## From my Seventh A-to-Z: Tiling (the accidental remake)

For the 2020 A-to-Z I took the suggestion to write about tiling. It’s a fun field with many interesting wrinkles. And I realized after publishing that I had already written about Tiling, just two years before. There was no scrambling together a replacement essay, so I had to let it stand as is.

The accidental remake allows for some interesting studies, though. The two essays have very similar structures, which probably reflects that I came to both essays with similar rough ideas what to write, and went to similar sources to fill in details. The second essay turned out longer. Also, I think, better. I did a bit more tracking down specifics, such as trying to find Hao Wang’s paper and see just what it says. And rewriting is often key to good writing. This offers lessons in preparing these essays for book publication.

Mr Wu, author of the Singapore Maths Tuition blog, had an interesting suggestion for the letter T: Talent. As in mathematical talent. It’s a fine topic but, in the end, too far beyond my skills. I could share some of the legends about mathematical talent I’ve received. But what that says about the culture of mathematicians is a deeper and more important question.

So I picked my own topic for the week. I do have topics for next week — U — and the week after — V — chosen. But the letters W and X? I’m still open to suggestions. I’m open to creative or wild-card interpretations of the letters. Especially for X and (soon) Z. Thanks for sharing any thoughts you care to.

# Tiling.

Think of a floor. Imagine you are bored. What do you notice?

What I hope you notice is that it is covered. Perhaps by carpet, or concrete, or something homogeneous like that. Let’s ignore that. My floor is covered in small pieces, repeated. My dining room floor is slats of wood, about three and a half feet long and two inches wide. The slats are offset from the neighbors so there’s a pleasant strong line in one direction and stippled lines in the other. The kitchen is squares, one foot on each side. This is a grid we could plot high school algebra functions on. The bathroom is more elaborate. It has white rectangles about two inches long, tan rectangles about two inches long, and black squares. Each rectangle is perpendicular to ones of the other color, and arranged to bisect those. The black squares fill the gaps where no rectangle would fit.

Move from my house to pure mathematics. It’s easy to turn the floor of a room into abstract mathematics. We start with something to tile. Usually this is the infinite, two-dimensional plane. The thing you get if you have a house and forget the walls. Sometimes we look to tile the hyperbolic plane, a different geometry that we of course represent with a finite circle. (Setting particular rules about how to measure distance makes this equivalent to a funny-shaped plane.) Or the surface of a sphere, or of a torus, or something like that. But if we don’t say otherwise, it’s the plane.

What to cover it with? … Smaller shapes. We have a mathematical tiling if we have a collection of not-overlapping open sets. And if those open sets, plus their boundaries, cover the whole plane. “Cover” here means what “cover” means in English, only using more technical words. These sets — these tiles — can be any shape. We can have as many or as few of them as we like. We can even add markings to the tiles, give them colors or patterns or such, to add variety to the puzzles.

(And if we want, we can do this in other dimensions. There are good “tiling” questions to ask about how to fill a three-dimensional space, or a four-dimensional one, or more.)

Having an unlimited collection of tiles is nice. But mathematicians learn to look for how little we need to do something. Here, we look for the smallest number of distinct shapes. As with tiling an actual floor, we can get all the tiles we need. We can rotate them, too, to any angle. We can flip them over and put the “top” side “down”, something kitchen tiles won’t let us do. Can we reflect them? Use the shape we’d get looking at the mirror image of one? That’s up to whoever’s writing this paper.

What shapes will work? Well, squares, for one. We can prove that by looking at a sheet of graph paper. Rectangles would work too. We can see that by drawing boxes around the squares on our graph paper. Two-by-one blocks, three-by-two blocks, 40-by-1 blocks, these all still cover the paper and we can imagine covering the plane. If we like, we can draw two-by-two squares. Squares made up of smaller squares. Or repeat this: draw two-by-one rectangles, and then group two of these rectangles together to make two-by-two squares.

We can take it on faith that, oh, rectangles π long by e wide would cover the plane too. These can all line up in rows and columns, the way our squares would. Or we can stagger them, like bricks or my dining room’s wood slats are.

How about parallelograms? Those, it turns out, tile exactly as well as rectangles or squares do. Grids or staggered, too. Ah, but how about trapezoids? Surely they won’t tile anything. Not generally, anyway. The slanted sides will, most of the time, only fit in weird winding circle-like paths.

Unless … take two of these trapezoid tiles. We’ll set them down so the parallel sides run horizontally in front of you. Rotate one of them, though, 180 degrees. And try setting them — let’s say so the longer slanted line of both trapezoids meet, edge to edge. These two trapezoids come together. They make a parallelogram, although one with a slash through it. And we can tile parallelograms, whether or not they have a slash.

OK, but if you draw some weird quadrilateral shape, and it’s not anything that has a more specific name than “quadrilateral”? That won’t tile the plane, will it?

It will! In one of those turns that surprises and impresses me every time I run across it again, any quadrilateral can tile the plane. It opens up so many home decorating options, if you get in good with a tile maker.

That’s some good news for quadrilateral tiles. How about other shapes? Triangles, for example? Well, that’s good news too. Take two of any identical triangle you like. Turn one of them around and match sides of the same length. The two triangles, bundled together like that, are a quadrilateral. And we can use any quadrilateral to tile the plane, so we’re done.

How about pentagons? … With pentagons, the easy times stop. It turns out not every pentagon will tile the plane. The pentagon has to be of the right kind to make it fit. If the pentagon is in one of these kinds, it can tile the plane. If not, not. There are fifteen families of tiling known. The most recent family was discovered in 2015. It’s thought that there are no other convex pentagon tilings. I don’t know whether the proof of that is generally accepted in tiling circles. And we can do more tilings if the pentagon doesn’t need to be convex. For example, we can cut any parallelogram into two identical pentagons. So we can make as many pentagons as we want to cover the plane. But we can’t assume any pentagon we like will do it.

Hexagons look promising. First, a regular hexagon tiles the plane, as strategy games know. There are also at least three families of irregular hexagons that we know can tile the plane.

And there the good times end. There are no convex heptagons or octagons or any other shape with more sides that tile the plane.

Not by themselves, anyway. If we have more than one tile shape we can start doing fine things again. Octagons assisted by squares, for example, will tile the plane. I’ve lived places with that tiling. Or something that looks like it. It’s easier to install if you have square tiles with an octagon pattern making up the center, and triangle corners a different color. These squares come together to look like octagons and squares.

And this leads to a fun avenue of tiling. Hao Wang, in the early 60s, proposed a sort of domino-like tiling. You may have seen these in mathematics puzzles, or in toys. Each of these Wang Tiles, or Wang Dominoes, is a square. But the square is cut along the diagonals, into four quadrants. Each quadrant is a right triangle. Each quadrant, each triangle, is one of a finite set of colors. Adjacent triangles can have the same color. You can place down tiles, subject only to the rule that the tile edge has to have the same color on both sides. So a tile with a blue right-quadrant has to have on its right a tile with a blue left-quadrant. A tile with a white upper-quadrant on its top has, above it, a tile with a white lower-quadrant.

In 1961 Wang conjectured that if a finite set of these tiles will tile the plane, then there must be a periodic tiling. That is, if you picked up the plane and slid it a set horizontal and vertical distance, it would all look the same again. This sort of translation is common. All my floors do that. If we ignore things like the bounds of their rooms, or the flaws in their manufacture or installation or where a tile broke in some mishap.

This is not to say you couldn’t arrange them aperiodically. You don’t even need Wang Tiles for that. Get two colors of square tile, a white and a black, and lay them down based on whether the next decimal digit of π is odd or even. No; Wang’s conjecture was that if you had tiles that you could lay down aperiodically, then you could also arrange them to set down periodically. With the black and white squares, lay down alternate colors. That’s easy.

In 1964, Robert Berger proved Wang’s conjecture was false. He found a collection of Wang Tiles that could only tile the plane aperiodically. In 1966 he published this in the Memoirs of the American Mathematical Society. The 1964 proof was for his thesis. 1966 was its general publication. I mention this because while doing research I got irritated at how different sources dated this to 1964, 1966, or sometimes 1961. I want to have this straightened out. It appears Berger had the proof in 1964 and the publication in 1966.

I would like to share details of Berger’s proof, but haven’t got access to the paper. What fascinates me about this is that Berger’s proof used a set of 20,426 different tiles. I assume he did not work this all out with shards of construction paper, but then, how to get 20,426 of anything? With computer time as expensive as it was in 1964? The mystery of how he got all these tiles is worth an essay of its own and regret I can’t write it.

Berger conjectured that a smaller set might do. Quite so. He himself reduced the set to 104 tiles. Donald Knuth in 1968 modified the set down to 92 tiles. In 2015 Emmanuel Jeandel and Michael Rao published a set of 11 tiles, using four colors. And showed by computer search that a smaller set of tiles, or fewer colors, would not force some aperiodic tiling to exist. I do not know whether there might be other sets of 11, four-colored, tiles that work. You can see the set at the top of Wikipedia’s page on Wang Tiles.

These Wang Tiles, all squares, inspired variant questions. Could there be other shapes that only aperiodically tile the plane? What if they don’t have to be squares? Raphael Robinson, in 1971, came up with a tiling using six shapes. The shapes have patterns on them too, usually represented as colored lines. Tiles can be put down only in ways that fit and that make the lines match up.

Among my readers are people who have been waiting, for 1800 words now, for Roger Penrose. It’s now that time. In 1974 Penrose published an aperiodic tiling, one based on pentagons and using a set of six tiles. You’ve never heard of that either, because soon after he found a different set, based on a quadrilateral cut into two shapes. The shapes, as with Wang Tiles or Robinson’s tiling, have rules about what edges may be put against each other. Penrose — and independently Robert Ammann — also developed another set, this based on a pair of rhombuses. These have rules about what edges may tough one another, and have patterns on them which must line up.

The Penrose tiling became, and stayed famous. (Ammann, an amateur, never had much to do with the mathematics community. He died in 1994.) Martin Gardner publicized it, and it leapt out of mathematicians’ hands into the popular culture. At least a bit. That it could give you nice-looking floors must have helped.

To show that the rhombus-based Penrose tiling is aperiodic takes some arguing. But it uses tools already used in this essay. Remember drawing rectangles around several squares? And then drawing squares around several of these rectangles? We can do that with these Penrose-Ammann rhombuses. From the rhombus tiling we can draw bigger rhombuses. Ones which, it turns out, follow the same edge rules that the originals do. So that we can go again, grouping these bigger rhombuses into even-bigger rhombuses. And into even-even-bigger rhombuses. And so on.

What this gets us is this: suppose the rhombus tiling is periodic. Then there’s some finite-distance horizontal-and-vertical move that leaves the pattern unchanged. So, the same finite-distance move has to leave the bigger-rhombus pattern unchanged. And this same finite-distance move has to leave the even-bigger-rhombus pattern unchanged. Also the even-even-bigger pattern unchanged.

Keep bundling rhombuses together. You get eventually-big-enough-rhombuses. Now, think of how far you have to move the tiles to get a repeat pattern. Especially, think how many eventually-big-enough-rhombuses it is. This distance, the move you have to make, is less than one eventually-big-enough rhombus. (If it’s not you aren’t eventually-big-enough yet. Bundle them together again.) And that doesn’t work. Moving one tile over without changing the pattern makes sense. Moving one-half a tile over? That doesn’t. So the eventually-big-enough pattern can’t be periodic, and so, the original pattern can’t be either. This is explained in graphic detail a nice Powerpoint slide set from Professor Alexander F Ritter, A Tour Of Tilings In Thirty Minutes.

It’s possible to do better. In 2010 Joshua E S Socolar and Joan M Taylor published a single tile that can force an aperiodic tiling. As with the Wang Tiles, and Robinson shapes, and the Penrose-Ammann rhombuses, markings are part of it. They have to line up so that the markings — in two colors, in the renditions I’ve seen — make sense. With the Penrose tilings, you can get away from the pattern rules for the edges by replacing them with little notches. The Socolar-Taylor shape can make a similar trade. Here the rules are complex enough that it would need to be a three-dimensional shape, one that looks like the dilithium housing of the warp core. You can see the tile — in colored, marked form, and also in three-dimensional tile shape — at the PDF here. It’s likely not coming to the flooring store soon.

It’s all wonderful, but is it useful? I could go on a few hundred words about, particularly, crystals and quasicrystals. These are important for materials science. Especially these days as we have harnessed slightly-imperfect crystals to be our computers. I don’t care. These are lovely to look at. If you see nothing appealing in a great heap of colors and polygons spread over the floor there are things we cannot communicate about. Tiling is a delight; what more do you need?

Thanks for your attention. This and all of my 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be at this link. See you next week, I hope.

## My All 2020 Mathematics A to Z: Tiling

Mr Wu, author of the Singapore Maths Tuition blog, had an interesting suggestion for the letter T: Talent. As in mathematical talent. It’s a fine topic but, in the end, too far beyond my skills. I could share some of the legends about mathematical talent I’ve received. But what that says about the culture of mathematicians is a deeper and more important question.

So I picked my own topic for the week. I do have topics for next week — U — and the week after — V — chosen. But the letters W and X? I’m still open to suggestions. I’m open to creative or wild-card interpretations of the letters. Especially for X and (soon) Z. Thanks for sharing any thoughts you care to.

# Tiling.

Think of a floor. Imagine you are bored. What do you notice?

What I hope you notice is that it is covered. Perhaps by carpet, or concrete, or something homogeneous like that. Let’s ignore that. My floor is covered in small pieces, repeated. My dining room floor is slats of wood, about three and a half feet long and two inches wide. The slats are offset from the neighbors so there’s a pleasant strong line in one direction and stippled lines in the other. The kitchen is squares, one foot on each side. This is a grid we could plot high school algebra functions on. The bathroom is more elaborate. It has white rectangles about two inches long, tan rectangles about two inches long, and black squares. Each rectangle is perpendicular to ones of the other color, and arranged to bisect those. The black squares fill the gaps where no rectangle would fit.

Move from my house to pure mathematics. It’s easy to turn the floor of a room into abstract mathematics. We start with something to tile. Usually this is the infinite, two-dimensional plane. The thing you get if you have a house and forget the walls. Sometimes we look to tile the hyperbolic plane, a different geometry that we of course represent with a finite circle. (Setting particular rules about how to measure distance makes this equivalent to a funny-shaped plane.) Or the surface of a sphere, or of a torus, or something like that. But if we don’t say otherwise, it’s the plane.

What to cover it with? … Smaller shapes. We have a mathematical tiling if we have a collection of not-overlapping open sets. And if those open sets, plus their boundaries, cover the whole plane. “Cover” here means what “cover” means in English, only using more technical words. These sets — these tiles — can be any shape. We can have as many or as few of them as we like. We can even add markings to the tiles, give them colors or patterns or such, to add variety to the puzzles.

(And if we want, we can do this in other dimensions. There are good “tiling” questions to ask about how to fill a three-dimensional space, or a four-dimensional one, or more.)

Having an unlimited collection of tiles is nice. But mathematicians learn to look for how little we need to do something. Here, we look for the smallest number of distinct shapes. As with tiling an actual floor, we can get all the tiles we need. We can rotate them, too, to any angle. We can flip them over and put the “top” side “down”, something kitchen tiles won’t let us do. Can we reflect them? Use the shape we’d get looking at the mirror image of one? That’s up to whoever’s writing this paper.

What shapes will work? Well, squares, for one. We can prove that by looking at a sheet of graph paper. Rectangles would work too. We can see that by drawing boxes around the squares on our graph paper. Two-by-one blocks, three-by-two blocks, 40-by-1 blocks, these all still cover the paper and we can imagine covering the plane. If we like, we can draw two-by-two squares. Squares made up of smaller squares. Or repeat this: draw two-by-one rectangles, and then group two of these rectangles together to make two-by-two squares.

We can take it on faith that, oh, rectangles π long by e wide would cover the plane too. These can all line up in rows and columns, the way our squares would. Or we can stagger them, like bricks or my dining room’s wood slats are.

How about parallelograms? Those, it turns out, tile exactly as well as rectangles or squares do. Grids or staggered, too. Ah, but how about trapezoids? Surely they won’t tile anything. Not generally, anyway. The slanted sides will, most of the time, only fit in weird winding circle-like paths.

Unless … take two of these trapezoid tiles. We’ll set them down so the parallel sides run horizontally in front of you. Rotate one of them, though, 180 degrees. And try setting them — let’s say so the longer slanted line of both trapezoids meet, edge to edge. These two trapezoids come together. They make a parallelogram, although one with a slash through it. And we can tile parallelograms, whether or not they have a slash.

OK, but if you draw some weird quadrilateral shape, and it’s not anything that has a more specific name than “quadrilateral”? That won’t tile the plane, will it?

It will! In one of those turns that surprises and impresses me every time I run across it again, any quadrilateral can tile the plane. It opens up so many home decorating options, if you get in good with a tile maker.

That’s some good news for quadrilateral tiles. How about other shapes? Triangles, for example? Well, that’s good news too. Take two of any identical triangle you like. Turn one of them around and match sides of the same length. The two triangles, bundled together like that, are a quadrilateral. And we can use any quadrilateral to tile the plane, so we’re done.

How about pentagons? … With pentagons, the easy times stop. It turns out not every pentagon will tile the plane. The pentagon has to be of the right kind to make it fit. If the pentagon is in one of these kinds, it can tile the plane. If not, not. There are fifteen families of tiling known. The most recent family was discovered in 2015. It’s thought that there are no other convex pentagon tilings. I don’t know whether the proof of that is generally accepted in tiling circles. And we can do more tilings if the pentagon doesn’t need to be convex. For example, we can cut any parallelogram into two identical pentagons. So we can make as many pentagons as we want to cover the plane. But we can’t assume any pentagon we like will do it.

Hexagons look promising. First, a regular hexagon tiles the plane, as strategy games know. There are also at least three families of irregular hexagons that we know can tile the plane.

And there the good times end. There are no convex heptagons or octagons or any other shape with more sides that tile the plane.

Not by themselves, anyway. If we have more than one tile shape we can start doing fine things again. Octagons assisted by squares, for example, will tile the plane. I’ve lived places with that tiling. Or something that looks like it. It’s easier to install if you have square tiles with an octagon pattern making up the center, and triangle corners a different color. These squares come together to look like octagons and squares.

And this leads to a fun avenue of tiling. Hao Wang, in the early 60s, proposed a sort of domino-like tiling. You may have seen these in mathematics puzzles, or in toys. Each of these Wang Tiles, or Wang Dominoes, is a square. But the square is cut along the diagonals, into four quadrants. Each quadrant is a right triangle. Each quadrant, each triangle, is one of a finite set of colors. Adjacent triangles can have the same color. You can place down tiles, subject only to the rule that the tile edge has to have the same color on both sides. So a tile with a blue right-quadrant has to have on its right a tile with a blue left-quadrant. A tile with a white upper-quadrant on its top has, above it, a tile with a white lower-quadrant.

In 1961 Wang conjectured that if a finite set of these tiles will tile the plane, then there must be a periodic tiling. That is, if you picked up the plane and slid it a set horizontal and vertical distance, it would all look the same again. This sort of translation is common. All my floors do that. If we ignore things like the bounds of their rooms, or the flaws in their manufacture or installation or where a tile broke in some mishap.

This is not to say you couldn’t arrange them aperiodically. You don’t even need Wang Tiles for that. Get two colors of square tile, a white and a black, and lay them down based on whether the next decimal digit of π is odd or even. No; Wang’s conjecture was that if you had tiles that you could lay down aperiodically, then you could also arrange them to set down periodically. With the black and white squares, lay down alternate colors. That’s easy.

In 1964, Robert Berger proved Wang’s conjecture was false. He found a collection of Wang Tiles that could only tile the plane aperiodically. In 1966 he published this in the Memoirs of the American Mathematical Society. The 1964 proof was for his thesis. 1966 was its general publication. I mention this because while doing research I got irritated at how different sources dated this to 1964, 1966, or sometimes 1961. I want to have this straightened out. It appears Berger had the proof in 1964 and the publication in 1966.

I would like to share details of Berger’s proof, but haven’t got access to the paper. What fascinates me about this is that Berger’s proof used a set of 20,426 different tiles. I assume he did not work this all out with shards of construction paper, but then, how to get 20,426 of anything? With computer time as expensive as it was in 1964? The mystery of how he got all these tiles is worth an essay of its own and regret I can’t write it.

Berger conjectured that a smaller set might do. Quite so. He himself reduced the set to 104 tiles. Donald Knuth in 1968 modified the set down to 92 tiles. In 2015 Emmanuel Jeandel and Michael Rao published a set of 11 tiles, using four colors. And showed by computer search that a smaller set of tiles, or fewer colors, would not force some aperiodic tiling to exist. I do not know whether there might be other sets of 11, four-colored, tiles that work. You can see the set at the top of Wikipedia’s page on Wang Tiles.

These Wang Tiles, all squares, inspired variant questions. Could there be other shapes that only aperiodically tile the plane? What if they don’t have to be squares? Raphael Robinson, in 1971, came up with a tiling using six shapes. The shapes have patterns on them too, usually represented as colored lines. Tiles can be put down only in ways that fit and that make the lines match up.

Among my readers are people who have been waiting, for 1800 words now, for Roger Penrose. It’s now that time. In 1974 Penrose published an aperiodic tiling, one based on pentagons and using a set of six tiles. You’ve never heard of that either, because soon after he found a different set, based on a quadrilateral cut into two shapes. The shapes, as with Wang Tiles or Robinson’s tiling, have rules about what edges may be put against each other. Penrose — and independently Robert Ammann — also developed another set, this based on a pair of rhombuses. These have rules about what edges may tough one another, and have patterns on them which must line up.

The Penrose tiling became, and stayed famous. (Ammann, an amateur, never had much to do with the mathematics community. He died in 1994.) Martin Gardner publicized it, and it leapt out of mathematicians’ hands into the popular culture. At least a bit. That it could give you nice-looking floors must have helped.

To show that the rhombus-based Penrose tiling is aperiodic takes some arguing. But it uses tools already used in this essay. Remember drawing rectangles around several squares? And then drawing squares around several of these rectangles? We can do that with these Penrose-Ammann rhombuses. From the rhombus tiling we can draw bigger rhombuses. Ones which, it turns out, follow the same edge rules that the originals do. So that we can go again, grouping these bigger rhombuses into even-bigger rhombuses. And into even-even-bigger rhombuses. And so on.

What this gets us is this: suppose the rhombus tiling is periodic. Then there’s some finite-distance horizontal-and-vertical move that leaves the pattern unchanged. So, the same finite-distance move has to leave the bigger-rhombus pattern unchanged. And this same finite-distance move has to leave the even-bigger-rhombus pattern unchanged. Also the even-even-bigger pattern unchanged.

Keep bundling rhombuses together. You get eventually-big-enough-rhombuses. Now, think of how far you have to move the tiles to get a repeat pattern. Especially, think how many eventually-big-enough-rhombuses it is. This distance, the move you have to make, is less than one eventually-big-enough rhombus. (If it’s not you aren’t eventually-big-enough yet. Bundle them together again.) And that doesn’t work. Moving one tile over without changing the pattern makes sense. Moving one-half a tile over? That doesn’t. So the eventually-big-enough pattern can’t be periodic, and so, the original pattern can’t be either. This is explained in graphic detail a nice Powerpoint slide set from Professor Alexander F Ritter, A Tour Of Tilings In Thirty Minutes.

It’s possible to do better. In 2010 Joshua E S Socolar and Joan M Taylor published a single tile that can force an aperiodic tiling. As with the Wang Tiles, and Robinson shapes, and the Penrose-Ammann rhombuses, markings are part of it. They have to line up so that the markings — in two colors, in the renditions I’ve seen — make sense. With the Penrose tilings, you can get away from the pattern rules for the edges by replacing them with little notches. The Socolar-Taylor shape can make a similar trade. Here the rules are complex enough that it would need to be a three-dimensional shape, one that looks like the dilithium housing of the warp core. You can see the tile — in colored, marked form, and also in three-dimensional tile shape — at the PDF here. It’s likely not coming to the flooring store soon.

It’s all wonderful, but is it useful? I could go on a few hundred words about, particularly, crystals and quasicrystals. These are important for materials science. Especially these days as we have harnessed slightly-imperfect crystals to be our computers. I don’t care. These are lovely to look at. If you see nothing appealing in a great heap of colors and polygons spread over the floor there are things we cannot communicate about. Tiling is a delight; what more do you need?

Thanks for your attention. This and all of my 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be at this link. See you next week, I hope.

## 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.

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.

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.

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.

## Reading the Comics, April 27, 2016: Closing The Month (April) Out Edition

I concede this isn’t a set of mathematically-themed comics that inspires deep discussions. That’s all right. It’s got three that I can give pictures for, which is important. Also it means I can wrap up April with another essay. This gives me two months in a row of posting something every day, and I’d have bet that couldn’t happen.

Ted Shearer’s Quincy for the 1st of March, 1977, rerun the 25th of April, is not actually a “mathematics is useless in the real world” comic strip. It’s more about the uselessness of any school stuff in the face of problems like the neighborhood bully. Arithmetic just fits on the blackboard efficiently. There’s some sadness in the setting. There’s also some lovely artwork, though, and it’s worth noticing it. The lines are nice and expressive, and the greyscale wash well-placed. It’s good to look at.

dro-mo for the 26th I admit I’m not sure what exactly is going on. I suppose it’s a contest to describe the most interesting geometric shape. I believe the fourth panel is meant to be a representation of the tesseract, the four-dimensional analog of the cube. This causes me to realize I don’t remember any illustrations of a five-dimensional hypercube. Wikipedia has a couple, but they’re a bit disappointing. They look like the four-dimensional cube with some more lines. Maybe it has some more flattering angles somewhere.

Bill Amend’s FoxTrot for the 26th (a rerun from the 3rd of May, 2005) poses a legitimate geometry problem. Amend likes to do this. It was one of the things that first attracted me to the comic strip, actually, that his mathematics or physics or computer science jokes were correct. “Determine the sum of the interior angles for an N-sided polygon” makes sense. The commenters at Gocomics.com are quick to say what the sum is. If there are N sides, the interior angles sum up to (N – 2) times 180 degrees. I believe the commenters misread the question. “Determine”, to me, implies explaining why the sum is given by that formula. That’s a more interesting question and I think still reasonable for a freshman in high school. I would do it by way of triangles.

David L Hoyt and Jeff Knurek’s Jumble for the 27th of April gives us another arithmetic puzzle. As often happens, you can solve the surprise-answer by looking hard at the cartoon and picking up the clues from there. And it gives us an anthropomorphic-numerals gag for this collection.

Bill Holbrook’s On The Fastrack for the 28th of April has the misanthropic Fi explain some of the glories of numbers. As she says, they can be reliable, consistent partners. If you have learned something about ‘6’, then it not only is true, it must be true, at least if we are using ‘6’ to mean the same thing. This is the sort of thing that transcends ordinary knowledge and that’s so wonderful about mathematics.

Fi describes ‘x’ and ‘y’ as “shifty little goobers”, which is a bit unfair. ‘x’ and ‘y’ are names we give to numbers when we don’t yet know what values they have, or when we don’t care what they have. We’ve settled on those names mostly in imitation of Réné Descartes. Trying to do without names is a mess. You can do it, but it’s rather like novels in which none of the characters has a name. The most skilled writers can carry that off. The rest of us make a horrid mess. So we give placeholder names. Before ‘x’ and ‘y’ mathematicians would use names like ‘the thing’ (well, ‘re’) or ‘the heap’. Anything that the quantity we talk about might measure. It’s done better that way.

## Reading the Comics, September 24, 2015: Yes, I Do So Edition

Yes, in this roundup of mathematically-themed comic strips I talk seriously about the educational techniques of the fictional Great Smokey Mountains community where the comic strip Barney Google and Snuffy Smith takes place. I accept the implications of this.

John Rose’s Barney Google And Snuffy Smith for the 23rd of September is your standard snarky-response joke. I’m a bit surprised to see that at whatever class level Jughaid’s in they’re using “x” to stand in for the not-yet-known number. I thought empty boxes or question marks were more common. But I also think Miz Prunelly’s not working most effectively by getting angry at Jughaid for not knowing what x is.

I would suggest trying this: can Jughaid find some possible values of x that are definitely too small? And some possible values that are certainly too big? Then what kinds of numbers are both not-too-small and not-too-big? One standard mathematician’s trick for finding an unknown quantity is to show that it can’t be smaller than some number, giving us a lower bound. And then show it can’t be larger than some number, giving us an upper bound. If the lower bound and the upper bound are the same number, we’re done. If they’re not the same number we might have to go looking, but at least we’ve got a better idea what a correct answer should look like. If the lower bound is a larger number than the upper bound, we have to go back and check whether there actually is an answer, or if we started off in the wrong direction.

Scott Adams’s Dilbert Classics for the 23rd of September (a rerun from the 30th of July, 1992) mentions “conversational geometry”. It’s built on a bit of geometry that somehow escaped into being a common allusion, and that occasionally riles up grammar nerds. The problem is trying to use “turned around 360 degrees” for “turned completely around”. 360 degrees is certainly turning something all the way around, but it leaves the thing back where it started, apparently unchanged. (Well, there are some oddball structures where you can rotate something 360 degrees and have it not back the way it started, but those only occur in abstract mathematical constructions and in some — not all! — subatomic particles. Yes, it’s weird. It’s like that.)

The grammar nerd will insist that what’s meant is to turn something 180 degrees, reversing its direction. Or maybe changed 90 degrees, looking perpendicular to whatever the original situation was. Personally I can’t get upset about a shorthand English phrase not making literal sense, because there are only about six shorthand English phrases that make even the slightest literal sense, and four of those are tapas orders. Eventually you have to stop with the rage and just say something already. And rotating 360 degrees is a different process from rotating not at all. You move, you break your focus, you break your attention. Even if you face the same things again you face them having refreshed your perceptions. You might now see something you had not before.

John Zakour and Scott Roberts’s Maria’s Day for the 23rd of September asserts that mathematics is important so that one can check one’s accountants. This is true, although it’s hardly everything mathematics is enjoyable for. And while I don’t often get to call attention to comic strip artwork, do look at the different papers; there’s some fun there.

Pab Sungenis’s New Adventures of Queen Victoria for the 24th of September — and the days around it — have seen Victoria and Nikola Tesla facing the end result of too much holiday creep: a holiday singularity. By a singularity a mathematician means a point where stuff gets weird: where a function isn’t defined, where a surface breaks off, where several independent solutions suddenly stop being independent, that sort of thing. It’ll often correspond with some measure becoming infinitely large (as a positive or a negative number), though I don’t think it’s safe to say that always happens.

We generally can’t say what’s happening at a singularity. But the existence of a singularity, and what it behaves like, can tell us something about what’s happening away from the singularity. It can happen, for example, that a singularity is removable. That is, if a function is undefined for some values, perhaps we can come up with a logically compelling definition for what it might do at those values. If you can remove a singularity then we call this a “removable singularity”. This serves to show you don’t necessarily need grad school to understand everything mathematicians are saying. Sometimes a singularity can’t be removed, and those are known as “nonremovable singularities” or “essential singularities” or sometimes some other nastier names.

Usually, if one has a singularity in a mathematical construct, then information about one side of the singularity isn’t enough to extrapolate what might be on the other side. This makes the literary use of a “singularity” as “something magical that does whatever the plot requires” justified enough. Tesla here is clearly using the idea of reaching an infinitely vast, or an infinitely dense, holiday concentration as a singularity. I grant that would be singular enough. The strip does make me think of a fun sequence in Walt Kelly’s Pogo where one year the Bun Rabbit decided to get all the holiday-celebrating done first thing in the year, to clear out the rest. He went about banging the drum and listing every holiday ever, which is what made me aware of the New Jersey Big Sea Day.

Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 24th of September includes a sequence identified as the “Catalan Series”. I’d have said “sequence” myself. The Catalan sequence describes (among other things) how many ways you can break down a regular polygon into a particular number of triangles. A square can be broken down into two triangles just two ways (if orientation counts, which for this problem, it does). A pentagon can be broken down into three triangles in five ways. A hexagon can be broken down into four triangles in fourteen ways, and so on. (The key is you break the polygon into a number of triangles that’s two less than the number of sides. So if you had a 9-sided polygon, you’d break it up into 7 triangles. If you had a 20-sided polygon, you’d break it up into 18 triangles.) The sequence describes more stuff than that, but this is an easy-to-understand application. As the name of the sequence suggests, it comes to us from the Belgian-French mathematician Eugène Charles Catalan (1814 – 1894).

Catalan’s name also might be faintly familiar for a conjecture he posed in 1844, which was finally proven true in 2002 by Preda Mihăilescu. His conjecture is based on observing that the number 2 raised to the third power is 8, while the number 3 raised to the second power is 9, quite close together. Catalan conjectured this was the only case of consecutive powers. That is, there’s nothing like 15 to the twentieth power being one less than 12 to the twenty-fourth power or anything like that. I’m afraid I don’t know enough of this kind of mathematics, known as number theory, to say whether that’s of use for anything more than settling curiosity on the point.

## Looking At Things Four-Dimensionally

I’d like to close out the month by pointing to 4D Visualization, a web site set up by … well, I’m not sure the person, but the contact e-mail address is 4d ( at ) eusebeia.dyndns.org for whatever that’s worth. (Worse, I can not remember what site led me to it; if you’re out there, referent, please say so so I can thank you properly. In the meantime, thank you.) The author takes eleven chapters to discuss ways to visualize four-dimensional structures, and does quite a nice job at it. The ways we visualize three-dimensional structures are used heavily for analogies, and the illustrations — static and animated — build what feels like an intuitive bridge to me, at least.

Eusebeia (if I may use that as a name) goes through cross-sections, which are generally simple to render but which tax the imagination to put together1, and projections, and the subtleties in rendering two-dimensional images of three-dimensional projections of four-dimensional structures so that they’re sensible. It’s all quite good and I’m just sorry that my belief in the promise “More chapters coming soon!” clashes with the notice, “Last updated 13 Oct 2008”.

The main page is still being updated regularly, including a Polytope Of The Month feature. A polytope is what people call a polygon or polyhedron if they don’t want their discussion to carry the connotation of being about a two- or three-dimensional figure. It’s kind of the way someone in celestial mechanics talking about the orbit of an object around another might say periapsis and apoapsis, instead of perigee and apogee or perihelion and aphelion, although as far as I can tell people in celestial mechanics are only that precise if they suspect someone pedantic is watching them. I’m not well-versed enough to say how much polytope is used compared to polyhedron.

Anyway, for those looking for the chance to poke around higher dimensions, consider giving this a try; it’s a good read.

[1: I knew that a three-dimensional cube has, on the right slice, a hexagonal cross-section. It’s something I discovered while fiddling around with the problem of charged particles on a conductive-particule sphere, believe it or not. ]

## Geometry the Old-Fashioned Way

I failed to keep note of where I got this link from, so I apologize to whatever fine person sent me over here.

Science Vs Magic.net has a splendid page that lets one have fun doing geometry in the classical Greek format, with tools that are the equivalent of straightedge and compass. The compass is used properly, too: there’s no cheating and copying distance by lifting the compass carefully and setting it back down. You have to draw from a center point and a chosen radius each time, the way classical constructions are supposed to be done.

There’s a package of forty challenges offered, things like drawing squares or making rosettes of circles or the like, and if that’s not enough there’s the challenge in beating a par for the number of moves required. Meanwhile I’m gratified to learn that, years after I had to do this stuff for school, I’ve still remembered how to do bisections of lines and angles, and how to drop perpendiculars to a point.

(I haven’t figured yet how to draw a circle to an arbitrary point — sometimes you don’t need to connect anywhere particular — but imagine if I read the instructions maybe that would be obvious or something.)