanwhile, follows a completely different path. That one traces to the Middle French “médian”, which traces to the Late Latin “medianus” and Latin “medius” and Proto-Indo-European “medhyo”. This appeared as a mathematical term in the late 19th century; Etymology Online claims 1883, but doesn’t give a manuscript citation.

The word “mode”, meanwhile, follows a completely different path. This one traces to the Old French “mode”, itself from the Latin “modus”, meaning the measure or melody or style. We get from music to common values by way of the “style” meaning. Think of something being done “á la mode”, that is, “in the [ fashionable or popular ] style”. I haven’t dug up a citation about when this word entered the mathematical parlance.

So “mean” and “median” don’t have much chance to do anything but alliterate. “Mode” is coincidence here. I agree, it might be nice if we spread out the words a little more.

Edison, pointing to a checkerboard: 'So Grandpa if you put one cookie on the first square, two on the second, four on the next, then eight, and you keep doubling them until you fill all 64 squares do you know what you'll end up with?' Grandpa: 'A stomachache for a month?'
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th of March, 2019. I’ve been talking about this strip a lot lately, it seems to me. Essays where I do discuss Edison Lee are at this link.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th has Edison introduce a sequence to his grandfather. Doubling the number of things for each square of a checkerboard is an ancient thought experiment. The notion, with grains of wheat rather than cookies, seems to be first recorded in 1256 in a book by the scholar Ibn Khallikan. One story has it that the inventor of chess requested from the ruler that many grains of wheat as reward for inventing the game.

If we followed Edison Lee’s doubling through all 64 squares we’d have, in total, need for 263-1 or 18,446,744,073,709,551,615 cookies. You can see why the inventor of chess didn’t get that reward, however popular the game was. It stands as a good display of how exponential growth eventually gets to be just that intimidatingly big.

Edison, like many a young nerd, is trying to stagger his grandfather with the enormity of this. I don’t know that it would work. Grandpa ponders eating all that many cookies, since he’s a comical glutton. I’d estimate eating all that many cookies, at the rate of one a second, eight hours a day, to take something like eighteen billion centuries. If I’m wrong? It doesn’t matter. It’s a while. But is that any more staggering than imagining a task that takes a mere ten thousand centuries to finish?

Toby, looking at his homework, and a calculator, and a textbook: '... Wow. Crazy ... Huh. How about that? ... Am I stupid if math *always* has a surprise ending?'
Greg Cravens’s The Buckets for the 19th of March, 2019. It also seems like I discuss The Buckets more these days, as seen at this link.

Greg Cravens’s The Buckets for the 19th sees Toby surprised by his mathematics homework. He’s surprised by how it turned out. I know the feeling. Everyone who does mathematics enough finds that. Surprise is one of the delights of mathematics. I had a great surprise last month, with a triangle theorem. Thomas Hobbes, the philosopher/theologian, entered his frustrating sideline of mathematics when he found the Pythagorean Theorem surprising.

Mathematics is, to an extent, about finding interesting true statements. What makes something interesting? That depends on the person surprised, certainly. A good guideline is probably “something not obvious before you’ve heard it, thatlooks inevitable after you have”. That is, a surprise. Learning mathematics probably has to be steadily surprising, and that’s good, because this kind of surprise is fun.

If it’s always a surprise there might be trouble. If you’re doing similar kinds of problems you should start to see them as pretty similar, and have a fair idea what the answers should be. So, from what Toby has said so far … I wouldn’t call him stupid. At most, just inexperienced.

Caption: Eric had good feelings about his date. It turned out, contrary to what he first thought ... [ Venn Diagram picture of two circles with a good amount of overlap ] ... They actually had quite a lot in common.
Eric the Circle for the 19th of March, 2019, this one by Janka. This and other essays with Eric the Circle, by any artist, should be at this link.

Eric the Circle for the 19th, by Janka, is the Venn Diagram joke for the week. Properly any Venn Diagram with two properties has an overlap like this. We’re supposed to place items in both circles, and in the intersection, to reflect how much overlap there is. Using the sizes of each circle to reflect the sizes of both sets, and the size of the overlap to represent the size of the intersection, is probably inevitable. The shorthand calls on our geometric intuition to convey information, anyway.

She: 'What time is it?' He: 'The clock in the hall says 7:56, the oven clock says 8:02, the DVD clock says 8:07, and the bed table clock says 8:13.' She: 'Which one's right?' He: 'I dunno.' She: 'Let's ee. Four clocks ... carry the one ... divided by four ... ' He: 'Whenever we want to know what time it is we have to do algebra.' She: 'We better hurry, it's 8:05 and five-eighths!'
Tony Murphy’s It’s All About You for the 19th of March, 2019. And the occasional time I discuss something from It’s All About You are here.

Tony Murphy’s It’s All About You for the 19th has a bunch of things going on. The punch line calls “algebra” what’s really a statistics problem, calculating the arithmetic mean of four results. The work done is basic arithmetic. But making work seem like a more onerous task is a good bit of comic exaggeration, and algebra connotes something harder than arithmetic. But Murphy exaggerates with restraint: the characters don’t rate this as calculus.

Then there’s what they’re doing at all. Given four clocks, what’s the correct time? The couple tries averaging them. Why should anyone expect that to work?

There’s reason to suppose this might work. We can suppose all the clocks are close to the correct time. If they weren’t, they would get re-set, or not looked at anymore. A clock is probably more likely to be a little wrong than a lot wrong. You’d let a clock that was two minutes off go about its business, in a way you wouldn’t let a clock that was three hours and 42 minutes off. A clock is probably as likely to show a time two minutes too early as it is two minutes too late. This all suggests that the clock errors are normally distributed, or something like that. So the error of the arithmetic mean of a bunch of clock measurements we can expect to be zero. Or close to zero, anyway.

There’s reasons this might not work. For example, a clock might systematically run late. My mantle clock, for example, usually drifts about a minute slow over the course of the week it takes to wind. Or the clock might be deliberately set wrong: it’s not unusual to set an alarm clock to five or ten or fifteen minutes ahead of the true time, to encourage people to think it’s later than it really is and they should hurry up. Similarly with watches, if their times aren’t set by Internet-connected device. I don’t know whether it’s possible to set a smart watch to be deliberately five minutes fast, or something like that. I’d imagine it should be possible, but also that the people programming watches don’t see why someone might want to set their clock to the wrong time. From January to March 2018, famously, an electrical grid conflict caused certain European clocks to lose around six minutes. The reasons for this are complicated and technical, and anyway The Doctor sorted it out. But that sort of systematic problem, causing all the clocks to be wrong in the same way, will foil this take-the-average scheme.

Murphy’s not thinking of that, not least because this comic’s a rerun from 2009. He was making a joke, going for the funnier-sounding “it’s 8:03 and five-eights” instead of the time implied by the average, 8:04 and a half. That’s all right. It’s a comic strip. Being amusing is what counts.


There were just enough mathematically-themed comic strips this past week for one more post. When that is ready, it should be at this link. I’ll likely post it Tuesday.

Reading the Comics, February 23, 2019: Numerals Edition


It’s happened again: another slow week around here. My supposition is that Comic Strip Master Command was snowed in about a month ago, and I’m seeing the effects only now. There’s obviously no other reason that more comic strips didn’t address my particular narrow interest in one seven-day span.

Samson’s Dark Side of the Horse for the 18th is a numerals joke. The mathematics content is slight, I admit, but I’ve always had a fondness for Dark Side of the Horse. (I know it sounds like I have a fondness for every comic strip out there. I don’t quite, but I grant it’s close.) Conflating numerals and letters, and finding words represented by numerals, is an old tradition. It was more compelling in ancient days when letters were used as numerals so that it was impossible not to find neat coincidences. I suppose these days it’s largely confined to typefaces that make it easy to conflate a letter and a numeral. I mean moreso than the usual trouble telling apart 1 and l, 0 and O, or 5 and S. Or to special cases like hexadecimal numbers where, for ease of representation, we use the letters A through F as numerals.

Samson, counting sheep in bed. #198 leaps over the fence; the numbers are written as might appear on a 14-segment LED. #199 follows. The next panel #200, makes the numerals look like the word ZOO, and an elephant marches through, uprooting the fence.
Samson’s Dark Side of the Horse for the 18th of February, 2019. I don’t always discussDark Side of the Horse but when I do, it should appear at this link.

Jef Mallett’s Frazz for the 18th is built on an ancient problem. I remember being frustrated with it. How is “questions 15 to 25” eleven questions when the difference between 15 and 25 is ten? The problem creeps into many fields. Most of the passion has gone out of the argument but around 1999 you could get a good fight going about whether the new millennium was to begin with January 2000 or 2001. The kind of problem is called a ‘fencepost error’. The name implies how often this has complicated someone’s work. Divide a line into ten segments. There are nine cuts on the interior of the line and the two original edges. I’m not sure I could explain to an elementary school student how the cuts and edges of a ten-unit-long strip match up to the questions in this assignment. I might ask how many birthdays someone’s had when they’re nine years old, though. And then flee the encounter.

Kid: 'If 25 minus 10 is 15, how is *doing* questions 15-25 eleven questions? I should get credit for answering one extra question.' Teacher: 'Two extra questions, if you can answer the one you just brought up.'
Jef Mallett’s Frazz for the 18th of February, 2019. Essays with a mention of Frazz appear at this link.

Mark Parisi’s Off The Mark for the 19th is another numerals joke. This one’s also the major joke to make about an ice skater doing a figure eight: write the eight some other way. (I’d have sworn there was an M-G-M Droopy cartoon in which Spike demonstrates his ability to skate a figure 8, and then Droopy upstages him by skating ‘4 + 4’. I seem to be imagining it; the only cartoon where this seems to possibly fit is 1950’s The Chump Champ, and the joke isn’t in that one. If someone knows the cartoon I am thinking of, please let me know.) Here, the robot is supposed to be skating some binary numeral. It’s nothing close to an ‘8’, but perhaps the robot figures it needs to demonstrate some impressive number to stand out.

An ice-staking woman does a figure 8. An ice-skating robot does a figure 1110101101.
Mark Parisi’s Off The Mark for the 19th of February, 2019. When I have discussed Off The Mark I’ve tried to tag it so the essays appear at this link.

Bud Blake’s Tiger for the 21st has Tiger trying to teach his brother arithmetic. Working it out with fingers seems like a decent path to try, given Punkinhead’s age and background. And Punkinhead has a good point: why is the demonstration the easy problem and the homework the hard problem? I haven’t taught in a while, but do know I would do that sort of thing. My rationalization, I think, would be that a hard problem is usually hard because it involves several things. If I want to teach a thing, then I want to highlight just that thing. So I would focus on a problem in which that thing is the only tricky part, and everything else is something the students are so familiar with they don’t notice it. The result is usually an easy problem. There isn’t room for toughness. I’m not sure if that’s a thing I should change, though. Demonstrations of how to work harder problems are worth doing. But I usually think of those as teaching “how to use these several things we already know”. Using a tough problem to show one new thing, plus several already-existing tricky things, seems dangerous. It might be worth it, though.

Tiger, holding up his fingers: 'Look, one plus one is two, two plus two is four. So then what's four plus four?' Punkinhead: 'How come YOU do the easy ones and you save the tough oens for me?'
Bud Blake’s Tiger rerun for the 21st of February, 2019. I have no information about when it first appeared. Essays inspired by Tiger should appear at this link.

This was not a busy week for comic strips. If it had been, I likely wouldn’t have brought in Dark Side of the Horse. Still there were a handful of comics too slight to get a write-up, even so. John Zakour and Scott Roberts’s Maria’s Day on the 19th just mentioned mathematics homework as hard, for example. Eric the Circle for the 22nd has a binary numeral written out. That one was written by ‘urwatuis’. Maybe that would have been a good, third, numeral comic strip to discuss.


That’s all the mathematically-themed comic strips for the week, though. Next Sunday I should have a fresh Reading the Comics post at this link.

Reading the Comics, February 13, 2019: Light Geometry Edition


Comic Strip Master Command decided this would be a light week, with about six comic strips worth discussing. I’ll go into four of them here, and in a day or two wrap up the remainder. There were several strips that didn’t quite rate discussion, and I’ll share those too. I never can be sure what strips will be best taped to someone’s office door.

Alex Hallatt’s Arctic Circle for the 10th was inspired by a tabular iceberg that got some attention in October 2018. It looked surprisingly rectangular. Smoother than we expect natural things to be. My first thought about this strip was to write about crystals. The ways that molecules can fit together may be reflected in how the whole structure looks. And this gets us to studying symmetries.

Ed Penguin 'Did you see the perfectly rectangular iceberg?' Lenny Lemming: 'Yes, but I've seen perfect triangles, rhomboids, and octagons, too.' (Oscar Penguin is startled. He walks over to Frank, who is chiseling out some kind of octagonal prism.) Oscar: 'Ok, Frank, I know I said you needed a hobby ... ' Frank: 'Let's see them explain THIS one with science.'
Alex Hallatt’s Arctic Circle for the 10th of February, 2019. Essays in which I discuss Arctic Circle should be at this link.

But I got to another thought. We’re surprised to see lines in nature. We know what lines are, and understand properties of them pretty well. Even if we don’t specialize in geometry we can understand how we expect them to work. I don’t know how much of this is a cultural artifact: in the western mathematics tradition lines and polygons and circles are taught a lot, and from an early age. My impression is that enough different cultures have similar enough geometries, though. (Are there any societies that don’t seem aware of the Pythagorean Theorem?) So what is it that has got so many people making perfect lines and circles and triangles and squares out of crooked timbers?

Broom Hilda: 'I'm a winner! I won $18 in the lottery!' Gaylord: 'How much did you spend on tickets?' Hilda: '$20.' Gaylord: 'So you're actually a loser!' Hilda: 'Well, I guess you could say that, but I wish you hadn't!'
Russell Myers’s Broom Hilda for the 13th of February, 2019. Essays inspired by Broom Hilda should be gathered at this link.

Russell Myers’s Broom Hilda for the 13th is a lottery joke. Also, really, an accounting joke. Most of the players of a lottery will not win, of course. Nearly none of them will win more than they’ve paid into the lottery. If they didn’t, there would be an official inquiry. So, yes, nearly all people, even those who win money at the lottery, would have had more money if they skipped playing altogether.

Where it becomes an accounting question is how much did Broom Hilda expect to have when the week was through? If she planned to spend $20 on lottery tickets, and got exactly that? It seems snobbish to me to say that’s a dumber way to spend twenty bucks than, say, buying twenty bucks worth of magazines that you’ll throw away in a month would be. Or having dinner at a fast-casual place. Or anything else that you like doing even though it won’t leave you, in the long run, any better off. Has she come out ahead? That depends where she figures she should be.

Caption: Transcendental Eric achieves a higher plane. It shows a shaded, spherical Eric in a three-dimensional space, while below him a square asks some other polygon, 'Where's Eric gone to?'
Eric the Circle for the 13th of February, 2019, this one by Alabama_Al. Appearances by Eric the Circle, whoever the writer, should be at this link.

Eric the Circle for the 13th, this one by Alabama_Al, is a plane- and solid-geometry joke. This gets it a bit more solidly on-topic than usual. But it’s still a strip focused on the connotations of mathematically-connected terms. There’s the metaphorical use of the ‘plane’ as in the thing people perceive as reality. There’s conflation between the idea of a ‘higher plane’ and ‘higher dimensions’. Also somewhere in here is the idea that ‘higher’ and ‘more’ dimensions of space are the same thing. ‘Transcendental’ here is used in the common English sense of surpassing something. ‘Transcendental’ has a mathematical definition too. That one relates to polynomials, because everything in mathematics is about polynomials. And, of course, one of the two numbers we know to be transcendental, and that people have any reason to care about, is π, which turns up all over circles.

Joey: 'Mom, can I have a cookie?' Mom: 'Joey, you had two cookies this morning, three at lunch, and one an hour ago! Now how many is that?' Joey: 'I changed my mind.' (Thinking) 'No cookie is worth a pop quiz in math.'
Larry Wright’s Motley for the 13th of February, 2019. It originally ran in 1988, I believe on the same date. When I have written about Motley the results should appear at this link. In transcribing the strip for the alt-text here I was getting all ready to grumble that I didn’t know the kid’s name, and the strip is so old and minor that nobody has a cast list on it. Then I noticed, oh, yes, Mom says what the kid’s name is.

Larry Wright’s Motley for the 13th riffs on the form of a story problem. Joey’s mother does ask something that seems like a plausible addition problem. I’m a bit surprised he hadn’t counted all the day’s cookies already, but perhaps he doesn’t dwell on past snacks.


This and all my Reading the Comics posts should appear at this link. Thanks for looking at my comments.

Reading the Comics, January 19, 2019: Not Making The Cut Edition


I’m trying to be a bit more rigorous about comic strips needing mathematical content before I talk about them. So, for example, Maria Scrivan had a Half Full that’s a Venn Diagram joke. But I feel like that isn’t quite enough for me to discuss at greater length. There was a Barney Google where Jughead explains why he didn’t do his mathematics homework. And I’m trying not to bring up Randolph Itch, since I’ve been through several circuits of the short-lived strip already. But it re-ran the one that renders Randolph as a string of numerals. If you haven’t seen that before, it’s a cute bit of symbols play.

Now here’s the comics that did make the cut:

A numeral 3 turns backwards to Fi, calling out 'Catch me!' Fi doesn't respond; the 3 falls on its back. Dethany: 'You're rejecting that department's expense report?' Fi: 'I don't trust its numbers.'
Bill Holbrook’s On The Fastrack for the 18th of January 2019. This and other essays mentioning On The Fastrack should be at this link.

Bill Holbrook’s On The Fastrack for the 18th is an anthropomorphic numerals joke. It’s part of Holbrook’s style to draw metaphors as literal happenings. It’s also a variation on a joke Holbrook used just last month, depicting then the phrase “accepting his numbers”. What I said about “accepting numbers” transfers over naturally to “trusting numbers”. It’s not that a number itself means anything. It’s that numbers are used to represent some narrative. If we can’t believe the narrative, we don’t believe the numbers. And the numbers used to represent something can give us reasons to trust, or reject, a narrative.

Caption; Eric didn't listen in class when teacher explained circ = 2 * pi * R. Picture is Eic trying to squeeze into too tight a pair of pants: 'These pants are too small ... can't fit'em.'
Eric the Circle for the 18th of January 2019. This one is credited to ‘Krisis’. Somehow I don’t write up every single Eric the Circle. But when I do, Eric the Circle talk is at this link.

Eric the Circle for the 18th I can dub an anthropomorphic geometry joke for the week. At least it brings up one of the handful of geometry facts that people remember outside school. The relationship between the circumference and the diameter (or radius, if you rather) of a circle has been known just forever. It has the advantage of going through π, supporting and being supported by that celebrity number. … I’m not quite sure about the logic of this joke, though. My experience is that guys at least are fairly good about knowing their waist size (if you don’t know, it’s 38, although a 40 can feel so comfortable, and they’re sure they can wear a 36). Radius is a harder thing to keep in mind. But maybe it’s different for circles.

Teacher pointing to 6 + 4 on the blackboard: 'Nerwin, if I add these two numbers together, what do I get?' Nerwin: 'A bigger number!' Nerwin, in the corner: 'I'm not a detail person!'
Russell Myers’s Broom Hilda for the 19th of January 2019. When I find something in Broom Hilda inspirational, what’s inspired should be at this link.

Russell Myers’s Broom Hilda for the 19th is a student-and-teacher problem. One thing is that Nerwin’s not wrong. It’s just that simply saying something true isn’t enough. We want to say things that are true and interesting.

But “you add two numbers and get a number” can be interesting. It depends on context. For example, in group theory, we will start by describing groups as a collection of things and an operation which works like addition. What does it mean to work like addition? Here, it means if you add two things from the collection, you get something from the collection. The collection of things is “closed” under your operation. And mathematical operations defined this abstractly — or defined this vaguely, if you don’t like the way it goes — can be great. We’re introduced to vectors, for example, as “ordered sets of numbers”. And that definition works all right. But when you start thinking of them instead as “things you can add to vectors and get other vectors out” you gain new power. You can use the mechanism developed for ordered sets of numbers to describe many things, including matrices and functions and shapes. But when we do that we’re saying things about how addition works, rather than what this particular addition is.

You know, on reflection, I’m not sure that Eric the Circle was more worthy of discussion than that Barney Google was. Hm.


And I should be back with more comics on Sunday. They should appear at this link when it’s all ready.

Reading the Comics, December 22, 2018: Christmas Break Edition


There were just enough mathematically-themed comic strips last week for me to make two posts out of it. This current week? Is looking much slower, at least as of Wednesday night. But that’s a problem for me to worry about on Sunday.

Eric the Circle for the 20th, this one by Griffinetsabine, mentions a couple of shapes. That’s enough for me, at least on a slow comics week. There is a fictional tradition of X marking the spot. It can be particularly credited to Robert Louis Stevenson’s Treasure Island. Any symbol could be used to note a special place on maps, certainly. Many maps are loaded with a host of different symbols to convey different information. Circles and crosses have the advantage of being easy to draw and difficult to confuse for one another. Squares, triangles, and stars are good too.

Eric on a treasure hunt: Eric asking, 'Wait ... triangle marks the spot? No ... rhombus marks the spot? Dodecahedron marks the spot?' A square sighs; an X coughs, 'ahem!'
Eric the Circle for the 20th of December, 2018, this one by Griffinetsabine. This and other essays featuring Eric the Circle are at this link.

Bill Whitehead’s Free Range for the 22nd spoofs Wheel of Fortune with “theoretical mathematics”. Making a game out of filling in parts of a mathematical expression isn’t ridiculous, although it is rather niche. I don’t see how the revealed string of mathematical expressions build to a coherent piece, but perhaps a few further pieces would help.

Wheel Of Theoretical Mathematics. Contestant: 'I'd like to buy a sqrt(x).' On the board are several mathematical expressions, including 'dx = sqrt{pi}', 'a^2 + b^2 = (a + b)', and 'dy/dx x^4 - (1 - x^2)^4$.
Bill Whitehead’s Free Range for the 22nd of December, 2018. Appearances by Free Range should be at this link.

The parts shown are all legitimate enough expressions. Well, like a^2 + b^2 = (a + b) is only true for some specific numbers ‘a’ and ‘b’, but you can find solutions. -b \pm \sqrt{b^2 - x^2y^2} is just an expression, not picking out any particular values of ‘b’ or ‘x’ or ‘y’ as interesting. But in conjunction with a^2 + b^2 = (a + b) or other expressions there might be something useful. On the second row is a graph, highlighting a region underneath a curve (and above the x-axis) between two vertical lines. This is often the sort of thing looked at in calculus. It also turns up in probability, as the area under a curve like this can show the chance that an experiment will turn up something in a range of values. And \frac{dy}{dx} = x^4 - \left(1 - x^2\right)^4 is a straightforward differential equation. Its solution is a family of similar-looking polynomials.

Scientist guy runs in to the Lucky Cow restaurant. The scientist begs of cashier Neil, the left half of Schrodinger's equation equals ?!? Neil thinks about it some and then provides the answer, earning the scientist's gratitude and the admiration of his coworkers. Later, out in back, Neil pays off the scientist.
Mark Pett’s Lucky Cow for the 22nd of December, 2018. This and other discussions inspired by Lucky Cow should be at this link.

Mark Pett’s Lucky Cow for the 22nd has run before. I’ve even made it the title strip for a Reading the Comics post back in 2014. So it’s probably time to drop this from my regular Reading the Comics reporting. The physicists comes running in with the left half of the time-dependent Schrödinger Equation. This is all over quantum mechanics. In this form, quantum mechanics contains information about how a system behaves by putting it into a function named \psi . Its value depends on space (‘x’). It can also depend on time (‘t’). The physicists pretends to not be able to complete this. Neil arranges to give the answer.

Schrödinger’s Equation looks very much like a diffusion problem. Normal diffusion problems don’t have that \imath which appears in the part of Neil’s answer. But this form of equation turns up a lot. If you have something that acts like a fluid — and heat counts — then a diffusion problem is likely important in understanding it.

And, yes, the setup reminds me of a mathematical joke that I only encounter in lists of mathematics jokes. That one I told the last time this strip came up in the rotation. You might chuckle, or at least be convinced that it is a correctly formed joke.


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, October 26, 2018: I Am Overloaded Edition


I’ve settled to a pace of about four comics each essay. It makes for several Reading the Comics posts each week. But none of them are monsters that eat up whole evenings to prepare. Except that last week there were enough comics which made my initial cut that I either have to write a huge essay or I have to let last week’s strips spill over to Sunday. I choose that option. It’s the only way to square it with the demands of the A to Z posts, which keep creeping above a thousand words each however much I swear that this next topic is a nice quick one.

Roy Schneider’s The Humble Stumble for the 25th has some mathematics in a supporting part. It’s used to set up how strange Tommy is. Mathematics makes a good shorthand for this. It’s usually compact to write, important for word balloons. And it’s usually about things people find esoteric if not hilariously irrelevant to life. Tommy’s equation is an accurate description of what centripetal force would be needed to keep the Moon in a circular orbit at about the distance it really is. I’m not sure how to take Tommy’s doubts. If he’s just unclear about why this should be so, all right. Part of good mathematical learning can be working out the logic of some claim. If he’s not sure that Newtonian mechanics is correct — well, fair enough to wonder how we know it’s right. Spoiler: it is right. (For the problem of the Moon orbiting the Earth it’s right, at least to any reasonable precision.)

Tommy: 'Newton's theory states that the centripetal force holding the moon in its orbit must equal mv^2/R = mv^2/(60 R_E), but I'm not sure I agree.' Molly stares at him a while, and then shouts, 'TAKE THE PENCILS OUTTA YOUR NOSE!'
Roy Schneider’s The Humble Stumble rerun for the 25th of October, 2018. It originally ran the 30th of January, 2007.

Stephan Pastis’s Pearls Before Swine for the 25th shows how we can use statistics to improve our lives. At least, it shows how tracking different things can let us find correlations. These correlations might give us information about how to do things better. It’s usually a shaky plan to act on a correlation before you have a working hypothesis about why the correlation should hold. But it can give you leads to pursue.

Pig, writing out on paper: 'Percentage of my problems that occur during my waking hours: 100%. Percentage of my problems that occur when I am asleep in bed: 0%' Next panel: Pig, in bed, explaining to Rat, 'Bed is mathematically correct.'
Stephan Pastis’s Pearls Before Swine for the 25th of October, 2018. And from this we learn that Pig is not yet of the age where sometimes your neck and back hurt for three weeks because your pillow was a quarter-inch off its normal position. Bodies are fun things and everyone should have one.

Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 25th is a joke about mathematics being hard. In this case even for a being that’s a natural mathematician. Relatable.

Virginia: 'Ms Delphi, can you calculate launch vectors?' Delphi: 'Do you think Madame Delphi's powers extend to doing MATHS?' Virginia: 'Not exactly. I think you're the analytic and predicative node of an incredibly advanced hive mind .. which, yes, does involve some math.' Delphi: 'Madame Delphi demands a pencil.'
Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 25th of October, 2018. Uh, the characters here are trapped inside an artificial intelligence and Virginia (the black-haired woman with clear glasses) has worked out the other characters are part of an intelligent swarm of bees and they’re trying to launch an escape, so this is why all the plot makes sense.

Eric the Circle for the 26th, this one by Vissoro, is a “two types of people in the world” joke. Given the artwork I believe it’s also riffing on the binary-arithmetic version of the joke. Which is, “there are 10 types of people in the world, those who understand binary and those who don’t”.

Eric, a circle, inside a field of 1's. Caption; 'There are two types of people in the world. The ones that know about Eric, and the ones that don't.'
Eric the Circle for the 26th of October, 2018, this one by Vissoro. It makes me think of that Futurama scene where Bender has a nightmare, dreaming of a 2.

If you’d like to see more Reading the Comics posts, try this link. Essays mentioning The Humble Stumble are at this link. Essays discussing by Pearls Before Swine are at this link. Essays with a mention of Skin Horse should be at this link. I’m surprised to learn there are others, too. I’d have thought it was a new tag. Posts about what’s brought up by Eric the Circle should be at link. And this month and the rest of this year my Fall 2018 Mathematics A-To-Z should continue. And it is open for requests for more of the alphabet.

Reading the Comics, 1 September 2018: Retirement Of A Tag Edition


I figure to do something rare, and retire one of my comic strip tags after today. Which strip am I going to do my best to drop from Reading the Comics posts? Given how many of the ones I read are short-lived comics that have been rerun three or four times since I started tracking them? Read on and see!

Bill Holbrook’s On The Fastrack for the 29th of August continues the sequence of Fi talking with kids about mathematics. My understanding was that she tried to give talks about why mathematics could be fun. That there are different ways to express the same number seems like a pretty fine-grain detail to get into. But this might lead into some bigger point. That there are several ways to describe the same thing can be surprising and unsettling to discover. That you have, when calculating, the option to switch between these ways freely can be liberating. But you have to know the option is there, and where to look for it. And how to see it’ll make something simpler.

Wendy: 'I never thought Fi would have a talent for teaching.' Dethany: 'It surprised her, too. But something about her demeanor appeals to kids.' (At the class.) Fi: 'See? Two-sixth of a zombie is the same as one-third ... '
Bill Holbrook’s On The Fastrack for the 29th of August, 2018. What … what graphic does she have on-screen?

Bill Holbrook’s On The Fastrack for the 30th of August gets onto a thread about statistics. The point of statistics is to describe something complicated with something simple. So detail must be lost. That said, there are something like 2,038 different things called “average”. Each of them has a fair claim to the term, too. In Fi’s example here, 73 degrees (Fahrenheit) could be called the average as in the arithmetic mean, or average as in the median. The distribution reflects how far and how often the temperature is from 73. This would also be reflected in a quantity called the variance, or the standard deviation. Variance and standard deviation are different things, but they’re tied together; if you know one you know the other. It’s just sometimes one quantity is more convenient than the other to work with.

Fi: 'Numbers don't lie ... but the unscrupulous can get them to say whatever they want. Like when the boss claims the average temperature in your office is 73 degrees when it's really kept at 63 degrees in the winter and 83 degrees in the summer.' Kid: 'Our principal does that.'
Bill Holbrook’s On The Fastrack for the 30th of August, 2018. Somebody nag me sometime to tell the story about when I used Skylab’s torn-away meteorite shield for a heat-flow problem in a differential equations class. Thank you.

Bill Holbrook’s On The Fastrack for the 1st of September has Fi argue that apparent irrelevance makes mathematics boring. It’s a common diagnosis. I think I’ve advanced the claim myself. I remember a 1980s probability textbook asking the chance that two transistors out of five had broken simultaneously. Surely in the earlier edition of the textbook, it was two vacuum tubes out of five. Five would be a reasonable (indeed, common) number of vacuum tubes to have in a radio. And it would be plausible that two might be broken at the same time.

Kid: 'I think math's boring.' Fi: 'That's because you've been taught with 75-year-old word problems. They just need a little updating. ... Like, if your tweet is retweeted by ten people ... who each share it with ten MORE people ... at what point can it be said to go viral?' (Everyone has hands up.)
Bill Holbrook’s On The Fastrack for the 1st of September, 2018. Not to question Holbrook’s writing, since he somehow maintains three successful daily comic strips and may be presumed to know what he’s doing, but shouldn’t this have come before the strip from the 29th?

It seems obvious that wanting to know an answer makes it easier to do the work needed to find it. I’m curious whether that’s been demonstrated true. Like, it seems obvious that a reference to a thing someone doesn’t know anything about would make it harder to work on. But does it? Does it distract someone trying to work out the height of a ziggurat based on its distance and apparent angle, if all they know about a ziggurat is their surmise that it’s a thing whose height we might wish to know?

Randolph's dream: a pirate at a schooldesk. 'Algebra pretty much put pirates out of business.' Teacher: 'If ax^2 + bx + c = 0, what is x?' (The pirate looks at the treasure map, marked x, sweating.' Footer joke: the teacher asks, '15 men on a dead man's chest, yo ho ho, and a bottle of rum equals what?'
Tom Toles’s Randolph Itch, 2 am rerun for the 30th of August, 2018. It originally ran the 13th of January, 2000.

Tom Toles’s Randolph Itch, 2 am rerun for the 30th of August is an old friend that’s been here a couple times. I suppose I do have to retire the strip from my Reading the Comics posts, at least, although I’m still amused enough by it to keep reading it daily. Simon Garfield’s On The Map, a book about the history of maps, notes that the X-marks-the-spot thing is an invention of the media. Robert Louis Stevenson’s Treasure Island particularly. Stevenson’s treasure map, Garfield notes, had to be redrawn from the manuscript and the author’s notes. The original went missing in the mail to the publishers. I just mention because I think that adds a bit of wonder to the treasure map. And since, I guess, I won’t have the chance to mention this again.

Kid: 'Mom, Dad, why do you have a giant inflatable Klein bottle hidden in the closet?' Mom: 'Compromise. I'll say nothing more. NOW GO WASH YOUR HANDS.' Underneath, a Venn diagram, with one bubble 'Having Sex Inside', the other 'Having Sex Outside', and the intersection 'Having Sex Near A Non-Orientable Surface'.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 30th of August, 2018. Interesting to me is that either the panel comic by itself, or the Venn diagram by itself, would be a sufficient joke. The panel would be a cryptic one, one that probably attracted ‘I don’t get it’ responses, but it’d be decipherable. The Venn Diagram one would be fine, but wouldn’t have the tension and energy of the full strip.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 30th of August satisfies the need for a Venn Diagram joke this time around. It’s also the strange-geometry joke for the week. Klein bottles were originally described by Felix Klein. They exist in four (or more) dimensions, in much the way that M&oum;bius strips exist in three. And like the M&oum;bius strip the surface defies common sense. You can try to claim some spot on the surface is inside and some other spot outside. But you can get from your inside to your outside spot in a continuous path, one you might trace out on the surface without lifting your stylus.

If you were four-dimensional. Or more. If we were to see one in three dimensions we’d see a shape that intersects itself. As beings of only three spatial dimensions we have to pretend that doesn’t happen. It’s the same we we pretend a drawing of a cube shows six squares all of equal size and connected at right angles to one another, even though the drawing is nothing like that. The bottle-like shape Weinersmith draws is, I think, the most common representation of the Klein bottle. It looks like a fancy bottle, and you can buy one as a novelty gift for a mathematician. I don’t need one but do thank you for thinking of me. MathWorld shows another representation, a figure-eight-based one which looks to me like an advanced pasta noodle. But it doesn’t look anything like a bottle.

Abstract's Bar and Grille. Once again, Eric the Circle's pick-up line backfires ... and he is left confused and speechless. Eric: 'You're acutey. What's your sign?' Triangle: 'Opposite over hypotenuse. What's yours?'
Eric the Circle for the 31st of August, 2018. … Shouldn’t this be with a right triangle?

Eric the Circle for the 31st of August, this one by JohnG, is a spot of wordplay. The pun here is the sine of an angle in a (right) triangle. That would be the length of the leg opposite the angle divided by the length of the hypotenuse. This is still stuff relevant to circles, though. One common interpretation of the cosine and sine of an angle is to look at the unit circle. That is, a circle with radius 1 and centered on the origin. Draw a line segment opening up an angle θ from the positive x-axis. Draw it counterclockwise. That is, if your angle is a very small number, you’re drawing a line segment that’s a little bit above the positive x-axis. Draw the line segment long enough that it touches the unit circle. That point where the line segment and the circle intersect? Look at its Cartesian coordinates. The y-coordinate will be the sine of θ. The x-coordinate will be the cosine of θ. The triangle you’re looking at has vertices at the origin; at x-coordinate cosine θ, y-coordinate 0; and at x-coordinate cosine θ, y-coordinate sine θ.

[ When Zippy was three, he said the darnedest things ] Zippy: 'X plus Y divided by Shirley Booth equals Soft Serve.' [ At the age of 11, he continued to amaze and impress his parents. ] 'If I had the powers of Spider-Man and the costume of Mighty Mouse, I could understand algebra!' [ He mellowed a bit at 16 and thought deep thoughts about the universe and stuff. ] 'If you lined up ALL the jars of Bosco ever manufactured, they'd form a ring all the way around the Earth and wind up conking Einstein on the bean in Newark!' [ As a ADULT, Zippy knows that not everyone appreciates his surreal spoutings, so he often muses to himself. ] Zippy, thinking: 'If I drew Mary Worth, she'd drive a 1958 two-tone Metro!'
Bill Griffith’s Zippy the Pinhead for the 1st of September, 2018. Somebody nag me sometime to save that last panel for my next What’s Going On In Mary Worth plot recap.

Bill Griffith’s Zippy the Pinhead for the 1st of September is its usual sort of nonsense, the kind that’s up my alley. It does spend two panels using arithmetic and algebra as signifiers of intelligence, or at least thoughtfulness.


My other Reading the Comics posts should appear at this link. Other essays with On The Fastrack are at this link. The essays that mentioned Randolph Itch, 2 am, are at this link, and I suppose this will be the last of them. We’ll see if I do succeed in retiring the tag. Other appearances by Saturday Morning Breakfast Cereal are at this link. The strip comes up here a lot. Eric the Circle comics should be at this link. And other essays with Zippy the Pinhead mentions should be at this link. Thank you.

Reading the Comics, June 19, 2018: Don’t Ask About The Hyperbolic Cosine Edition


Although the hyperbolic cosine is interesting and I could go on about it.

Eric the Circle for the 18th of June is a bit of geometric wordplay for the week. A secant is — well, many things. One of the important things is it’s a line that cuts across a circle. It intersects the circle in two points. This is as opposed to a tangent, which touch it in one. Or missing it altogether, which I think hasn’t got any special name. “Secant” also appears as one of the six common trig functions out there.

Small circle: 'Hey, Eric! There's a line on you!' Medium circle: 'Get it off!' Eric, with a line across his side: 'See? Can't!'
Eric the Circle for the 18th of June, 2018. This one was composed by Griffinetsabine. It originally appeared sometime in 2012.

In value the secant of an angle is just the reciprocal of the cosine of that angle. Where the cosine is never smaller than -1 nor larger than 1, the secant is always either greater than 1 or smaller than -1. It’s a useful function to have by name. We can write “the secant of angle θ” as sec(\theta) . The otherwise sensible-looking \cos^{-1}(\theta) is unavailable, because we use that to mean “the angle whose cosine is θ”. We need to express that idea, the “arc-cosine” or “inverse cosine”, quite a bit too. And \cos(\theta)^{-1} would look like we wanted the cosine of one divided by θ. Ultimately, we have a lot of ideas we’d like to write down, and only so many convenient quick shorthand ways to write them. And by using secant as its own function we can let the arc-cosine have a convenient shorthand symbol. These symbols are a point where you see the messy, human, evolutionary nature of mathematical symbols at work.

We can understand the cosine of an angle θ by imagining a right triangle with hypotenuse of length 1. Set that so the hypotenuse makes angle θ with respect to the x-axis. Then the opposite leg of that right triangle will be the cosine of θ away from the origin. The secant, now, that works differently. Again here imagine a right triangle, but this time one of the legs has length 1. And that leg is at an angle θ with respect to the x-axis. Then the far leg of that right triangle is going to cross the x-axis. And it’ll do that at a point that’s the secant of θ away from the origin.

Debbie: 'In this soap opera, Kimberly is trying to hide her past from Renaldo ... who has hired a detective to find out how many times (x) Kimberly has made love to how many lovers (y). ... Who says algebra has no use outside the classroom?'
Larry Wright’s Motley Classics for the 19th of June, 2018. It originally ran sometime in 1997.

Larry Wright’s Motley Classics for the 19th speaks of algebra as the way to explain any sufficiently complicated thing. Algebra’s probably not the right tool to analyze a soap opera, or any drama really. The interactions of characters are probably more a matter for graph theory. That’s the field that studies groups of things and the links between them. Occasionally you’ll see analyses of, say, which characters on some complicated science fiction show spend time with each other and which ones don’t. I’m not aware of any that were done on soap operas proper. I suspect most mathematics-oriented nerds view the soaps as beneath their study. But most soap operas do produce a lot of show to watch, and to summarize; I can’t blame them for taking a smaller, easier-to-summarize data set to study. (Also I’m not sure any of these graphs reveal anything more enlightening than, “Huh, really thought The Doctor met Winston Churchill more often than that”.)

Teacher: 'You two making progress on the math problem?' Nancy: 'We're making progress on *A* math problem.' (Nancy and Esther's paper: 'number of seconds left in school, 24 x 5 x 60 x 60'.)
Olivia Jaimes’s Nancy for the 19th of June, 2018. This one originally appeared in June of 2018.

Olivia Jaimes’s Nancy for the 19th is a joke on getting students motivated to do mathematics. Set a problem whose interest people see and they can do wonderful things.

Circle in the bar, speaking to another circle: 'You wanna get out of here, come back to my place and create a Venn diagram?' ... Squirrel in the corner, adding commentary: 'It'll never work ... they have nothing in common.'
Dave Whamond’s Reality Check for the 19th of June, 2018. Those seem like small drinks for circles that large.

Dave Whamond’s Reality Check for the 19th is our Venn Diagram strip for the week. I say the real punch line is the squirrel’s, though. Properly, yes, the Venn Diagram with the two having nothing in common should still have them overlap in space. There should be a signifier inside that there’s nothing in common, such as the null symbol or an x’d out intersection. But not overlapping at all is so commonly used that it might as well be standard.

Cardinal: 'Whatever you're thinking, don't say it.' Other bird has a thought balloon full of arithmetic expressions.
Teresa Bullitt’s Frog Applause for the 21st of June, 2018. It’s a Dadaist comic strip; embrace the bizarreness.

Teresa Bullitt’s Frog Applause for the 21st uses a thought balloon full of mathematical symbols as icon for far too much deep thinking to understand. I would like to give my opinion about the meaningfulness of the expressions. But they’re too small for me to make out, and GoComics doesn’t allow for zooming in on their comics anymore. I looks like it’s drawn from some real problem, based on the orderliness of it all. But I have no good reason to believe that.


If you’d like more of these Reading the Comics posts, you can find them in reverse chronological order at this link. If you’re interested in the comics mentioned particularly here, Eric the Circle strips are here. Frog Applause comics are on that link. Motley strips are on that link. Nancy comics are on that page. And And Reality Check strips are here.

Reading the Comics, April 19, 2018: Late Because Of Pinball Edition


Hi, all. I apologize for being late in posting this, but my Friday and Saturday were eaten up by pinball competition. Pinball At The Zoo, particularly, in Kalamazoo, Michigan. There, Friday, I stepped up first thing and put in four games on the Classics, pre-1985, tournament bank and based on my entry scores was ranked the second-best player there. And then over the day my scores dwindled lower and lower on the list of what people had entered until, in the last five minutes of qualifying, they dropped off the roster altogether and I was knocked out. Meanwhile in the main tournament, I was never even close to making playoffs. But I did have a fantastic game of Bally/Midway’s World Cup Soccer, a game based on how much the United States went crazy for soccer that time we hosted the World Cup for some reason. The game was interrupted by one of the rubber straps around one of the kickers (the little triangular table just past the flippers that you would think would be called the bumpers) breaking, and then by the drain breaking in a way that later knocked the game entirely out of the competition. So anyway besides that glory I’ve been very busy trying to figure out what’s gone wrong and stepping outside to berate the fox squirrels out back, and that’s why I’m late with all this. I’m sure you relate.

Danielle Rabbit as a lion tamer whipping a 2. Danielle as orchestra conductor leading a 4 playing violin. As a puppet-master holding up an 8 and 3 as marionettes. Juggling the numerals 0 through 9. Nursing a 7. Then reality: Kevin saying, 'Danielle, thanks for doing our taxes.' Danielle: 'Well, you just have to know how to handle numbers.'
Bill Holbrook’s Kevin and Kell rerun for the 15th of April, 2018. The strip is this enormously tall format because at the time it originally ran (in 2012) the strip appeared in print in the Atlanta Journal-Constitution, sharing the page with Wiley Miller’s similarly-formatted Non Sequitur. The strip has since resumed more normal dimensions.

Bill Holbrook’s Kevin and Kell rerun for the 15th is the anthropomorphic numerals strip for the week. Also the first of the anthropomorphic strips for the week. Calculating taxes has always been one of the compelling social needs for mathematics, arithmetic especially. If we consider the topic to be “accounting” then that might be the biggest use of mathematics in society. At least by humans; I’m not sure how to rate the arithmetic that computers do even for not explicitly mathematical tasks like sending messages back and forth. New comic strip tag for around here, too.

Fauna, to her brother Tucker: 'I learned a valuable lesson in trigonometry class today. The next time I sign up for a class, it will have nothing to do with numbers.'
Bill Schorr’s The Grizzwells for the 17th of April, 2018. Yeah, people say that, but then they get into Abstract Algebra and then they see any proof whatsoever that involves ideals of rings.

Bill Schorr’s The Grizzwells for the 17th sees Fauna not liking trigonometry class. I’m sympathetic. I remember it as seeming to be a lot of strange new definitions put to vague purposes. On the bright side, when you get into calculus trigonometry starts solving more problems than it creates. On the dim side, at least when I took it they tried to pass off “trigonometric substitution” as a thing we might need. (OK, it’s come in useful sometimes, but not as often as the presentation made it look.) Also a new comic strip tag.

A two-circle Venn diagram. In one circle: 'Eric's friends'. In the other: 'Eric's enemies'. In the intersection: 'Eric's cat'.
Eric the Circle for the 18th of April, 2018, this one by sdhardie. It’s a rerun, yes, although I don’t know just from when. The copyright date of 2012 suggests I’ve probably already covered this in a Reading the Comics post before. (If I have I can’t find it.)

Eric the Circle for the 18th, this one by sdhardie, is a joke in the Venn Diagram mode. The strip’s a little unusual for not having one of the circles be named Eric. Not a new comic strip tag.

A trophy room. Behind the adult are the heads of an elephant and a tiger . Behind the child are Maths Teacher Year 1 and Maths Teacher Year 2.
Ham’s Life on Earth for the 19th of April, 2018. I suppose that Ham is a pseudonym but I have no information about the cartoonist other than that I guess she’s not American.

Ham’s Life on Earth for the 19th leaves me feeling faintly threatened. Maybe it’s just me. Also not a new comic strip tag, somehow.

Mostly a list of '6 Daydreams That Will Immediately Improve Your Mood'. Relevant is #3, 'Oh hey professor who failed me in college math I'm doing pretty well thanks MATH SLAP.'
Lord Birthday’s Dumbwitch Castle for the 19th of April, 2018. I … I would swear when this comic first started appearing it was by a less absurd pseudonym. I don’t remember, though.

Lord Birthday’s Dumbwitch Castle for the 19th is a small sketch and mostly a list of jokes. This is the normal format for this strip, which tests the idea of what makes something a comic strip. I grant it’s a marginal inclusion, but I am tickled by the idea of a math slap so here you go. This one’s another new comic strip tag.

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, November 4, 2017: Slow, Small Week Edition


It was a slow week for mathematically-themed comic strips. What I have are meager examples. Small topics to discuss. The end of the week didn’t have anything even under loose standards of being on-topic. Which is fine, since I lost an afternoon of prep time to thunderstorms that rolled through town and knocked out power for hours. Who saw that coming? … If I had, I’d have written more the day before.

Mac King and Bill King’s Magic in a Minute for the 29th of October looks like a word problem. Well, it is a word problem. It looks like a problem about extrapolating a thing (price) from another thing (quantity). Well, it is an extrapolation problem. The fun is in figuring out what quantities are relevant. Now I’ve spoiled the puzzle by explaining it all so.

Olivia Walch’s Imogen Quest for the 30th doesn’t say it’s about a mathematics textbook. But it’s got to be. What other kind of textbook will have at least 28 questions in a section and only give answers to the odd-numbered problems in back? You never see that in your social studies text.

Eric the Circle for the 30th, this one by Dennill, tests how slow a week this was. I guess there’s a geometry joke in Jane Austen? I’ll trust my literate readers to tell me. My doing the world’s most casual search suggests there’s no mention of triangles in Pride and Prejudice. The previous might be the most ridiculously mathematics-nerdy thing I have written in a long while.

Tony Murphy’s It’s All About You for the 31st does some advanced-mathematics name-dropping. In so doing, it’s earned a spot taped to the door of two people in any mathematics department with more than 24 professors across the country. Or will, when they hear there was a gap unification theory joke in the comics. I’m not sure whether Murphy was thinking of anything particular in naming the subject “gap unification theory”. It sounds like a field of mathematical study. But as far as I can tell there’s just one (1) paper written that even says “gap unification theory”. It’s in partition theory. Partition theory is a rich and developed field, which seems surprising considering it’s about breaking up sets of the counting numbers into smaller sets. It seems like a time-waster game. But the game sneaks into everything, so the field turns out to be important. Gap unification, in the paper I can find, is about studying the gaps between these smaller sets.

There’s also a “band-gap unification” problem. I could accept this name being shortened to “gap unification” by people who have to say its name a lot. It’s about the physics of semiconductors, or the chemistry of semiconductors, as you like. The physics or chemistry of them is governed by the energies that electrons can have. Some of these energies are precise levels. Some of these energies are bands, continuums of possible values. When will bands converge? When will they not? Ask a materials science person. Going to say that’s not mathematics? Don’t go looking at the papers.

Whether partition theory or materials since it seems like a weird topic. Maybe Murphy just put together words that sounded mathematical. Maybe he has a friend in the field.

Bill Amend’s FoxTrot Classics for the 1st of November is aiming to be taped up to the high school teacher’s door. It’s easy to show how the square root of two is irrational. Takes a bit longer to show the square root of three is. Turns out all the counting numbers are either perfect squares — 1, 4, 9, 16, and so on — or else have irrational square roots. There’s no whole number with a square root of, like, something-and-three-quarters or something-and-85-117ths. You can show that, easily if tediously, for any particular whole number. What’s it look like to show for all the whole numbers that aren’t perfect squares already? (This strip originally ran the 8th of November, 2006.)

Guy Gilchrist’s Nancy for the 1st does an alphabet soup joke, so like I said, it’s been a slow week around here.

John Zakour and Scott Roberts’s Maria’s Day for the 2nd is really just mathematics being declared hated, so like I said, it’s been a slow week around here.

Reading the Comics, August 26, 2017: Dragon Edition


It’s another week where everything I have to talk about comes from GoComics.com. So, no pictures. The Comics Kingdom and the Creators.com strips are harder for non-subscribers to read so I feel better including those pictures. There’s not an overarching theme that I can fit to this week’s strips either, so I’m going to name it for the one that was most visually interesting to me.

Charlie Pondrebarac’s CowTown for the 22nd I just knew was a rerun. It turned up the 26th of August, 2015. Back then I described it as also “every graduate students’ thesis defense anxiety dream”. Now I wonder if I have the possessive apostrophe in the right place there. On reflection, if I have “every” there, then “graduate student” has to be singular. If I dropped the “every” then I could talk about “graduate students” in the plural and be sensible. I guess that’s all for a different blog to answer.

Mike Thompson’s Grand Avenue for the 22nd threatened to get me all cranky again, as Grandmom decided the kids needed to do arithmetic worksheets over the summer. The strip earned bad attention from me a few years ago when a week, maybe more, of the strip was focused on making sure the kids drudged their way through times tables. I grant it’s a true attitude that some people figure what kids need is to do a lot of arithmetic problems so they get better at arithmetic problems. But it’s hard enough to convince someone that arithmetic problems are worth doing, and to make them chores isn’t helping.

John Zakour and Scott Roberts’s Maria’s Day for the 22nd name-drops fractions as a worse challenge than dragon-slaying. I’m including it here for the cool partial picture of the fire-breathing dragon. Also I take a skeptical view of the value of slaying the dragons anyway. Have they given enough time for sanctions to work?

Maria’s Day pops back in the 24th. Needs more dragon-slaying.

Eric the Circle for the 24th, this one by Dennill, gets in here by throwing some casual talk about arcs around. That and π. The given formula looks like nonsense to me. \frac{pi}{180}\cdot 94 - sin 94\deg has parts that make sense. The first part will tell you what radian measure corresponds to 94 degrees, and that’s fine. Mathematicians will tend to look for radian measures rather than degrees for serious work. The sine of 94 degrees they might want to know. Subtracting the two? I don’t see the point. I dare to say this might be a bunch of silliness.

Cathy Law’s Claw for the 25th writes off another Powerball lottery loss as being bad at math and how it’s like algebra. Seeing algebra in lottery tickets is a kind of badness at mathematics, yes. It’s probability, after all. Merely playing can be defended mathematically, though, at least for the extremely large jackpots such as the Powerball had last week. If the payout is around 750 million dollars (as it was) and the chance of winning is about one in 250 million (close enough to true), then the expectation value of playing a ticket is about three dollars. If the ticket costs less than three dollars (and it does; I forget if it’s one or two dollars, but it’s certainly not three), then, on average you could expect to come out slightly ahead. Therefore it makes sense to play.

Except that, of course, it doesn’t make sense to play. On average you’ll lose the cost of the ticket. The on-average long-run you need to expect to come out ahead is millions of tickets deep. The chance of any ticket winning is about one in 250 million. You need to play a couple hundred million times to get a good enough chance of the jackpot for it to really be worth it. Therefore it makes no sense to play.

Mathematical logic therefore fails us: we can justify both playing and not playing. We must study lottery tickets as a different thing. They are (for the purposes of this) entertainment, something for a bit of disposable income. Are they worth the dollar or two per ticket? Did you have other plans for the money that would be more enjoyable? That’s not my ruling to make.

Samson’s Dark Side Of The Horse for the 25th just hurts my feelings. Why the harsh word, Samson? Anyway, it’s playing on the typographic similarity between 0 and O, and how we bunch digits together.

Grouping together three decimal digits as a block is as old, in the Western tradition, as decimal digits are. Leonardo of Pisa, in Liber Abbaci, groups the thousands and millions and thousands of millions and such together. By 1228 he had the idea to note this grouping with an arc above the set of digits, like a tie between notes on a sheet of music. This got cut down, part of the struggle in notation to write as little as possible. Johannes de Sacrobosco in 1256 proposed just putting a dot every third digit. In 1636 Thomas Blundeville put a | mark after every third digit. (I take all this, as ever, from Florian Cajori’s A History Of Mathematical Notations, because it’s got like everything in it.) We eventually settled on separating these stanzas of digits with a , or . mark. But that it should be three digits goes as far back as it could.

Reading the Comics, August 5, 2017: Lazy Summer Week Edition


It wasn’t like the week wasn’t busy. Comic Strip Master Command sent out as many mathematically-themed comics as I might be able to use. But they were again ones that don’t leave me much to talk about. I’ll try anyway. It was looking like an anthropomorphic-symboles sort of week, too.

Tom Thaves’s Frank and Ernest for the 30th of July is an anthropomorphic-symbols joke. The tick marks used for counting make an appearance and isn’t that enough? Maybe.

Dan Thompson’s Brevity for the 31st is another entry in the anthropomorphic-symbols joke contest. This one sticks to mathematical symbols, so if the Frank and Ernest makes the cut this week so must this one.

Eric the Circle for the 31st, this installment by “T daug”, gives the slightly anthropomorphic geometric figure a joke that at least mentions a radius, and isn’t that enough? What catches my imagination about this panel particularly is that the “fractured radius” is not just a legitimate pun but also resembles a legitimate geometry drawing. Drawing a diameter line is sensible enough. Drawing some other point on the circle and connecting that to the ends of the diameter is also something we might do.

Scott Hilburn’s The Argyle Sweater for the 1st of August is one of the logical mathematics jokes you could make about snakes. The more canonical one runs like this: God in the Garden of Eden makes all the animals and bids them to be fruitful. And God inspects them all and finds rabbits and doves and oxen and fish and fowl all growing in number. All but a pair of snakes. God asks why they haven’t bred and they say they can’t, not without help. What help? They need some thick tree branches chopped down. The bemused God grants them this. God checks back in some time later and finds an abundance of baby snakes in the Garden. But why the delay? “We’re adders,” explain the snakes, “so we need logs to multiply”. This joke absolutely killed them in the mathematics library up to about 1978. I’m told.

John Deering’s Strange Brew for the 1st is a monkeys-at-typewriters joke. It faintly reminds me that I might have pledged to retire mentions of the monkeys-at-typewriters joke. But I don’t remember so I’ll just have to depend on saying I don’t think I retired the monkeys-at-typewriters jokes and trust that someone will tell me if I’m wrong.

Dana Simpson’s Ozy and Millie rerun for the 2nd name-drops multiplication tables as the sort of thing a nerd child wants to know. They may have fit the available word balloon space better than “know how to diagram sentences” would.

Mark Anderson’s Andertoons for the 3rd is the reassuringly normal appearance of Andertoons for this week. It is a geometry class joke about rays, line segments with one point where there’s an end and … a direction where it just doesn’t. And it riffs on the notion of the existence of mathematical things. At least I can see it that way.

Dad: 'How many library books have you read this summer, Hammie?' Hammie: 'About 47.' Zoe: 'HA!' Dad: 'Hammie ... ' Hammie: 'Okay ... two.' Dad: 'Then why did you say 47?' Hammie: 'I was rounding up.' Zoe: 'NOW he understands math!'
Rick Kirkman and Jerry Scott’s Baby Blues for the 5th of August, 2017. Hammie totally blew it by saying “about forty-seven”. Too specific a number to be a plausible lie. “About forty” or “About fifty”, something you can see as the result of rounding off, yes. He needs to know there are rules about how to cheat.

Rick Kirkman and Jerry Scott’s Baby Blues for the 5th is a rounding-up joke that isn’t about herds of 198 cattle.

Stephen Bentley’s Herb and Jamaal for the 5th tosses off a mention of the New Math as something well out of fashion. There are fashions in mathematics, as in all human endeavors. It startles many to learn this.

Reading the Comics, May 31, 2017: Feast Week Edition


You know we’re getting near the end of the (United States) school year when Comic Strip Master Command orders everyone to clear out their mathematics jokes. I’m assuming that’s what happened here. Or else a lot of cartoonists had word problems on their minds eight weeks ago. Also eight weeks ago plus whenever they originally drew the comics, for those that are deep in reruns. It was busy enough to split this week’s load into two pieces and might have been worth splitting into three, if I thought I had publishing dates free for all that.

Larry Wright’s Motley Classics for the 28th of May, a rerun from 1989, is a joke about using algebra. Occasionally mathematicians try to use the the ability of people to catch things in midair as evidence of the sorts of differential equations solution that we all can do, if imperfectly, in our heads. But I’m not aware of evidence that anyone does anything that sophisticated. I would be stunned if we didn’t really work by a process of making a guess of where the thing should be and refining it as time allows, with experience helping us make better guesses. There’s good stuff to learn in modeling how to catch stuff, though.

Michael Jantze’s The Norm Classics rerun for the 28th opines about why in algebra you had to not just have an answer but explain why that was the answer. I suppose mathematicians get trained to stop thinking about individual problems and instead look to classes of problems. Is it possible to work out a scheme that works for many cases instead of one? If it isn’t, can we at least say something interesting about why it’s not? And perhaps that’s part of what makes algebra classes hard. To think about a collection of things is usually harder than to think about one, and maybe instructors aren’t always clear about how to turn the specific into the general.

Also I want to say some very good words about Jantze’s graphical design. The mock textbook cover for the title panel on the left is so spot-on for a particular era in mathematics textbooks it’s uncanny. The all-caps Helvetica, the use of two slightly different tans, the minimalist cover art … I know shelves stuffed full in the university mathematics library where every book looks like that. Plus, “[Mathematics Thing] And Their Applications” is one of the roughly four standard approved mathematics book titles. He paid good attention to his references.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 28th deploys a big old whiteboard full of equations for the “secret” of the universe. This makes a neat change from finding the “meaning” of the universe, or of life. The equations themselves look mostly like gibberish to me, but Wise and Aldrich make good uses of their symbols. The symbol \vec{B} , a vector-valued quantity named B, turns up a lot. This symbol we often use to represent magnetic flux. The B without a little arrow above it would represent the intensity of the magnetic field. Similarly an \vec{H} turns up. This we often use for magnetic field strength. While I didn’t spot a \vec{E} — electric field — which would be the natural partner to all this, there are plenty of bare E symbols. Those would represent electric potential. And many of the other symbols are what would naturally turn up if you were trying to model how something is tossed around by a magnetic field. Q, for example, is often the electric charge. ω is a common symbol for how fast an electromagnetic wave oscillates. (It’s not the frequency, but it’s related to the frequency.) The uses of symbols is consistent enough, in fact, I wonder if Wise and Aldrich did use a legitimate sprawl of equations and I’m missing the referenced problem.

John Graziano’s Ripley’s Believe It Or Not for the 28th mentions how many symbols are needed to write out the numbers from 1 to 100. Is this properly mathematics? … Oh, who knows. It’s just neat to know.

Mark O’Hare’s Citizen Dog rerun for the 29th has the dog Fergus struggle against a word problem. Ordinary setup and everything, but I love the way O’Hare draws Fergus in that outfit and thinking hard.

The Eric the Circle rerun for the 29th by ACE10203040 is a mistimed Pi Day joke.

Bill Amend’s FoxTrot Classicfor the 31st, a rerun from the 7th of June, 2006, shows the conflation of “genius” and “good at mathematics” in everyday use. Amend has picked a quixotic but in-character thing for Jason Fox to try doing. Euclid’s Fifth Postulate is one of the classic obsessions of mathematicians throughout history. Euclid admitted the thing — a confusing-reading mess of propositions — as a postulate because … well, there’s interesting geometry you can’t do without it, and there doesn’t seem any way to prove it from the rest of his geometric postulates. So it must be assumed to be true.

There isn’t a way to prove it from the rest of the geometric postulates, but it took mathematicians over two thousand years of work at that to be convinced of the fact. But I know I went through a time of wanting to try finding a proof myself. It was a mercifully short-lived time that ended in my humbly understanding that as smart as I figured I was, I wasn’t that smart. We can suppose Euclid’s Fifth Postulate to be false and get interesting geometries out of that, particularly the geometries of the surface of the sphere, and the geometry of general relativity. Jason will surely sometime learn.

Reading the Comics, May 20, 2017: Major Computer Malfunction Week Edition


I was hit by a massive computer malfunction this week, the kind that forced me to buy a new computer and spend half a week copying stuff over from a limping hard drive and hoping it would maybe work if I held things just right. Mercifully, Comic Strip Master Command gave me a relatively easy week. No huge rush of mathematically-themed comic strips and none that are going to take a thousand words of writing to describe. Let’s go.

Sam Hepburn’s Questionable Quotebook for the 14th includes this week’s anthropomorphic geometry sketch underneath its big text block.

Eric the Circle for the 15th, this one by “Claire the Square”, is the rare Eric the Circle to show off properties of circles. So maybe that’s the second anthropomorphic geometry sketch for the week. If the week hadn’t been dominated by my computer woes that might have formed the title for this edition.

Werner Wejp-Olsen’s Inspector Danger’s Crime Quiz for the 15th puts a mathematician in mortal peril and leaves him there to die. As is traditional for this sort of puzzle the mathematician left a dying clue. (Mathematicians were similarly kind to their investigators on the 4th of July, 2016 and the 9th of July, 2012. I was expecting the answer to be someone with a four-letter and an eight-letter name, none of which anybody here had. Doesn’t matter. It’ll never stand up in court.

John Graziano’s Ripley’s Believe It Or Not for the 17th features one of those astounding claims that grows out of number theory. Graziano asserts that there are an astounding 50,613,244,155,051,856 ways to score exactly 100 points in (ten-pin) bowling. I won’t deny that this seems high to me. But partitioning a number — that is, taking a (positive) whole number and writing down the different ways one can add up (positive) whole numbers to get that sum — often turns up a lot of possibilities. That there should be many ways to get a score of 100 by adding between ten and twenty numbers that could be between zero and ten each, plus the possibility of adding pairs of the numbers (for spares) or trios of numbers (for strikes) makes this less astonishing.

Wikipedia led me to this page, from Balmoral Software, about all the different ways there are to score different numbers. The most surprising thing it reveals to me is that 100 isn’t even the score with the greatest number of possible scores. 77 is. There are 172,542,309,343,731,946 ways to score exactly 77 points. I agree this ought to make me feel better about my game. It doesn’t. It turns out there are, altogether, something like 5,726,805,883,325,784,576 possible different outcomes for a bowling game. And how we can tell that, given there’s no practical way to go and list all of them, is described at the end of the page.

The technique is called “divide and conquer”. There’s no way to list all the outcomes of ten frames of bowling, but there’s certainly a way to list all the outcomes of one. Or two. Or three. So, work out how many possible scores there would be in few enough frames you can handle that. Then combine these shortened games into one that’s the full ten frames. There’s some trouble in matching up the ends of the short games. A spare or a strike in the last frame of a shortened game means one has to account for the first or first two frames of the next one. But this is still an easier problem than the one we started with.

Bill Amend’s FoxTrot Classics for the 18th (rerun from the 25th of May, 2006) is your standard percentages and infinities joke. Really would have expected Paige’s mother to be wise to this game by now, but this sort of thing happens.

Reading the Comics, February 15, 2017: SMBC Cuts In Line Edition


It’s another busy enough week for mathematically-themed comic strips that I’m dividing the harvest in two. There’s a natural cutting point since there weren’t any comics I could call relevant for the 15th. But I’m moving a Saturday Morning Breakfast Cereal of course from the 16th into this pile. That’s because there’s another Saturday Morning Breakfast Cereal of course from after the 16th that I might include. I’m still deciding if it’s close enough to on topic. We’ll see.

John Graziano’s Ripley’s Believe It Or Not for the 12th mentions the “Futurama Theorem”. The trivia is true, in that writer Ken Keeler did create a theorem for a body-swap plot he had going. The premise was that any two bodies could swap minds at most one time. So, after a couple people had swapped bodies, was there any way to get everyone back to their correct original body? There is, if you bring two more people in to the body-swapping party. It’s clever.

From reading comment threads about the episode I conclude people are really awestruck by the idea of creating a theorem for a TV show episode. The thing is that “a theorem” isn’t necessarily a mind-boggling piece of work. It’s just the name mathematicians give when we have a clearly-defined logical problem and its solution. A theorem and its proof can be a mind-wrenching bit of work, like Fermat’s Last Theorem or the Four-Color Map Theorem are. Or it can be on the verge of obvious. Keeler’s proof isn’t on the obvious side of things. But it is the reasoning one would have to do to solve the body-swap problem the episode posited without cheating. Logic and good story-telling are, as often, good partners.

Teresa Burritt’s Frog Applause is a Dadaist nonsense strip. But for the 13th it hit across some legitimate words, about a 14 percent false-positive rate. This is something run across in hypothesis testing. The hypothesis is something like “is whatever we’re measuring so much above (or so far below) the average that it’s not plausibly just luck?” A false positive is what it sounds like: our analysis said yes, this can’t just be luck, and it turns out that it was. This turns up most notoriously in medical screenings, when we want to know if there’s reason to suspect a health risk, and in forensic analysis, when we want to know if a particular person can be shown to have been a particular place at a particular time. A 14 percent false positive rate doesn’t sound very good — except.

Suppose we are looking for a rare condition. Say, something one person out of 500 will have. A test that’s 99 percent accurate will turn up positives for the one person who has got it and for five of the people who haven’t. It’s not that the test is bad; it’s just there are so many negatives to work through. If you can screen out a good number of the negatives, though, the people who haven’t got the condition, then the good test will turn up fewer false positives. So suppose you have a cheap or easy or quick test that doesn’t miss any true positives but does have a 14 percent false positive rate. That would screen out 430 of the people who haven’t got whatever we’re testing for, leaving only 71 people who need the 99-percent-accurate test. This can make for a more effective use of resources.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 13th is an algebra-in-real-life joke and I can’t make something deeper out of that.

Mike Shiell’s The Wandering Melon for the 13th is a spot of wordplay built around statisticians. Good for taping to the mathematics teacher’s walls.

Eric the Circle for the 14th, this one by “zapaway”, is another bit of wordplay. Tans and tangents.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 16th identifies, aptly, a difference between scientists and science fans. Weinersmith is right that loving trivia is a hallmark of a fan. Expertise — in any field, not just science — is more about recognizing patterns of problems and concepts, ways to bring approaches from one field into another, this sort of thing. And the digits of π are great examples of trivia. There’s no need for anyone to know the 1,681st digit of π. There’s few calculations you could ever do when you needed more than three dozen digits. But if memorizing digits seems like fun then π is a great set to learn. e is the only other number at all compelling.

The thing is, it’s very hard to become an expert in something without first being a fan of it. It’s possible, but if a field doesn’t delight you why would you put that much work into it? So even though the scientist might have long since gotten past caring how many digits of π, it’s awfully hard to get something memorized in the flush of fandom out of your head.

I know you’re curious. I can only remember π out to 3.14158926535787962. I might have gotten farther if I’d tried, but I actually got a digit wrong, inserting a ‘3’ before that last ’62’, and the effort to get that mistake out of my head obliterated any desire to waste more time memorizing digits. For e I can only give you 2.718281828. But there’s almost no hope I’d know that far if it weren’t for how e happens to repeat that 1828 stanza right away.

Mid-April 2012 Comics Review


I’ve gotten enough comics, I think, to justify a fresh roundup of mathematics appearances in the comic strips. Unfortunately the first mathematics-linked appearance since my most recent entry is also the most badly dated. Pab Sugenis’s The New Adventures of Queen Victoria took (the appropriate) day to celebrate the birthday of Tom Lehrer, but fails to mention his actual greatest contribution to American culture, the “Silent E” song for The Electric Company. He’s also author of the humorous song “Lobachevsky”, which is pretty much the only place to go if you need a mathematics-based song and can’t use They Might Be Giants for some reason. (I regard Lehrer’s “New Math” song as not having a strong enough melody to count.)

Continue reading “Mid-April 2012 Comics Review”

Early April’s Math Comics


I had started to think the mathematics references in the comics pages were fading out and I might not have an installment to offer anytime soon. Then, on April 3, Pab Sugenis’s The New Adventures of Queen Victoria — a clip art comic strip which supposes the reader will recognize an illustration of King Edward VI — skipped its planned strip for the day (Sugenis’s choice, he says) and ran a Fuzzy Bunny Time strip calling on pretty much the expected rabbits and mathematics comic strip. (Some people in the Usenet group alt.fan.cecil-adams, which I read reliably and write to occasionally, say Sugenis was briefly a regular; perhaps so, but I don’t remember.) This would start a bumper crop of math strips for the week.

Continue reading “Early April’s Math Comics”