## The Summer 2017 Mathematics A To Z: Cohomology

Today’s A To Z topic is another request from Gaurish, of the For The Love Of Mathematics blog. Also part of what looks like a quest to make me become a topology blogger, at least for a little while. It’s going to be exciting and I hope not to faceplant as I try this.

Also, a note about Thomas K Dye, who’s drawn the banner art for this and for the Why Stuff Can Orbit series: the publisher for collections of his comic strip is having a sale this weekend.

# Cohomology.

The word looks intimidating, and faintly of technobabble. It’s less cryptic than it appears. We see parts of it in non-mathematical contexts. In biology class we would see “homology”, the sharing of structure in body parts that look superficially very different. We also see it in art class. The instructor points out that a dog’s leg looks like that because they stand on their toes. What looks like a backward-facing knee is just the ankle, and if we stand on our toes we see that in ourselves. We might see it in chemistry, as many interesting organic compounds differ only in how long or how numerous the boring parts are. The stuff that does work is the same, or close to the same. And this is a hint to what a mathematician means by cohomology. It’s something in shapes. It’s particularly something in how different things might have similar shapes. Yes, I am using a homology in language here.

I often talk casually about the “shape” of mathematical things. Or their “structures”. This sounds weird and abstract to start and never really gets better. We can get some footing if we think about drawing the thing we’re talking about. Could we represent the thing we’re working on as a figure? Often we can. Maybe we can draw a polygon, with the vertices of the shape matching the pieces of our mathematical thing. We get the structure of our thing from thinking about what we can do to that polygon without changing the way it looks. Or without changing the way we can do whatever our original mathematical thing does.

This leads us to homologies. We get them by looking for stuff that’s true even if we moosh up the original thing. The classic homology comes from polyhedrons, three-dimensional shapes. There’s a relationship between the number of vertices, the number of edges, and the number of faces of a polyhedron. It doesn’t change even if you stretch the shape out longer, or squish it down, for that matter slice off a corner. It only changes if you punch a new hole through the middle of it. Or if you plug one up. That would be unsporting. A homology describes something about the structure of a mathematical thing. It might even be literal. Topology, the study of what we know about shapes without bringing distance into it, has the number of holes that go through a thing as a homology. This gets feeling like a comfortable, familiar idea now.

But that isn’t a cohomology. That ‘co’ prefix looks dangerous. At least it looks significant. When the ‘co’ prefix has turned up before it’s meant something is shaped by how it refers to something else. Coordinates aren’t just number lines; they’re collections of number lines that we can use to say where things are. If ‘a’ is a factor of the number ‘x’, its cofactor is the number you multiply ‘a’ by in order to get ‘x’. (For real numbers that’s just x divided by a. For other stuff it might be weirder.) A codomain is a set that a function maps a domain into (and must contain the range, at least). Cosets aren’t just sets; they’re ways we can divide (for example) the counting numbers into odds and evens.

So what’s the ‘co’ part for a homology? I’m sad to say we start losing that comfortable feeling now. We have to look at something we’re used to thinking of as a process as though it were a thing. These things are morphisms: what are the ways we can match one mathematical structure to another? Sometimes the morphisms are easy. We can match the even numbers up with all the integers: match 0 with 0, match 2 with 1, match -6 with -3, and so on. Addition on the even numbers matches with addition on the integers: 4 plus 6 is 10; 2 plus 3 is 5. For that matter, we can match the integers with the multiples of three: match 1 with 3, match -1 with -3, match 5 with 15. 1 plus -2 is -1; 3 plus -6 is -9.

What happens if we look at the sets of matchings that we can do as if that were a set of things? That is, not some human concept like ‘2’ but rather ‘match a number with one-half its value’? And ‘match a number with three times its value’? These can be the population of a new set of things.

And these things can interact. Suppose we “match a number with one-half its value” and then immediately “match a number with three times its value”. Can we do that? … Sure, easily. 4 matches to 2 which goes on to 6. 8 matches to 4 which goes on to 12. Can we write that as a single matching? Again, sure. 4 matches to 6. 8 matches to 12. -2 matches to -3. We can write this as “match a number with three-halves its value”. We’ve taken “match a number with one-half its value” and combined it with “match a number with three times its value”. And it’s given us the new “match a number with three-halves its value”. These things we can do to the integers are themselves things that can interact.

This is a good moment to pause and let the dizziness pass.

It isn’t just you. There is something weird thinking of “doing stuff to a set” as a thing. And we have to get a touch more abstract than even this. We should be all right, but please do not try not to use this to defend your thesis in category theory. Just use it to not look forlorn when talking to your friend who’s defending her thesis in category theory.

Now, we can take this collection of all the ways we can relate one set of things to another. And we can combine this with an operation that works kind of like addition. Some way to “add” one way-to-match-things to another and get a way-to-match-things. There’s also something that works kind of like multiplication. It’s a different way to combine these ways-to-match-things. This forms a ring, which is a kind of structure that mathematicians learn about in Introduction to Not That Kind Of Algebra. There are many constructs that are rings. The integers, for example, are also a ring, with addition and multiplication the same old processes we’ve always used.

And just as we can sort the integers into odds and evens — or into other groupings, like “multiples of three” and “one plus a multiple of three” and “two plus a multiple of three” — so we can sort the ways-to-match-things into new collections. And this is our cohomology. It’s the ways we can sort and classify the different ways to manipulate whatever we started on.

I apologize that this sounds so abstract as to barely exist. I admit we’re far from a nice solid example such as “six”. But the abstractness is what gives cohomologies explanatory power. We depend very little on the specifics of what we might talk about. And therefore what we can prove is true for very many things. It takes a while to get there, is all.

• #### mathtuition88 3:40 pm on Saturday, 5 August, 2017 Permalink | Reply

Very nice article. I have written a post on De Rham Cohomology at https://mathtuition88.com/2017/03/15/de-rham-cohomology/, also a talk relating De Rham Cohomology to Physics: https://mathtuition88.files.wordpress.com/2017/03/de-rham-cohomology.pdf.

I find De Rham Cohomology one of the most concrete examples of Cohomology, though alas it is mostly accessible to advanced undergraduates or graduate students…

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• #### Joseph Nebus 5:04 pm on Sunday, 13 August, 2017 Permalink | Reply

Thank you!

I have to admit looking over those posts I’m trying to think just when as an undergraduate I would be up to the topic. It’s a pity so much interesting stuff takes so much preliminary work.

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• #### gaurish 9:51 am on Saturday, 12 August, 2017 Permalink | Reply

Finally, this is a nice write-up which I can ask people to see when they ask me to give an idea of cohomology!

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• #### Joseph Nebus 5:07 pm on Sunday, 13 August, 2017 Permalink | Reply

Thank you again, and I hope that I haven’t lead anyone too badly wrong with it.

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## The End 2016 Mathematics A To Z: Jordan Curve

I realize I used this thing in one of my Theorem Thursday posts but never quite said what it was. Let me fix that.

## Jordan Curve

Get a rubber band. Well, maybe you can’t just now, even if you wanted to after I gave orders like that. Imagine a rubber band. I apologize to anyone so offended by my imperious tone that they’re refusing. It’s the convention for pop mathematics or science.

Anyway, take your rubber band. Drop it on a table. Fiddle with it so it hasn’t got any loops in it and it doesn’t twist over any. I want the whole of one edge of the band touching the table. You can imagine the table too. That is a Jordan Curve, at least as long as the rubber band hasn’t broken.

This may not look much like a circle. It might be close, but I bet it’s got some wriggles in its curves. Maybe it even curves so much the thing looks more like a kidney bean than a circle. Maybe it pinches so much that it looks like a figure eight, a couple of loops connected by a tiny bridge on the interior. Doesn’t matter. You can bring out the circle. Put your finger inside the rubber band’s loops and spiral your finger around. Do this gently and the rubber band won’t jump off the table. It’ll round out to as perfect a circle as the limitations of matter allow.

And for that matter, if we wanted, we could take a rubber band laid down as a perfect circle. Then nudge it here and push it there and wrinkle it up into as complicated a figure as you like. Either way is as possible.

A Jordan Curve is a closed curve, a curve that loops around back to itself. And it’s simple. That is, it doesn’t cross over itself at any point. However weird and loopy this figure is, as long as it doesn’t cross over itself, it’s got in a sense the same shape as a circle. We can imagine a function that matches every point on a true circle to a point on the Jordan Curve. A set of points in order on the original circle will match to points in the same order on the Jordan Curve. There’s nothing missing and there’s no jumps or ambiguous points. And no point on the Jordan Curve matches to two or more on the original circle. (This is why we don’t let the curve to cross over itself.)

When I wrote about the Jordan Curve Theorem it was about how to tell how a curve divides a plane into two pieces, an inside and an outside. You can have some pretty complicated-looking figures. I have an example on the Jordan Curve Theorem essay, but you can make your own by doodling. And we can look at it as a circle, as a rubber band, twisted all around.

This all dips into topology, the study of how shapes connect when we don’t care about distance. But there are simple wondrous things to find about them. For example. Draw a Jordan Curve, please. Any that you like. Now draw a triangle. Again, any that you like.

There is some trio of points in your Jordan Curve which connect to a triangle the same shape as the one you drew. It may be bigger than your triangle, or smaller. But it’ll look similar. The angles inside will all be the same as the ones you started with. This should help make doodling during a dull meeting even more exciting.

There may be four points on your Jordan Curve that make a square. I don’t know. Nobody knows for sure. There certainly are if your curve is convex, that is, if no line between any two points on the curve goes outside the curve. And it’s true even for curves that aren’t complex if they are smooth enough. But generally? For an arbitrary curve? We don’t know. It might be true. It might be impossible to find a square in some Jordan Curve. It might be the Jordan Curve you drew. Good luck looking.

• #### gaurish 3:52 am on Thursday, 24 November, 2016 Permalink | Reply

Jordan curve theorem is again in news: http://wp.me/p3qzP-2tV

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• #### Joseph Nebus 11:11 pm on Friday, 25 November, 2016 Permalink | Reply

Ooh, thank you, that’s interesting stuff. And on that conjecture about squares, too, which is so neat.

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• #### Joseph Nebus 6:00 pm on Sunday, 18 September, 2016 Permalink | Reply Tags: animation ( 8 ), arithmetic ( 69 ), calculus ( 85 ), notation ( 18 ), pizza ( 2 ), symbols ( 10 ), topology

As though to reinforce how nothing was basically wrong, Comic Strip Master Command sent a normal number of mathematically themed comics around this past week. They bunched the strips up in the first half of the week, but that will happen. It was a fun set of strips in any event.

Rob Harrell’s Adam @ Home for the 11th tells of a teacher explaining division through violent means. I’m all for visualization tools and if we are going to use them, the more dramatic the better. But I suspect Mrs Clark’s students will end up confused about what exactly they’ve learned. If a doll is torn into five parts, is that communicating that one divided by five is five? If the students were supposed to identify the mass of the parts of the torn-up dolls as the result of dividing one by five, was that made clear to them? Maybe it was. But there’s always the risk in a dramatic presentation that the audience will misunderstand the point. The showier the drama the greater the risk, it seems to me. But I did only get the demonstration secondhand; who knows how well it was done?

Greg Cravens’ The Buckets for the 11th has the kid, Toby, struggling to turn a shirt backwards and inside-out without taking it off. As the commenters note this is the sort of problem we get into all the time in topology. The field is about what can we say about shapes when we don’t worry about distance? If all we know about a shape is the ways it’s connected, the number of holes it has, whether we can distinguish one side from another, what else can we conclude? I believe Gocomics.com commenter Mike is right: take one hand out the bottom of the shirt and slide it into the other sleeve from the outside end, and proceed from there. But I have not tried it myself. I haven’t yet started wearing long-sleeve shirts for the season.

Bill Amend’s FoxTrot for the 11th — a new strip — does a story problem featuring pizzas cut into some improbable numbers of slices. I don’t say it’s unrealistic someone might get this homework problem. Just that the story writer should really ask whether they’ve ever seen a pizza cut into sevenths. I have a faint memory of being served a pizza cut into tenths by same daft pizza shop, which implies fifths is at least possible. Sevenths I refuse, though.

Mark Tatulli’s Heart of the City for the 12th plays on the show-your-work directive many mathematics assignments carry. I like Heart’s showiness. But the point of showing your work is because nobody cares what (say) 224 divided by 14 is. What’s worth teaching is the ability to recognize what approaches are likely to solve what problems. What’s tested is whether someone can identify a way to solve the problem that’s likely to succeed, and whether that can be carried out successfully. This is why it’s always a good idea, if you are stumped on a problem, to write out how you think this problem should be solved. Writing out what you mean to do can clarify the steps you should take. And it can guide your instructor to whether you’re misunderstanding something fundamental, or whether you just missed something small, or whether you just had a bad day.

Norm Feuti’s Gil for the 12th, another rerun, has another fanciful depiction of showing your work. The teacher’s got a fair complaint in the note. We moved away from tally marks as a way to denote numbers for reasons. Twelve depictions of apples are harder to read than the number 12. And they’re terrible if we need to depict numbers like one-half or one-third. Might be an interesting side lesson in that.

Brian Basset’s Red and Rover for the 14th is a rerun and one I’ve mentioned in these parts before. I understand Red getting fired up to be an animator by the movie. It’s been a while since I watched Donald Duck in Mathmagic Land but my recollection is that while it was breathtaking and visually inventive it didn’t really get at mathematics. I mean, not at noticing interesting little oddities and working out whether they might be true always, or sometimes, or almost never. There is a lot of play in mathematics, especially in the exciting early stages where one looks for a thing to prove. But it’s also in seeing how an ingenious method lets you get just what you wanted to know. I don’t know that the short demonstrates enough of that.

Bud Blake’s Tiger rerun for the 15th of September, 2016. I don’t get to talking about the art of the comics here, but, I quite like Julian’s expressions here. And Bud Blake drew fantastic rumpled clothes.

Bud Blake’s Tiger rerun for the 15th gives Punkinhead the chance to ask a question. And it’s a great question. I’m not sure what I’d say arithmetic is, not if I’m going to be careful. Offhand I’d say arithmetic is a set of rules we apply to a set of things we call numbers. The rules are mostly about how we can take two numbers and a rule and replace them with a single number. And these turn out to correspond uncannily well with the sorts of things we do with counting, combining, separating, and doing some other stuff with real-world objects. That it’s so useful is why, I believe, arithmetic and geometry were the first mathematics humans learned. But much of geometry we can see. We can look at objects and see how they fit together. Arithmetic we have to infer from the way the stuff we like to count works. And that’s probably why it’s harder to do when we start school.

What’s not good about that as an answer is that it actually applies to a lot of mathematical constructs, including those crazy exotic ones you sometimes see in science press. You know, the ones where there’s this impossibly complicated tangle with ribbons of every color and a headline about “It’s Revolutionary. It’s 46-Dimensional. It’s Breaking The Rules Of Geometry. Is It The Shape That Finally Quantizes Gravity?” or something like that. Well, describe a thing vaguely and it’ll match a lot of other things. But also when we look to new mathematical structures, we tend to look for things that resemble arithmetic. Group theory, for example, is one of the cornerstones of modern mathematical thought. It’s built around having a set of things on which we can do something that looks like addition. So it shouldn’t be a surprise that many groups have a passing resemblance to arithmetic. Mathematics may produce universal truths. But the ones we see are also ones we are readied to see by our common experience. Arithmetic is part of that common experience.

Jerry Scott and Jim Borgman’s Zits for the 14th of September, 2016. Properly speaking that is ink on his face, but I suppose saying it’s calculus pins down where it came from. Just observing.

Also Jerry Scott and Jim Borgman’s Zits for the 14th I think doesn’t really belong here. It’s just got a cameo appearance by the concept of mathematics. Dave Whamond’s Reality Check for the 17th similarly just mentions the subject. But I did want to reassure any readers worried after last week that Pierce recovered fine. Also that, you know, for not having a stomach for mathematics he’s doing well carrying on. Discipline will carry one far.

• #### ivasallay 3:44 am on Monday, 19 September, 2016 Permalink | Reply

You said, “Twelve depictions of apples are harder to read than the number 12.” It might be a little difficult to see at first, but the twelve apples were arranged to form the numerals 1 and 2. I thought it was rather clever.

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## 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.)

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.

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.

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.

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• #### 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.)

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• #### 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.

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• #### 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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## Theorem Thursday: The Jordan Curve Theorem

There are many theorems that you have to get fairly far into mathematics to even hear of. Often they involve things that are so abstract and abstruse that it’s hard to parse just what we’re studying. This week’s entry is not one of them.

# The Jordan Curve Theorem.

There are a couple of ways to write this. I’m going to fall back on the version that Richard Courant and Herbert Robbins put in the great book What Is Mathematics?. It’s a theorem in the field of topology, the study of how shapes interact. In particular it’s about simple, closed curves on a plane. A curve is just what you figure it should be. It’s closed if it … uh … closes, makes a complete loop. It’s simple if it doesn’t cross itself or have any disconnected bits. So, something you could draw without lifting pencil from paper and without crossing back over yourself. Have all that? Good. Here’s the theorem:

A simple closed curve in the plane divides that plane into exactly two domains, an inside and an outside.

It’s named for Camille Jordan, a French mathematician who lived from 1838 to 1922, and who’s renowned for work in group theory and topology. It’s a different Jordan from the one named in Gauss-Jordan Elimination, which is a matrix thing that’s important but tedious. It’s also a different Jordan from Jordan Algebras, which I remember hearing about somewhere.

The Jordan Curve Theorem is proved by reading its proposition and then saying, “Duh”. This is compelling, although it lacks rigor. It’s obvious if your curve is a circle, or a slightly squished circle, or a rectangle or something like that. It’s less obvious if your curve is a complicated labyrinth-type shape.

A generic complicated maze shape. Can you pick out which part is the inside and which the outside? Pretend you don’t notice that little peninsula thing in the upper right corner. I didn’t mean the line to overlap itself but I was using too thick a brush in ArtRage and didn’t notice before I’d exported the image.

It gets downright hard if the curve has a lot of corners. This is why a completely satisfying rigorous proof took decades to find. There are curves that are nowhere differentiable, that are nothing but corners, and those are hard to deal with. If you think there’s no such thing, then remember the Koch Snowflake. That’s that triangle sticking up from the middle of a straight line, that itself has triangles sticking up in the middle of its straight lines, and littler triangles still sticking up from the straight lines. Carry that on forever and you have a shape that’s continuous but always changing direction, and this is hard to deal with.

Still, you can have a good bit of fun drawing a complicated figure, then picking a point and trying to work out whether it’s inside or outside the curve. The challenging way to do that is to view your figure as a maze and look for a path leading outside. The easy way is to draw a new line. I recommend doing that in a different color.

In particular, draw a line from your target point to the outside. Some definitely outside point. You need the line to not be parallel to any of the curve’s line segments. And it’s easier if you don’t happen to intersect any vertices, but if you must, we’ll deal with that two paragraphs down.

A red dot that turns out to be outside the labyrinth, based on the number of times the testing line, in blue, crosses the curve. I learned doing this that I should have drawn the dot and blue line first and then fit a curve around it so I wouldn’t have to work so hard to find one lousy point and line segment that didn’t have some problems.

So draw your testing line here from the point to something definitely outside. And count how many times your testing line crosses the original curve. If the testing line crosses the original curve an even number of times then the original point was outside the curve. If the testing line crosses the original an odd number of times then the original point was inside of the curve. Done.

If your testing line touches a vertex, well, then it gets fussy. It depends whether the two edges of the curve that go into that vertex stay on the same side as your testing line. If the original curve’s edges stay on the same side of your testing line, then don’t count that as a crossing. If the edges go on opposite sides of the testing line, then that does count as one crossing. With that in mind, carry on like you did before. An even number of crossings means your point was outside. An odd number of crossings means your point was inside.

This? Doesn’t count as the blue testing line crossing the black curve.

This? This counts as the blue testing line crossing the black curve.

So go ahead and do this a couple times with a few labyrinths and sample points. It’s fun and elevates your doodling to the heights of 19th-century mathematics. Also once you’ve done that a couple times you’ve proved the Jordan curve theorem.

Well, no, not quite. But you are most of the way to proving it for a special case. If the curve is a polygon, a shape made up of a finite number of line segments, then you’ve got almost all the proof done. You have to finish it off by choosing a ray, a direction, that isn’t parallel to any of the polygon’s line segments. (This is one reason this method only works for polygons, and fails for stuff like the Koch Snowflake. It also doesn’t work well with space-filling curves, which are things that exist. Yes, those are what they sound like: lines that squiggle around so much they fill up area. Some can fill volume. I swear. It’s fractal stuff.) Imagine all the lines that are parallel to that ray. There’s definitely some point along that line that’s outside the curve. You’ll need that for reference. Classify all the points on that line by whether there’s an even or an odd number of crossings between a starting point and your reference definitely-outside point. Keep doing that for all these many parallel lines.

And that’s it. The mess of points that have an odd number of intersections are the inside. The mess of points that have an even number of intersections are the outside.

You won’t be surprised to know there’s versions of the Jordan curve theorem for solid objects in three-dimensional space. And for hyperdimensional spaces too. You can always work out an inside and an outside, as long as space isn’t being all weird. But it might sound like it’s not much of a theorem. So you can work out an inside and an outside; so what?

But it’s one of those great utility theorems. It pops in to places, the perfect tool for a problem you were just starting to notice existed. If I can get my rhetoric organized I hope to show that off next week, when I figure to do the Five-Color Map Theorem.

• #### howardat58 7:00 pm on Thursday, 7 July, 2016 Permalink | Reply

Richard Courant and Herbert Robbins: What Is Mathematics?.

My bedside book, since 1961.

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• #### Joseph Nebus 4:10 am on Saturday, 9 July, 2016 Permalink | Reply

I’d first read it as an undergraduate and it was one of my first online book purchases. I do keep dipping into it and finding things I feel like I should write about here. But then I have to think of something to add to it. In my case, that’s jokes, mostly.

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• #### mathtuition88 4:45 am on Friday, 8 July, 2016 Permalink | Reply

Very interesting. Jordan Curve Theorem shows the rigor of math in action.

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• #### Joseph Nebus 4:15 am on Saturday, 9 July, 2016 Permalink | Reply

I like it for being the sort of theorem that seems too obvious to be useful. I have got it scheduled to be used in next Thursday’s post.

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• #### Mark Jackson 12:18 am on Sunday, 17 July, 2016 Permalink | Reply

“You won’t be surprised to know there’s versions of the Jordan curve theorem for solid objects in three-dimensional space.” Not that I ought to doubt this, but the counterintuitive discovery that the 3-sphere can be everted sprang to mind, and now I’m worried.

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• #### Joseph Nebus 4:42 pm on Wednesday, 20 July, 2016 Permalink | Reply

It’s a good worry and I’ll admit this is getting deeper into topology than I’m trained in. My suspicion is that the possible self-intersections of a sphere being turned inside-out cause it to fall outside the bounds of the Jordan-Brouwer Separation Theorem. I don’t have a good argument that has to be the case though; that’s just where I would start looking.

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## The Poincaré Homology Sphere, and Thinking What I’ll Do Next

Yenergy was good enough to write a comment about this, but people might have missed it. “Dodecahedral construction of the Poincaré homology sphere, part II” is up. The post is an illustration trying to describe several pages of the 1979 paper Eight Faces Of The Poincaré Homology 3-Sphere by R C Kirby and M G Sharlemann.

I admit I have to read it almost the same way a non-mathematician would. My education never took me into topology deep enough to be fluent in the notation or the working assumptions behind the paper. I may work my way farther than a non-mathematician, since I’ve been exposed to some of the symbols. The grammar of the argument is familiar. And many points of it are common to fields I did study. Nevertheless, even if you just skim the text, skipping over anything that seems too hard to follow, and look at the illustrations you’ll get something from it.

Past that, I wanted to thank everyone for seeing me into the start of May. I am figuring to give up the post-a-day schedule. It’s exciting to have three thousand-word and four posts of more variable lengths each week, but I need to relax that schedule some. I am considering, based on the conversation I got into with Elke Stangl about the Yukawa Potential, whether to do a string of essays about closed orbits. That would almost surely involve many more equations than is normal around here. But it could make for a nice change of pace.

• #### howardat58 3:29 pm on Sunday, 1 May, 2016 Permalink | Reply

The more years spent studying maths the happier one becomes as one realises that on at least the first reading one can skip ALL the “math” and jst read the text. If that doesn’t make any sense the symbols will only make things worse!

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• #### Joseph Nebus 3:48 pm on Wednesday, 4 May, 2016 Permalink | Reply

That is the piece that took me longest to learn, really. I think the bad habit comes from textbooks where, so often, the real work is done in a string of equations and the text surrounding it is meaningless or at least uninsightful. (“Now we move x to the left side” … thanks, saw that, but why ‘x’ and why the left side?)

Reading the text, at least when it explains what the thinking is, gets the lay of the land. Then the details have something to cling to.

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## Reading the Comics, February 11, 2016: Apples And Pointing Things Out Edition

I didn’t expect quite so many mathematically themed comic strips so soon after the last round. Most of them just highlight one or another familiar joke. So this edition is mostly just noting that yeah, the joke is there and has been successfully made. There’s an exception, though. Enjoy.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 7th of February is a cute chart. It’s got an unusual label to the x-axis. Now that I’ve seen it, I’m surprised not to see more jokes constructed this way.

Ruben Bolling’s Super-Fun-Pak Comix for the 7th of February was this essay’s Schrödinger’s Cat mention. I’m considering putting a moratorium on Schrödinger’s Cat strips, at least for a little while. I need to find something fresh to say about them.

Russell Myers’s Broom Hilda for the 8th of February inspires a Fermi problem. These are named for the great physicist Enrico Fermi, who often asked problems of estimation and order of magnitude. Given a few pieces of information, can you say about how big something might be? In this case, how many hours of work are spent peeling labels off grocery store apples? If we had the right information it would be easy to answer. How long does the average label take to peel off? How many apples get peeled each year? I admit not knowing either offhand. I would guess the average label-peeling time to be under five seconds, but if I wanted to be exact I’d get a bag, a stopwatch, and a sheet of paper for notes.

How many apples get peeled each year? That’s tougher. We might be able to get the total number of apples sold. But not every apple is sold with a label on it. A bag of apples doesn’t need individual labels, after all. But we might estimate what fraction of apples are sold loose and thus with labels by looking in local supermarkets. That requires assuming the turnover of apple stock is about the same whether the apple’s labelled or unlabelled. It also assumes our local supermarket is representative of the whole nation’s. But if we’re just looking for an idea of how big the number should be, or if we’re looking for what further information we have to determine, that’s good enough.

Wikipedia says the United States produced 4,100,046 metric tons of apples in 2012, the last year they have records for. If an apple is about a fifth of a kilogram, then, that implies something like 2,050,230,000 apples got sold in the United States that year. Let’s guess that three-quarters of them go right to industrial uses, into the hands of the Apple Pie Trusts and other corporate uses that don’t need labelling, while the remaining quarter go to consumers. That’s a wild guess on my part, but, industry is big. And of those, I’ll guess two-fifths get sold individually, with labels on. The rest can be sold in bags or whatnot. I’m basing that on what I kind of remember from my last trip to the farmer’s market with the free coffee bar and the bag-your-own candies.

So this implies something like 205,023,000 apples could be sold with labels. And if each label takes an average of five seconds, then this implies a total of 17,085,250 minutes spent unpeeling apple labels. That sounds like a big number, but it’s really only over 284,754 hours, or not quite 11,865 days. Of course, divided up among all the apple-eaters it’s not so much per year.

My number is wrong. I picked important bits of information out of thin air. But if I want to be more precise, I have an idea of what I need to learn. And I have an idea of how big I should expect the right answer to be. I can go from this to a better estimate, if I think now it’s worth being more exact.

Stephan Pastis’s Pearls Before Swine tries picking a fight with mathematicians on the 8th of February, with Rat boasting how he’s never used algebra. I’m not sure why bragging about not using algebra is supposed to be funny. The strip says it’s cathartic. I suppose. But it’s a joke that’s been told many times over and this doesn’t feel like a fresh use.

Rick Stromoski’s Soup To Nutz for the 8th of February is a fractions joke. Royboy perceives a difference between one-half of an orange and four-eighths of an orange. I can’t say there isn’t a difference in connotation between the two representations.

Percy Crosby’s Skippy for the 9th of February (a rerun from sometime 1928) shows Sookie with a ball. Well, a ball with a hole in it. A topologist would agree. If you’re interested in how the points on, or inside, an object connect to each other then a hoop like this is the same as a ball with a hole through it or a doughnut or bagel. This is my favorite for this group, because of the wonderful convergence of kid logic and serious mathematics.

Larry Wright’s Motley Classics for the 10th of February (a rerun from that date in 1988) is a joke about the terrors of word problems. I’m not convinced an authentic child would have trouble adding up all those cookies.

Hector D Cantu and Carlos Castellanos’s Baldo for the 11th of February reveals they have a week’s more lead time than most of the comics on the page.

• #### Barb Knowles 1:02 am on Saturday, 13 February, 2016 Permalink | Reply

I love Broom Hilda. And I am very happy that I can understand your explanation. 😃

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• #### Joseph Nebus 3:52 am on Tuesday, 16 February, 2016 Permalink | Reply

Oh, my, I’m glad I could explain the reasoning in a way that makes sense. Thank you.

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## Reading the Comics, August 14, 2015: Name-Dropping Edition

There have been fewer mathematically-themed comic strips than usual the past week, but they have been coming in yet. This week seems to have included a fair number of name-drops of interesting mathematical concepts.

David L Hoyt and Jeff Knurek’s Jumble (August 10) name-drops the abacus. It has got me wondering about how abacuses were made in the pre-industrial age. On the one hand they could in principle be made by anybody who has beads and rods. On the other hand, a skillfully made abacus will make the tool so much more effective. Who made and who sold them? I honestly don’t know.

David L Hoyt and Jeff Knurek’s Jumble for the 10th of August, 2015. The link will likely expire around the 10th of September.

Mick Mastroianni and Mason Mastroianni’s Dogs of C Kennel (August 11) has Tucker reveal that most of the mathematics he scrawls is just to make his work look harder. I suspect Tucker overdid his performance. My experience is you can get the audience’s eyes to glaze over with much less mathematics on the board.

Leigh Rubin’s Rubes (August 11) mentions chaos theory. It’s not properly speaking a Chaos Butterfly comic strip. But certainly it’s in the vicinity.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (August 11) name-drops Banach-Tarski. This is a reference to a famous-in-some-circles theorem, or paradox. The theorem, published in 1924 by Stefan Banach and Alfred Tarski, shows something astounding. It’s possible to take a ball, and disassemble it into a number of pieces. Then, doing nothing more than sliding and rotating the pieces, one can reassemble the pieces to get two balls each with the same volume of the original. If that doesn’t sound ridiculous enough, consider that it’s possible to do this trick by cutting the ball into as few as five pieces. (Four, if you’re willing to exclude the exact center of the original ball.) So you can see why this is called a paradox, and why this joke works for people who know the background.

Scott Hilburn’s The Argyle Sweater (August 12) illustrates that joke about rounding up the cattle you might have seen going around.

• #### sarcasticgoat 3:50 pm on Saturday, 15 August, 2015 Permalink | Reply

Are the scrambled words mathematical? Because I do this kind of thing a lot, and I could only get the first one ‘Query’??

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• #### Joseph Nebus 4:47 am on Tuesday, 18 August, 2015 Permalink | Reply

They’re not mathematical. Well, the third one I could stretch to be mathematical, if I tried, and I might use it if my Reading the Comics post was a little short that week. Only the punch line has a fairly direct mathematical link. The Jumble words don’t tend to be thematically linked. ‘QUERY’ comes up a lot in the puzzles, too.

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• #### sarcasticgoat 1:36 pm on Tuesday, 18 August, 2015 Permalink | Reply

I still haven’t got any of them….

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• #### Joseph Nebus 9:01 pm on Saturday, 22 August, 2015 Permalink | Reply

I find it often helps if you write out the letters of the jumbled word in alphabetical order and then try combinations from there. This destroys accidental patterns (like the appearance of ‘SUN’ in the third word) that might keep you from getting to the real word.

And if that doesn’t help they did print the answers the next day. And if that link’s expired since then there’s the Internet Anagram Server to the rescue.

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## Reading the Comics, April 22, 2015: April 21, 2015 Edition

I try to avoid doing Reading The Comics entries back-to-back since I know they can get a bit repetitive. How many ways can I say something is a student-resisting-the-word-problem joke? But if Comic Strip Master Command is going to send a half-dozen strips at least mentioning mathematical topics in a single day, how can I resist the challenge? Worse, what might they have waiting for me tomorrow? So here’s a bunch of comic strips from the 21st of April, 2015:

Mark Anderson’s Andertoons plays on the idea of a number being used up. I’m most tickled by this one. I have heard that the New York Yankees may be running short on uniform numbers after having so many retired. It appears they’ve only retired 17 numbers, but they do need numbers for a 40-player roster as well as managers and coaches and other participants. Also, and this delights me, two numbers are retired for two people each. (Number 8, for Yogi Berra and Bill Dickey, and Number 42, for Jackie Robinson and Mariano Rivera.)

• #### ivasallay 3:03 am on Sunday, 26 April, 2015 Permalink | Reply

Andertoons was really funny!

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• #### Joseph Nebus 8:20 pm on Monday, 27 April, 2015 Permalink | Reply

And there’s more on the way! Sometimes I wonder if I’m his underpaid publicity agent.

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## Reading The Comics, September 24, 2014: Explained In Class Edition

I’m a fan of early 20th century humorist Robert Benchley. You might not be yourself, but it’s rather likely that among the humorists you do like are a good number of people who are fans of his. He’s one of the people who shaped the modern American written-humor voice, and as such his writing hasn’t dated, the way that, for example, a 1920s comic strip will often seem to come from a completely different theory of what humor might be. Among Benchley’s better-remembered quotes, and one of those striking insights into humanity, not to mention the best productivity tip I’ve ever encountered, was something he dubbed the Benchley Principle: “Anyone can do any amount of work, provided it isn’t the work he is supposed to be doing at the moment.” One of the comics in today’s roundup of mathematics-themed comics brought the Benchley Principle to mind, and I mean to get to how it did and why.

Eric The Circle (by ‘Griffinetsabine’ this time) (September 18) steps again into the concerns of anthropomorphized shapes. It’s also got a charming-to-me mention of the trapezium, the geometric shape that’s going to give my mathematics blog whatever immortality it shall have.

Bill Watterson’s Calvin and Hobbes (September 20, rerun) dodged on me: I thought after the strip from the 19th that there’d be a fresh round of explanations of arithmetic, this time including imaginary numbers like “eleventeen” and “thirty-twelve” and the like. Not so. After some explanation of addition by Calvin’s Dad,
Spaceman Spiff would take up the task on the 22nd of smashing together Mysterio planets 6 and 5, which takes a little time to really get started, and finally sees the successful collision of the worlds. Let this serve as a reminder: translating a problem to a real-world application can be a fine way to understand what is wanted, but you have to make sure that in the translation you preserve the result you wanted from the calculation.

Rick Detorie’s One Big Happy for the 21st of September, 2014. I confess ignorance as to whether these odds are accurate.

It’s Rick DeTorie’s One Big Happy (September 21) which brought the Benchley Principle to my mind. Here, Joe is shown to know extremely well the odds of poker hands, but to have no chance at having learned the multiplication table. It seems like something akin to Benchley’s Principle is at work here: Joe memorizing the times tables might be socially approved, but it isn’t what he wants to do, and that’s that. But inspiring the desire to know something is probably the one great challenge facing everyone who means to teach, isn’t it?

Jonathan Lemon’s Rabbits Against Magic (September 21) features a Möbius strip joke that I imagine was a good deal of fun to draw. The Möbius strip is one of those concepts that really catches the imagination, since it seems to defy intuition that something should have only the one side. I’m a little surprise that topology isn’t better-popularized, as it seems like it should be fairly accessible — you don’t need equations to get some surprising results, and you can draw pictures — but maybe I just don’t understand the field well enough to understand what’s difficult about bringing it to a mass audience.

Hector D. Cantu and Carlos Castellanos’s Baldo (September 23) tells a joke about percentages and students’ self-confidence about how good they are with “numbers”. In strict logic, yes, the number of people who say they are and who say they aren’t good at numbers should add up to something under 100 percent. But people don’t tend to be logically perfect, and are quite vulnerable to the way questions are framed, so the scenario is probably more plausible in the real world than the writer intended.

Steve Moore’s In The Bleachers (September 23) falls back on the most famous of all equations as representative of “something it takes a lot of intelligence to understand”.

• #### Angie Mc 4:10 am on Thursday, 25 September, 2014 Permalink | Reply

Joe could be my sons, except instead of poker odds they know baseball stats! Multiplication table? Not real good :) Thanks.

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• #### fluffy 5:48 pm on Thursday, 25 September, 2014 Permalink | Reply

The various statistics tables I can find show that those stats are in the ballpark (within the margin of error allowed for most probability tables being given in percentages instead of ratios). I’m just bored enough to work them out. For all hand counts I’ll assume the cards have been sorted in some order, so permutations don’t matter.

There are 52!/(47!5!)=2598960 possible hands.

Royal flush: 4 possible hands = 649740:1
Straight flush (assuming ace can be low): 40 hands, of which 4 are also royal flushes, so 36 hands = 72193:1
Four of a kind: 13*48=624 possible hands = 4165:1
Full house: I am having a surprisingly difficult time working this one out in a way that matches any of the probability charts I’m finding.
Flush: 52*12*11*10*9/5! – 40 = 5108 hands = 509:1

I suspect the cartoonist’s persistent off-by-one errors are a sign that he’s removing the 1 from the odds, forgetting that 1:1 odds is the same as 100% (and not 50%).

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• #### Joseph Nebus 8:21 pm on Sunday, 28 September, 2014 Permalink | Reply

Thank you for the work put into this! And I suspect that you’re right about the off-by-one errors, although that makes me wonder why he has got it wrong then. If Detorie had been used some reference book listing card odds, why would the book have gotten it a little wrong? But if he’d worked it out himself, why go to that much work?

I wonder if he got it from a probability textbook that used the card problems as a good natural problem. The textbook writer could have made the one-off mistake from not understanding the odds notation, and the writing could easily not be read by someone who’d catch the error before publication.

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• #### elkement 9:21 pm on Saturday, 27 September, 2014 Permalink | Reply

Baldo’s joke is a variation of the classic ‘There are three kinds of people – those who can count, and those who cannot!’.

It’s interesting what you said about humor – true… often old movies and dated books that most likely were intended to be funny just strikes us odd. So I guess humor is always constructed in relation to what is important and ‘serious’ at a certain point of time.

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• #### Joseph Nebus 8:26 pm on Sunday, 28 September, 2014 Permalink | Reply

You’re right about Baldo’s joke, yeah.

Trav S D — Travalanche, on WordPress — has a magnificent book about the history of Vaudeville (No Applause, Just Throw Money: The Book That Made Vaudeville Famous) which points out that a major shift in American humor, at least, came from radio and talking pictures adapting the vaudevillian style of comedy and then trimming it way down. Part of this was that the performers didn’t have to worry about being heard and understood even in the way back seats (so they could speak quicker, and more punchily, and didn’t have to repeat setups quite so much as they could be more sure people heard them the first time), and this did a lot for the pacing of comic writing.

The process, like all processes, was surely more complicated than that. But it is, typically, a lot easier to see why something written as comedy in (say) 1935 was funny than something from 1915. (Though not slapstick; that stays nice and clear and easy to understand.)

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