What I’ve Been Reading, Mid-March 2018

So here’s some of the stuff I’ve noticed while being on the Internet and sometimes noticing interesting mathematical stuff.

Here from the end of January is a bit of oddball news. A story problem for 11-year-olds in one district of China set up a problem that couldn’t be solved. Not exactly, anyway. The question — “if a ship had 26 sheep and 10 goats onboard, how old is the ship’s captain?” — squares nicely with that Gil comic strip I discussed the other day. After seeing 26 (something) and 10 (something else) it’s easy to think of what answers might be wanted: 36 (total animals) or 16 (how many more sheep there are than goats) or maybe 104 (how many hooves there are, if they all have the standard four hooves). That the question doesn’t ask anything that the given numbers matter for barely registers unless you read the question again. I like the principle of reminding people not to calculate until you know what you want to do and why that. And it’s possible to give partial answers: the BBC News report linked above includes a mention from one commenter that allowed a reasonable lower bound to be set on the ship’s captain’s age.

In something for my mathematics majors, here’s A Regiment of Monstrous Functions as assembled by Rob J Low. This is about functions with a domain and a range that are both real numbers. There’s many kinds of these functions. They match nicely to the kinds of curves you can draw on a sheet of paper. So take a sheet of paper and draw a curve. You’ve probably drawn a continuous curve, one that can be drawn without lifting your pencil off the paper. Good chance you drew a differentiable one, one without corners. But most functions aren’t continuous. And aren’t differentiable. Of those few exceptions that are, many of them are continuous or differentiable only in weird cases. Low reviews some of the many kinds of functions out there. Functions discontinuous at a point. Functions continuous only on one point, and why that’s not a crazy thing to say. Functions continuous on irrational numbers but discontinuous on rational numbers. This is where mathematics majors taking real analysis feel overwhelmed. And then there’s stranger stuff out there.

Zachary Abel finds large primes which when written out in large rectangles, produce recognizable images: https://t.co/YvdNAq7iJ6

— Henry Segerman (@henryseg) March 6, 2018

Here’s a neat one. It’s about finding recognizable, particular, interesting pictures in long enough prime numbers. The secret to it is described in the linked paper. The key is that the eye is very forgiving of slightly imperfect images. This fact should reassure people learning to draw, but will not. And there’s a lot of prime numbers out there. If an exactly-correct image doesn’t happen to be a prime number that’s all right. There’s a number close enough to it that will be. That latter point is something that anyone interested in number theory “knows”, in that we know some stuff about the biggest possible gaps between prime numbers. But that fact isn’t the same as seeing it.

Here is a clearly written page on solving differential equations using symmetry methods. https://t.co/RmjV6YzWHQ

— math prof (@mathematicsprof) March 2, 2018

And finally there’s something for mathematics majors. Differential equations are big and important. They appear whenever you want to describe something that changes based on its current state. And this is so much stuff. Finding solutions to differential equations is a whole major field of mathematics. The linked PDF is a slideshow of notes about one way to crack these problems: find symmetries. The only trouble is it’s a PDF of a Powerpoint presentation, one of those where each of the items gets added on in sequence. So each slide appears like eight times, each time with one extra line on it. It’s still good, interesting stuff.

Reading the Comics, June 25, 2016: Busy Week Edition

I had meant to cut the Reading The Comics posts back to a reasonable one a week. Then came the 23rd, which had something like six hundred mathematically-themed comic strips. So I could post another impossibly long article on Sunday or split what I have. And splitting works better for my posting count, so, here we are.

Charles Brubaker’s Ask A Cat for the 19th is a soap-bubbles strip. As ever happens with comic strips, the cat blows bubbles that can’t happen without wireframes and skillful bubble-blowing artistry. It happens that a few days ago I linked to a couple essays showing off some magnificent surfaces that the right wireframe boundary might inspire. The mathematics describing how a soap bubbles’s shape should be made aren’t hard; I’m confident I could’ve understood the equations as an undergraduate. Finding exact solutions … I’m not sure I could have done. (I’d still want someone looking over my work if I did them today.) But numerical solutions, that I’d be confident in doing. And the real thing is available when you’re ready to get your hands … dirty … with soapy water.

Rick Stromoski’s Soup To Nutz for the 19th Shows RoyBoy on the brink of understanding symmetry. To lose at rock-paper-scissors is indeed just as hard as winning is. Suppose we replaced the names of the things thrown with letters. Suppose we replace ‘beats’ and ‘loses to’ with nonsense words. Then we could describe the game: A flobs B. B flobs C. C flobs A. A dostks C. C dostks B. B dostks A. There’s no way to tell, from this, whether A is rock or paper or scissors, or whether ‘flob’ or ‘dostk’ is a win.

Bill Whitehead’s Free Range for the 20th is the old joke about tipping being the hardest kind of mathematics to do. Proof? There’s an enormous blackboard full of symbols and the three guys in lab coats are still having trouble with it. I have long wondered why tips are used as the model of impossibly difficult things to compute that aren’t taxes. I suppose the idea of taking “fifteen percent” (or twenty, or whatever) of something suggests a need for precision. And it’ll be fifteen percent of a number chosen without any interest in making the calculation neat. So it looks like the worst possible kind of arithmetic problem. But the secret, of course, is that you don’t have to have “the” right answer. You just have to land anywhere in an acceptable range. You can work out a fraction — a sixth, a fifth, or so — of a number that’s close to the tab and you’ll be right. So, as ever, it’s important to know how to tell whether you have a correct answer before worrying about calculating it.

Allison Barrows’s Preeteena rerun for the 20th is your cheerleading geometry joke for this week.

Bill Holbrook’s On The Fastrack for the 22nd of June, 2016. There are so many bloggers wondering if Holbrook is talking about them.

I am sure Bill Holbrook’s On The Fastrack for the 22nd is not aimed at me. He hangs around Usenet group rec.arts.comics.strips some, as I do, and we’ve communicated a bit that way. But I can’t imagine he thinks of me much or even at all once he’s done with racs for the day. Anyway, Dethany does point out how a clear identity helps one communicate mathematics well. (Fi is to talk with elementary school girls about mathematics careers.) And bitterness is always a well-received pose. Me, I’m aware that my pop-mathematics brand identity is best described as “I guess he writes a couple things a week, doesn’t he?” and I could probably use some stronger hook, somewhere. I just don’t feel curmudgeonly most of the time.

Darby Conley’s Get Fuzzy rerun for the 22nd is about arithmetic as a way to be obscure. We’ve all been there. I had, at first, read Bucky’s rating as “out of 178 ¹/₃ π” and thought, well, that’s not too bad since one-third of π is pretty close to 1. But then, Conley being a normal person, probably meant “one-hundred seventy-eight and a third”, and π times that is a mess. Well, it’s somewhere around 550 or so. Octave tells me it’s more like 560.251 and so on.

Fibonacci’s Biased Scarf

Here is a neat bit of crochet work with a bunch of nice recreational-mathematics properties. The first is that the distance between yellow rows, or between blue rows, represents the start of the Fibonacci sequence of numbers. I’m not sure if the Fibonacci sequence is the most famous sequence of whole numbers but it’s certainly among the most famous, and it’s got interesting properties and historical context.

The second recreational-mathematics property is that the pattern is rotationally symmetric. Rotate it 180 degrees and you get back the original pattern, albeit with blue and yellow swapped. You can form a group out of the ways that it’s possible to rotate an object and get back something that looks like the original. Symmetry groups can be things of simple aesthetic beauty, describing scarf patterns and ways to tile floors and the like. They can also describe things of deep physical significance. Much of the ability of quantum chromodynamics to describe nuclear physics comes from these symmetry groups.

The logo at top of the page is of a trefoil knot, which I’d mentioned a couple weeks back. A trefoil knot isn’t perfectly described by its silhouette. Where the lines intersect you have to imagine the string (or whatever makes up the knot) passing twice, once above and once below itself. If you do that crossing-over and crossing-under consistently you get the trefoil knot, the simplest loop that isn’t an unknot, that can’t be shaken loose into a simple circle.

Knot Theorist

This scarf is totally biased. That’s not to say that it’s prejudiced, but that it was worked in the diagonal direction of the cloth.

My project was made from Julie Blagojevich’s free pattern Fibonacci’s Biased using Knit Picks Curio. The number of rows in each stripe is according to the numbers of the Fibonacci sequence up to 34. In other words, if you start at the blue side of the scarf and work your way right, the sequence of the number of yellow rows is 1, 1, 2, 3, 5, 8, 13, 21, 34. The sequence of the blue stripes are the same, but in the opposite direction. The effect is a rotationally symmetric scarf with few color changes at the edges and frequent color changes in the center. As I frequently tell my friends, math is beautiful.

If my geekiness hasn’t scared you away yet, here’s a random fun…

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Reading the Comics, April 15, 2015: Tax Day Edition

Since it is mid-April, and most of the comic strips at Comics Kingdom and GoComics.com are based in the United States, Comic Strip Master Command ordered quite a few comics about taxes. Most of those are simple grumbling, but the subject naturally comes around to arithmetic and calculation and sometimes even logic. Thus, this is a Tax Day edition, though it’s bookended with Mutt and Jeff.

Bud Fisher’s Mutt And Jeff (April 11) — a rerun rom goodness only knows when, and almost certainly neither written nor drawn by Bud Fisher at that point — recounts a joke that has the form of a word problem in which a person’s age is deduced from information about the age. It’s an old form, but jokes about cutting the Gordion knot are probably always going to be reliable. I’m reminded there’s a story of Thomas Edison giving a new hire, mathematician, the problem of working out the volume of a light bulb. Edison got impatient with the mathematician treating it as a calculus problem — the volume of a rotationally symmetric object like a bulb is the sort of thing you can do by the end of Freshman Calculus — and instead filling a bulb with water, pouring the water into a graduated cylinder, and reading it off that.

Sandra Bell-Lundy’s Between Friends for the 12th of April, 2015. The link will likely expire around the 12th of May.

Sandra Bell-Lundy’s Between Friends (April 12) uses Calculus as the shorthand for “the hardest stuff you might have to deal with”. The symbols on the left-hand side are fair enough, although I’d think of them more as precalculus or linear algebra or physics, but they do parse well enough as long as I suppose that what sure looks like a couple of extraneous + signs are meant to refer to “t”. But “t” is a common enough variable in calculus problems, usually representing time, sometimes just representing “some parameter whose value we don’t really care about, but we don’t want it to be x”, and it looks an awful lot like a plus sign there too. On the right side, I have no idea what a root of forty minutes on a treadmill might be. It’s symbolic.

Playing With Tiles

The Math Less Travelled, one of the blogs that I read, posted yesterday a link to another web site, a Tiling Database created by Brian Wichmann and Tony Lee. The database is exactly what it says on the label: a collection of patterns which one could put on a flat surface and extend outward in both directions as far as you like. In principle, you could get any of them to spruce up your kitchen, although some of them would be a bit staggering to face in the morning, even in other color schemes.