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  • Joseph Nebus 6:00 pm on Sunday, 28 August, 2016 Permalink | Reply
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    Reading the Comics, August 27, 2016: Calm Before The Term Edition 

    Here in the United States schools are just lurching back into the mode where they have students come in and do stuff all day. Perhaps this is why it was a routine week. Comic Strip Master Command wants to save up a bunch of story problems for us. But here’s what the last seven days sent into my attention.

    Jeff Harris’s Shortcuts educational feature for the 21st is about algebra. It’s got a fair enough blend of historical trivia and definitions and examples and jokes. I don’t remember running across the “number cruncher” joke before.

    Mark Anderson’s Andertoons for the 23rd is your typical student-in-lecture joke. But I do sympathize with students not understanding when a symbol gets used for different meanings. It throws everyone. But sometimes the things important to note clearly in one section are different from the needs in another section. No amount of warning will clear things up for everybody, but we try anyway.

    Tom Thaves’s Frank and Ernest for the 23rd tells a joke about collapsing wave functions, which is why you never see this comic in a newspaper but always see it on a physics teacher’s door. This is properly physics, specifically quantum mechanics. But it has mathematical import. The most practical model of quantum mechanics describes what state a system is in by something called a wave function. And we can turn this wave function into a probability distribution, which describes how likely the system is to be in each of its possible states. “Collapsing” the wave function is a somewhat mysterious and controversial practice. It comes about because if we know nothing about a system then it may have one of many possible values. If we observe, say, the position of something though, then we have one possible value. The wave functions before and after the observation are different. We call it collapsing, reflecting how a universe of possibilities collapsed into a mere fact. But it’s hard to find an explanation for what that is that’s philosophically and physically satisfying. This problem leads us to Schrödinger’s Cat, and to other challenges to our sense of how the world could make sense. So, if you want to make your mark here’s a good problem for you. It’s not going to be easy.

    John Allison’s Bad Machinery for the 24th tosses off a panel full of mathematics symbols as proof of hard thinking. In other routine references John Deering’s Strange Brew for the 26th is just some talk about how hard fractions are.

    While it’s outside the proper bounds of mathematics talk, Tom Toles’s Randolph Itch, 2 am for the 23rd is a delight. My favorite strip of this bunch. Should go on the syllabus.

  • Joseph Nebus 6:00 pm on Thursday, 25 August, 2016 Permalink | Reply
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    Why Stuff Can Orbit, Part 3: It Turns Out Spinning Matters 

    Way previously:

    Before the big distractions of Theorem Thursdays and competitive pinball events and all that I was writing up the mathematics of orbits. Last time I’d got to establishing that there can’t be such a thing as an orbit. This seems to disagree with what a lot of people say we can observe. So I want to resolve that problem. Yes, I’m aware I’m posting this on a Thursday, which I said I wasn’t going to do because it’s too hard on me to write. I don’t know how it worked out like that.

    Let me get folks who didn’t read the previous stuff up to speed. I’m using as model two things orbiting each other. I’m going to call it a sun and a planet because it’s way too confusing not to give things names. But they don’t have to be a sun and a planet. They can be a planet and moon. They can be a proton and an electron if you want to pretend quantum mechanics isn’t a thing. They can be a wood joist and a block of rubber connected to it by a spring. That’s a legitimate central force. They can even be stuff with completely made-up names representing made-up forces. So far I’m supposing the things are attracted or repelled by a force with a strength that depends on how far they are from each other but on nothing else.

    Also I’m supposing there are only two things in the universe. This is because the mathematics of two things with this kind of force is easy to do. An undergraduate mathematics or physics major can do it. The mathematics of three things is too complicated to do. I suppose somewhere around two-and-a-third things the mathematics hard enough you need an expert but the expert can do it.

    Mathematicians and physicists will call this sort of problem a “central force” problem. We can make it easier by supposing the sun is at the center of the universe, or at least our coordinate system. So we don’t have to worry about it moving. It’s just there at the center, the “origin”, and it’s only the planet that moves.

    Forces are tedious things to deal with. They’re vectors. In this context that makes them bundles of three quantities each related to the other two. We can avoid a lot of hassle by looking at potential energy instead. Potential energy is a scalar, a single number. Numbers are nice and easy. Calculus tells us how to go from potential energy to forces, in case we need the forces. It also tells us how to go from forces to potential energy, so we can do the easier problem instead. So we do.

    To write about potential energy mathematical physicists use exactly the letter you would guess they’d use if every other letter were unavailable for some reason: V. Or U, if they prefer. I’ll stick with V. Right now I don’t want to say anything about what rule determines the values of V. I just want to allow that its value changes as the planet’s distance from the star — the radius ‘r’ of its orbit — changes. So we make that clear by writing the potential energy is V = V(r). (The potential energy might change with the mass of the planet or sun, or the strength of gravity in the universe, or whatever. But we’re going to pretend those don’t change, not for the problem we’re doing, so we don’t have to write them out.)

    If you draw V(r) versus r you can discover right away circular orbits. They’re ones that are local maximums or local minimums of V(r). Physical intuition will help us here. Imagine the graph of the potential energy as if it were a smooth bowl. Drop a marble into it. Where would the marble come to rest? That’s a local minimum. The radius of that minimum is a circular orbit. (Oh, a local maximum, where the marble is at the top of a hill and doesn’t fall to either side, could be a circular orbit. But it isn’t going to be stable. The marble will roll one way or another given the slightest chance.)

    The potential energy for a force like gravity or electric attraction looks like the distance, r, raised to a power. And then multiplied by some number, which is where we hide gravitational constants and masses and all that stuff. Generally, it looks like V(r) = C rn where C is some number and n is some other number. For gravity and electricity that number is -1. For two particles connected by a spring that number n is +2. Could be anything.

    The trouble is if you draw these curves you realize that a marble dropped in would never come to a stop. It would roll down to the center, the planet falling into the sun. Or it would roll away forever, the planet racing into deep space. Either way it doesn’t orbit or do anything near orbiting. This seems wrong.

    It’s not, though. Suppose the force is repelling, that is, the potential energy gets to be smaller and smaller numbers as the distance increases. Then the two things do race away from each other. Physics students are asked to imagine two positive charges let loose next to each other. Physics students understand they’ll go racing away from each other, even though we don’t see stuff in the real world that does that very often. We suppose the students understand, though. These days I guess you can make an animation of it and people will accept that as if it’s proof of anything.

    Suppose the force is attracting. Imagine just dropping a planet out somewhere by a sun. Set it carefully just in place and let it go and get out of the way before happens. This is what we do in physics and mathematics classes, so that’s the kind of fun stuff you skipped if you majored in something else. But then we go on to make calculations about it. But that’ll orbit, right? It won’t just drop down into the sun and get melted or something?

    Not so, the way I worded it. If we set the planet into space so it was holding still, not moving at all, then it will fall. Plummet, really. The planet’s attracted to the sun, and it moves in that direction, and it’s just going to keep moving that way. If it were as far from the center as the Earth is from the Sun it’ll take its time, yes, but it’ll fall into the sun and not do anything remotely like orbiting. And yet there’s still orbits. What’s wrong?

    What’s wrong is a planet isn’t just sitting still there waiting to fall into the sun. Duh, you say. But why isn’t it just sitting still? That’s because it’s moving. Might be moving in any direction. We can divide that movement up into two pieces. One is the radial movement, how fast it’s moving towards or away from the center, that is, along the radius between sun and planet. If it’s a circular orbit this speed is zero; the planet isn’t moving any closer or farther away. If this speed isn’t zero it might affect how fast the planet falls into the sun, but it won’t affect the fact of whether it does or not. No more than how fast you toss a ball up inside a room changes whether it’ll eventually hit the floor. </p.

    It’s the other part, the transverse velocity, that matters. This is the speed the thing is moving perpendicular to the radius. It’s possible that this is exactly zero and then the planet does drop into the sun. It’s probably not. And what that means is that the planet-and-sun system has an angular momentum. Angular momentum is like regular old momentum, only for spinning. And as with regular momentum, the total is conserved. It won’t change over time. When I was growing up this was always illustrated by thinking of ice skaters doing a spin. They pull their arms in, they spin faster. They put their arms out, they spin slower.

    (Ice skaters eventually slow down, yes. That’s for the same reasons they slow down if they skate in a straight line even though regular old momentum, called “linear momentum” if you want to be perfectly clear, is also conserved. It’s because they have to get on to the rest of their routine.)

    The same thing has to happen with planets orbiting a sun. If the planet moves closer to the sun, it speeds up; if it moves farther away, it slows down. To fall into the exact center while conserving angular momentum demands the planet get infinitely fast. This they don’t typically do.

    There was a tipoff to this. It’s from knowing the potential energy V(r) only depends on the distance between sun and planet. If you imagine taking the system and rotating it all by any angle, you wouldn’t get any change in the forces or the way things move. It would just change the values of the coordinates you used to describe this. Mathematical physicists describe this as being “invariant”, which means what you’d imagine, under a “continuous symmetry”, which means a change that isn’t … you know, discontinuous. Rotating thing as if they were on a pivot, that is, instead of (like) reflecting them through a mirror.

    And invariance under a continuous symmetry like this leads to a conservation law. This is known from Noether’s Theorem. You can find explained quite well on every pop-mathematics and pop-physics blog ever. It’s a great subject for pop-mathematics/physics writing. The idea, that the geometry of a problem tells us something about its physics and vice-versa, is important. It’s a heady thought without being so exotic as to seem counter-intuitive. And its discoverer was Dr Amalie Emmy Noether. She’s an early-20th-century demonstration of the first-class work that one can expect women to do when they’re not driven out of mathematics. You see why the topic is so near irresistible.

    So we have to respect the conservation of angular momentum. This might sound like we have to give up on treating circular orbits as one-variable problems. We don’t have to just yet. We will, eventually, want to look at not just how far the planet is from the origin but also in what direction it is. We don’t need to do that yet. We have a brilliant hack.

    We can represent the conservation of angular momentum as a slight repulsive force. It’s not very big if the angular momentum is small. It’s not going to be a very big force unless the planet gets close to the origin, that is, until r gets close to zero. But it does grow large and acts as if the planet is being pushed away. We consider that a pseudoforce. It appears because our choice of coordinates would otherwise miss some important physics. And that’s fine. It’s not wrong any more than, say, a hacksaw is the wrong tool to cut through PVC pipe just because you also need a vise.

    This pseudoforce can be paired with a pseduo-potential energy. One of the great things about the potential-energy view of physics is that adding two forces together is as easy as adding their potential energies together. We call the sum of the original potential energy and the angular-momentum-created pseudopotential the “effective potential energy”. Far from the origin, for large radiuses r, this will be almost identical to the original potential energy. Close to the origin, this will be a function that rises up steeply. And as a result there can suddenly be a local minimum. There can be a circular orbit.

    Spring potential, which is a parabola growing with the distance r from the origin. And the effective potential, which grows to a vertical asymptote where the radius is zero.

    Figure 1. The potential energy of a spring — the red line — and the effective potential energy — the blue line — when the angular momentum is added as a pseudoforce. Without angular momentum in consideration the only equilibrium is at the origin. With angular momentum there’s some circular orbit, somewhere. Don’t pay attention to the numbers on the axes. They don’t mean anything.

    Gravitational potential, with a vertical asymptote at the radius equalling zero going down to negative infinitely great numbers and a horizontal asymptote at the radius going off to infinity. And the effective potential, with the vertical asymptote at radius of zero going to positive infinitely great numbers, forcing there to be some minimum: a circular orbit.

    Figure 2. The potential energy of a gravitational attraction — the red line — and the effective potential energy — the blue line — when the angular momentum is added as a pseudoforce. Without angular momentum in consideration there’s no equilibrium. The thing, a planet, falls into the center, the sun. With angular momentum there’s some circular orbit. As before the values of the numbers don’t matter and you should just ignore them.

    The location of the minimum — the radius of the circular orbit — will depend on the original potential, of course. It’ll also depend on the angular momentum. The smaller the angular momentum the closer to the origin will be the circular orbit. If the angular momentum is zero we have the original potential and the planet dropping into the center again. If the angular momentum is large enough there might not even be a minimum anymore. That matches systems where the planet has escape velocity and can go plunging off into deep space. And we can see this by looking at the plot of the effective velocity even before we calculate things.

    Gravitational potential, with a vertical asymptote at the radius equalling zero going down to negative infinitely great numbers and a horizontal asymptote at the radius going off to infinity. And then the effective potential for a very large angular momentum. There's a vertical asymptote at radius of zero going to positive infinitely great numbers, and so large that there isn't any local minimum except at an infinitely large radius.

    Figure 3. Gravitational potential energy — the red line — and the effective potential energy — the blue line — when angular momentum is considered. In this case the angular momentum is so large, that is, the planet is moving so fast, that there are no orbits. The planet’s reached escape velocity and can go infinitely far away from the sun.

    This only goes so far as demonstrating a circular orbit should exist. Or giving some conditions for which a circular orbit wouldn’t. We might want to know something more, like where that circular orbit is. Or if it’s possible for there to be an elliptic orbit. Or other shapes. I imagine it’s possible to work this out with careful enough drawings. But at some point it gets easier to just calculate things. We’ll get to that point soon.

  • Joseph Nebus 6:00 pm on Sunday, 21 August, 2016 Permalink | Reply
    Tags: cheese, , , , , ,   

    Reading the Comics, August 19, 2016: Mathematics Signifier Edition 

    I know it seems like when I write these essays I spend the most time on the first comic in the bunch and give the last ones a sentence, maybe two at most. I admit when there’s a lot of comics I have to write up at once my energy will droop. But Comic Strip Master Command apparently wants the juiciest topics sent out earlier in the week. I have to follow their lead.

    Stephen Beals’s Adult Children for the 14th uses mathematics to signify deep thinking. In this case Claremont, the dog, is thinking of the Riemann Zeta function. It’s something important in number theory, so longtime readers should know this means it leads right to an unsolved problem. In this case it’s the Riemann Hypothesis. That’s the most popular candidate for “what is the most important unsolved problem in mathematics right now?” So you know Claremont is a deep-thinking dog.

    The big Σ ordinary people might recognize as representing “sum”. The notation means to evaluate, for each legitimate value of the thing underneath — here it’s ‘n’ — the value of the expression to the right of the Sigma. Here that’s \frac{1}{n^s} . Then add up all those terms. It’s not explicit here, but context would make clear, n is positive whole numbers: 1, 2, 3, and so on. s would be a positive number, possibly a whole number.

    The big capital Pi is more mysterious. It’s Sigma’s less popular brother. It means “product”. For each legitimate value of the thing underneath it — here it’s “p” — evaluate the expression on the right. Here that’s \frac{1}{1 - \frac{1}{p^s}} . Then multiply all that together. In the context of the Riemann Zeta function, “p” here isn’t just any old number, or even any old whole number. It’s only the prime numbers. Hence the “p”. Good notation, right? Yeah.

    This particular equation, once shored up with the context the symbols live in, was proved by Leonhard Euler, who proved so much you sometimes wonder if later mathematicians were needed at all. It ties in to how often whole numbers are going to be prime, and what the chances are that some set of numbers are going to have no factors in common. (Other than 1, which is too boring a number to call a factor.) But even if Claremont did know that Euler got there first, it’s almost impossible to do good new work without understanding the old.

    Charlos Gary’s Working It Out for the 14th is this essay’s riff on pie charts. Or bar charts. Somewhere around here the past week I read that a French idiom for the pie chart is the “cheese chart”. That’s a good enough bit I don’t want to look more closely and find out whether it’s true. If it turned out to be false I’d be heartbroken.

    Ryan North’s Dinosaur Comics for the 15th talks about everyone’s favorite physics term, entropy. Everyone knows that it tends to increase. Few advanced physics concepts feel so important to everyday life. I almost made one expression of this — Boltzmann’s H-Theorem — a Theorem Thursday post. I might do a proper essay on it yet. Utahraptor describes this as one of “the few statistical laws of physics”, which I think is a bit unfair. There’s a lot about physics that is statistical; it’s often easier to deal with averages and distributions than the mass of real messy data.

    Utahraptor’s right to point out that it isn’t impossible for entropy to decrease. It can be expected not to, in time. Indeed decent scientists thinking as philosophers have proposed that “increasing entropy” might be the only way to meaningfully define the flow of time. (I do not know how decent the philosophy of this is. This is far outside my expertise.) However: we would expect at least one tails to come up if we simultaneously flipped infinitely many coins fairly. But there is no reason that it couldn’t happen, that infinitely many fairly-tossed coins might all come up heads. The probability of this ever happening is zero. If we try it enough times, it will happen. Such is the intuition-destroying nature of probability and of infinitely large things.

    Tony Cochran’s Agnes on the 16th proposes to decode the Voynich Manuscript. Mathematics comes in as something with answers that one can check for comparison. It’s a familiar role. As I seem to write three times a month, this is fair enough to say to an extent. Coming up with an answer to a mathematical question is hard. Checking the answer is typically easier. Well, there are many things we can try to find an answer. To see whether a proposed answer works usually we just need to go through it and see if the logic holds. This might be tedious to do, especially in those enormous brute-force problems where the proof amounts to showing there are a hundred zillion special cases and here’s an answer for each one of them. But it’s usually a much less hard thing to do.

    Johnny Hart and Brant Parker’s Wizard of Id Classics for the 17th uses what seems like should be an old joke about bad accountants and nepotism. Well, you all know how important bookkeeping is to the history of mathematics, even if I’m never that specific about it because it never gets mentioned in the histories of mathematics I read. And apparently sometime between the strip’s original appearance (the 20th of August, 1966) and my childhood the Royal Accountant character got forgotten. That seems odd given the comic potential I’d imagine him to have. Sometimes a character’s only good for a short while is all.

    Mark Anderson’s Andertoons for the 18th is the Andertoons representative for this essay. Fair enough. The kid speaks of exponents as a kind of repeating oneself. This is how exponents are inevitably introduced: as multiplying a number by itself many times over. That’s a solid way to introduce raising a number to a whole number. It gets a little strained to describe raising a number to a rational number. It’s a confusing mess to describe raising a number to an irrational number. But you can make that logical enough, with effort. And that’s how we do make the idea rigorous. A number raised to (say) the square root of two is something greater than the number raised to 1.4, but less than the number raised to 1.5. More than the number raised to 1.41, less than the number raised to 1.42. More than the number raised to 1.414, less than the number raised to 1.415. This takes work, but it all hangs together. And then we ask about raising numbers to an imaginary or complex-valued number and we wave that off to a higher-level mathematics class.

    Nate Fakes’s Break of Day for the 18th is the anthropomorphic-numerals joke for this essay.

    Lachowski’s Get A Life for the 18th is the sudoku joke for this essay. It’s also a representative of the idea that any mathematical thing is some deep, complicated puzzle at least as challenging as calculating one’s taxes. I feel like this is a rerun, but I don’t see any copyright dates. Sudoku jokes like this feel old, but comic strips have been known to make dated references before.

    Samson’s Dark Side Of The Horse for the 19th is this essay’s Dark Side Of The Horse gag. I thought initially this was a counting-sheep in a lab coat. I’m going to stick to that mistaken interpretation because it’s more adorable that way.

  • Joseph Nebus 6:00 pm on Wednesday, 17 August, 2016 Permalink | Reply
    Tags: , , , sheep, , Utopia, ,   

    Some Mathematical Tweets To Read 

    Can’t deny that I will sometimes stockpile links of mathematics stuff to talk about. Sometimes I even remember to post it. Sometimes it’s a tweet like this, which apparently I’ve been carrying around since April:

    I admit I do not know whether the claim is true. It’s plausible enough. English has many variants in England alone, and any trade will pick up its own specialized jargon. The words are fun as it is.

    From the American Mathematical Society there’s this:

    I talk a good bit about knot theory. It captures the imagination and it’s good for people who like to doodle. And it has a lot of real-world applications. Tangled wires, protein strands, high-energy plasmas, they all have knots in them. Some work by Paul Sutcliffe and Fabian Maucher, both of Durham University, studies tangled vortices. These are vortices that are, er, tangled together, just like you imagine. Knot theory tells us much about this kind of vortex. And it turns out these tangled vortices can untangle themselves and smooth out again, even without something to break them up and rebuild them. It gives hope for power cords everywhere.

    Nerds have a streak which compels them to make blueprints of things. It can be part of the healthier side of nerd culture, the one that celebrates everything. The side that tries to fill in the real-world things that the thing-celebrated would have if it existed. So here’s a bit of news about doing that:

    I like the attempt to map Sir Thomas More’s Utopia. It’s a fun exercise in matching stuff to a thin set of data. But as mentioned in the article, nobody should take it too seriously. The exact arrangement of things in Utopia isn’t the point of the book. More probably didn’t have a map for it himself.

    (Although maybe. I believe I got this from Simon Garfield’s On The Map: A Mind-Expanding Exploration Of The Way The World Looks and apologize generally if I’ve got it wrong. My understanding is Robert Louis Stevenson drew a map of Treasure Island and used it to make sure references in the book were consistent. Then the map was lost in the mail to his publishers. He had to read his text and re-create it as best he could. Which, if true, makes the map all the better. It makes it so good a lost-map story that I start instinctively to doubt it; it’s so colorfully perfect, after all.)

    And finally there’s this gem from the Magic Realism Bot:

    Happy reading.

  • Joseph Nebus 6:00 pm on Sunday, 14 August, 2016 Permalink | Reply
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    Reading the Comics, August 12, 2016: Skipping Saturday Edition 

    I have no idea how many or how few comic strips on Saturday included some mathematical content. I was away most of the day. We made a quick trip to the Michigan’s Adventure amusement park and then to play pinball in a kind-of competitive league. The park turned out to have every person in the world there. If I didn’t wave to you from the queue on Shivering Timbers I apologize but it hasn’t got the greatest lines of sight. The pinball stuff took longer than I expected too and, long story short, we got back home about 4:15 am. So I’m behind on my comics and here’s what I did get to.

    Tak Bui’s PC and Pixel for the 8th depicts the classic horror of the cleaning people wiping away an enormous amount of hard work. It’s a primal fear among mathematicians at least. Boards with a space blocked off with the “DO NOT ERASE” warning are common. At this point, though, at least, the work is probably savable. You can almost always reconstruct work, and a few smeared lines like this are not bad at all.

    The work appears to be quantum mechanics work. The tell is in the upper right corner. There’s a line defining E (energy) as equal to something including \imath \hbar \frac{\partial}{\partial t}\phi(r, t) . This appears in the time-dependent Schrödinger Equation. It describes how probability waveforms look when the potential energies involved may change in time. These equations are interesting and impossible to solve exactly. We have to resort to approximations, including numerical approximations, all the time. So that’s why the computer lab would be working on this.

    Mark Anderson’s Andertoons! Where would I be without them? Besides short on content. The strip for the 10th depicts a pollster saying to “put the margin of error at 50%”, guaranteeing the results are right. If you follow elections polls you do see the results come with a margin of error, usually of about three percent. But every sampling technique carries with it a margin of error. The point of a sample is to learn something about the whole without testing everything in it, after all. And probability describes how likely it is the quantity measured by a sample will be far from the quantity the whole would have. The logic behind this is independent of the thing being sampled. It depends on what the whole is like. It depends on how the sampling is done. It doesn’t matter whether you’re sampling voter preferences or whether there are the right number of peanuts in a bag of squirrel food.

    So a sample’s measurement will almost never be exactly the same as the whole population’s. That’s just requesting too much of luck. The margin of error represents how far it is likely we’re off. If we’ve sampled the voting population fairly — the hardest part — then it’s quite reasonable the actual vote tally would be, say, one percent different from our poll. It’s implausible that the actual votes would be ninety percent different. The margin of error is roughly the biggest plausible difference we would expect to see.

    Except. Sometimes we do, even with the best sampling methods possible, get a freak case. Rarely noticed beside the margin of error is the confidence level. This is what the probability is that the actual population value is within the sampling error of the sample’s value. We don’t pay much attention to this because we don’t do statistical-sampling on a daily basis. The most normal people do is read election polling results. And most election polls settle for a confidence level of about 95 percent. That is, 95 percent of the time the actual voting preference will be within the three or so percentage points of the survey. The 95 percent confidence level is popular maybe because it feels like a nice round number. It’ll be off only about one time out of twenty. It also makes a nice balance between a margin of error that doesn’t seem too large and that doesn’t need too many people to be surveyed. As often with statistics the common standard is an imperfectly-logical blend of good work and ease of use.

    For the 11th Mark Anderson gives me less to talk about, but a cute bit of wordplay. I’ll take it.

    Anthony Blades’s Bewley for the 12th is a rerun. It’s at least the third time this strip has turned up since I started writing these Reading The Comics posts. For the record it ran also the 27th of April, 2015 and on the 24th of May, 2013. It also suggests mathematicians have a particular tell. Try this out next time you do word problem poker and let me know how it works for you.

    Julie Larson’s The Dinette Set for the 12th I would have sworn I’d seen here before. I don’t find it in my archives, though. We are meant to just giggle at Larson’s characters who bring their penny-wise pound-foolishness to everything. But there is a decent practical mathematics problem here. (This is why I thought it had run here before.) How far is it worth going out of one’s way for cheaper gas? How much cheaper? It’s simple algebra and I’d bet many simple Javascript calculator tools. The comic strip originally ran the 4th of October, 2005. Possibly it’s been rerun since.

    Bill Amend’s FoxTrot Classics for the 12th is a bunch of gags about a mathematics fighting game. I think Amend might be on to something here. I assume mathematics-education contest games have evolved from what I went to elementary school on. That was a Commodore PET with a game where every time you got a multiplication problem right your rocket got closer to the ASCII Moon. But the game would probably quickly turn into people figuring how to multiply the other person’s function by zero. I know a game exploit when I see it.

    The most obscure reference is in the third panel one. Jason speaks of “a z = 0 transform”. This would seem to be some kind of z-transform, a thing from digital signals processing. You can represent the amplification, or noise-removal, or averaging, or other processing of a string of digits as a polynomial. Of course you can. Everything is polynomials. (OK, sometimes you must use something that looks like a polynomial but includes stuff like the variable z raised to a negative power. Don’t let that throw you. You treat it like a polynomial still.) So I get what Jason is going for here; he’s processing Peter’s function down to zero.

    That said, let me warn you that I don’t do digital signal processing. I just taught a course in it. (It’s a great way to learn a subject.) But I don’t think a “z = 0 transform” is anything. Maybe Amend encountered it as an instructor’s or friend’s idiosyncratic usage. (Amend was a physics student in college, and shows his comfort with mathematics-major talk often. He by the way isn’t even the only syndicated cartoonist with a physics degree. Bud Grace of The Piranha Club was also a physics major.) I suppose he figured “z = 0 transform” would read clearly to the non-mathematician and be interpretable to the mathematician. He’s right about that.

  • Joseph Nebus 6:00 pm on Thursday, 11 August, 2016 Permalink | Reply
    Tags: alliteration, , , , ,   

    Finally, What I Learned Doing Theorem Thursdays 

    Here’s the index to the stuff I posted on them.

    The biggest thing I learned from my Theorem Thursdays project was: don’t do this for Thursdays. The appeal is obvious. If things were a little different I’d have no problem with Thursdays. But besides being a slightly-read pop-mathematics blogger I’m also a slightly-read humor blogger. And I try to have a major piece, about seven hundred words that are more than simply commentary on how a comic strip’s gone wrong, ready for Thursday evenings my time.

    That’s all my doing. It’s a relic of my thinking that the humor blog should run at least a bit like a professional syndicated columnist’s, with a fixed deadline for bigger pieces. While I should be writing more ahead of deadline than this, what I would do is get to Wednesday realizing I have two major things to write in a day. I’d have an idea for one of them, the mathematics thing, since I would pick a topic the previous Thursday. And once I’ve picked an idea the rest is easy. (Part of the process of picking is realizing whether there’s any way to make seven hundred words about something.) But that’s a lot of work for something that’s supposed to be recreational. Plus Wednesdays are, two weeks a month, a pinball league night.

    So Thursday is right out, unless I get better about having first drafts of stuff done Monday night. So Thursday is right out. This has problems for future appearances of the gimmick. The alliterative pull is strong. The only remotely compelling alternative is Theorems on the Threes, maybe one the 3rd, 13th, and 23rd of the month. That leaves the 30th and 31st unaccounted for, and room for a good squabble about whether they count in an “on the threes” scheme.

    There’s a lot of good stuff to say about the project otherwise. The biggest is that I had fun with it. The Theorem Thursday pieces sprawled into for-me extreme lengths, two to three thousand words. I had space to be chatty and silly and autobiographic in ways that even the A To Z projects don’t allow. Somehow those essays didn’t get nearly as long, possibly because I was writing three of them a week. I didn’t actually write fewer things in July than I did in, say, May. But it was fewer kinds of things; postings were mostly Theorem Thursdays and Reading the Comics posts. Still, overall readership didn’t drop and people seemed to quite like what I did write. It may be fewer but longer-form essays are the way I should go.

    Also I found that people like stranger stuff. There’s an understandable temptation in doing pop-mathematics to look for topics that are automatically more accessible. People are afraid enough of mathematics. They have good reason to be terrified of some topic even mathematics majors don’t encounter until their fourth year. So there’s a drive to simpler topics, or topics that have fewer prerequisites, and that’s why every mathematics blogger has an essay about how the square root of two is irrational and how there’s different sizes to infinitely large sets. And that’s produced some excellent writing about topics like those, which are great topics. They have got the power to inspire awe without requiring any warming up. That’s special.

    But it also means they’re hard to write anything new or compelling about if you’re like me, and in somewhere like the second hundred billion of mathematics bloggers. I can’t write anything better than what’s already gone about that. Liouville’s Theorem? That’s something I can be a good writer about. With that, I can have a blog personality. It’s like having a real personality but less work.

    As I did with the Leap Day 2016 A To Z project, I threw the topics open to requests. I didn’t get many. Possibly the form gave too much freedom. Picking something to match a letter, as in the A to Z, gives a useful structure for choosing something specific. Pick a theorem from anywhere in mathematics? Something from algebra class? Something mentioned in a news report about a major breakthrough the reporter doesn’t understand but had an interesting picture? Something that you overheard the name of once without any context? How should people know what the scope of it is, before they’ve even seen a sample? And possibly people don’t actually remember the names of theorems unless they stay in mathematics or mathematics-related fields. Those folks hardly need explained theorems with names they remember. This is a hard problem to imagine people having, but it’s something I must consider.

    So this is what I take away from the two-month project. There’s a lot of fun digging into the higher-level mathematics stuff. There’s an interest in it, even if it means I write longer and therefore fewer pieces. Take requests, but have a structure for taking them that makes it easy to tell what requests should look like. Definitely don’t commit to doing big things for Thursday, not without a better scheme for getting the humor blog pieces done. Free up some time Wednesday and don’t put up an awful score on Demolition Man like I did last time again. Seriously, I had a better score on The Simpsons Pinball Party than I did on Demolition Man and while you personally might not find this amusing there’s at least two people really into pinball who know how hilarious that is. (The games have wildly different point scorings. This like having a basketball score be lower than a hockey score.) That isn’t so important to mathematics blogging but it’s a good lesson to remember anyway.

    • elkement (Elke Stangl) 6:21 am on Monday, 22 August, 2016 Permalink | Reply

      You are such a prolific writer – kudos! Sorry that I am hardly able to catch up in some months ;-)


      • Joseph Nebus 8:48 pm on Sunday, 28 August, 2016 Permalink | Reply

        Aw, well, thank you, trusting that prolific is a good thing. I doubt I have time to read myself myself, as my problem with comments should prove.

        It happens I’ve gotten into a slow stretch the past few weeks. I’m hoping that with the start of a new season I’ll be able to get to a better balance between twice-a-week and daily.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Tuesday, 9 August, 2016 Permalink | Reply
    Tags: , teachers,   

    Reading the Comics, August 5, 2016: Word Problems Edition 

    And now to close out the rest of last week’s comics, those from between the 1st and the 6th of the month. It’s a smaller set. Take it up with the traffic division of Comic Strip Master Command.

    Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 2nd is mostly a word problem joke. It’s boosted some by melting into it a teacher complaining about her pay. It does make me think some about what the point of a story problem is. That is, why is the story interesting? Often it isn’t. The story is just an attempt to make a computation problem look like the sort of thing someone might wonder in the real world. This is probably why so many word problems are awful as stories and as incentive to do a calculation. There’s a natural interest that one might have in, say, the total distance travelled by a rubber ball dropped and bouncing until it finally comes to a rest. But that’s only really good for testing how one understands a geometric series. It takes more storytelling to work out why you might want to find a cube root of x2 minus eight.

    Dave Whamond’s Reality Check for the 3rd uses mathematics on the blackboard as symbolic for all the problems one might have. Also a solution, if you call it that. It wouldn’t read so clearly if Ms Haversham had an English problem on the board.

    Mark Anderson’s Andertoons for the 5th keeps getting funnier to me. At first reading I didn’t connect the failed mathematics problem of 2 x 0 with the caption. Once I did, I realized how snugly fit the comic is.

    Greg Curfman’s Meg Classics for the 5th ran originally the 23rd of May, 1998. The application of mathematics to everyday sports was a much less developed thing back then. It’s often worthwhile to methodically study what you do, though, to see what affects the results. Here Mike has found the team apparently makes twelve missed shots for each goal. This might not seem like much of a formula, but these are kids. We shouldn’t expect formulas with a lot of variables under consideration. Since Meg suggests Mike needed to account for “the whiff factor” I have to suppose she doesn’t understand the meaning of the formula. Or perhaps she wonders why missed kicks before getting to the goal don’t matter. Well, every successful model starts out as a very simple thing to which we add complexity, and realism, as we’re able to handle them. If lucky we end up with a good balance between a model that describes what we want to know and yet is simple enough to understand.

  • Joseph Nebus 6:00 pm on Sunday, 7 August, 2016 Permalink | Reply
    Tags: , discovery, divisors, ,   

    Reading the Comics, August 1, 2016: Kalends Edition 

    The last day of July and first day of August saw enough mathematically-themed comic strips to fill a standard-issue entry. The rest of the week wasn’t so well-stocked. But I’ll cover those comics on Tuesday if all goes well. This may be a silly plan, but it is a plan, and I will stick to that.

    Johnny Hart’s Back To BC reprints the venerable and groundbreaking comic strip from its origins. On the 31st of July it reprinted a strip from February 1959 in which Peter discovers mathematics. The work’s elaborate, much more than we would use to solve the problem today. But it’s always like that. Newly-discovered mathematics is much like any new invention or innovation, a rickety set of things that just barely work. With time we learn better how the idea should be developed. And we become comfortable with the cultural assumptions going into the work. So we get more streamlined, faster, easier-to-use mathematics in time.

    The early invention of mathematics reappears the 1st of August, in a strip from earlier in February 1959. In this case it’s the sort of word problem confusion strip that any comic with a student could do. That’s a bit disappointing but Hart had much less space than he’d have for the Sunday strip above. One must do what one can.

    Mac King and Bill King’s Magic in a Minute for the 31st maybe isn’t really mathematics. I guess there’s something in the modular-arithmetic implied by it. But it depends on a neat coincidence. Follow the directions in the comic about picking a number from one to twelve and counting out the letters in the word for that number. And then the letters in the word for the number you’re pointing to, and then once again. It turns out this leads to the same number. I’d never seen this before and it’s neat that it does.

    Rick Detorie’s One Big Happy rerun for the 31st features Ruthie teaching, as she will. She mentions offhand the “friendlier numbers”. By this she undoubtedly means the numbers that are attractive in some way, like being nice to draw. There are “friendly numbers”, though, as number theorists see things. These are sets of numbers. For each number in this set you get the same index if you add together all its divisors (including 1 and the original number) and divide it by the original number. For example, the divisors of six are 1, 2, 3, and 6. Add that together and you get 12; divide that by the original 6 and you get 2. The divisors of 28 are 1, 2, 4, 7, 14, and 28. Add that pile of numbers together and you get 56; divide that by the original 28 and you get 2. So 6 and 28 are friendly numbers, each the friend of the other.

    As often happens with number theory there’s a lot of obvious things we don’t know. For example, we know that 1, 2, 3, 4, and 5 have no friends. But we do not know whether 10 has. Nor 14 nor 20. I do not know if it is proved whether there are infinitely many sets of friendly numbers. Nor do I know if it is proved whether there are infinitely many numbers without friends. Those last two sentences are about my ignorance, though, and don’t reflect what number theory people know. I’m open to hearing from people who know better.

    There are also things called “amicable numbers”, which are easier to explain and to understand than “friendly numbers”. A pair of numbers are amicable if the sum of one number’s divisors is the other number. 220 and 284 are the smallest pair of amicable numbers. Fermat found that 17,296 and 18,416 were an amicable pair; Descartes found that 9,363,584 and 9,437,056 were. Both pairs were known to Arab mathematicians already. Amicable pairs are easy enough to produce. From the tenth century we’ve had Thâbit ibn Kurrah’s rule, which lets you generate sets of numbers. Ruthie wasn’t thinking of any of this, though, and was more thinking how much fun it is to write a 7.

    Terry Border’s Bent Objects for the 1st just missed the anniversary of John Venn’s birthday and all the joke Venn Diagrams that were going around at least if your social media universe looks anything like mine.

    Jon Rosenberg’s Scenes from a Multiverse for the 1st is set in “Mathpinion City”, in the “Numerically Flexible Zones”. And I appreciate it’s a joke about the politicization of science. But science and mathematics are human activities. They are culturally dependent. And especially at the dawn of a new field of study there will be long and bitter disputes about what basic terms should mean. It’s absurd for us to think that the question of whether 1 + 1 should equal 2 or 3 could even arise.

    But we think that because we have absorbed ideas about what we mean by ‘1’, ‘2’, ‘3’, ‘plus’, and ‘equals’ that settle the question. There was, if I understand my mathematics history right — and I’m not happy with my reading on this — a period in which it was debated whether negative numbers should be considered as less than or greater than the positive numbers. Absurd? Thermodynamics allows for the existence of negative temperatures, and those represent extremely high-energy states, things that are hotter than positive temperatures. A thing may get hotter, from 1 Kelvin to 4 Kelvin to a million Kelvin to infinitely many Kelvin to -1000 Kelvin to -6 Kelvin. If there are intuition-defying things to consider about “negative six” then we should at least be open to the proposition that the universal truths of mathematics are understood by subjective processes.

  • Joseph Nebus 6:00 pm on Saturday, 6 August, 2016 Permalink | Reply
    Tags: , , , themes, ,   

    How July 2016 Treated My Mathematics Blog 

    I’m not unhappy. Of course not; I can find something cheery to say about whatever my readership in a given month was like. But for a month in which I spent nearly two weeks away from my normal Internet routines of visiting blog friends and belatedly answering comments and the like it wasn’t bad at all.

    Readership Numbers:

    So there were 1,057 page views in July. That’s down from June’s 1,099, but only a touch, and it’s up from May’s 981. And it’s above a thousand which makes me feel secure about being at least tolerated in these parts. The number of unique visitors was down to 585 from June’s 598 and May’s 627. But the June-to-July drop I can’t imagine is significant.

    The number of likes rose to 177, from June’s 155 and May’s 133. I can’t hide it: I’m hoping for 199 in August and I don’t know where it’ll go from there. Comments were down a touch to 33, from June’s 39. But some of that is my failing to respond to other people because I was away. My own comments should count, shouldn’t they?

    I am considering making one of those big changes and switching away from the theme — “P2 Classic” — that I have. I like its look, especially that it lets comments appear on the front page around here. But I’ve realized that the theme is a disaster on mobile devices. I don’t want to be needlessly difficult.

    At the top of my WordPress theme is a box saying, 'Hi, Joseph. Whatcha up to?' encouraging me to make quick little informal posts which I never ever do.

    I don’t know, worrying about what I should post? I’m sorry, I can’t use a slangy informal posting mechanism like this. I’m far too pompous. Also you have no idea how disorienting it is to have this image on my page.

    Also while it’s got a nice friendly “Whatcha up to?” panel up top for me, to quickly add a post, I have never used it except when I wanted to search for something and the cursor was in the wrong field. If someone knows of an updated P2 Classic that you can read on a hand phone please let me know. I’d be glad for it.

    Popular Posts:

    To posts! The most popular stuff around here in July was a fair split between Reading the Comics posts and Theorem Thursday posts, plus a note that something I started back in May would too be returning. I hope to get to that soon again, maybe this week. That’s also comforting. They’re the things I put the most effort into and I’m glad people like them and don’t find much terribly wrong about them. The top five articles in July according to WordPress were:

    Listing Countries:

    What countries like me? … You know what? Bullet lists are so reportedly popular I’ll just try listing everybody and we’ll see what that does for drumming up interest. Readership by country, per WordPress’s data, were:

    Country Readers
    United States 616
    Canada 57
    India 52
    United Kingdom 36
    Philippines 30
    Australia 27
    Germany 26
    Slovenia 22
    Singapore 20
    Austria 15
    Brazil 15
    Spain 13
    Thailand 11
    Pakistan 10
    Puerto Rico 7
    Indonesia 6
    Ireland 6
    Italy 6
    Croatia 5
    France 5
    Hong Kong SAR China 5
    New Zealand 5
    Sweden 5
    China 4
    Mexico 4
    South Korea 4
    Finland 3
    Greece 3
    Portugal 3
    Russia 3
    Venezuela 3
    Argentina 2
    Czech Republic 2
    European Union 2
    Jordan 2
    Netherlands 2
    Norway 2
    South Africa 2
    United Arab Emirates 2
    Belgium 1
    Chile 1
    Denmark 1
    Dominican Republic 1
    Ecuador 1
    Latvia 1
    Lithuania 1
    Malaysia 1
    Oman 1
    Saudi Arabia 1
    Serbia 1
    Tunisia 1
    Turkey 1
    Ukraine 1 (*)

    Ukraine is the only country to have been a single-reader country in June too. This is the nearest clean sweep I’ve noticed. The European Union reader, after seven months being alone, found a friend too. I hope they get along.

    Search Term Non-Poetry:

    Whew. It’s back.

    • origin is the gateway to your entire gaming universe.
    • what is the average number of grooves on one side of an lp record (if “1” doesn’t satisfy you)
    • arithmetic sequences and series joke 48 (the punch line I’d heard was “why did they laugh so much at 15,268?” “Well, you see, we’d never heard that one before!”)
    • example of convergent boundaries komiks stris (honestly now tempted to commission a comic strip artist just to do something about convergent boundaries.)
    • comics about arithmetic sequence / arithmetic sequence comics (probably I should also commission one about sequences)

    Counting Readers:

    If I have this right August started with the blog having had 39,394 page views — curse that leap second! — and 16,083 unique viewers. (Because the leap second would give time for one more page view, keeping me from 39,393. If there were a leap second, and if it were at the end of July instead of the end of June. Trust me, if you share a long sequence of assumptions with me then it’s funny.)

    WordPress reports me as starting with 610 followers, which feels way up from the start of July’s 597. If you want to join me as a follower there ought to be a button in the upper-right corner, a bit below and to the right of my blog name and above the “Or Follow By Way Of RSS” tag. There’s also a Follow Blog Via Email option and don’t think it doesn’t bother me there’s no dash in E-mail there. More reasons to change the theme I suppose.

    I’d wondered last month about WordPress reporting the most popular dates and times around here. So that’s why I moved my default posting time from 11 am Eastern to 2 pm Eastern. But just as in July the most popular day is Sunday (22 percent of page views). Comics posts I suppose. The most popular hour remains 3:00 pm (9 percent of page views). It kind of suggests the time of posting doesn’t matter to people. We’ll see, as I start trying 6 am or if I try something really wild like eleventy-q pm.

    See you, I expect, tomorrow with comic strips.

    • mathtuition88 7:34 am on Sunday, 7 August, 2016 Permalink | Reply

      Congrats for your increase in views! Math bloggers have a tougher time getting views than say, food bloggers. My most popular posts ironically have the least mathematical content..

      Liked by 1 person

      • Joseph Nebus 7:44 pm on Tuesday, 9 August, 2016 Permalink | Reply

        Thank you. Yeah, mathematics has a tougher time getting readers. Not enough pictures, at least when you get away from strange topological constructs. This is surely why Baking And Math is doing well, or ought to be.

        There’s really no guessing what’s going to be popular. It usually turns out to be a trifle, and something with a slight but humiliating-to-yourself error in it.

        Liked by 1 person

    • breathmath 12:52 pm on Sunday, 7 August, 2016 Permalink | Reply

      Yep! Owning educational sites and getting views/getting unique visitors of minimum 250/day is tough tie! My highest count of unique visitors was 189.
      Hoping for the best :) Let’s grow together :) All the best.. keep posting!


      • Joseph Nebus 7:52 pm on Tuesday, 9 August, 2016 Permalink | Reply

        Oh, I don’t even know what my highest visitor count on a day was. It would have been in November of last year, though, when I got a lot of spillover curiosity from visitors to my humor blog, which was covering the bizarre collapse of the comic strip Apartment 3-G. And, well thank you, and I hope you enjoy good posting and good reading too.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Thursday, 4 August, 2016 Permalink | Reply
    Tags: reviews, ,   

    What Did I Post On Theorem Thursdays? 

    Hi again folks. I don’t want you to think I forgot about my little blog here. I’ve just been off for a pretty major competitive-pinball event the past week and that and a visit to the Kennywood amusement park slurped up all my writing time. Should be recovered soon enough.

    I’ll be back to posting original stuff soon enough, and to posting links to other people’s stuff. But for now I wanted to gather links to all the Theorem Thursday now that the project’s safely at its pre-announced conclusion. I’ve have thoughts about what it all meant soon, too.

    Over June and July I put up rather extended posts about:

    I do figure on returning to these long-form explanations of theorems, so that Mean Value Theorem and Fixed Point Theorem stuff shouldn’t be left dangling forever. I don’t know just when I will, though. I’ll discuss why that is in my “what-did-I-learn” post, when I have the chance.

    • vagabondurges 4:57 pm on Friday, 5 August, 2016 Permalink | Reply

      So good! I’m torn between wanting to hear more about major pinball competitions and how you picked a fight with all of New England… One thing at a time.


      • Joseph Nebus 7:41 pm on Tuesday, 9 August, 2016 Permalink | Reply

        Well, thank you. The New England thing is just all the jokes I tossed off along the way to explaining the five-color map theorem. There’s a lot of contended boundaries there and erasing them for one reason or another is good for stirring up trouble. (I’m not above this sort of thing. I’m from New Jersey, so there is a very long and rather sad history of fights over the state’s basic dignity, mostly against New York. But there’s room for other squabbles, as seen by the land border Delaware has the gall to claim from the Garden State.)

        I’d love to talk more about the mathematics of pinball competitions although I’ve already used the best discussion topic, detailed balance. There’s probably more, though.


  • Joseph Nebus 6:00 pm on Sunday, 31 July, 2016 Permalink | Reply
    Tags: , , electronics, , mnemonics, rock-paper-scissors   

    Reading the Comics, July 30, 2016: Learning Tools Edition 

    I thank Comic Strip Master Command for the steady pace of mathematically-themed comics this past week. It fit quite nicely with my schedule, which you might get hints about in weeks to come. Depends what I remember to write about. I did have to search a while for any unifying motif of this set. The idea of stuff you use to help learn turned up several times over, and that will do.

    Steve Breen and Mike Thompson’s Grand Avenue threatened on the 24th to resume my least-liked part of reading comics for mathematics themes. This would be Grandma’s habit of forcing the kids to spend their last month of summer vacation doing arithmetic drills. I won’t say that computing numbers isn’t fun because I know what it’s like to work out how many seconds are in 50 years in your head. But that’s never what this sort of drill is about. The strip’s diverted from that subject, but it might come back to spoil the end of summer vacation. (I’m not positive what my least-liked part of the comics overall is. I suspect it might be the weird anti-participation-trophy bias comic strip writers have.)

    Ryan North’s Dinosaur Comics reprint for the 25th is about the end of the universe. We’ve got several competing theories about how the universe is likely to turn out, several trillion years down the road. The difference between them is in the shape of space and how that shape is changing. I’ve mentioned sometimes the wonder of being able to tell something about a whole shape from local information, things we can tell without being far from a single point. The fate of the universe must be the greatest example of this. Considering how large the universe is and how little of it we will ever be able to send an instrument to, we measure the shape of space from a single point. And we can realistically project what will happen in unimaginably distant times. Admittedly, if we get it wrong, we’ll never know, which takes off some of the edge.

    Dinosaur Comics reappears the 28th with some talk about number bases. It’s all fine and accurate enough, except for the suggestion that anyone would use base five for something other than explaining how bases work. I like learning about bases. When I was a kid this concept explained much to me about how our symbols for numbers work. It also helped appreciate that symbols are not these fixed or universal things. They’re our creations and ours to adapt for whatever reason we find convenient. In the past we’ve found bases as high as sixty to be convenient. (The division of angles into 360 degrees each of 60 minutes, each of those of 60 seconds, is an echo of that.) But when I was a kid doing alternate-base problems nobody knew what I was doing or why, except the mathematics teacher who said I might like the optional sections in the book. We only really need base ten, base two, and base sixteen, which might as well be base two written more compactly. The rest are toys, good for instruction and for fun. Sorry, base seven.

    Scott Meyer’s Basic Instructions rerun for the 27th is about everyone’s favorite bit of intransitivity. Rock-Paper-Scissors and its related games are all about systems in which any two results can be decisive but any three might not be. This prospect turns up whenever there are three or more possible outcomes. And it doesn’t require a system to be irrational or random. Chaos and counterintuitive results just happen when there’s three of a thing.

    I remember, and possibly you remember too, learning of a computer system that can consistently beat humans at Rock-Paper-Scissors. It manages to do that by the oldest of game theory exploits, cheating. Its sensors look for the twitches suggesting what a person is going to throw and then it changes its throw to beat that. I don’t know what that’s supposed to prove since anyone who’s played a Sid Meier’s Civilization game knows that computers already know how to cheat.

    Thom Bluemel’s Birdbrains, yes, you can be in my Reading The Comics post this week too. Don’t beg.

    Bill Schorr’s The Grizzwells for the 28th is a resisted word problem joke. It doesn’t use the classic railroad or airplane forms, but it’s the same joke anyway.

    'My drive to solve this [ a pie chart of the relationships between current, voltage, resistance, and power in an electrical circuit ] keeps getting interrupted by my desire to order a PIZZA'.

    Benita Epstein’s Six Chix for the 29th of July, 2016. The pie chart’s valid, in case you need it, in which case you’re doing the mathematics of electric circuits. Current, voltage, resistance, and power all relate to one another in ways the chart makes clear once you know how to read it. Each of the quantities — I, V, R, or P — is equal to each of the expressions outside it. Pizza, meanwhile, is just a naturally funny word and thus appears in comic strips to amuse you.

    Benita Epstein’s Six Chix for the 29th is probably familiar to the folks taking electronics. The chart is a compact map used as a mnemonic for the different relationships between the current (I), the voltage (V), the resistance (R), and the power (P) in a circuit. When I was a student we got this as two separate circles, one for current-voltage-resistance and one for power-current-voltage. Each was laid out like the T-and-O maps which pre-Renaissance Western Europe used to diagram the world. While I now see that as a convenient and useful tool, as a student, I was skeptical that it was any easier to use the mnemonic aid than it was to just remember “voltage equals current times resistance” and “power equals voltage times current”. I’ve always had an irrational suspicion of mnemonic devices. I’m trying to do better.

    Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 30th is a return of the whiteboard full of symbols to represent deep thinking. The symbols don’t mean anything as equations, though that might be my limited perspective. And that also might represent the sketchy, shorthand way serious work is done. As an idea is sketched out weird bundles of symbols that don’t literally parse do appear. In a publishable paper this is all turned into neatly formatted and standard stuff. Or we introduce symbols with clear explanations of what they mean so that others can learn to read what we write. But for ourselves, in the heat of work, we’ll produce what looks like gibberish to others and that’s all right as long as we don’t forget what the gibberish means. Sometimes we do, but the gibberish typically helps us recapture a lost idea. (I offer the tale of a mathematician with pages of notes for a brilliant insight which she has to reconstruct from a lost memory to would-be short story writers looking for a Romantic hook.)

    • tziviaeadler 9:43 pm on Tuesday, 2 August, 2016 Permalink | Reply

      That image is worse than Venn diagrams for making someone want pizza.
      At least i had some pie on pi day :)


      • Joseph Nebus 7:34 pm on Tuesday, 9 August, 2016 Permalink | Reply

        Ah, well, good. I’m glad you could have some pie. I’ve refrained lately — watching my weight as it goes up against what I wanted to be a hard boundary — but I know that’s just a prelude to me inhaling the Chinese buffet sometime soon.


  • Joseph Nebus 6:00 pm on Thursday, 28 July, 2016 Permalink | Reply
    Tags: , , , , , matchings, ,   

    Theorem Thursday: Tutte’s Theorem, Magic, And Happy Endings 

    To wrap up my two-month Theorem Thursday project I have another request to fill. And it’s another from Gaurish, whose requests for the Leap Day A To Z inspired so many interesting posts.

    Tutte’s Theorem

    I admit I had not heard of Tutte’s Theorem before Gaurish’s request and I had to spend time working up to knowing what it was and why it was useful. I also admit I’m not completely sure I have it. But I’m happy to try to accept with grace corrections from the people in graph theory who know better.

    It comes back to graphs. These are a bunch of points, “vertices”, which have curves, “edges” connecting pairs of them. This describes a lot of systems. Bunches of things that have to connect are naturally graphs. Connecting utilities to houses gave the first example I called up last week. The Internet’s made people vaguely familiar with the idea of bunches of computers linked to bunches of other computers. Social networks can be seen as graphs; each person is a vertex, her friendships edges connecting to other vertices.

    In a graph not every vertex has to be connected to every other vertex. Not everyone needs to be friends with everyone else, which is a relief. I don’t want to point fingers but some of your friends are insufferable. I don’t know how you put up with them. You, you’re great. But the friends of a friend … yeeeeeeesh.

    So we — mathematicians, anyway — get to wondering. Give me a graph. It’s got some vertices and some edges. Let’s look at only some of these edges. Each edge links to two vertices. Is there a subset of the edges that touch every one of the vertices exactly once? Sometimes there are; sometimes there aren’t. If there are, we have a “perfect matching”.

    We look at this sort of thing because mathematicians learn to look at coverings. Coverings are what they sound like. What’s the smallest set of some standard item you need to … cover … all the stuff you’re interested in this problem? I think we’re bred to look for coverings in Real Analysis, because the idea of covering space with discs gives us measure. This gives us a rigorous idea of what length is, and what dimensions are, and lets us see there have to be more irrational than rational numbers and other interesting results like that. Mathematicians get used to looking for this sort of thing in other fields.

    Tutte’s Theorem is about perfect matchings. It says what conditions a graph has to have to have a perfect matching. It’s based on striking subsets of the vertices from the original graph. The accompanying edges go too. What’s left might be connected, which means just what you think. You can get from any vertex in the decimated graph to any other vertex by following its edges. Or it might be disconnected, which means again just what you think. Mathematics is not always about complicated lingo.

    Take the survivors. Count how many of the remaining connected components have an odd number of vertices. Is that less than or equal to the number of vertices you struck out? If it is, and if it is no matter how many vertices you struck out, and no matter how you arranged those vertices, then there’s a perfect matching.

    This isn’t one for testing. I mean, consider what’s easier to do: start from your original graph and test out coloring in some edges to see if you can touch every edge the one time? Take every possible sub-graph of the original graph and count the number of connected sub-pieces with an odd number of vertices? … Well, maybe that isn’t so bad, if you set a computer to do the boring work. It’s going to take forever for your graph with 102 vertices, though, unless it’s a boring network. You have to test every possible removal all the way up to striking 102 vertices. (OK, it’s easy to show it’s true if 102 of 102 vertices are removed from the graph. Also if 101 of 102 vertices are removed. And 100 of 102 is also not hard. But that’s only a couple easy cases left.)

    I don’t know the real work the theorem does. It has some neat and implications good for pop mathematics. Take a standard deck of 52 well-shuffled cards. Deal them out into thirteen piles of four cards each. Is it possible to select exactly one card from each pile so that, when done, there’s exactly one of each face value — Ace, Two, Three, Four, up through Queen and King — in the selected set? Indeed it is. I leave it to the Magic In A Minute cartoonists to turn this into a way to annoy a compliant monkey.

    Another appealing use for it is in marriage problems. Well, marriage looked to mathematicians like a good source of bipartite graph problems. Society’s since noticed that nobody really hit on a compelling reason for “why does a woman have to marry a man, exactly”. I imagine the change to filter into mathematics textbooks sometime in the next seventy years. But please accept that this problem was formed and solved in the 1930s.

    Suppose you have a group of women and a group of men, of equal size. Each woman would be happy married to some of the men in that group. Each man would be happy married to some of the women in that group. Is it possible to match women and men up in a way that everybody is married to at least someone they’d be happy with? We’re not guaranteeing that anyone gets their best possible pairing. We promise just that everyone marries someone they’ll be glad to.

    It depends. Maybe this is best done by testing. Work through every possible subset of women. That is, look at every group of one woman, every group of two women, every group of three women, and so on. By group I mean what normal people mean by group, not what mathematicians mean. So look at your group of women. Count how many of the men at least one woman would be content marrying. Is that number equal to or larger than the number of women? Is that number equal to or larger than the number of women, however many women you picked and whichever women you did pick? If it did, great: everybody can marry someone they’re happy with.

    Parlor tricks, I admit, but pleasant ones. What are its real uses? At this point I am really synthesizing my readings on the subject rather than speaking from things I’m confident about. If I understand right, though, Tutte’s Theorem is a foundation for the study of graph toughness. That is what you’d guess from the name. It’s about how easy it is to break up a graph into disconnected pieces. It’s easy to imagine real networks with strength or weakness. Image the person that holds a complicated group of friends together, for example, and whose removal devastates it. Consider the electrical network with a few vulnerable points that allow small problems to become nationwide blackouts. I believe it’s relevant to the study of proteins. Those long strands of blocks of molecules that then fold back on themselves. (I haven’t seen a source that says this, but it can’t imagine why it shouldn’t. I am open to correction from sneering protein researchers.) I would be surprised if the theorem has nothing to say about when a strand of Christmas tree lights will get too tangled to fix.

    Let me close with a puzzle, a delightful little one. It regards one of the graphs we met last week. K5 is a complete graph with five vertices. Each vertex is connected to all four of its siblings. It can’t have a perfect matching. Only a graph with an even number of vertices can. Each edge connects to two vertices, after all. So — what is the subset that breaks Tutte’s theorem? It must be possible to strike some set of vertices from K5 so that the number of stricken vertices is smaller than the number of remaining connected components with an odd number of vertices. What’s that set?

    Go ahead and ponder it a while. If you give up I don’t blame you. The answer is at the other end of this link. If you’d like a hint, let me offer this, which you might be able to mouse over to reveal.

    It is obvious once you’ve heard what the subset of vertices is, and it is not K5. The rest of this paragraph is padding so that the size of the spoiler doesn’t give matters away. And by the way I’d like to thank the world at large for doing such a great job not spoiling Star Wars: The Force Awakens. So why could you not give some similar consideration for Star Trek Beyond? I stopped reading TrekBBS for a month partly to avoid spoilers and then I started getting them everywhere I looked. Not cool, world.

    Good luck!

    • gaurish 3:27 pm on Friday, 29 July, 2016 Permalink | Reply

      Thanks! I wish my teacher could present the topics like you.


      • Joseph Nebus 7:48 am on Sunday, 31 July, 2016 Permalink | Reply

        You’re quite kind, and I’m glad I can give you something satisfying.

        I do have an advantage over a teacher in this, though. I haven’t got an obligation to explain reasons and processes in enough detail that you could recreate them. That’s difficult stuff to make lighthearted and entertaining. I can choose to skip as much instructional text as I want, and keep to just what entertains me or what I think will entertain readers. So it’d be strange if I weren’t a more fun presentation.


  • Joseph Nebus 6:00 pm on Tuesday, 26 July, 2016 Permalink | Reply
    Tags: Plinko, ,   

    Something To Read: Galton Boards 

    I do need to take another light week of writing I’m afraid. There’ll be the Theorem Thursday post and all that. But today I’d like to point over to Gaurish4Math’s WordPress Blog, and a discussion of the Galton Board. I’m not familiar with it by that name, but it is a very familiar concept. You see it as Plinko boards on The Price Is Right and as a Boardwalk or amusement-park game. Set an array of pins on a vertical board and drop a ball or a round chip or something that can spin around freely on it. Where will it fall?

    It’s random luck, it seems. At least it is incredibly hard to predict where, underneath all the pins, the ball will come to rest. Some of that is ignorance: we just don’t know the weight distribution of the ball, the exact way it’s dropped, the precise spacing of pins well enough to predict it all. We don’t care enough to do that. But some of it is real randomness. Ideally we make the ball bounce so many times that however well we estimated its drop, the tiny discrepancy between where the ball is and where we predict it is, and where it is going and where we predict it is going, will grow larger than the Plinko board and our prediction will be meaningless.

    (I am not sure that this literally happens. It is possible, though. It seems more likely the more rows of pins there are on the board. But I don’t know how tall a board really needs to be to be a chaotic system, deterministic but unpredictable.)

    But here is the wonder. We cannot predict what any ball will do. But we can predict something about what every ball will do, if we have enormously many of them. Gaurish writes some about the logic of why that is, and the theorems in probability that tell us why that should be so.

    • gaurish 6:11 pm on Tuesday, 26 July, 2016 Permalink | Reply

      Thanks for pointing to my blog post. I would like to quote Tim Gowers (A very short introduction to Mathematics, pp. 6) regarding the classical die throwing experiment (“the model”) of probability theory:
      “One might object to this model on the grounds that the dice, when rolled, are obeying Newton’s laws, at least to a very high degree of precision, so the way they land is anything but random…”


      • Joseph Nebus 6:02 am on Wednesday, 27 July, 2016 Permalink | Reply

        Quite welcome. I’m happy to pass along interesting writing.

        Granted that falling dice, or balls in a Plinko board like this, are moving deterministically. I do wonder if we get to chaotic behavior, in which the toss is nevertheless random. I’m not well-versed enough in the mechanics of this sort of problem to be really sure about my answer. For the balls falling off pins I would imagine that something like twenty rebounds, on either pin or other balls, would be enough to effectively randomize the result.

        (If each rebound doubles the discrepancy between the direction of the ball’s actual velocity and our representation of its direction, then after twenty rebounds the error is about a million times what it started as, and it seems hard to know the direction of a ball’s travel to within a millionth of two-pi radians. But that’s a very rough argument, supposing that randomizing the direction of travel is all we need to have a random ball drop. And maybe two-pi-over-a-million radians is a reasonable precision; maybe we need thirty rebounds, or forty, to be quite sure.)

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Sunday, 24 July, 2016 Permalink | Reply
    Tags: , , ,   

    Reading the Comics, July 23, 2016: Familiar Friends Week Edition 

    This past week was refreshing. The mathematics comics appeared at a regular, none-too-excessive pace. And some old familiar friends reappeared. Some were comic strips that haven’t been around in a while. Some were jokes that haven’t been. Enjoy.

    Bill Whitehead’s Free Range for the 17th is the first use of the meth/math lab pun to appear in the comics since September 2014 by my reckoning. And only the second in my Reading the Comics series. I’m surprised too. For all this goes around Twitter and other social media I’d imagine it to make the comics more.

    Scott Hilburn’s The Argyle Sweater gets back in my review here for the first time since April, to my amazement. Used to be you couldn’t go two weeks without Hilburn looking for my attention. And here’s the first Roman Numerals joke since … I don’t quite feel up to checking just now. I’m going to go ahead and suppose it’s the first one since the last time Samson’s Dark Side Of The Horse was here.

    It’s anachronistic to speak of Ancient Roman students getting ‘C’ grades. Of course it is; it could hardly be otherwise. It’s a joke; how much is that to be worried about? But if I haven’t been mislead the use of letters, A through E-or-F, in student evaluations is an American innovation of the late 19th century. It developed over the 20th century and took over at least American education, in conjunction with the 100-through-0 points evaluation scale. And in parallel to the Grade Point Average, typically with 4.0 as its highest score.

    Samson’s Dark Side Of The Horse makes a comfortable visit back here on the 20th. It’s another counting-sheep and number-representation gag. I love the third panel’s artwork.

    Mark Anderson’s Andertoons for the 22nd is a joke about motivating mathematics study. I believe I’ve mentioned this before, but there was a lovely bit on The Mary Tyler Moore Show along these lines once. Fantastically stupid newsman Ted Baxter was struggling to do some arithmetic until Murray Slaughter gave him the advice: “put a dollar sign in front of it”. Then he had the answer instantly.

    Nat Fakes’s Break of Day for the 22nd brings back mathematics as signifier of the hardest homework a kid can have, or the toughest thing someone can have to think about. Fine enough stuff, although it isn’t really that stunning to think a parent might not understand what the kid’s homework is about. Often the point of an assignment is not to learn how to do something, but to encourage thinking about ways one could do something. That’s a hard assignment to create, and a harder one to do, and a very hard one to help with. As adults we get used to looking at problems as calculations to identify and do as swiftly as possible. That there is value in wandering around the slow routes needs remembering.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 22nd riffs on … I’m not sure exactly. The idea that the sort of meaningless nonsense that makes for good late-night dorm conversations when you’re 20 comes back around to being the cutting edge of theoretical physics, I suppose. It’s funny enough. A complaint often brought against the most cutting edge of theoretical physics is that it’s so abstract that there aren’t any conceivable tests that would say whether a calculation is right or not. In that condition mathematics and theoretical physics merge back into a thread of philosophy and its question of how can we know what it is for something to be true. Once we have a way of discerning whether an idea might happen to be true we’re ejected again from philosophy and into a science. And then the scientist makes a smug, snarky comment about the impossibility of testing philosophical conclusions.

    Since the late 19th century much cutting-edge physics has involved counter-intuitive results. Often they have premises that strain intuition, as see relativity, or that seem to violate it altogether, as see quantum mechanics. But they turn out so very right so very often it’s hard not to feel excited and encouraged by this. Who wouldn’t look for a surprising and counter-intuitive explanation for the world as thrilling and maybe even right idea? I don’t blame anyone for looking to a wild idea like “what if the universe is made of math”. I don’t know what that would mean exactly unless we suppose we do live in a universe of Platonic Forms, in which case perfection runs counter-intuitively to me. I do understand being excited by the question. But the answers probably won’t be that much fun.

  • Joseph Nebus 6:00 pm on Saturday, 23 July, 2016 Permalink | Reply
    Tags: , derivations,   

    Anatomizing An Error 

    Though it’s the summer months I’m happy to say the Carnot Cycle thermodynamics blog is still posting. He had been writing about Jacobus Henricus van ‘t Hoff, first recipient of the Nobel Prize in Chemistry. In the 1880s van ‘t Hoff was studying the osmosis. In April’s essay Carnot Cycle described the problem, and how van ‘t Hoff passed up a correct formula describing osmotic pressure in favor of an attractive but wrong alternative.

    In this month’s essay Carnot Cycle continues the topic. It particularly goes over just how van ‘t Hoff got to his mistaken idea. It’s not that he started out wrong. He began from a good start and derived a mistaken formula. The derivation involved a string of assumptions and simplifications and approximations, of the kind that must be made to go from starting principles to a specific problem. He was guided by an idea of what the answer ought to look like, though, and that led him astray. The blog describes what he did and why it would look reasonable in the circumstance. It’s worth reading to see what actual mathematics, the kind that doesn’t have known answers, is like.

    • davekingsbury 11:20 pm on Sunday, 24 July, 2016 Permalink | Reply

      So mathematics is essentially exploration of the unknown?


      • Joseph Nebus 4:36 am on Tuesday, 26 July, 2016 Permalink | Reply

        I’m not sure exactly how I would describe mathematics. Part of it does feel like an exploration of the unknown: we set out basic rules and find implications that aren’t obvious. A lot of work does feel like experimentation and discovery, just as one might do in a science. But it does seem bizarre to imagine that the logical consequences of our chosen premises are unknown; it seems like saying that a chess move might need discovery. I’m not sure how to represent it all. Possibly there’s no representing it all as one thing; there are several strands of thought that run through mathematics, I believe.

        Liked by 1 person

        • davekingsbury 7:24 am on Tuesday, 26 July, 2016 Permalink | Reply

          Thank you for this honest and lucid response. Strikes me it’s a language which avoids the pitfalls of imprecision and emotionality.


          • Joseph Nebus 5:54 am on Wednesday, 27 July, 2016 Permalink | Reply

            I think avoiding imprecision and emotionality are considered ideals, yes. And a fully mature, cleaned-up mathematical field has got its important work set up and defined in ways that are precise and avoid emotional appeals. But when working out a problem, especially a new and exciting one, there are many provisional definitions and ambiguities discovered late in the paper and all that. Mathematicians are humans and their lives are all over their work, necessarily. We try to look good when strangers peek in, which is again a most human thing to do.


    • davekingsbury 8:15 am on Wednesday, 27 July, 2016 Permalink | Reply

      Thanks for humanising the world of mathematics for me. You have the skills of a natural teacher.


  • Joseph Nebus 6:00 pm on Thursday, 21 July, 2016 Permalink | Reply
    Tags: , , , , , utilities   

    Theorem Thursday: Kuratowski’s Reduction Theorem and Playing With Gas Pipelines 

    I’m doing that thing again. Sometime for my A To Z posts I’ve used one essay to explain something, but also to introduce ideas and jargon that will come up later. Here, I want to do a theorem in graph theory. But I don’t want to overload one essay. So I’m going to do a theorem that’s interesting and neat in itself. But my real interest in next week’s piece. This is just so recurring readers are ready for it.

    The Kuratowski Reduction Theorem.

    It starts with a children’s activity book-type puzzle. A lot of real mathematics does. In the traditional form that I have a faint memory of ever actually seeing it’s posed as a problem of hooking up utilities to houses. There are three utilities, usually gas, water, and electricity. There are three houses. Is it possible to connect pipes from each of the three utility starting points to each of the three houses?

    Of course. We do it all the time. We do this with many more utilities and many more than three buildings. The underground of Manhattan island is a network of pipes and tunnels and subways and roads and basements and buildings so complicated I doubt humans can understand it all. But the problem isn’t about that. The problem is about connecting these pipes all in the same plane, by drawing lines on a sheet of paper, without ever going into the third dimension. Nor making that little semicircular hop that denotes one line going over the other.

    That’s a little harder. By that I mean it’s impossible. You can try and it’s fun to try a while. Draw three dots that are the houses and three dots that are the utilities. Try drawing three lines, one from each utility to each of the houses. Or one leading into each house that comes from each of the utilities. The lines don’t have to be straight. They can have extra jogs, too. Soon you’ll hit on the possibilities of lines that go way out, away from the dots, in the quest to avoid crossing over one another. It doesn’t matter. The attempt’s doomed to failure.

    You’ll be sure of this by at latest the twelfth attempt at finding an arrangement. But that leaves open the possibility you weren’t clever enough to find an arrangement. To close that possibility guess what theorem is sitting there ready to answer your question, just like I told you it would be?

    This is a problem in graph theory. I’ve talked about graph theory before. It’s the field of mathematics most comfortable to people who like doodling. A graph is a bunch of points, which we call vertices, connected by arcs or lines, which we call edges. For this utilities graph problem, the houses are the vertices. The pipes are the edges. An edge has to start at one vertex and end at a vertex. These may be the same vertex. We’re not judging. A vertex can have one edge connecting it to something else, or two edges, or three edges. It can have no edges. It can have any number of edges. We’re even allowed to have two or more edges connecting a vertex to the same vertex. My experience is we think of that last, forgetting that it is a possibility, but it’s there.

    This is a “nonplanar” graph. This means you can’t draw it in a plane, like a sheet of paper, without having at least two edges crossing each other. We draw this on paper by making one of the lines wiggle in a little half-circle to show it’s going over, or to fade out and back in again to show it’s going under. There are planar graphs. Try the same problem with two houses and two utilities, for example. Or three houses and two utilities. Or three houses and three utilities, but one of the houses doesn’t get one of the utilities. Your choice which. It can be a little surprise to the homeowners.

    This utilities graph is an example of a “bipartite” graph. The “bi” maybe suggests where things are going. You can always divide the vertices in a graph into two groups for the same reason you can always divide a pile of change into two piles. As long as you have at least two vertices or pieces of change. But a graph is bipartite if, once you’ve divided the vertices up, each edge has one vertex in the first set and the other vertex in the second. For the utilities graph these sets are easy to find. Each edge, each pipe, connects one utility to one house. There’s our division: vertices representing houses and vertices representing utilities.

    This graph turns up a lot. Graph theorists have a shorthand way of writing it. It’s written as K3,3. This means it’s a bipartite graph. It has three vertices in the first set. There’s three vertices in the second set. There’s an edge connecting everything in the first set to everything in the second. Go ahead now and guess what K2, 2 is. Or K3,5. The K — I’ve never heard what the K stands for, although while writing this essay I started to wonder if it’s for “Kuratowski”. That seems too organized, somehow.

    Not every graph is bipartite. You might say “of course; why else would we have a name `bipartite’ if there weren’t such a thing as `non-bipartite’?” Well, we have the name “graph” for everything that’s a graph in graph theory. But there are non-bipartite graphs. They just don’t look like the utility-graph problem. Imagine three vertices, each of them connected to the other two. If you aren’t imagining a triangle you’re overthinking this. But this is a perfectly good non-bipartite graph. There’s no way to split the vertices into two sets with every edge connecting something in one set to something in the other. No, that isn’t inevitable once you have an odd number of vertices. Look above at the utilities problem where there’s three houses and two utilities. That’s nice and bipartite.

    Non-bipartite graphs can be planar. The one with three vertices, each connected to each other, is. The one with four vertices, each vertex connected to each other, is also planar. But if you have five vertices, each connected to each other — well, that’s a lovely star-in-pentagon shape. It’s also not planar. There’s no connecting each vertex to each other one without some line crossing another or jumping out of the plane.

    This shape, five vertices each connected to one another, shows up a lot too. And it has a shorthand notation. It’s K5. That is, it’s five points, all connected to each other. This makes it a “complete” graph: every set of two vertices has an edge connecting them. If you’ve leapt to the supposition that K3 is that circle and K4 is that square with diagonals drawn in you’re right. K6 is six vertices, each one connected to the five other vertices.

    It may seem intolerably confusing that we have two kinds of graphs and describe them both with K and a subscript. But they’re completely different. The bipartite graphs have a subscript that’s two numbers separated by a comma: p, q. The p is the number of vertices in one of the subsets. The q is the number of vertices in the other subset. There’s an edge connecting every point in p to every point in q, and vice-versa. The points in the p subset aren’t connected to one another, though. And the points in the q subset aren’t connected to one another. That they don’t mean this isn’t a complete graph.

    The others, K with a single number r in the subscript, are complete graphs, ones that aren’t bipartite. They have r vertices, and each vertex is connected to the (r – 1) other vertices. So there’s (1/2) times r times (r – 1) edges all told.

    Not every graph is either Kp, q or Kr. There’s a lot of kinds of graphs out there. Some are planar, some are not. But here’s an amazing thing, and it’s Kuratowski’s Reduction Theorem. If a graph is not planar, then it has to have, somewhere inside it, K3, 3 or K5 or both. Maybe several of them.

    A graph that’s hidden within another is called a “subgraph”. This follows the same etymological reasoning that gives us “subsets” and “subgroups” and many other mathematics words beginning with “sub”. And these subgraphs turn up whenever you have a nonplanar graph. A subgraph uses some set of the vertices and edges of the original graph; it doesn’t need all of them. A nonplanar graph has a subgraph that’s K3, 3 or K5 or both.

    Sometimes it’s easy to find one of these. K4, 4 obviously has K3, 3 inside it. Pick three of the four vertices on one side and three of the four vertices on the other, and look at the edges connecting them up. There’s your K3, 3. Or on the other side, K6 obviously has K5 inside it. Pick any five of the vertices inside K6 and the edges connecting those vertices. There’s your K5.

    Sometimes it’s hard to find one of these. We can make a graph look more complicated without changing whether it’s planar or not. Take your K3, 3 again. Go to each edge and draw another dot, another vertex, inside it. Well, now it’s a graph that’s got twelve vertices in it. It’s not obvious whether this is bipartite still. (Play with it a while.) But it hasn’t become planar, not because of this. It won’t be.

    This is because we can make graphs more complicated, or more simple, without changing whether they’re planar. The process is a lot like what we did last week with the five-color map theorem, making a map simpler until it was easy enough to color. Suppose there’s a little section of the graph that’s a vertex connected by one edge to a middle vertex connected by one edge to a third vertex. Do we actually need that middle vertex for anything? Besides padding our vertex count? Nah. We can drop that whole edge-middle vertex-edge sequence and replace it all with a single edge. And there’s other rules that let us turn a vertex-edge-vertex set into a single vertex. That sort of thing. It won’t change a planar graph to a nonplanar one or vice-versa.

    So it can be hard to find the K3, 3 or the K5 hiding inside a nonplanar graph. A big enough graph can have so much going on it’s hard to find the pattern. But they’ll be there, in some group of five or six vertices with the right paths between them.

    It would make a good activity puzzle, if you could explain what to look for to kids.

  • Joseph Nebus 6:00 pm on Wednesday, 20 July, 2016 Permalink | Reply
    Tags: , , intuition, , reruns   

    Reading the Comics, July 16, 2016: More To Life Than Mathematics Edition 

    I know, it’s impolitic for me to say something like my title. But I noticed a particular rerun in this set of mathematically-themed comics. And it left me wondering if I should drop that from my daily routine. There are strips I read more out of a fear of missing out than anything else. Most of them are in perpetual reruns, though some of them are so delightful I wouldn’t dare drop them. (Here I mean Cul de Sac and Peanuts.) An individual comic takes typically little time to read, but add that up and it does take a while, especially on vacation or the like. I won’t actually change anything; I’m too stubborn in lazy ways for that. But it crosses my mind.

    Tim Lachowski’s Get A Life for the 14th is what set me off. Lachowski’s rerun this before, and I’ve mentioned it before, back in March of 2015 and back in November 2012. Given this I wonder if there’s a late-2013 or early-2014 reuse of the strip I failed to note around here. Or just missed, possibly because I was on vacation.

    Nicholas Gurewitch’s Perry Bible Fellowship reprint for the 14th gives me the title for this edition. It uses symbols and diagrams of mathematics for their graphical artistry, the sort of thing I’m surprised doesn’t get done more. Back in college the creative-writing-and-arts editor for the unread leftist weekly asked me to do a page of physics calculations as an aesthetic composition and I was glad to do it. Good notation has a beauty to it; I wonder if people would like mathematics more if they got to spend time at play with its shapes.

    Morrie Turner’s Wee Pals rerun for the 14th name-checks the New Math. The New Math was this attempt to reform mathematics in the 1970s. It was great for me, and my love remembers only liking or understanding mathematics while in New Math-guided classes. But it was an attempt at educational reform that didn’t promise that people at the cash registers would make change fast enough, and so was doomed to failure. (I am being reductive here. Much about the development of New Math went wrong, and it’s unfair to blame it all on the resistance of parents to new teaching methods. But educational reform always crashes hard against parents’ reasonable question, “Why should my child be your test case?”)

    Many of the New Math ideas grew out of the work of Nicholas Bourbaki, and the attempt to explain mathematics on completely rigorous logical foundations, as free from intuition as possible to get. That sounds like an odd thing to do; intuition is a guide to useful ways to spend one’s time and energy. But that supposes the intuition is good.

    Much of late 19th and early 20th century mathematics was spent discovering cases in which intuitive understandings of things were wrong. Deterministic systems can be unpredictable. A curve can be continuous at a single point and nowhere else in space. Infinitely large sets can be bigger or smaller than other sets. A line can wriggle around so much that it has a volume, it fills space. In that context wanting to ditch intuition a a once-useful but now-unreliable guide is not a bad idea.

    I like the New Math. I suppose we always like the way we first learned things. But I still think it’s got a healthy focus. The idea that mathematics is built on rules we agree to use, and that we are free to change if we find they’re not doing things we need, is true. It’s one easy to forget considering mathematics’ primary job, which has always been making trade, accounting, and record-keeping go smoothly. Changing those systems are perilous. But we should know something about how to pick tools to use.

    Zoe Piel’s At The Zoo for the 15th uses the blackboard-full-of-mathematics image to suggest deep thinking. (Toby the lion’s infatuated with the vet, which is why he’s thinking how to get her to visit again.) Really there’s a bunch of iconic cartoon images of deep thinking, including a mid-century-esque big-tin-box computer with reel-to-reel memory tape. Modern computers are vastly more powerful than that sort of 50s/60s contraption, but they’re worthless artistically if you want to suggest any deep thinking going on. You need stuff with moving parts for that, even in a still image.

    Scott Adams’s Dilbert Classics for the 16th originally ran the 21st of May, 1993. And it comes back to a practical use for mathematics and the sort of thing we do need to know how to calculate. It also uses the image of mathematics as obscurant nonsense.

    That tweet’s interesting in itself, although one of the respondents wonders if William meant astrology, often called “mathematics” at the time. That would be a fairer thing to call magic. But it would be only a century after William of Malmesbury’s death that Arabic numerals would become familiar in Europe. They would bring suspicions that merchants and moneylenders were trying to cheat their customers, by using these exotic specialist notations with unrecognizable rules, instead of the traditional and easy-to-follow Roman numerals. If this particular set of mathematics comics were mostly reruns, that’s all right; sometimes life is like that.

  • Joseph Nebus 6:00 pm on Monday, 18 July, 2016 Permalink | Reply
    Tags: , , birds, , ,   

    Reading the Comics, July 13, 2016: Catching Up On Vacation Week Edition 

    I confess I spent the last week on vacation, away from home and without the time to write about the comics. And it was another of those curiously busy weeks that happens when it’s inconvenient. I’ll try to get caught up ahead of the weekend. No promises.

    Art and Chip Samson’s The Born Loser for the 10th talks about the statistics of body measurements. Measuring bodies is one of the foundations of modern statistics. Adolphe Quetelet, in the mid-19th century, found a rough relationship between body mass and the square of a person’s height, used today as the base for the body mass index.Francis Galton spent much of the late 19th century developing the tools of statistics and how they might be used to understand human populations with work I will describe as “problematic” because I don’t have the time to get into how much trouble the right mind at the right idea can be.

    No attempt to measure people’s health with a few simple measurements and derived quantities can be fully successful. Health is too complicated a thing for one or two or even ten quantities to describe. Measures like height-to-waist ratios and body mass indices and the like should be understood as filters, the way temperature and blood pressure are. If one or more of these measurements are in dangerous ranges there’s reason to think there’s a health problem worth investigating here. It doesn’t mean there is; it means there’s reason to think it’s worth spending resources on tests that are more expensive in time and money and energy. And similarly just because all the simple numbers are fine doesn’t mean someone is perfectly healthy. But it suggests that the person is more likely all right than not. They’re guides to setting priorities, easy to understand and requiring no training to use. They’re not a replacement for thought; no guides are.

    Jeff Harris’s Shortcuts educational panel for the 10th is about zero. It’s got a mix of facts and trivia and puzzles with a few jokes on the side.

    I don’t have a strong reason to discuss Ashleigh Brilliant’s Pot-Shots rerun for the 11th. It only mentions odds in a way that doesn’t open up to discussing probability. But I do like Brilliant’s “Embrace-the-Doom” tone and I want to share that when I can.

    John Hambrock’s The Brilliant Mind of Edison Lee for the 13th of July riffs on the world’s leading exporter of statistics, baseball. Organized baseball has always been a statistics-keeping game. The Olympic Ball Club of Philadelphia’s 1837 rules set out what statistics to keep. I’m not sure why the game is so statistics-friendly. It must be in part that the game lends itself to representation as a series of identical events — pitcher throws ball at batter, while runners wait on up to three bases — with so many different outcomes.

    'Edison, let's discuss stats while we wait for the opening pitch.' 'Statistics? I have plenty of those. A hot dog has 400 calories and costs five dollars. A 12-ounce root beer has 38 grams of sugar.' 'I mean *player* stats.' 'Oh'. (To his grandfather instead) 'Did you know the average wait time to buy nachos is eight minutes and six seconds?'

    John Hambrock’s The Brilliant Mind of Edison Lee for the 13th of July, 2016. Properly speaking, the waiting time to buy nachos isn’t a player statistic, but I guess Edison Lee did choose to stop talking to his father for it. Which is strange considering his father’s totally natural and human-like word emission ‘Edison, let’s discuss stats while we wait for the opening pitch’.

    Alan Schwarz’s book The Numbers Game: Baseball’s Lifelong Fascination With Statistics describes much of the sport’s statistics and record-keeping history. The things recorded have varied over time, with the list of things mostly growing. The number of statistics kept have also tended to grow. Sometimes they get dropped. Runs Batted In were first calculated in 1880, then dropped as an inherently unfair statistic to keep; leadoff hitters were necessarily cheated of chances to get someone else home. How people’s idea of what is worth measuring changes is interesting. It speaks to how we change the ways we look at the same event.

    Dana Summers’s Bound And Gagged for the 13th uses the old joke about computers being abacuses and the like. I suppose it’s properly true that anything you could do on a real computer could be done on the abacus, just, with a lot ore time and manual labor involved. At some point it’s not worth it, though.

    Nate Fakes’s Break of Day for the 13th uses the whiteboard full of mathematics to denote intelligence. Cute birds, though. But any animal in eyeglasses looks good. Lab coats are almost as good as eyeglasses.

    LERBE ( O O - O - ), GIRDI ( O O O - - ), TACNAV ( O - O - O - ), ULDNOA ( O O O - O - ). When it came to measuring the Earth's circumference, there was a ( - - - - - - - - ) ( - - - - - ).

    David L Hoyt and Jeff Knurek’s Jumble for the 13th of July, 2016. The link will be gone sometime after mid-August I figure. I hadn’t thought of a student being baffled by using the same formula for an orange and a planet’s circumference because of their enormous difference in size. It feels authentic, though.

    David L Hoyt and Jeff Knurek’s Jumble for the 13th is about one of geometry’s great applications, measuring how large the Earth is. It’s something that can be worked out through ingenuity and a bit of luck. Once you have that, some clever argument lets you work out the distance to the Moon, and its size. And that will let you work out the distance to the Sun, and its size. The Ancient Greeks had worked out all of this reasoning. But they had to make observations with the unaided eye, without good timekeeping — time and position are conjoined ideas — and without photographs or other instantly-made permanent records. So their numbers are, to our eyes, lousy. No matter. The reasoning is brilliant and deserves respect.

  • Joseph Nebus 6:00 pm on Sunday, 17 July, 2016 Permalink | Reply
    Tags: , , , , , ,   

    Bourbaki and How To Write Numbers, A Trifle 

    So my attempt at keeping the Reading the Comics posts to Sunday has crashed and burned again. This time for a good reason. As you might have read between the lines on my humor blog, I spent the past week on holiday and just didn’t have time to write stuff. I barely had time to read my comics. I’ll get around to it this week.

    In the meanwhile then I’d like to point people to the MathsByAGirl blog. The blog recently had an essay on Nicolas Bourbaki, who’s among the most famous mathematicians of the 20th century. Bourbaki is also someone with a tremendous and controversial legacy, one that I expect to touch on as I catch up on last week’s comics. If you don’t know the secret of Bourbaki then do go over and learn it. If you do, well, go over and read anyway. The author’s wondering whether to write more about Bourbaki’s mathematics and while I’m all in favor of that more people should say.

    And as I promised a trifle, let me point to something from my own humor blog. How To Write Out Numbers is an older trifle based on everyone’s love for copy-editing standards. I had forgotten I wrote it before digging it up for a week of self-glorifying posts last week. I hope folks around here like it too.

    Oh, one more thing: it’s the anniversary of the publishing of an admirable but incorrect proof of the four-color map theorem. It would take another century to get right. As I said Thursday, the five-color map theorem is easy. it’s that last color that’s hard.

    Vacations are grand but there is always that comfortable day or two once you’re back home.

  • Joseph Nebus 6:00 pm on Thursday, 14 July, 2016 Permalink | Reply
    Tags: coloring, , , , , New England, states, ,   

    Theorem Thursday: The Five-Color Map Theorem 

    People think mathematics is mostly counting and arithmetic. It’s what we get at when we say “do the math[s]”. It’s why the mathematician in the group is the one called on to work out what the tip should be. Heck, I attribute part of my love for mathematics to a Berenstain Bears book which implied being a mathematician was mostly about adding up sums in a base on the Moon, which is an irresistible prospect. In fact, usually counting and arithmetic are, at least, minor influences on real mathematics. There are legends of how catastrophically bad at figuring mathematical genius can be. But usually isn’t always, and this week I’d like to show off a case where counting things and adding things up lets us prove something interesting.

    The Five-Color Map Theorem.

    No, not four. I imagine anyone interested enough to read a mathematics blog knows the four-color map theorem. It says that you only need four colors to color a map. That’s true, given some qualifiers. No discontiguous chunks that need the same color. Two regions with the same color can touch at a point, they just can’t share a line or curve. The map is on a plane or the surface of a sphere. Probably some other requirements. I’m not going to prove that. Nobody has time for that. The best proofs we’ve figured out for it amount to working out how every map fits into one of a huge number of cases, and trying out each case. It’s possible to color each of those cases with only four colors, so, we’re done. Nice but unenlightening and way too long to deal with.

    The five-color map theorem is a lot like the four-color map theorem, with this difference: it says that you only need five colors to color a map. Same qualifiers as before. Yes, it’s true because the four-color map theorem is true and because five is more than four. We can do better than that. We can prove five colors are enough even without knowing whether four colors will do. And it’s easy. The ease of the five-color map theorem gave people reason to think four colors would be maybe harder but still manageable.

    The proof I want to show uses one of mathematicians’ common tricks. It employs the same principle which Hercules used to slay the Hydra, although it has less cauterizing lake-monster flesh with flaming torches, as that’s considered beneath the dignity of the Academy anymore except when grading finals for general-requirements classes. The part of the idea we do use is to take a problem which we might not be able to do and cut it down to one we can do. Properly speaking this is a kind of induction proof. In those we start from problems we can do and show that if we can do those, we can do all the complicated problems. But we come at it by cutting down complicated problems and making them simple ones.

    So suppose we start with a map that’s got some huge number of territories to color. I’m going to start with the United States states which were part of the Dominion of New England. As I’m sure I don’t need to remind any readers, American or otherwise, this was a 17th century attempt by the English to reorganize their many North American colonies into something with fewer administrative irregularities. It lasted almost long enough for the colonists to hear about it. At that point the Glorious Revolution happened (not involving the colonists) and everybody went back to what they were doing before.

    Please enjoy my little map of the place. It gives all the states a single color because I don’t really know how to use QGIS and it would probably make my day job easier if I did. (Well, QGIS is open-source software, so its interface is a disaster and its tutorials gibberish. The only way to do something with it is to take flaming torches to it.)

    Map showing New York, New Jersey, and New England (Connecticut, Rhode Island, Massachusetts, Vermont, New Hampshire, and Maine) in a vast white space.

    States which, in their 17th-century English colonial form, were part of the Dominion of New England (1685-1689). More or less. If I’ve messed up don’t tell me as it doesn’t really matter for this problem.

    There’s eight regions here, eight states, so it’s not like we’re at the point we can’t figure how to color this with five different colors. That’s all right. I’m using this for a demonstration. Pretend the Dominion of New England is so complicated we can’t tell whether five colors are enough. Oh, and a spot of lingo: if five colors are enough to color the map we say the map is “colorable”. We say it’s “5-colorable” if we want to emphasize five is enough colors.

    So imagine that we erase the border between Maine and New Hampshire. Combine them into a single state over the loud protests of the many proud, scary Mainers. But if this simplified New England is colorable, so is the real thing. There’s at least one color not used for Greater New Hampshire, Vermont, or Massachusetts. We give that color to a restored Maine. If the simplified map can be 5-colored, so can the original.

    Maybe we can’t tell. Suppose the simplified map is still too complicated to make it obvious. OK, then. Cut out another border. How about we offend Roger Williams partisans and merge Rhode Island into Massachusetts? Massachusetts started out touching five other states, which makes it a good candidate for a state that needed a sixth color. With Rhode Island reduced to being a couple counties of the Bay State, Greater Massachusetts only touches four other states. It can’t need a sixth color. There’s at least one of our original five that’s free.

    OK, but, how does that help us find a color for Rhode Island? Maine it’s easy to see why there’s a free color. But Rhode Island?

    Well, it’ll have to be the same color as either Greater New Hampshire or Vermont or New York. At least one of them has to be available. Rhode Island doesn’t touch them. Connecticut’s color is out because Rhode Island shares a border with it. Same with Greater Massachusetts’s color. But we’ve got three colors for the taking.

    But is our reduced map 5-colorable? Even with Maine part of New Hampshire and Rhode Island part of Massachusetts it might still be too hard to tell. There’s six territories in it, after all. We can simplify things a little. Let’s reverse the treason of 1777 and put Vermont back into New York, dismissing New Hampshire’s claim on the territory as obvious absurdity. I am never going to be allowed back into New England. This Greater New York needs one color for itself, yes. And it touches four other states. But these neighboring states don’t touch each other. A restored Vermont could use the same color as New Jersey or Connecticut. Greater Massachusetts and Greater New Hampshire are unavailable, but there’s still two choices left.

    And now look at the map we have remaining. There’s five states in it: Greater New Hampshire, Greater Massachusetts, Greater New York, Regular Old Connecticut, and Regular old New Jersey. We have five colors. Obviously we can give the five territories different colors.

    This is one case, one example map. That’s all we need. A proper proof makes things more abstract, but uses the same pattern. Any map of a bunch of territories is going to have at least one territory that’s got at most five neighbors. Maybe it will have several. Look for one of them. If you find a territory with just one neighbor, such as Maine had, remove that border. You’ve got a simpler map and there must be a color free for the restored territory.

    If you find a territory with just two neighbors, such as Rhode Island, take your pick. Merge it with either neighbor. You’ll still have at least one color free for the restored territory. With three neighbors, such as Vermont or Connecticut, again you have your choice. Merge it with any of the three neighbors. You’ll have a simpler map and there’ll be at least one free color.

    If you have four neighbors, the way New York has, again pick a border you like and eliminate that. There is a catch. You can imagine one of the neighboring territories reaching out and wrapping around to touch the original state on more than one side. Imagine if Massachusetts ran far out to sea, looped back through Canada, and came back to touch New Jersey, Vermont from the north, and New York from the west. That’s more of a Connecticut stunt to pull, I admit. But that’s still all right. Most of the colonies tried this sort of stunt. And even if Massachusetts did that, we would have colors available. It would be impossible for Vermont and New Jersey to touch. We’ve got a theorem that proves it.

    Yes, it’s the Jordan Curve Theorem, here to save us right when we might get stuck. Just like I promised last week. In this case some part of the border of New York and Really Big Massachusetts serves as our curve. Either Vermont or New Jersey is going to be inside that curve, and the other state is outside. They can’t touch. Thank you.

    If you have five neighbors, the way Massachusetts has, well, maybe you’re lucky. We are here. None of its neighboring states touches more than two others. We can cut out a border easily and have colors to spare. But we could be in trouble. We could have a map in which all the bordering states touch three or four neighbors and that seems like it would run out of colors. Let me show a picture of that.

    The map shows a pentagonal region A which borders five regions, B, C, D, E, and F. Each of those regions borders three or four others. B is entirely enclosed by regions A, C, and D, although from B's perspective they're all enclosed by it.

    A hypothetical map with five regions named by an uninspired committee.

    So this map looks dire even when you ignore that line that looks like it isn’t connected where C and D come together. Flood fill didn’t run past it, so it must be connected. It just doesn’t look right. Everybody has four neighbors except the province of B, which has three. The province of A has got five. What can we do?

    Call on the Jordan Curve Theorem again. At least one of the provinces has to be landlocked, relative to the others. In this case, the borders of provinces A, D, and C come together to make a curve that keeps B in the inside and E on the outside. So we’re free to give B and E the same color. We treat this in the proof by doing a double merger. Erase the boundary between provinces A and B, and also that between provinces A and E. (Or you might merge B, A, and F together. It doesn’t matter. The Jordan Curve Theorem promises us there’ll be at least one choice and that’s all we need.)

    So there we have it. As long as we have a map that has some provinces with up to five neighbors, we can reduce the map. And reduce it again, if need be, and again and again. Eventually we’ll get to a map with only five provinces and that has to be 5-colorable.

    Just … now … one little nagging thing. We’re relying on there always being some province with at most five neighbors. Why can’t there be some horrible map where every province has six or more neighbors?

    Counting will tell us. Arithmetic will finish the job. But we have to get there by way of polygons.

    That is, the easiest way to prove this depends on a map with boundaries that are all polygons. That’s all right. Polygons are almost the polynomials of geometry. You can make a polygon that looks so much like the original shape the eye can’t tell the difference. Look at my Dominion of New England map. That’s computer-rendered, so it’s all polygons, and yet all those shore and river boundaries look natural.

    But what makes up a polygon? Well, it’s a bunch of straight lines. We call those ‘edges’. Each edge starts and ends at a corner. We call those ‘vertices’. These edges come around and close together to make a ‘face’, a territory like we’ve been talking about. We’re going to count all the regions that have a certain number of neighboring other regions.

    Specifically, F2 will represent however many faces there are that have two sides. F3 will represent however many faces there are that have three sides. F4 will represent however many faces there are that have four sides. F10 … yeah, you got this.

    One thing you didn’t get. The outside counts as a face. We need this to make the count come out right, so we can use some solid-geometry results. In my map that’s the vast white space that represents the Atlantic Ocean, the other United States, the other parts of Canada, the Great Lakes, all the rest of the world. So Maine, for example, belongs to F2 because it touches New Hampshire and the great unknown void of the rest of the universe. Rhode Island belongs to F3 similarly. New Hampshire’s in F4.

    Any map has to have at least one thing that’s in F2, F3, F4, or F5. They touch at most two, three, four or five neighbors. (If they touched more, they’d represent a face that was a polygon of even more sides.)

    How do we know? It comes from Euler’s Formula, which starts out describing the ways corners and edges and faces of a polyhedron fit together. Our map, with its polygon on the surface of the sphere, turns out to be just as good as a polyhedron. It looks a little less blocky, but that doesn’t show.

    By Euler’s Formula, there’s this neat relationship between the number of vertices, the number of edges, and the number of faces in a polyhedron. (This is the same Leonhard Euler famous for … well, everything in mathematics, really. But in this case it’s for his work with shapes.) It holds for our map too. Call the number of vertices V. Call the number of edges E. Call the number of faces F. Then:

    V - E + F = 2

    Always true. Try drawing some maps yourself, using simple straight lines, and see if it works. For that matter, look at my Really Really Simplified map and see if it doesn’t hold true still.

    One of those blocky diagrams of New York, New Jersey, and New England, done in that way transit maps look, only worse because I'm not so good at this.

    A very simplified blocky diagram of my Dominion of New England, with the vertices and edges highlighted so they’re easy to count if you want to do that.

    Here’s one of those insights that’s so obvious it’s hard to believe. Every edge ends in two vertices. Three edges meet at every vertex. (We don’t have more than three territories come together at a point. If that were to happen, we’d change the map a little to find our coloring and then put it back afterwards. Pick one of the territories and give it a disc of area from the four or five or more corners. The troublesome corner is gone. Once we’ve done with our proof, shrink the disc back down to nothing. Coloring done!) And therefore 2E = 3V .

    A polygon has the same number of edges as vertices, and if you don’t believe that then draw some and count. Every edge touches exactly two regions. Every vertex touches exactly three edges. So we can rework Euler’s formula. Multiply it by six and we get 6V - 6E + 6F = 12 . And from doubling the equation about edges and vertices equation in the last paragraph, 4E = 6V . So if we break up that 6E into 4E and 2E we can rewrite that Euler’s formula again. It becomes 6V - 4E - 2E + 6F = 12. 6V – 4E is zero, so, -2E + 6F = 12 .

    Do we know anything about F itself?

    Well, yeah. F = F_2 + F_3 + F_4 + F_5 + F_6 + \cdots . The number of faces has to equal the sum of the number of faces of two edges, and of three edges, and of four edges, and of five edges, and of six edges, and on and on. Counting!

    Do we know anything about how E and F relate?

    Well, yeah. A polygon in F2 has two edges. A polygon in F3 has three edges. A polygon in F4 has four edges. And each edge runs up against two faces. So therefore 2E = 2F_2 + 3F_3 + 4F_4 + 5F_5 + 6F_6 + \cdots . This goes on forever but that’s all right. We don’t need all these terms.

    Because here’s what we do have. We know that -2E + 6F = 12 . And we know how to write both E and F in terms of F2, F3, F4, and so on. We’re going to show at least one of these low-subscript Fsomethings has to be positive, that is, there has to be at least one of them.

    Start by just shoving our long sum expressions into the modified Euler’s Formula we had. That gives us this:

    -(2F_2 + 3F_3 + 4F_4 + 5F_5 + 6F_6 + \cdots) + 6(F_2 + F_3 + F_4 + F_5 + F_6 + \cdots) = 12

    Doesn’t look like we’ve got anywhere, does it? That’s all right. Multiply that -1 and that 6 into their parentheses. And then move the terms around, so that we group all the terms with F2 together, and all the terms with F3 together, and all the terms with F4 together, and so on. This gets us to:

    (-2 + 6) F_2 + (-3 + 6) F_3 + (-4 + 6) F_4 + (-5 + 6) F_5  + (-6 + 6) F_6 + (-7 + 6) F_7 + (-8 + 6) F_8 + \cdots = 12

    I know, that’s a lot of parentheses. And it adds negative numbers to positive which I guess we’re allowed to do but who wants to do that? Simplify things a little more:

    4 F_2 + 3 F_3 + 2 F_4 + 1 F_5 + 0 F_6 - 1 F_7 - 2 F_8 - \cdots = 12

    And now look at that. Each Fsubscript has to be zero or a positive number. You can’t have a negative number of shapes. If you can I don’t want to hear about it. Most of those Fsubscript‘s get multiplied by a negative number before they’re added up. But the sum has to be a positive number.

    There’s only one way that this sum can be a positive number. At least one of F2, F3, F4, or F5 has to be a positive number. So there must be at least one region with at most five neighbors. And that’s true without knowing anything about our map. So it’s true about the original map, and it’s true about a simplified map, and about a simplified-more map, and on and on.

    And that is why this hydra-style attack method always works. We can always simplify a map until it obviously can be colored with five colors. And we can go from that simplified map back to the original map, and color it in just fine. Formally, this is an existence proof: it shows there must be a way to color a map with five colors. But it does so the devious way, by showing a way to color the map. We don’t get enough existence proofs like that. And, at its critical point, we know the proof is true because we can count the number of regions and the number of edges and the number of corners they have. And we can add and subtract those numbers in the right way. Just like people imagine mathematicians do all day.

    Properly this works only on the surface of a sphere. Euler’s Formula, which we use for the proof, depends on that. We get away with it on a piece of paper because we can pretend this is just a part of the globe so small we don’t see how flat it is. The vast white edge we suppose wraps around the whole world. And that’s fine since we mostly care about maps on flat surfaces or on globes. If we had a map that needed three dimensions, like one that looked at mining and water and overflight and land-use rights, things wouldn’t be so easy. Nor would they work at all if the map turned out to be on an exotic shape like a torus, a doughnut shape.

    But this does have a staggering thought. Suppose we drew boundary lines. And suppose we found an arrangement of them so that we needed more than five colors. This would tell us that we have to be living on a surface such as a torus, the doughnut shape. We could learn something about the way space is curved by way of an experiment that never looks at more than where two regions come together. That we can find information about the whole of space, global information, by looking only at local stuff amazes me. I hope it at least surprises you.

    From fiddling with this you probably figure the four-color map theorem should follow right away. Maybe involve a little more arithmetic but nothing too crazy. I agree, it so should. It doesn’t. Sorry.

    • FlowCoef 7:41 am on Monday, 1 August, 2016 Permalink | Reply

      How awful: I want to follow along the math, but my overriding interest in geography blocks my mind.


      • Joseph Nebus 7:32 pm on Tuesday, 9 August, 2016 Permalink | Reply

        Aw, I knew there’d be trouble when I tossed in so many state boundary jokes in one essay. But there’s something so compelling in looking at maps, isn’t there? I ultimately have no regrets about this.

        (I wonder what other mathematics stuff I could spin out of maps, come to think of it. My day job does take me into Geographic Information Services, I should be able to make that work alongside with my fun.)


        • FlowCoef 11:21 pm on Tuesday, 9 August, 2016 Permalink | Reply

          I agree that maps – of any kind – are compelling. You are lucky to be involved into GIS, that was always amazing to me. Computers and maps, together.


          • Joseph Nebus 2:20 am on Friday, 12 August, 2016 Permalink | Reply

            I admit I should appreciate my position better. It’s well-suited for me in a great many ways, although I’d like to be back in academia if I could. I haven’t yet had enough experience with students to be fed up with them.


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