## Reading the Comics, May 20, 2019: I Guess I Took A Week Off Edition

I’d meant to get back into discussing continuous functions this week, and then didn’t have the time. I hope nobody was too worried.

Bill Amend’s FoxTrot for the 19th is set up as geometry or trigonometry homework. There are a couple of angles that we use all the time, and they do correspond to some common unit fractions of a circle: a quarter, a sixth, an eighth, a twelfth. These map nicely to common cuts of circular pies, at least. Well, it’s a bit of a freak move to cut a pie into twelve pieces, but it’s not totally out there. If someone cuts a pie into 24 pieces, flee.

Tom Batiuk’s vintage Funky Winkerbean for the 19th of May is a real vintage piece, showing off the days when pocket electronic calculators were new. The sales clerk describes the calculator as having “a floating decimal”. And here I must admit: I’m poorly read on early-70s consumer electronics. So I can’t say that this wasn’t a thing. But I suspect that Batiuk either misunderstood “floating-point decimal”, which would be a selling point, or shortened the phrase in order to make the dialogue less needlessly long. Which is fine, and his right as an author. The technical detail does its work, for the setup, by existing. It does not have to be an actual sales brochure. Reducing “floating point decimal” to “floating decimal” is a useful artistic shorthand. It’s the dialogue equivalent to the implausibly few, but easy to understand, buttons on the calculator in the title panel.

Floating point is one of the ways to represent numbers electronically. The storage scheme is much like scientific notation. That is, rather than think of 2,038, think of 2.038 times 103. In the computer’s memory are stored the 2.038 and the 3, with the “times ten to the” part implicit in the storage scheme. The advantage of this is the range of numbers one can use now. There are different ways to implement this scheme; a common one will let one represent numbers as tiny as 10-308 or as large as 10308, which is enough for most people’s needs.

The disadvantage is that floating point numbers aren’t perfect. They have only around (commonly) sixteen digits of significance. That is, the first sixteen or so nonzero numbers in the number you represent mean anything; everything after that is garbage. Most of the time, that trailing garbage doesn’t hurt. But most is not always. Trying to add, for example, a tiny number, like 10-20, to a huge number, like 1020 won’t get the right answer. And there are numbers that can’t be represented correctly anyway, including such exotic and novel numbers as $\frac{1}{3}$. A lot of numerical mathematics is about finding ways to compute that avoid these problems.

Back when I was a grad student I did have one casual friend who proclaimed that no real mathematician ever worked with floating point numbers, because of the limitations they impose. I could not get him to accept that no, in fact, mathematicians are fine with these limitations. Every scheme for representing numbers on a computer has limitations, and floating point numbers work quite well. At some point, you have to suspect some people would rather fight for a mistaken idea they already have than accept something new.

Mac King and Bill King’s Magic in a Minute for the 19th does a bit of stage magic supported by arithmetic: forecasting the sum of three numbers. The trick is that all eight possible choices someone would make have the same sum. There’s a nice bit of group theory hidden in the “Howdydoit?” panel, about how to do the trick a second time. Rotating the square of numbers makes what looks, casually, like a different square. It’s hard for human to memorize a string of digits that don’t have any obvious meaning, and the longer the string the worse people are at it. If you’ve had a person — as directed — black out the rows or columns they didn’t pick, then it’s harder to notice the reused pattern.

The different directions that you could write the digits down in represent symmetries of the square. That is, geometric operations that would replace a square with something that looks like the original. This includes rotations, by 90 or 180 or 270 degrees clockwise. Mac King and Bill King don’t mention it, but reflections would also work: if the top row were 4, 9, 2, for example, and the middle 3, 5, 7, and the bottom 8, 1, 6. Combining rotations and reflections also works.

If you do the trick a second time, your mark might notice it’s odd that the sum came up 15 again. Do it a third time, even with a different rotation or reflection, and they’ll know something’s up. There are things you could do to disguise that further. Just double each number in the square, for example: a square of 4/18/8, 14/10/6, 12/2/16 will have each row or column or diagonal add up to 30. But this loses the beauty of doing this with the digits 1 through 9, and your mark might grow suspicious anyway. The same happens if, say, you add one to each number in the square, and forecast a sum of 18. Even mathematical magic tricks are best not repeated too often, not unless you have good stage patter.

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead’s marveling at what seems at first like an asymmetry, about squares all being rhombuses yet rhombuses not all being squares. There are similar results with squares and rectangles. Still, it makes me notice something. Nobody would write a strip where the kid marvelled that all squares were polygons but not all polygons were squares. It seems that the rhombus connotes something different. This might just be familiarity. Polygons are … well, if not a common term, at least something anyone might feel familiar. Rhombus is a more technical term. It maybe never quite gets familiar, not in the ways polygons do. And the defining feature of a rhombus — all four sides the same length — seems like the same thing that makes a square a square.

There should be another Reading the Comics post this coming week, and it should appear at this link. I’d like to publish it Tuesday but, really, Wednesday is more probable.

## Reading the Comics, May 11, 2019: I Concede I Am Late Edition

I concede I am late in wrapping up last week’s mathematically-themed comics. But please understand there were important reasons for my not having posted this earlier, like, I didn’t get it written in time. I hope you understand and agree with me about this.

Bill Griffith’s Zippy the Pinhead for the 9th brings up mathematics in a discussion about perfection. The debate of perfection versus “messiness” begs some important questions. What I’m marginally competent to discuss is the idea of mathematics as this perfect thing. Mathematics seems to have many traits that are easy to think of as perfect. That everything in it should follow from clearly stated axioms, precise definitions, and deductive logic, for example. This makes mathematics seem orderly and universal and fair in a way that the real world never is. If we allow that this is a kind of perfection then … does mathematics reach it?

Even the idea of a “precise definition” is perilous. If it weren’t there wouldn’t be so many pop mathematics articles about why 1 isn’t a prime number. It’s difficult to prove that any particular set of axioms that give us interesting results are also logically consistent. If they’re not consistent, then we can prove absolutely anything, including that the axioms are false. That seems imperfect. And few mathematicians even prepare fully complete, step-by-step proofs of anything. It takes ridiculously long to get anything done if you try. The proofs we present tend to show, instead, the reasoning in enough detail that we’re confident we could fill in the omitted parts if we really needed them for some reason. And that’s fine, nearly all the time, but it does leave the potential for mistakes present.

Zippy offers up a perfect parallelogram. Making it geometry is of good symbolic importance. Everyone knows geometric figures, and definitions of some basic ideas like a line or a circle or, maybe, a parallelogram. Nobody’s ever seen one, though. There’s never been a straight line, much less two parallel lines, and even less the pair of parallel lines we’d need for a parallellogram. There can be renderings good enough to fool the eye. But none of the lines are completely straight, not if we examine closely enough. None of the pairs of lines are truly parallel, not if we extend them far enough. The figure isn’t even two-dimensional, not if it’s rendered in three-dimensional things like atoms or waves of light or such. We know things about parallelograms, which don’t exist. They tell us some things about their shadows in the real world, at least.

Mark Litzler’s Joe Vanilla for the 9th is a play on the old joke about “a billion dollars here, a billion dollars there, soon you’re talking about real money”. As we hear more about larger numbers they seem familiar and accessible to us, to the point that they stop seeming so big. A trillion is still a massive number, at least for most purposes. If you aren’t doing combinatorics, anyway; just yesterday I was doing a little toy problem and realized it implied 470,184,984,576 configurations. Which still falls short of a trillion, but had I made one arbitrary choice differently I could’ve blasted well past a trillion.

Ruben Bolling’s Super-Fun-Pak Comix for the 9th is another monkeys-at-typewriters joke, that great thought experiment about probability and infinity. I should add it to my essay about the Infinite Monkey Theorem. Part of the joke is that the monkey is thinking about the content of the writing. This doesn’t destroy the prospect that a monkey given enough time would write any of the works of William Shakespeare. It makes the simple estimates of how unlikely that is, and how long it would take to do, invalid. But the event might yet happen. Suppose this monkey decided there was no credible way to delay Hamlet’s revenge to Act V, and tried to write accordingly. Mightn’t the monkey make a mistake? It’s easy to type a letter you don’t mean to. Or a word you don’t mean to. Why not a sentence you don’t mean to? Why not a whole act you don’t mean to? Impossible? No, just improbable. And the monkeys have enough time to let the improbable happen.

Eric the Circle for the 10th, this one by Kingsnake, declares itself set in “the 20th dimension, where shape has no meaning”. This plays on a pop-cultural idea of dimensions as a kind of fairyland, subject to strange and alternate rules. A mathematician wouldn’t think of dimensions that way. 20-dimensional spaces — and even higher-dimensional spaces — follow rules just as two- and three-dimensional spaces do. They’re harder to draw, certainly, and mathematicians are not selected for — or trained in — drawing, at least not in United States schools. So attempts at rendering a high-dimensional space tend to be sort of weird blobby lumps, maybe with a label “N-dimensional”.

And a projection of a high-dimensional shape into lower dimensions will be weird. I used to have around here a web site with a rotatable tesseract, which would draw a flat-screen rendition of what its projection in three-dimensional space would be. But I can’t find it now and probably it ran as a Java applet that you just can’t get to work anymore. Anyway, non-interactive videos of this sort of thing are common enough; here’s one that goes through some of the dimensions of a tesseract, one at a time. It’ll give some idea how something that “should” just be a set of cubes will not look so much like that.

Steve Kelly and Jeff Parker’s Dustin for the 11th is a variation on the “why do I have to learn this” protest. This one is about long division and the question of why one needs to know it when there’s cheap, easily-available tools that do the job better. It’s a fair question and Hayden’s answer is a hard one to refute. I think arithmetic’s worth knowing how to do, but I’ll also admit, if I need to divide something by 23 I’m probably letting the computer do it.

And a couple of the comics that week seemed too slight to warrant discussion. You might like them anyway. Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 5th featured a poorly-written numeral. Charles Schulz’s Peanuts Begins rerun for the 6th has Violet struggling with counting. Glenn McCoy and Gary McCoy’s The Flying McCoys for the 8th has someone handing in mathematics homework. Henry Scarpelli and Craig Boldman’s Archie rerun for the 9th talks about Jughead sleeping through mathematics class. All routine enough stuff.

This and other Reading the Comics posts should appear at this link. I mean to have a post tomorrow, although it might make my work schedule a little easier to postpone that until Monday. We’ll see.

## Reading the Comics, May 8, 2019: Strips With Art I Like Edition

Of course I like all the comics. … Well, that’s not literally true; but I have at least some affection for nearly all of the syndicated comics. This essay I bring up some strips, partly, because I just like them. This is my content hole. If you want a blog not filled with comic strips, go start your own and don’t put these things on it.

Mark Anderson’s Andertoons for the 5th is the Mark Anderson’s Andertoons for the week. Also a bit of a comment on the ability of collective action to change things. Wavehead is … well, he’s just wrong about making the number four plus the number four equal to the number seven. Not based on the numbers we mean by the words “four” and “seven”, and based on the operation we mean by “plus” and the relationship we mean by “equals”. The meaning of those things is set by, ultimately, axioms and deductive reasoning and the laws of deductive reasoning and there’s no changing the results.

But. The thing we’re referring to when we say “seven”? Or when we write the symbol “7”? That is convention. That is a thing we’ve agreed on as a reference for this concept. And that we can change, if we decide we want to. We’ve done this. Look at a thousand-year-old manuscript and the symbol that looks like ‘4’ may represent the number we call five. And the names of numbers are just common words. They’re subject to change the way every other common word is. Which is, admittedly, not very subject. It would be almost as much bother to change the word ‘four’ as it would be to change the word ‘mom’. But that’s not impossible. Just difficult.

Juba’s Viivi and Wagner for the 5th is a bit of a percentage joke. The characters also come to conclude that a thing either happens or it does not; there’s no indefinite states. This principle, the “excluded middle”, is often relied upon for deductive logic, and fairly so. It gets less clear that this can be depended on for predictions of the future, or fears for the future. And real-world things come in degrees that a mathematical concept might not. Like, your fear of the home catching fire comes true if the building burns down. But it’s also come true if a quickly-extinguished frying pan fire leaves the wall scorched, embarrassing but harmless. Anyway, relaxing someone else’s anxiety takes more than a quick declaration of statistics. Show sympathy.

Harry Bliss and Steve Martin’s Bliss for the 6th is a cute little classroom strip, with arithmetic appearing as the sort of topic that students feel overwhelmed and baffled by. It could be anything, but mathematics uses the illustration space efficiently. The strip may properly be too marginal to include, but I like Bliss’s art style and want more people to see it.

Will Henry’s Wallace the Brave for the 7th puts up what Spud calls a sadistic math problem. And, well, it is a story problem happening in their real life. You could probably turn this into an actual exam problem without great difficulty.

Rick Detorie’s One Big Happy for the 8th is a bit of wordplay built around geometry, as Ruthie plays teacher. She’s a bit dramatic, but she always has been.

I’ll read some more comics for later in this week. That essay, and all similar comic strip talk, should appear at this link. Thank you.

## Reading the Comics, April 24, 2019: Mic Drop Edition Edition

I can’t tell you how hard it is not to just end this review of last week’s mathematically-themed comic strips after the first panel here. It really feels like the rest is anticlimax. But here goes.

John Deering’s Strange Brew for the 20th is one of those strips that’s not on your mathematics professor’s office door only because she hasn’t seen it yet. The intended joke is obvious, mixing the tropes of the Old West with modern research-laboratory talk. “Theoretical reckoning” is a nice bit of word juxtaposition. “Institoot” is a bit classist in its rendering, but I suppose it’s meant as eye-dialect.

What gets it a place on office doors is the whiteboard, though. They’re writing out mathematics which looks legitimate enough to me. It doesn’t look like mathematics though. What’s being written is something any mathematician would recognize. It’s typesetting instructions. Mathematics requires all sorts of strange symbols and exotic formatting. In the old days, we’d handle this by giving the typesetters hazard pay. Or, if you were a poor grad student and couldn’t afford that, deal with workarounds. Maybe just leave space in your paper and draw symbols in later. If your university library has old enough papers you can see them. Maybe do your best to approximate mathematical symbols using ASCII art. So you get expressions that look something like this:

  / 2 pi
|   2
|  x cos(theta) dx - 2 F(theta) == R(theta)
|
/ 0


This gets old real fast. Mercifully, Donald Knuth, decades ago, worked out a great solution. It uses formatting instructions that can all be rendered in standard, ASCII-available text. And then by dark incantations and summoning of Linotype demons, re-renders that as formatted text. It handles all your basic book formatting needs — much the way HTML, used for web pages, will — and does mathematics much more easily. For example, I would enter a line like:

\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)


And this would be rendered in print as:

$\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)$

There are many, many expansions available to this, to handle specialized needs, hardcore mathematics among them.

Anyway, the point that makes me realize John Deering was aiming at everybody with an advanced degree in mathematics ever with this joke, using a string of typesetting instead of the usual equations here?

The typesetting language is named TeX.

Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for the week. It’s about one of those questions that nags at you as a kid, and again periodically as an adult. The perimeter is the boundary around a shape. The circumference is the boundary around a circle. Why do we have two words for this? And why do we sound all right talking about either the circumference or the perimeter of a circle, while we sound weird talking about the circumference of a rhombus? We sound weird talking about the perimeter of a rhombus too, but that’s the rhombus’s fault.

The easy part is why there’s two words. Perimeter is a word of Greek origin; circumference, of Latin. Perimeter entered the English language in the early 15th century; circumference in the 14th. Why we have both I don’t know; my suspicion is either two groups of people translating different geometry textbooks, or some eager young Scholastic with a nickname like ‘Doctor Magnifico Triangulorum’ thought Latin sounded better. Perimeter stuck with circules early; every etymology I see about why we use the symbol π describes it as shorthand for the perimeter of the circle. Why `circumference’ ended up the word for circles or, maybe, ellipses and ovals and such is probably the arbitrariness of language. I suspect that opening “circ” sound cues people to think of it for circles and circle-like shapes, in a way that perimeter doesn’t. But that is my speculation and should not be mistaken for information.

Steve McGarry’s KidTown for the 21st is a kids’s information panel with a bit of talk about representing numbers. And, in discussing things like how long it takes to count to a million or a billion, or how long it would take to type them out, it gets into how big these numbers can be. Les Stewart typed out the English names of numbers, in words, by the way. He’d also broken the Australian record for treading water, and for continuous swimming.

Gary Delainey and Gerry Rasmussen’s Betty for the 24th is a sudoku comic. Betty makes the common, and understandable, conflation of arithmetic with mathematics. But she’s right in identifying sudoku as a logical rather than an arithmetic problem. You can — and sometimes will see — sudoku type puzzles rendered with icons like stars and circles rather than numerals. That you can make that substitution should clear up whether there’s arithmetic involved. Commenters at GoComics meanwhile show a conflation of mathematics with logic. Certainly every mathematician uses logic, and some of them study logic. But is logic mathematics? I’m not sure it is, and our friends in the philosophy department are certain it isn’t. But then, if something that a recognizable set of mathematicians study as part of their mathematics work isn’t mathematics, then we have a bit of a logic problem, it seems.

Come Sunday I should have a fresh Reading the Comics essay available at this link.

## Six Or Arguably Four Things For Pi Day

I hope you’ll pardon me for being busy. I haven’t had the chance to read all the Pi Day comic strips yet today. But I’d be a fool to let the day pass without something around here. I confess I’m still not sure that Pi Day does anything lasting to encourage people to think more warmly of mathematics. But there is probably some benefit if people temporarily think more fondly of the subject. Certainly I’ll do more foolish things than to point at things and say, “pi, cool, huh?” this week alone.

I’ve got a couple of essays that discuss π some. The first noteworthy one is Calculating Pi Terribly, discussing a way to calculate the value of π using nothing but a needle, a tile floor, and a hilariously excessive amount of time. Or you can use an HTML5-and-JavaScript applet and slightly less time, and maybe even experimentally calculate the digits of π to two decimal places, if you get lucky.

In Calculating Pi Less Terribly I showed a way to calculate π that’s … well, you see where that sentence was going. This is a method that uses an alternating series. To get π exactly correct you have to do an infinite amount of work. But if you just want π to a certain precision, all right. This will even tell you how much work you have to do. There are other formulas that will get you digits of π with less work, though, and maybe I’ll write up one of those sometime.

And the last of the relevant essays I’ve already written is an A To Z essay about normal numbers. I don’t know whether π is a normal number. No human, to the best of my knowledge, does. Well, anyone with an opinion on the matter would likely say, of course it’s normal. There’s fantastic reasons to think it is. But none of those amount to a proof it is.

That’s my three items. After that I’d like to share … I don’t know whether to classify this as one or three pieces. They’re YouTube videos which a couple months ago everybody in the world was asking me if I’d seen. Now it’s your turn. I apologize if you too got this, a couple months ago, but don’t worry. You can tell people you watched and not actually do it. I’ll alibi you.

It’s a string of videos posted on youTube by 3Blue1Brown. The first lays out the matter with a neat physics problem. Imagine you have an impenetrable wall, a frictionless floor, and two blocks. One starts at rest. The other is sliding towards the first block and the wall. How many times will one thing collide with another? That is, will one block collide with another block, or will one block collide with a wall?

The answer seems like it should depend on many things. What it actually depends on is the ratio of the masses of the two blocks. If they’re the same mass, then there are three collisions. You can probably work that sequence out in your head and convince yourself it’s right. If the outer block has ten times the mass of the inner block? There’ll be 31 collisions before all the hits are done. You might work that out by hand. I did not. You will not work out what happens if the outer block has 100 times the mass of the inner block. That’ll be 314 collisions. If the outer block has 1,000 times the mass of the inner block? 3,141 collisions. You see where this is going.

The second video in the sequence explains why the digits of π turn up in this. And shows how to calculate this. You could, in principle, do this all using Newtonian mechanics. You will not live long enough to finish that, though.

The video shows a way that saves an incredible load of work. But you save on that tedious labor by having to think harder. Part of it is making use of conservation laws, that energy and linear momentum are conserved in collisions. But part is by recasting the problem. Recast it into “phase space”. This uses points in an abstract space to represent different configurations of a system. Like, how fast blocks are moving, and in what direction. The recasting of the problem turns something that’s impossibly tedious into something that’s merely … well, it’s still a bit tedious. But it’s much less hard work. And it’s a good chance to show off you remember the Inscribed Angle Theorem. You do remember the Inscribed Angle Theorem, don’t you? The video will catch you up. It’s a good show of how phase spaces can make physics problems so much more manageable.

The third video recasts the problem yet again. In this form, it’s about rays of light reflecting between mirrors. And this is a great recasting. That blocks bouncing off each other and walls should have anything to do with light hitting mirrors seems ridiculous. But set out your phase space, and look hard at what collisions and reflections are like, and you see the resemblance. The sort of trick used to make counting reflections easy turns up often in phase spaces. It also turns up in physics problems on toruses, doughnut shapes. You might ask when do we ever do anything on a doughnut shape. Well, real physical doughnuts, not so much. But problems where there are two independent quantities, and both quantities are periodic? There’s a torus lurking in there. There might be a phase space using that shape, and making your life easier by doing so.

That’s my promised four or maybe six items. Pardon, please, now, as I do need to get back to reading the comics.

## Proving That Disturbing Triangle Theorem That Isn’t Morley’s Somehow

I couldn’t leave people just hanging on that triangle theorem from the other day. Tthis was a compass-and-straightedge method to split a triangle into two shapes of equal area. The trick was you could split it along any point on one of the three legs of the triangle.

The theorem unsettled me, yes. But proving that it does work is not so bad and I thought to do that today.

The process: start with a triangle ABC. Pick a point P on one of the legs. We’ll say it’s on leg AB. Draw the line segment from the other vertex, C, to point P.

Now from the median point S on leg AB, draw the line parallel to PC and that intersects either leg AC or leg BC. Label that point R. The line segment RP cuts the triangle ABC into one triangle and another shape, normally a quadrilateral. Both shapes have the same area, half that of the original triangle.

To prove it nicely will involve one extra line, and the identification of one point. Construct the line SC. Lines SC and PC intersect at some point; call that Q. I’ve actually made a diagram of this, just below. I’ve put the intersection point R on the leg AC. All that would change if the point R were on BC instead would be some of the labels.

Here’s how the proof will go. I want to show triangle APR has half the area of triangle ABC. The area of triangle ARP has to be equal to the area of triangle ASC, plus the area of triangle SPQ, minus the area of triangle QCR. So the first step is proving that triangle ASC has half the area of triangle ABC. The second step is showing triangle SPQ has the same area as does triangle QCR. When that’s done, we know triangle APR has the same area as triangle ASC, which is half that of triangle ABC.

First. That ASC has half the area of triangle ABC. The area of a triangle is one-half times the length of a base times its height. The base is any of the three legs which connect two points. The height is the perpendicular distance from the third point to the line that first leg is on. Here, take the base of triangle ABC to be the line segment AC. Also take the base of triangle ASC to be the line segment AC. They have the same base. Point S is the median of the line segment AB. So point S is half as far from the base AC as the point B is. Triangle ASC has half the height of triangle ABC. Same base, half the height. So triangle ASC has half the area of triangle ABC.

Second. That triangle SPQ has the same area as triangle QCR. This is going to be most easily done by looking at two other triangles, SPC and PCR. They’re relevant to triangles SPQ and QCR. Triangle SPC has the same area as triangle PCR. Take as the base for both of them the leg PC. Point S and point R are both on the line SR. SR was created parallel to the line PC. So the perpendicular distance from point S to line PC has to be the same as the perpendicular distance from point R to the line PC. Triangle SPC has the same base and same height as does triangle PCR. So they have the same area.

Now. Triangle SPC is made up of two smaller triangles: triangle SPQ and triangle PCQ. Its area is split, somehow, between those two. Triangle PCR is also made of two smaller triangles: triangle PCQ and triangle QCR. Its area is split between those two.

The area of triangle SPQ plus the area of triangle PCQ is the same as the area of triangle SPC. This is equal to the area of triangle PCR. The area of triangle PCR is the area of triangle PCQ plus the area of triangle QCR.

And that all adds up only if the area of triangle SPQ is the same as the area of triangle QCR.

So. We had that area of triangle APR is equal to the area of triangle ASC plus the area of triangle SPQ minus the area of triangle QCR. That’s the area of triangle ASC plus zero. And that’s half the area of triangle ABC. Whatever shape is left has to have the remaining area, half the area of triangle ABC.

It’s still such a neat result.

Morley’s theorem, by the way, says this: take any triangle. Trisect each of its three interior angles. That is, for each vertex, draw the two lines that cut the interior angle into three equal spans. This creates six lines. Take the three points where these lines for adjacent angles intersect. (That is, draw the obvious intersection points.) This creates a new triangle. It’s equilateral. What business could an equilateral triangle possibly have in all this? Exactly.

## In Which I Am Disturbed By A Triangle Theorem That Isn’t Morley’s Somehow

I’ve been reading Alfred S Posamentier and Ingmar Lehmann’s The Secrets of Triangles: A Mathematical Journey. It is exactly what you’d think: 365 pages, plus endnotes and an index, describing what we as a people have learned about triangles. It’s almost enough to make one wonder if we maybe know too many things about triangles. I admit letting myself skim over the demonstration of how, using straightedge and compass, to construct a triangle when you’re given one interior angle, the distance from another vertex to its corresponding median point, and the radius of the triangle’s circumscribed circle.

But there are a bunch of interesting theorems to find. I wanted to share one. When I saw it I felt creeped out. The process seemed like a bit of dark magic, a result starting enough that it seemed to come from nowhere. Here it is.

Start with any old triangle ABC. Without loss of generality, select a point along the leg AB (other than the vertices). Call that point P. (This same technique would work if you put your point on another leg, but I would have to change the names of the vertices and line segments from here on. But it doesn’t matter what the names of the vertices are. So I can suppose that I was lucky enough that whatever leg you put your point P on I happened to name AB.)

Now. Pick the midpoint of the leg AB. This median is a point we’ll label S.

Draw the line PC.

Draw the line parallel to the line PC and which passes through S. This will intersect either the line segment BC or the line segment AC. Whichever it is, label this point of intersection R.

Draw the line from R to P.

The line RP divides the triangle ABC into two shapes, a triangle and (unless your P was the median point S) a quadrilateral.

The punch line: both shapes have half the area of the original triangle.

I usually read while eating. This was one of those lines that made me put the fork down and stare, irrationally angry, until I could work through the proof. It didn’t help that you can use a technique like this to cut the triangle into any whole number you like of equal-area wedges.

I’m sure this is old news to a fair number of readers. I don’t care. I haven’t noticed this before. And yes, it’s not as scary weird magic as Morley’s Theorem. But I’ve seen that one before, long enough ago I kind of accept it.

## Reading the Comics, January 30, 2019: Interlude Edition

I think there are just barely enough comic strips from the past week to make three essays this time around. But one of them has to be a short group, only three comics. That’ll be for the next essay when I can group together all the strips that ran in February. One strip that I considered but decided not to write at length about was Ed Allison’s dadaist Unstrange Phenomena for the 28th. It mentions Roman Numerals and the idea of sneaking message in through them. But that’s not really mathematics. I usually enjoy the particular flavor of nonsense which Unstrange Phenomena uses; you might, too.

John McPherson’s Close to Home for the 29th uses an arithmetic problem as shorthand for an accomplished education. The problem is solvable. Of course, you say. It’s an equation with quadratic polynomial; it can hardly not be solved. Yes, fine. But McPherson could easily have thrown together numbers that implied x was complex-valued, or had radicals or some other strange condition. This is one that someone could do in their heads, at least once they practiced in mental arithmetic.

I feel reasonably confident McPherson was just having a giggle at the idea of putting knowledge tests into inappropriate venues. So I’ll save the full rant. But there is a long history of racist and eugenicist ideology that tried to prove certain peoples to be mentally incompetent. Making an arithmetic quiz prerequisite to something unrelated echoes that. I’d have asked McPherson to rework the joke to avoid that.

(I’d also want to rework the composition, since the booth, the swinging arm, and the skirted attendant with the clipboard don’t look like any tollbooth I know. But I don’t have an idea how to redo the layout so it’s more realistic. And it’s not as if that sort of realism would heighten the joke.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th riffs on the problem of squaring the circle. This is one of three classical problems of geometry. The lecturer describes it just fine: is it possible to make a square that’s got the same area as a given circle, using only straightedge and compass? There are shapes it’s easy to do this for, such as rectangles, parallelograms, triangles, and (why not?) this odd crescent-moon shaped figure called the lune. Circles defied all attempts. In the 19th century mathematicians found ways to represent the operations of classical geometry with algebra, and could use the tools of algebra to show squaring the circle was impossible. The squaring would be equivalent to finding a polynomial, with integer coefficients, that has $\sqrt{\pi}$ as a root. And we know from the way algebra works that this can’t be done. So squaring the circle can’t be done.

The lecturer’s hack, modifying the compass and straightedge, lets you in principle do whatever you want. The hack isn’t new either. Modifying the geometric tools changes what you can and can’t do. The Ancient Greeks recognized that adding some specialized tools would make the problem possible. But that falls outside the scope of the problem.

Which feeds to the secondary joke, of making the philosophers sad. Often philosophy problems test one’s intuition about an idea by setting out a problem, often with unpleasant choices. A common problem with students that I’m going ahead and guessing are engineers is then attacking the setup of the question, trying to show that the problem couldn’t actually happen. You know, as though there were ever a time significant numbers of people were being tied to trolley tracks. (By the way, that thing about silent movie villains tying women to railroad tracks? Only happened in comedies spoofing Victorian melodramas. It’s always been a parody.) Attacking the logic of a problem may make for good movie drama. But it makes for a lousy student and a worse class discussion.

Ted Shearer’s Quincy rerun for the 30th uses a bit of mathematics and logic talk. It circles the difference between the feeling one can have about the rational meaning of a situation and how the situation feels to someone. It seems like a jump that Quincy goes from being asked about logic to talking about arithmetic. Possibly Quincy’s understanding of logic doesn’t start from the sort of very abstract concept that makes arithmetic hard to get to, though.

There should be another Reading the Comics post this week. It should be here, when it appears. There should also be one on Sunday, as usual.

## My 2018 Mathematics A To Z: Witch of Agnesi

Nobody had a suggested topic starting with ‘W’ for me! So I’ll take that as a free choice, and get lightly autobiogrpahical.

# Witch of Agnesi.

I know I encountered the Witch of Agnesi while in middle school. Eighth grade, if I’m not mistaken. It was a footnote in a textbook. I don’t remember much of the textbook. What I mostly remember of the course was how much I did not fit with the teacher. The only relief from boredom that year was the month we had a substitute and the occasional interesting footnote.

It was in a chapter about graphing equations. That is, finding curves whose points have coordinates that satisfy some equation. In a bit of relief from lines and parabolas the footnote offered this:

$y = \frac{8a^3}{x^2 + 4a^2}$

In a weird tantalizing moment the footnote didn’t offer a picture. Or say what an ‘a’ was doing in there. In retrospect I recognize ‘a’ as a parameter, and that different values of it give different but related shapes. No hint what the ‘8’ or the ‘4’ were doing there. Nor why ‘a’ gets raised to the third power in the numerator or the second in the denominator. I did my best with the tools I had at the time. Picked a nice easy boring ‘a’. Picked out values of ‘x’ and found the corresponding ‘y’ which made the equation true, and tried connecting the dots. The result didn’t look anything like a witch. Nor a witch’s hat.

It was one of a handful of biographical notes in the book. These were a little attempt to add some historical context to mathematics. It wasn’t much. But it was an attempt to show that mathematics came from people. Including, here, from Maria Gaëtana Agnesi. She was, I’m certain, the only woman mentioned in the textbook I’ve otherwise completely forgotten.

We have few names of ancient mathematicians. Those we have are often compilers like Euclid whose fame obliterated the people whose work they explained. Or they’re like Pythagoras, credited with discoveries by people who obliterated their own identities. In later times we have the mathematics done by, mostly, people whose social positions gave them time to write mathematics results. So we see centuries where every mathematician is doing it as their side hustle to being a priest or lawyer or physician or combination of these. Women don’t get the chance to stand out here.

Today of course we can name many women who did, and do, mathematics. We can name Emmy Noether, Ada Lovelace, and Marie-Sophie Germain. Challenged to do a bit more, we can offer Florence Nightingale and Sofia Kovalevskaya. Well, and also Grace Hopper and Margaret Hamilton if we decide computer scientists count. Katherine Johnson looks likely to make that cut. But in any case none of these people are known for work understandable in a pre-algebra textbook. This must be why Agnesi earned a place in this book. She’s among the earliest women we can specifically credit with doing noteworthy mathematics. (Also physics, but that’s off point for me.) Her curve might be a little advanced for that textbook’s intended audience. But it’s not far off, and pondering questions like “why $8a^3$? Why not $a^3$?” is more pleasant, to a certain personality, than pondering what a directrix might be and why we might use one.

The equation might be a lousy way to visualize the curve described. The curve is one of that group of interesting shapes you get by constructions. That is, following some novel process. Constructions are fun. They’re almost a craft project.

For this we start with a circle. And two parallel tangent lines. Without loss of generality, suppose they’re horizontal, so, there’s lines at the top and the bottom of the curve.

Take one of the two tangent points. Again without loss of generality, let’s say the bottom one. Draw a line from that point over to the other line. Anywhere on the other line. There’s a point where the line you drew intersects the circle. There’s another point where it intersects the other parallel line. We’ll find a new point by combining pieces of these two points. The point is on the same horizontal as wherever your line intersects the circle. It’s on the same vertical as wherever your line intersects the other parallel line. This point is on the Witch of Agnesi curve.

Now draw another line. Again, starting from the lower tangent point and going up to the other parallel line. Again it intersects the circle somewhere. This gives another point on the Witch of Agnesi curve. Draw another line. Another intersection with the circle, another intersection with the opposite parallel line. Another point on the Witch of Agnesi curve. And so on. Keep doing this. When you’ve drawn all the lines that reach from the tangent point to the other line, you’ll have generated the full Witch of Agnesi curve. This takes more work than writing out $y = \frac{8a^3}{x^2 + 4a^2}$, yes. But it’s more fun. It makes for neat animations. And I think it prepares us to expect the shape of the curve.

It’s a neat curve. Between it and the lower parallel line is an area four times that of the circle that generated it. The shape is one we would get from looking at the derivative of the arctangent. So there’s some reasons someone working in calculus might find it interesting. And people did. Pierre de Fermat studied it, and found this area. Isaac Newton and Luigi Guido Grandi studied the shape, using this circle-and-parallel-lines construction. Maria Agnesi’s name attached to it after she published a calculus textbook which examined this curve. She showed, according to people who present themselves as having read her book, the curve and how to find it. And she showed its equation and found the vertex and asymptote line and the inflection points. The inflection points, here, are where the curve chances from being cupped upward to cupping downward, or vice-versa.

It’s a neat function. It’s got some uses. It’s a natural smooth-hill shape, for example. So this makes a good generic landscape feature if you’re modeling the flow over a surface. I read that solitary waves can have this curve’s shape, too.

And the curve turns up as a probability distribution. Take a fixed point. Pick lines at random that pass through this point. See where those lines reach a separate, straight line. Some regions are more likely to be intersected than are others. Chart how often any particular line is the new intersection point. That chart will (given some assumptions I ask you to pretend you agree with) be a Witch of Agnesi curve. This might not surprise you. It seems inevitable from the circle-and-intersecting-line construction process. And that’s nice enough. As a distribution it looks like the usual Gaussian bell curve.

It’s different, though. And it’s different in strange ways. Like, for a probability distribution we can find an expected value. That’s … well, what it sounds like. But this is the strange probability distribution for which the law of large numbers does not work. Imagine an experiment that produces real numbers, with the frequency of each number given by this distribution. Run the experiment zillions of times. What’s the mean value of all the zillions of generated numbers? And it … doesn’t … have one. I mean, we know it ought to, it should be the center of that hill. But the calculations for that don’t work right. Taking a bigger sample makes the sample mean jump around more, not less, the way every other distribution should work. It’s a weird idea.

Imagine carving a block of wood in the shape of this curve, with a horizontal lower bound and the Witch of Agnesi curve as the upper bound. Where would it balance? … The normal mathematical tools don’t say, even though the shape has an obvious line of symmetry. And a finite area. You don’t get this kind of weirdness with parabolas.

(Yes, you’ll get a balancing point if you actually carve a real one. This is because you work with finitely-long blocks of wood. Imagine you had a block of wood infinite in length. Then you would see some strange behavior.)

It teaches us more strange things, though. Consider interpolations, that is, taking a couple data points and fitting a curve to them. We usually start out looking for polynomials when we interpolate data points. This is because everything is polynomials. Toss in more data points. We need a higher-order polynomial, but we can usually fit all the given points. But sometimes polynomials won’t work. A problem called Runge’s Phenomenon can happen, where the more data points you have the worse your polynomial interpolation is. The Witch of Agnesi curve is one of those. Carl Runge used points on this curve, and trying to fit polynomials to those points, to discover the problem. More data and higher-order polynomials make for worse interpolations. You get curves that look less and less like the original Witch. Runge is himself famous to mathematicians, known for “Runge-Kutta”. That’s a family of techniques to solve differential equations numerically. I don’t know whether Runge came to the weirdness of the Witch of Agnesi curve from considering how errors build in numerical integration. I can imagine it, though. The topics feel related to me.

I understand how none of this could fit that textbook’s slender footnote. I’m not sure any of the really good parts of the Witch of Agnesi could even fit thematically in that textbook. At least beyond the fact of its interesting name, which any good blog about the curve will explain. That there was no picture, and that the equation was beyond what the textbook had been describing, made it a challenge. Maybe not seeing what the shape was teased the mathematician out of this bored student.

And next is ‘X’. Will I take Mr Wu’s suggestion and use that to describe something “extreme”? Or will I take another topic or suggestion? We’ll see on Friday, barring unpleasant surprises. Thanks for reading.

## My 2018 Mathematics A To Z: Volume

Ray Kassinger, of the popular web comic Housepets!, had a silly suggestion when I went looking for topics. In one episode of Mystery Science Theater 3000, Crow T Robot gets the idea that you could describe the size of a space by the number of turkeys which fill it. (It’s based on like two minor mentions of “turkeys” in the show they were watching.)

I liked that episode. I’ve got happy memories of the time when I first saw it. I thought the sketch in which Crow T Robot got so volume-obsessed was goofy and dumb in the fun-nerd way.

I accept Mr Kassinger’s challenge only I’m going to take it seriously.

# Volume.

How big is a thing?

There is a legend about Thomas Edison. He was unimpressed with a new hire. So he hazed the college-trained engineer who deeply knew calculus. He demanded the engineer tell him the volume within a light bulb. The engineer went to work, making measurements of the shape of the bulb’s outside. And then started the calculations. This involves a calculus technique called “volumes of rotation”. This can tell the volume within a rotationally symmetric shape. It’s tedious, especially if the outer edge isn’t some special nice shape. Edison, fed up, took the bulb, filled it with water, poured that out into a graduated cylinder and said that was the answer.

I’m skeptical of legends. I’m skeptical of stories about the foolish intellectual upstaged by the practical man-of-action. And I’m skeptical of Edison because, jeez, I’ve read biographies of the man. Even the fawning ones make him out to be yeesh.

But the legend’s Edison had a point. If the volume of a shape is not how much stuff fits inside the shape, what is it? And maybe some object has too complicated a shape to find its volume. Can we think of a way to produce something with the same volume, but that is easier? Sometimes we can. When we do this with straightedge and compass, the way the Ancient Greeks found so classy, we call this “quadrature”. It’s called quadrature from its application in two dimensions. It finds, for a shape, a square with the same area. For a three-dimensional object, we find a cube with the same volume. Cubes are easy to understand.

Straightedge and compass can’t do everything. Indeed, there’s so much they can’t do. Some of it is stuff you’d think it should be able to, like, find a cube with the same volume as a sphere. Integration gives us a mathematical tool for describing how much stuff is inside a shape. It’s even got a beautiful shorthand expression. Suppose that D is the shape. Then its volume V is:

$V = \int\int\int_D dV$

Here “dV” is the “volume form”, a description of how the coordinates we describe a space in relate to the volume. The $\int\int\int$ is jargon, meaning, “integrate over the whole volume”. The subscript “D” modifies that phrase by adding “of D” to it. Writing “D” is shorthand for “these are all the points inside this shape, in whatever coordinate system you use”. If we didn’t do that we’d have to say, on each $\int$ sign, what points are inside the shape, coordinate by coordinate. At this level the equation doesn’t offer much help. It says the volume is the sum of infinitely many, infinitely tiny pieces of volume. True, but that doesn’t give much guidance about whether it’s more or less than two cups of water. We need to get more specific formulas, usually. We need to pick coordinates, for example, and say what coordinates are inside the shape. A lot of the resulting formulas can’t be integrated exactly. Like, an ellipsoid? Maybe you can integrate that. Don’t try without getting hazard pay.

We can approximate this integral. Pick a tiny shape whose volume is easy to know. Fill your shape with duplicates of it. Count the duplicates. Multiply that count by the volume of this tiny shape. Done. This is numerical integration, sometimes called “numerical quadrature”. If we’re being generous, we can say the legendary Edison did this, using water molecules as the tiny shape. And working so that he didn’t need to know the exact count or the volume of individual molecules. Good computational technique.

It’s hard not to feel we’re begging the question, though. We want the volume of something. So we need the volume of something else. Where does that volume come from?

Well, where does an inch come from? Or a centimeter? Whatever unit you use? You pick something to use as reference. Any old thing will do. Which is why you get fascinating stories about choosing what to use. And bitter arguments about which of several alternatives to use. And we express the length of something as some multiple of this reference length.

Volume works the same way. Pick a reference volume, something that can be one unit-of-volume. Other volumes are some multiple of that unit-of-volume. Possibly a fraction of that unit-of-volume.

Usually we use a reference volume that’s based on the reference length. Typically, we imagine a cube that’s one unit of length on each side. The volume of this cube with sides of length 1 unit-of-length is then 1 unit-of-volume. This seems all nice and orderly and it’s surely not because mathematicians have paid off by six-sided-dice manufacturers.

Does it have to be?

That we need some reference volume seems inevitable. We can’t very well say the area of something is ten times nothing-in-particular. Does that reference volume have to be a cube? Or even a rectangle or something else? It seems obvious that we need some reference shape that tiles, that can fill up space by itself … right?

What if we don’t?

I’m going to drop out of three dimensions a moment. Not because it changes the fundamentals, but because it makes something easier. Specifically, it makes it easier if you decide you want to get some construction paper, cut out shapes, and try this on your own. What this will tell us about area is just as true for volume. Area, for a two-dimensional sapce, and volume, for a three-dimensional, describe the same thing. If you’ll let me continue, then, I will.

So draw a figure on a clean sheet of paper. What’s its area? Now imagine you have a whole bunch of shapes with reference areas. A bunch that have an area of 1. That’s by definition. That’s our reference area. A bunch of smaller shapes with an area of one-half. By definition, too. A bunch of smaller shapes still with an area of one-third. Or one-fourth. Whatever. Shapes with areas you know because they’re marked on them.

Here’s one way to find the area. Drop your reference shapes, the ones with area 1, on your figure. How many do you need to completely cover the figure? It’s all right to cover more than the figure. It’s all right to have some of the reference shapes overlap. All you need is to cover the figure completely. … Well, you know how many pieces you needed for that. You can count them up. You can add up the areas of all these pieces needed to cover the figure. So the figure’s area can’t be any bigger than that sum.

Can’t be exact, though, right? Because you might get a different number if you covered the figure differently. If you used smaller pieces. If you arranged them better. This is true. But imagine all the possible reference shapes you had, and all the possible ways to arrange them. There’s some smallest area of those reference shapes that would cover your figure. Is there a more sensible idea for what the area of this figure would be?

And put this into three dimensions. If we start from some reference shapes of volume 1 and maybe 1/2 and 1/3 and whatever other useful fractions there are? Doesn’t this covering make sense as a way to describe the volume? Cubes or rectangles are easy to imagine. Tetrahedrons too. But why not any old thing? Why not, as the Mystery Science Theater 3000 episode had it, turkeys?

This is a nice, flexible, convenient way to define area. So now let’s see where it goes all bizarre. We know this thanks to Giuseppe Peano. He’s among the late-19th/early-20th century mathematicians who shaped modern mathematics. They did this by showing how much of our mathematics broke intuition. Peano was (here) exploring what we now call fractals. And noted a family of shapes that curl back on themselves, over and over. They’re beautiful.

And they fill area. Fill volume, if done in three dimensions. It seems impossible. If we use this covering scheme, and try to find the volume of a straight line, we get zero. Well, we find that any positive number is too big, and from that conclude that it has to be zero. Since a straight line has length, but not volume, this seems fine. But a Peano curve won’t go along with this. A Peano curve winds back on itself so much that there is some minimum volume to cover it.

This unsettles. But this idea of volume (or area) by covering works so well. To throw it away seems to hobble us. So it seems worth the trade. We allow ourselves to imagine a line so long and so curled up that it has a volume. Amazing.

And now I get to relax and unwind and enjoy a long weekend before coming to the letter ‘W’. That’ll be about some topic I figure I can whip out a nice tight 500 words about, and instead, produce some 1541-word monstrosity while I wonder why I’ve had no free time at all since August. Tuesday, give or take, it’ll be available at this link, as are the rest of these glossary posts. Thanks for reading.

## My 2018 Mathematics A To Z: Tiling

For today’s a to Z topic I again picked one nominated by aajohannas. This after I realized I was falling into a never-ending research spiral on Mr Wu, of Mathtuition’s suggested “torus”. I do have an older essay describing the torus, as a set. But that does leave out a lot of why a torus is interesting. Well, we’ll carry on.

# Tiling.

Here is a surprising thought for the next time you consider remodeling the kitchen. It’s common to tile the floor. Perhaps some of the walls behind the counter. What patterns could you use? And there are infinitely many possibilities. You might leap ahead of me and say, yes, but they’re all boring. A tile that’s eight inches square is different from one that’s twelve inches square and different from one that’s 12.01 inches square. Fine. Let’s allow that all square tiles are “really” the same pattern. The only difference between a square two feet on a side and a square half an inch on a side is how much grout you have to deal with. There are still infinitely many possibilities.

You might still suspect me of being boring. Sure, there’s a rectangular tile that’s, say, six inches by eight inches. And one that’s six inches by nine inches. Six inches by ten inches. Six inches by one millimeter. Yes, I’m technically right. But I’m not interested in that. Let’s allow that all rectangular tiles are “really” the same pattern. So we have “squares” and “rectangles”. There are still infinitely many tile possibilities.

Let me shorten the discussion here. Draw a quadrilateral. One that doesn’t intersect itself. That is, there’s four corners, four lines, and there’s no X crossings. If you have that, then you have a tiling. Get enough of these tiles and arrange them correctly and you can cover the plane. Or the kitchen floor, if you have a level floor. It might not be obvious how to do it. You might have to rotate alternating tiles, or set them in what seem like weird offsets. But you can do it. You’ll need someone to make the tiles for you, if you pick some weird pattern. I hope I live long enough to see it become part of the dubious kitchen package on junk home-renovation shows.

Let me broaden the discussion here. What do I mean by a tiling if I’m allowing any four-sided figure to be a tile? We start with a surface. Usually the plane, a flat surface stretching out infinitely far in two dimensions. The kitchen floor, or any other mere mortal surface, approximates this. But the floor stops at some point. That’s all right. The ideas we develop for the plane work all right for the kitchen. There’s some weird effects for the tiles that get too near the edges of the room. We don’t need to worry about them here. The tiles are some collection of open sets. No two tiles overlap. The tiles, plus their boundaries, cover the whole plane. That is, every point on the plane is either inside exactly one of the open sets, or it’s on the boundary between one (or more) sets.

There isn’t a requirement that all these sets have the same shape. We usually do, and will limit our tiles to one or two shapes endlessly repeated. It seems to appeal to our aesthetics and our installation budget. Using a single pattern allows us to cover the plane with triangles. Any triangle will do. Similarly any quadrilateral will do. For convex pentagonal tiles — here things get weird. There are fourteen known families of pentagons that tile the plane. Each member of the family looks about the same, but there’s some room for variation in the sides. Plus there’s one more special case that can tile the plane, but only that one shape, with no variation allowed. We don’t know if there’s a sixteenth pattern. But then until 2015 we didn’t know there was a 15th, and that was the first pattern found in thirty years. Might be an opening for someone with a good eye for doodling.

There are also exciting opportunities in convex hexagons. Anyone who plays strategy games knows a regular hexagon will tile the plane. (Regular hexagonal tilings fit a certain kind of strategy game well. Particularly they imply an equal distance between the centers of any adjacent tiles. Square and triangular tiles don’t guarantee that. This can imply better balance for territory-based games.) Irregular hexagons will, too. There are three known families of irregular hexagons that tile the plane. You can treat the regular hexagon as a special case of any of these three families. No one knows if there’s a fourth family. Ready your notepad at the next overlong, agenda-less meeting.

There aren’t tilings for identical convex heptagons, figures with seven sides. Nor eight, nor nine, nor any higher figure. You can cover them if you have non-convex figures. See any Tetris game where you keep getting the ‘s’ or ‘t’ shapes. And you can cover them if you use several shapes.

There’s some guidance if you want to create your own periodic tilings. I see it called the Conway Criterion. I don’t know the field well enough to say whether that is a common term. It could be something one mathematics popularizer thought of and that other popularizers imitated. (I don’t find “Conway Criterion” on the Mathworld glossary, but that isn’t definitive.) Suppose your polygon satisfies a couple of rules about the shapes of the edges. The rules are given in that link earlier this paragraph. If your shape does, then it’ll be able to tile the plane. If you don’t satisfy the rules, don’t despair! It might yet. The Conway Criterion tells you when some shape will tile the plane. It won’t tell you that something won’t.

(The name “Conway” may nag at you as familiar from somewhere. This criterion is named for John H Conway, who’s famous for a bunch of work in knot theory, group theory, and coding theory. And in popular mathematics for the “Game of Life”. This is a set of rules on a grid of numbers. The rules say how to calculate a new grid, based on this first one. Iterating them, creating grid after grid, can make patterns that seem far too complicated to be implicit in the simple rules. Conway also developed an algorithm to calculate the day of the week, in the Gregorian calendar. It is difficult to explain to the non-calendar fan how great this sort of thing is.)

This has all gotten to periodic tilings. That is, these patterns might be complicated. But if need be, we could get them printed on a nice square tile and cover the floor with that. Almost as beautiful and much easier to install. Are there tilings that aren’t periodic? Aperiodic tilings?

Well, sure. Easily. Take a bunch of tiles with a right angle, and two 45-degree angles. Put any two together and you have a square. So you’re “really” tiling squares that happen to be made up of a pair of triangles. Each pair, toss a coin to decide whether you put the diagonal as a forward or backward slash. Done. That’s not a periodic tiling. Not unless you had a weird run of luck on your coin tosses.

All right, but is that just a technicality? We could have easily installed this periodically and we just added some chaos to make it “not work”. Can we use a finite number of different kinds of tiles, and have it be aperiodic however much we try to make it periodic? And through about 1966 mathematicians would have mostly guessed that no, you couldn’t. If you had a set of tiles that would cover the plane aperiodically, there was also some way to do it periodically.

And then in 1966 came a surprising result. No, not Penrose tiles. I know you want me there. I’ll get there. Not there yet though. In 1966 Robert Berger — who also attended Rensselaer Polytechnic Institute, thank you — discovered such a tiling. It’s aperiodic, and it can’t be made periodic. Why do we know Penrose Tiles rather than Berger Tiles? Couple reasons, including that Berger has to use 20,426 distinct tile shapes. In 1971 Raphael M Robinson simplified matters a bit and got that down to six shapes. Roger Penrose in 1974 squeezed the set down to two, although by adding some rules about what edges may and may not touch one another. (You can turn this into a pure edges thing by putting notches into the shapes.) That really caught the public imagination. It’s got simplicity and accessibility to combine with beauty. Aperiodic tiles seem to relate to “quasicrystals”, which are what the name suggests and do happen in some materials. And they’ve got beauty. Aperiodic tiling embraces our need to have not too much order in our order.

I’ve discussed, in all this, tiling the plane. It’s an easy surface to think about and a popular one. But we can form tiling questions about other shapes. Cylinders, spheres, and toruses seem like they should have good tiling questions available. And we can imagine “tiling” stuff in more dimensions too. If we can fill a volume with cubes, or rectangles, it’s natural to wonder what other shapes we can fill it with. My impression is that fewer definite answers are known about the tiling of three- and four- and higher-dimensional space. Possibly because it’s harder to sketch out ideas and test them. Possibly because the spaces are that much stranger. I would be glad to hear more.

I’m hoping now to have a nice relaxing weekend. I won’t. I need to think of what to say for the letter ‘U’. On Tuesday I hope that it will join the rest of my A to Z essays at this link.

## My 2018 Mathematics A To Z: Manifold

Two commenters suggested the topic for today’s A to Z post. I suspect I’d have been interested in it if only one had. (Although Dina Yagoditch’s suggestion of the Menger Sponge is hard to resist.) But a double domination? The topic got suggested by Mr Wu, author of MathTuition88, and by John Golden, author of Math Hombre. My thanks to all for interesting things to think about.

# Manifold.

So you know how in the first car you ever owned the alternator was always going bad? If you’re lucky, you reach a point where you start owning cars good enough that the alternator is not the thing always going bad. Once you’re there, congratulations. Now the thing that’s always going bad in your car will be the manifold. That one’s for my dad.

Manifolds are a way to do normal geometry on weird shapes. What’s normal geometry? It’s … you know, the way shapes work on your table, or in a room. The Euclidean geometry that we’re so used to that it’s hard to imagine it not working. Why worry about weird shapes? They’re interesting, for one. And they don’t have to be that weird to count as weird. A sphere, like the surface of the Earth, can be weird. And these weird shapes can be useful. Mathematical physics, for example, can represent the evolution of some complicated thing as a path drawn on a weird shape. Bringing what we know about geometry from years of study, and moving around rooms, to a problem that abstract makes our lives easier.

We use language that sounds like that of map-makers when discussing manifolds. We have maps. We gather together charts. The collection of charts describing a surface can be an atlas. All these words have common meanings. Mercifully, these common meanings don’t lead us too far from the mathematical meanings. We can even use the problem of mapping the surface of the Earth to understand manifolds.

If you love maps, the geography kind, you learn quickly that there’s no making a perfect two-dimensional map of the Earth’s surface. Some of these imperfections are obvious. You can distort shapes trying to make a flat map of the globe. You can distort sizes. But you can’t represent every point on the globe with a point on the paper. Not without doing something that really breaks continuity. Like, say, turning the North Pole into the whole line at the top of the map. Like in the Equirectangular projection. Or skipping some of the points, like in the Mercator projection. Or adding some cuts into a surface that doesn’t have them, like in the Goode homolosine projection. You may recognize this as the one used in classrooms back when the world had first begun.

But what if we don’t need the whole globedone in a single map? Turns out we can do that easy. We can make charts that cover a part of the surface. No one chart has to cover the whole of the Earth’s surface. It only has to cover some part of it. It covers the globe with a piece that looks like a common ordinary Euclidean space, where ordinary geometry holds. It’s the collection of charts that covers the whole surface. This collection of charts is an atlas. You have a manifold if it’s possible to make a coherent atlas. For this every point on the manifold has to be on at least one chart. It’s okay if a point is on several charts. It’s okay if some point is on all the charts. Like, suppose your original surface is a circle. You can represent this with an atlas of two charts. Each chart maps the circle, except for one point, onto a line segment. The two charts don’t both skip the same point. All but two points on this circle are on all the maps of this chart. That’s cool. What’s not okay is if some point can’t be coherently put onto some chart.

This sad fate can happen. Suppose instead of a circle you want to chart a figure-eight loop. That won’t work. The point where the figure crosses itself doesn’t look, locally, like a Euclidean space. It looks like an ‘x’. There’s no getting around that. There’s no atlas that can cover the whole of that surface. So that surface isn’t a manifold.

But many things are manifolds nevertheless. Toruses, the doughnut shapes, are. Möbius strips and Klein bottles are. Ellipsoids and hyperbolic surfaces are, or at least can be. Mathematical physics finds surfaces that describe all the ways the planets could move and still conserve the energy and momentum and angular momentum of the solar system. That cheesecloth surface stretched through 54 dimensions, is a manifold. There are many possible atlases, with many more charts. But each of those means we can, at least locally, for particular problems, understand them the same way we understand cutouts of triangles and pentagons and circles on construction paper.

So to get back to cars: no one has ever said “my car runs okay, but I regret how I replaced the brake covers the moment I suspected they were wearing out”. Every car problem is easier when it’s done as soon as your budget and schedule allow.

This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. What will I choose for ‘N’, later this week? I really should have decided that by now.

## My 2018 Mathematics A To Z: Hyperbolic Half-Plane

Today’s term was one of several nominations I got for ‘H’. This one comes from John Golden, @mathhobre on Twitter and author of the Math Hombre blog on Blogspot. He brings in a lot of thought about mathematics education and teaching tools that you might find interesting or useful or, better, both.

# Hyperbolic Half-Plane.

The half-plane part is easy to explain. By the “plane” mathematicians mean, well, the plane. What you’d get if a sheet of paper extended forever. Also if it had zero width. To cut it in half … well, first we have to think hard what we mean by cutting an infinitely large thing in half. Then we realize we’re overthinking this. Cut it by picking a line on the plane, and then throwing away everything on one side or the other of that line. Maybe throw away everything on the line too. It’s logically as good to pick any line. But there are a couple lines mathematicians use all the time. This is because they’re easy to describe, or easy to work with. At least once you fix an origin and, with it, x- and y-axes. The “right half-plane”, for example, is everything in the positive-x-axis direction. Every point with coordinates you’d describe with positive x-coordinate values. Maybe the non-negative ones, if you want the edge included. The “upper half plane” is everything in the positive-y-axis direction. All the points whose coordinates have a positive y-coordinate value. Non-negative, if you want the edge included. You can make guesses about what the “left half-plane” or the “lower half-plane” are. You are correct.

The “hyperbolic” part takes some thought. What is there to even exaggerate? Wrong sense of the word “hyperbolic”. The word here is the same one used in “hyperbolic geometry”. That takes explanation.

The Western mathematics tradition, as we trace it back to Ancient Greece and Ancient Egypt and Ancient Babylon and all, gave us “Euclidean” geometry. It’s a pretty good geometry. It describes how stuff on flat surfaces works. In the Euclidean formation we set out a couple of axioms that aren’t too controversial. Like, lines can be extended indefinitely and that all right angles are congruent. And one axiom that is controversial. But which turns out to be equivalent to the idea that there’s only one line that goes through a point and is parallel to some other line.

And it turns out that you don’t have to assume that. You can make a coherent “spherical” geometry, one that describes shapes on the surface of a … you know. You have to change your idea of what a line is; it becomes a “geodesic” or, on the globe, a “great circle”. And it turns out that there’s no lines geodesics that go through a point and that are parallel to some other line geodesic. (I know you want to think about globes. I do too. You maybe want to say the lines of latitude are parallel one another. They’re even called parallels, sometimes. So they are. But they’re not geodesics. They’re “little circles”. I am not throwing in ad hoc reasons I’m right and you’re not.)

There is another, though. This is “hyperbolic” geometry. This is the way shapes work on surfaces that mathematicians call saddle-shaped. I don’t know what the horse enthusiasts out there call these shapes. My guess is they chuckle and point out how that would be the most painful saddle ever. Doesn’t matter. We have surfaces. They act weird. You can draw, through a point, infinitely many lines parallel to a given other line.

That’s some neat stuff. That’s weird and interesting. They’re even called “hyperparallel lines” if that didn’t sound great enough. You can see why some people would find this worth studying. The catch is that it’s hard to order a pad of saddle-shaped paper to try stuff out on. It’s even harder to get a hyperbolic blackboard. So what we’d like is some way to represent these strange geometries using something easier to work with.

The hyperbolic half-plane is one of those approaches. This uses the upper half-plane. It works by a move as brilliant and as preposterous as that time Q told Data and LaForge how to stop that falling moon. “Simple. Change the gravitational constant of the universe.”

What we change here is the “metric”. The metric is a function. It tells us something about how points in a space relate to each other. It gives us distance. In Euclidean geometry, plane geometry, we use the Euclidean metric. You can find the distance between point A and point B by looking at their coordinates, $(x_A, y_A)$ and $(x_B, y_B)$. This distance is $\sqrt{\left(x_B - x_A\right)^2 + \left(y_B - y_A\right)^2}$. Don’t worry about the formulas. The lines on a sheet of graph paper are a reflection of this metric. Each line is (normally) a fixed distance from its parallel neighbors. (Yes, there are polar-coordinate graph papers. And there are graph papers with logarithmic or semilogarithmic spacing. I mean graph paper like you can find at the office supply store without asking for help.)

But the metric is something we choose. There are some rules it has to follow to be logically coherent, yes. But those rules give us plenty of room to play. By picking the correct metric, we can make this flat plane obey the same geometric rules as the hyperbolic surface. This metric looks more complicated than the Euclidean metric does, but only because it has more terms and takes longer to write out. What’s important about it is that the distance your thumb put on top of the paper covers up is bigger if your thumb is near the bottom of the upper-half plane than if your thumb is near the top of the paper.

So. There are now two things that are “lines” in this. One of them is vertical lines. The graph paper we would make for this has a nice file of parallel lines like ordinary paper does. The other thing, though … well, that’s half-circles. They’re half-circles with a center on the edge of the half-plane. So our graph paper would also have a bunch of circles, of different sizes, coming from regularly-spaced sources on the bottom of the paper. A line segment is a piece of either these vertical lines or these half-circles. You can make any polygon you like with these, if you pick out enough line segments. They’re there.

There are many ways to represent hyperbolic surfaces. This is one of them. It’s got some nice properties. One of them is that it’s “conformal”. Angles that you draw using this metric are the same size as those on the corresponding hyperbolic surface. You don’t appreciate how sweet that is until you’re working in non-Euclidean geometries. Circles that are entirely within the hyperbolic half-plane match to circles on a hyperbolic surface. Once you’ve got your intuition for this hyperbolic half-plane, you can step into hyperbolic half-volumes. And that lets you talk about the geometry of hyperbolic spaces that reach into four or more dimensions of human-imaginable spaces. Isometries — picking up a shape and moving it in ways that don’t change distance — match up with the Möbius Transformations. These are a well-understood set of altering planes that comes from a different corner of geometry. Also from that fellow with the strip, August Ferdinand Möbius. It’s always exciting to find relationships like that in mathematical structures.

Pictures often help. I don’t know why I don’t include them. But here is a web site with pages, and pictures, that describe much of the hyperbolic half-plane. It includes code to use with the Geometer Sketchpad software, which I have never used and know nothing about. That’s all right. There’s at least one page there showing a wondrous picture. I hope you enjoy.

This and other essays in the Fall 2018 A-To-Z should be at this link. And I’ll start paneling for more letters soon.

## Playful Mathematics Education Blog Carnival #121

Greetings one and all! Come, gather round! Wonder and spectate and — above all else — tell your friends of the Playful Mathematics Blog Carnival! Within is a buffet of delights and treats, fortifications for the mind and fire for the imagination.

121 is a special number. When I was a mere tot, growing in the wilds of suburban central New Jersey, it stood there. It held a spot of privilege in the multiplication tables on the inside front cover of composition books. On the forward diagonal, yet insulated from the borders. It anchors the safe interior. A square number, eleventh of that set in the positive numbers.

## The First Tent

The first wonder to consider is Iva Sallay’s Find the Factors blog. She brings each week a sequence of puzzles, all factoring challenges. The result of each, done right, is a scrambling of the multiplication tables; it’s up to you the patron to find the scramble. She further examines each number in turn, finding its factors and its interesting traits. And furthermore, usually, when beginning a new century of digits opens a horserace, to see which of the numbers have the greatest number of factorizations. She furthermore was the host of this Playful Mathematics Education Carnival for August of 2018.

121 is more than just a square. It is the lone square known to be the sum of the first several powers of a prime number: it is $1 + 3 + 3^2 + 3^3 + 3^4$, a fantastic combination. If there is another square that is such a sum of primes, it is unknown to any human — and must be at least 35 digits long.

We look now for a moment at some astounding animals. From the renowned Dr Nic: Introducing Cat Maths cards, activities, games and lessons — a fine collection of feline companions, such toys as will enterain them. A dozen attributes each; twenty-seven value cards. These cats, and these cards, and these activity puzzles, promise games and delights, to teach counting, subtraction, statistics, and inference!

Next and no less incredible is the wooly Mathstodon. Christian Lawson-Perfect hosts this site, an instance of the open-source Twitter-like service Mastodon. Its focus: a place for people interested in mathematicians to write of what they know. To date over 1,300 users have joined, and have shared nearly 25,000 messages. You need not join to read many of these posts — your host here has yet to — but may sample its wares as you like.

## The Second Tent

121 is one of only two perfect squares known to be four less than the cube of a whole number. The great Fermat conjectured that 4 and 121 are the only such numbers; no one has found a counter-example. Nor a proof.

Friends, do you know the secret to popularity? There is an astonishing truth behind it. Elias Worth of the MathSection blog explains the Friendship Paradox. This mind-warping phenomenon tells us your friends have more friends than you do. It will change forever how you look at your followers and following accounts.

And now to thoughts of learning. Stepping forward now is Monica Utsey, @Liveonpurpose47 of Chocolate Covered Boy Joy. Her declaration: “I incorporated Montessori Math materials with my right brain learner because he needed literal representations of the work we were doing. It worked and we still use it.” See now for yourself the representations, counting and comparing and all the joys of several aspects of arithmetic.

Take now a moment for your own fun. Blog Carnival patron and organizer Denise Gaskins wishes us to know: “The fun of mathematical coloring isn’t limited to one day. Enjoy these coloring resources all year ’round!” Happy National Coloring Book Day offers the title, and we may keep the spirit of National Coloring Book Day all the year round.

Confident in that? Then take on a challenge. Can you scroll down faster than Christian Lawson-Perfect’s web site can find factors? Prove your speed, prove your endurance, and see if you can overcome this infinite scroll.

## The Third Tent

121 is a star number, the fifth of that select set. 121 identical items can be tiled to form a centered hexagon. You may have seen it in the German game of Chinese Checkers, as the board of that has 121 holes.

We come back again to teaching. “Many homeschoolers struggle with teaching their children math. Here are some tips to make it easier”, offers Denise Gaskins. Step forth and benefit from this FAQ: Struggling with Arithmetic, a collection of tips and thoughts and resources to help make arithmetic the more manageable.

Step now over to the arcade, and to the challenge of Pac-Man. This humble circle-inspired polygon must visit the entirety of a maze, and avoid ghosts as he does. Matthew Scroggs of Chalk Dust Magazine here seeks and shows us Optimal Pac-Man. Graph theory tells us there are thirteen billion different paths to take. Which of them is shortest? Which is fastest? Can it be known, and can it help you through the game?

And now a recreation, one to become useful if winter arrives. Think of the mysteries of the snowball rolling down a hill. How does it grow in size? How does it speed up? When does it stop? Rodolfo A Diaz, Diego L Gonzalez, Francisco Marin, and R Martinez satisfy your curiosity with Comparative kinetics of the snowball respect to other dynamical objects. Be warned! This material is best suited for the college-age student of the mathematical snow sciences.

## The Fourth Tent

121 is furthermore the sixth of the centered octagonal numbers. 121 of a thing may be set into six concentric octagons of one, then two, then three, then four, then five, and then six of them on a side.

To teach is to learn! And we have here an example of such learning. James Sheldon writing for the American Mathematical Society Graduate Student blog offers Teaching Lessons from a Summer of Taking Mathematics Courses. What secrets has Sheldon to reveal? Come inside and learn what you may.

And now step over to the games area. The game Entanglement wraps you up in knots, challenging you to find the longest knot possible. David Richeson of Division By Zero sees in this A game for budding knot theorists. What is the greatest score that could be had in this game? Can it ever be found? Only Richeson has your answer.

Step now back to the amazing Mathstodon. Gaze in wonder at the account @dudeney_puzzles. Since the September of 2017 it has brought out challenges from Henry Ernest Dudeney’s Amusements in Mathematics. Puzzles given, yes, with answers that follow along. The impatient may find Dudeney’s 1917 book on Project Gutenberg among other places.

## The Fifth Tent

Sum the digits of 121; you will find that you have four. Take its prime factors, 11 and 11, and sum their digits; you will find that this is four again. This makes 121 a Smith number. These marvels of the ages were named by Albert Wilansky, in honor of his brother-in-law, a man known to history as Harold Smith, and whose telephone number of 4,937,775 was one such.

Now let us consider terror. What is it to enter a PhD program? Many have attempted it; some have made it through. Mathieu Besançon gives to you a peek behind academia’s curtain. A year in PhD describes some of this life.

And now to an astounding challenge. Imagine an assassin readies your death. Can you protect yourself? At all? Tai-Danae Bradley invites you to consider: Is the Square a Secure Polygon? This question takes you on a tour of geometries familiar and exotic. Learn how mathematicians consider how to walk between places on a torus — and the lessons this has for a square room. The fate of the universe itself may depend on the methods described herein — the techniques used to study it relate to those that study whether a physical system can return to its original state. And then J2kun turned this into code, Visualizing an Assassin Puzzle, for those who dare to program it.

Have you overcome this challenge? Then step into the world of linear algebra, and this delight from the Mathstodon account of Christian Lawson-Perfect. The puzzle is built on the wonders of eigenvectors, those marvels of matrix multiplication. They emerge from multiplication longer or shorter but unchanged in direction. Lawson-Perfect uses whole numbers, represented by Scrabble tiles, and finds a great matrix with a neat eigenvalue. Can you prove that this is true?

## The Sixth Tent

Another wonder of the digits of 121. Take them apart, then put them together again. Contorted into the form 112 they represent the same number. 121 is, in the base ten commonly used in the land, a Friedman Number, second of that line. These marvels, in the Arabic, the Roman, or even the Mayan numerals schemes, are named for Erich Friedman, a figure of mystery from the Stetson University.

We draw closer to the end of this carnival’s attractions! To the left I show a tool for those hoping to write mathematics: Donald E Knuth, Tracy Larrabee, and Paul M Roberts’s Mathematical Writing. It’s a compilation of thoughts about how one may write to be understood, or to avoid being misunderstood. Either would be a marvel for the ages.

To the right please see Gregory Taylor’s web comic Any ~Qs. Taylor — @mathtans on Twitter — brings a world of math-tans, personifications of mathematical concepts, together for adventures and wordplay. And if the strip is not to your tastes, Taylor is working on ε Project, a serialized written story with new installments twice a month.

If you will look above you will see the marvels of curved space. On YouTube, Eigenchris hopes to learn differential geometry, and shares what he has learned. While he has a series under way he suggested Episode 15, ‘Geodesics and Christoffel Symbols as one that new viewers could usefully try. Episode 16, ‘Geodesic Examples on Plane and Sphere, puts this work to good use.

And as we reach the end of the fairgrounds, please take a moment to try Find the Factors Puzzle number 121, a challenge from 2014 that still speaks to us today!

And do always stop and gaze in awe at the fantastic and amazing geometrical constructs of Robert Loves Pi. You shall never see stellations of its like elsewhere!

## The Concessions Tent

With no thought of the risk to my life or limb I read the newspaper comics for mathematical topics they may illuminate! You may gape in awe at the results here. And furthermore this week and for the remainder of this calendar year of 2018 I dare to explain one and only one mathematical concept for each letter of our alphabet! I remind the sensitive patron that I have already done not one, not two, not three, but four previous entries all finding mathematical words for the letter “X” — will there be one come December? There is but one way you might ever know.

Denise Gaskins coordinates the Playful Mathematics Education Blog Carnival. Upcoming scheduled carnivals, including the chance to volunteer to host it yourself, or to recommend your site for mention, are listed here. And October’s 122nd Playful Mathematics Education Blog Carnival is scheduled to be hosted by Arithmophobia No More, and may this new host have the best of days!

## Reading the Comics, August 18, 2018: Ragged Ends Edition

I apologize for the ragged nature of this entry, but I’ve had a ragged sort of week and it’s all I can do to keep up. Alert calendar-watchers might have figured out I would have rather had this posted on Thursday or Friday, but I couldn’t make that work. I’m trying. Thanks for your patience.

Mark Anderson’s Andertoons for the 17th feeds rumors that I just reflexively include Mark Anderson’s Andertoons in these posts whenever I see one. But it features the name of something dear to me, so that’s worthwhile. And I love etymology, although not enough to actually learn anything substantive about it. I just enjoy trivia about where some words come from, and sometimes how they change over time. (The average English word meant the exact opposite thing about two hundred years ago, and it meant something hilariously unrelated two centuries before that.)

So I’m not sure how real word-studyers would regard the “geo” in “geometry”. The word is more or less Ancient Greek, given a bit of age and worn down into common English forms. It’s fair enough to describe it as originally meaning “land survey” or “land measure”. This might seem eccentric. But much of the early use of geometry was to figure out where things were, and how far they were from each other. It seems likely the earliest uses, for example, of the Pythagorean Theorem dealt with how to draw right angles on the surface of the Earth. And how to draw boundaries. The Greek fascination with compass-and-straightedge construction — work done without a ruler, so that you know distance only as a thing relative to other things in your figure — obscures how much of the field is about measurement.

Brett Koth’s Diamond Lil for the 17th is another geometry joke, and a much clearer one. And if there’s one thing we can say about parallel lines it’s that they don’t meet. There are some corners of geometry in which it’s convenient to say they “meet at infinity”, that is, they intersect at some point an infinite distance away. I don’t recommend bringing this up in casual conversation. I’m not sure I wanted to bring it up here.

Johnny Hart’s Back to BC for the 18th is … hm. Well, I’ll call it a numerals joke. It’s part of the continuum of jokes made about ice skating in figure-eights.

Other essays about comic strips are at this link. When I’ve talked about Andertoons I’ve tried to make sure it turns up at this link. Essays in which I’ve discussed Diamond Lil should be at this link when there are other ones. Turns out this is a new tag. The times I’ve discussed B.C., old or new, should be at this link.

## In Which I Learn A Thing About Non-Euclidean Geometries

I got a book about the philosophy of mathematics, Stephan Körner’s The Philosophy of Mathematics: An Introductory Essay. It’s a subject I’m interested in, despite my lack of training. Made it to the second page before I got to something that I had to stop and ponder. I thought to share that point and my reflections with you, because if I had to think I may as well get an essay out of it. He lists some pure-mathematical facts and some applied-mathematical counterparts, among them:

• (2) any (Euclidean) triangle which is equiangular is also equilateral
• (5) if the angles of a triangular piece of paper are equal then its sides are also equal

So where I stopped was: what is the (Euclidean) doing in that first proposition there? Or, its counterpart, about being pieces of paper?

I’m not versed in non-Euclidean geometry. My training brought me to applied-physics applications right away. I never needed a full course in non-Euclidean geometries and have never picked up much on my own. It’s an oversight I’m embarrassed by and I sometimes think to take a proper class. So this bit about equiangular-triangles not necessarily being equilateral was new to me.

Euclidean geometry everyone knows; it’s the way space works on table tops and in normal rooms. Non-Euclidean geometries are harder to understand. It was surprisingly late that mathematicians understood they were legitimate. There are two classes of non-Euclidean geometries. One is “spherical geometries”, the way geometry works … on the surface of a sphere or a squished-out ball. This is familiar enough to people who navigate or measure large journeys on the surface of the Earth. Well. The other non-Euclidean geometry is “hyperbolic geometry”. This is how shapes work on the surface of a saddle shape. It’s familiar enough to … some mathematicians who work in non-Euclidean geometries and people who ride horses. Maybe also the horses.

But! Could someone as amateur as I am in this field think of an equiangular but not equilateral triangle? Hyperbolic geometries seemed sure to produce one. But it’s hard to think about shapes on saddles so I figured to use that only if I absolutely had to. How about on a spherical geometry? And there I got to one of the classic non-Euclidean triangles. Imagine the surface of the Earth. Imagine a point at the North Pole. (Or the South Pole, if you’d rather.) Imagine a point on the equator at longitude 0 degrees. And imagine another point on the equator at longitude 90 degrees, east or west as you like. Draw the lines connecting those three points. That’s a triangle with three right angles on its interior, which is exactly the sort of thing you can’t have in Euclidean geometry.

(Which gives me another question that proves how ignorant I am of the history of mathematics. This is an easy-to-understand example. You don’t even need to be an Age of Exploration navigator to understand it. You only need a globe. Or a ball. So why did it take so long for mathematicians to accept the existence of non-Euclidean geometries? My guess is that maybe they understood this surface stuff as a weird feature of solid geometries, rather than an internally consistent geometry. But I defer to anyone who actually knows something about the history of non-Euclidean geometries to say.)

And that’s fine, but it’s also an equilateral triangle. I can imagine smaller equiangular triangles. Ones with interior angles nearer to 60 degrees each. They have to be smaller, but that’s all right. They all seem to be equilateral, though. The closer to 60 degree angles the smaller the triangle is and the more everything looks like it’s on a flat surface. Like a piece of paper.

So. Hyperbolic geometry, and the surface of a saddle, after all? Oh dear I hope not. Maybe I could look at something else.

So while I, and many people, talk about spherical geometry, it doesn’t have to be literally the geometry of the surface of a sphere. It can be other nice, smooth shapes. Ellipsoids, for example, spheres that have got stretched out in one direction or other. For example, what if we took that globe and stretched it out some? Leave the equatorial diameter at (say) twelve inches. But expand it so that the distance from North Pole to South Pole is closer to 480 miles. This may seem extreme. But one of the secrets of mathematicians is to consider cartoonishly extreme exaggerations. They’re often useful in getting your intuition to show that something must be so.

Ah, now. Consider that North Pole-Equator-Equator triangle I had before. The North-Pole-to-equator-point distance is right about 240 miles. The equator-point-to-other-equator-point distance is more like nine and a half inches. Definitely not equilateral. But it’s equiangular; all the interior angles are 90 degrees still.

My doubts refuted! And I didn’t have to consider the saddle shape. Very good. “Euclidean” is doing some useful work in that proposition. And the specification that the triangles are pieces of paper does the same work.

And yes, I know that all the real mathematicians out there are giggling at me. This has to be pretty near the first thing one learns in non-Euclidean geometry. It’s so easy to run across, and it so defies ordinary-world intuition. I ought to take a class.

## Reading the Comics, May 12, 2018: New Nancy Artist Edition

And now, closer to deadline than I like, let me wrap up last week’s mathematically-themed comic strips. I had a lot happening, that’s all I can say.

Glenn McCoy and Gary McCoy’s The Flying McCoys for the 10th is another tragic moment in the mathematics department. I’m amused that white lab coats are taken to read as “mathematician”. There are mathematicians who work in laboratories, naturally. Many interesting problems are about real-world things that can be modelled and tested and played with. It’s hardly the mathematics-department uniform, but then, I’m not sure mathematicians have a uniform. We just look like academics is all.

It also shows off that motif of mathematicians as doing anything with numbers in a more complicated way than necessary. I can’t imagine anyone in an emergency trying to evoke 9-1-1 by solving any kind of puzzle. But comic strip characters are expected to do things at least a bit ridiculously. I suppose.

Mark Litzler’s Joe Vanilla for the 11th is about random numbers. We need random numbers; they do so much good. Getting them is hard. People are pretty lousy at picking random numbers in their head. We can say what “lousy” random numbers look like. They look wrong. There’s digits that don’t get used as much as the others do. There’s strings of digits that don’t get used as much as other strings of the same length do. There are patterns, and they can be subtle ones, that just don’t look right.

And yet we have a terrible time trying to say what good random numbers look like. Suppose we want to have a string of random zeroes and ones: is 101010 better or worse than 110101? Or 000111? Well, for a string of digits that short there’s no telling. It’s in big batches that we should expect to see no big patterns. … Except that occasionally randomness should produce patterns. How often should we expect patterns, and of what size? This seems to depend on what patterns we’ve found interesting enough to look for. But how can the cultural quirks that make something seem interesting be a substantial mathematical property?

Olivia Jaimes’s Nancy for the 11th uses mathematics-assessment tests for its joke. It’s of marginal relevance, yes, but it does give me a decent pretext to include the new artist’s work here. I don’t know how long the Internet is going to be interested in Nancy. I have to get what attention I can while it lasts.

Scott Hilburn’s The Argyle Sweater for the 12th is the anthropomorphic-geometry joke for the week. Unless there was one I already did Sunday that I already forgot. Oh, no, that was anthropomorphic-numerals. It’s easy to see why a circle might be labelled irrational: either its radius or its area has to be. Both can be. The triangle, though …

Well, that’s got me thinking. Obviously all the sides of a triangle can be rational, and so its perimeter can be too. But … the area of an equilateral triangle is $\frac{1}{2}\sqrt{3}$ times the square of the length of any side. It can have a rational side and an irrational area, or vice-versa. Just as the circle has. If it’s not an equilateral triangle?

Can you have a triangle that has three rational sides and a rational area? And yes, you can. Take the right triangle that has sides of length 5, 12, and 13. Or any scaling of that, larger or smaller. There is indeed a whole family of triangles, the Heronian Triangles. All their sides are integers, and their areas are integers too. (Sides and areas rational are just as good as sides and areas integers. If you don’t see why, now you see why.) So there’s that at least. The name derives from Heron/Hero, the ancient Greek mathematician whom we credit with that snappy formula that tells us, based on the lengths of the three sides, what the area of the triangle is. Not the Pythagorean formula, although you can get the Pythagorean formula from it.

Still, I’m going to bet that there’s some key measure of even a Heronian Triangle that ends up being irrational. Interior angles, most likely. And there are many ways to measure triangles; they can’t all end up being rational at once. There are over two thousand ways to define a “center” of a triangle, for example. The odds of hitting a rational number on all of them at once? (Granted, most of these triangle centers are unknown except to the center’s discoverer/definer and that discoverer’s proud but baffled parents.)

Carla Ventresca and Henry Beckett’s On A Claire Day for the 12th mentions taking classes in probability and statistics. They’re the classes nobody doubts are useful in the real world. It’s easy to figure probability is more likely to be needed than functional analysis on some ordinary day outside the university. I can’t even compose that last sentence without the language of probability.

I’d kind of agree with calling the courses intense, though. Well, “intense” might not be the right word. But challenging. Not that you’re asked to prove anything deep. The opposite, really. An introductory course in either provides a lot of tools. Many of them require no harder arithmetic work than multiplication, division, and the occasional square root. But you do need to learn which tool to use in which scenario. And there’s often not the sorts of proofs that make it easy to understand which tool does what. Doing the proofs would require too much fussing around. Many of them demand settling finicky little technical points that take you far from the original questions. But that leaves the course as this archipelago of small subjects, each easy in themselves. But the connections between them are obscured. Is that better or worse? It must depend on the person hoping to learn.

## Reading the Comics, February 24, 2018: My One Boring Linear Algebra Anecdote Edition

Wait for it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st mentions mathematics — geometry, primarily — as something a substitute teacher has tried teaching with the use of a cucumber and condom. These aren’t terrible examples to use to make concrete the difference between volumes and surface areas. There are limitations, though. It’s possible to construct a shape that has a finite volume but an infinitely large surface area, albeit not using cucumbers.

There’s also a mention of the spring constant, and physics. This isn’t explicitly mathematical. But the description of movement on a spring are about the first interesting differential equation of mathematical physics. The solution is that of simple harmonic motion. I don’t think anyone taking the subject for the first time would guess at the answer. But it’s easy enough to verify it’s right. And this motion — sine waves — just turns up everywhere in mathematical physics.

Bud Blake’s Tiger rerun for the 23rd just mentions mathematics as a topic Hugo finds challenging, and what’s challenging about it. So a personal story: when I took Intro to Linear Algebra my freshman year one day I spaced on the fact we had an exam. So, I put the textbook on the shelf under my desk, and then forgot to take it when I left. The book disappeared, of course, and the professor never heard of it being turned in to lost-and-found or anything. Fortunately the homework was handwritten questions passed out on photocopies (ask your parents), so I could still do the assignments, but for all those, you know, definitions and examples I had to rely on my own notes. I don’t know why I couldn’t ask a classmate. Shyness, probably. Came through all right, though.

Cathy Law’s Claw for the 23rd technically qualifies as an anthropomorphic-numerals joke, in this panel about the smothering of education by the infection of guns into American culture.

Jim Meddick’s Monty for the 23rd has wealthy child Wedgwick unsatisfied with a mere ball of snow. He instead has a snow Truncated Icosahedron (the hyphens in Jarvis’s word balloon may baffle the innocent reader). This is a real shape, one that’s been known for a very long time. It’s one of the Archimedean Solids, a set of 13 solids that have convex shapes (no holes or indents or anything) and have all vertices the same, the identical number of edges coming in to each point in the same relative directions. The truncated icosahedron you maybe also know as the soccer ball shape, at least for those old-style soccer balls made of patches that were hexagons and pentagons. An actual truncated icosahedron needs twelve pentagons, so the figure drawn in the third panel isn’t quite right. At least one pentagonal face would be visible. But that’s also tricky to draw. The aerodynamics of a truncated icosahedron are surely different from those of a sphere. But in snowball-fight conditions, probably not different enough to even notice.

Mark Litzler’s Joe Vanilla for the 24th uses a blackboard full of formulas to represent an overcomplicated answer. The formulas look, offhand, like gibberish to me. But I’ll admit uncertainty since the odd capitalization of “iG(p)” at the start makes me think of some deeper group theory or knot theory symbols. And to see an “m + p” and an “m – p” makes me think of quantum mechanics of atomic orbitals. (But then an “m – p2” is weird.) So if this were anything I’d say it was some quantum chemistry formula. But my gut says if Litzler did take the blackboard symbols from anything, it was without going back to references. (Which he has no need to do, I should point out; the joke wouldn’t be any stronger — or weaker — if the blackboard meant anything.)

## Reading the Comics, November 25, 2017: Shapes and Probability Edition

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

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

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

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

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

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

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

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

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

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

## Reading the Comics, October 14, 2017: Physics Equations Edition

So that busy Saturday I promised for the mathematically-themed comic strips? Here it is, along with a Friday that reached the lowest non-zero levels of activity.

Stephan Pastis’s Pearls Before Swine for the 13th is one of those equations-of-everything jokes. Naturally it features a panel full of symbols that, to my eye, don’t parse. There are what look like syntax errors, for example, with the one that anyone could see the { mark that isn’t balanced by a }. But when someone works rough they will, often, write stuff that doesn’t quite parse. Think of it as an artist’s rough sketch of a complicated scene: the lines and anatomy may be gibberish, but if the major lines of the composition are right then all is well.

Most attempts to write an equation for everything are really about writing a description of the fundamental forces of nature. We trust that it’s possible to go from a description of how gravity and electromagnetism and the nuclear forces go to, ultimately, a description of why chemistry should work and why ecologies should form and there should be societies. There are, as you might imagine, a number of assumed steps along the way. I would accept the idea that we’ll have a unification of the fundamental forces of physics this century. I’m not sure I would believe having all the steps between the fundamental forces and, say, how nerve cells develop worked out in that time.

Mark Anderson’s Andertoons makes it overdue appearance for the week on the 14th, with a chalkboard word-problem joke. Amusing enough. And estimating an answer, getting it wrong, and refining it is good mathematics. It’s not just numerical mathematics that will look for an approximate solution and then refine it. As a first approximation, 15 minus 7 isn’t far off 10. And for mental arithmetic approximating 15 minus 7 as 10 is quite justifiable. It could be made more precise if a more exact answer were needed.

Maria Scrivan’s Half Full for the 14th I’m going to call the anthropomorphic geometry joke for the week. If it’s not then it’s just wordplay and I’d have no business including it here.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 14th tosses in the formula describing how strong the force of gravity between two objects is. In Newtonian gravity, which is why it’s the Newton Police. It’s close enough for most purposes. I’m not sure how this supports the cause of world peace.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th names Riemann’s Quaternary Conjecture. I was taken in by the panel, trying to work out what the proposed conjecture could even mean. The reason it works is that Bernhard Riemann wrote like 150,000 major works in every field of mathematics, and about 149,000 of them are big, important foundational works. The most important Riemann conjecture would be the one about zeroes of the Riemann Zeta function. This is typically called the Riemann Hypothesis. But someone could probably write a book just listing the stuff named for Riemann, and that’s got to include a bunch of very specific conjectures.