## Mathematical Recreations

by Ian Stewart

I've popped the popcorn, Charlotte's brought along the soda-did you manage to rent the video?" Boris asked.
"Of course I did," Alison replied.
"Exterminator 4. "Oh, that Ernie Scrambledegger," Charlotte sighed. Boris put the tape on. The screen showed a pattern of irregular white lines, and then the title sequence came up. They stared at the screen. "Uh- maybe it's an ad," said Alison hopefully.
"I don't think so," Boris said.
"Alison, what the devil is Not Knot?"
"Whatever it is," said Charlotte in menacing tones, "it certainly isn't Exterminator 4. Alison, did you forget your glasses again?"
"Well, there was a bit of a mix-up at the checkout-"
"Hold it, guys," Boris said. "The video store's closed now. We'll just have to make do with whatever it is that Alison got by mistake."
On the screen a mass of multicolored wormlike tubes writhed around like a creature in its last death throes.
"Aliens," said Charlotte firmly. "Okay, I'll settle for aliens."
"I don't think so," Alison said. "In fact, I'm pretty sure I recognize this. It's a math video produced by the Geometry Center at the University of Minnesota."
"A math video?" said Boris in horror.
"Well, we do live in the age of math communicathionth," Charlotte lisped. "They got a big research grant to set the center up; this is just one of the things they've done. Hey, the graphics are good. Look at those transparent boxes. Alison, what's it all about?"
"Some amazing discoveries about the topology of knots," Alison said. "Topology?" asked Boris, at the same time as Charlotte asked, "Knots?"
"Um. Imagine tying a knot in a length of tubing and fusing the ends together so that the knot can't escape. The question is, Can you recognize when two such knots are equivalent? That is, can you deform one of them into the other by bending or twisting the surrounding space, without cutting or tearing it?"
"How do you tear Space?" Charlotte asked. "Or bend it?"
"Imagine it's filled with some kind of very soft, stretchy, squashy Jell-O, and bend or tear that."
"And this is math? Squashy Jell-O?"
"Not only is the universe stranger than we know," Alison said, "but it is stranger than we can know. Math especially. Be thankful we're only filling space with Jell-O."

 The Borromean rings

The screen changed to show three linked rings [see illustration on this page]. "That's a link," Alison said. "like a knot, but with more tubes. Those are the Borromean rings, which are famous because no two of them are linked, but all three are.I mean, if you cut any one of them, the whole thing falls apart, but if you don't, it hangs together."
"Oh."
They watched for a while.
"They're showing you that if you forget about the knotted tubes and just look at the spaces outside them-their complements, that's the word-then inequivalent knots have inequivalent complements." Charlotte thought for a moment.
"Isn't that obvious? If I follow you correctly, the complement is like the whole of space filled with Jell-O but with a tunnel burrowed through it where the knotted tube would be. If you can bend the space outside one knot to look like the space outside the other, doesn't the knot kind of get carried along?"
"Yes, but it might get kind of twisted up, I suppose," Alison said. "Anyway, it can't be obvious, because the same statement is false for links. You can find links whose outsides are the same but whose insides are different [see box on opposite page].

 Inequivalent Links with Equivalent Complements The two links at the top are inequivalent. In this topologically equivalent representation (bottom), one link has been stretched to create a thick cylindrical tube. Cut the right-hand picture across the top of the cylinder (dotted line). If you rotate the links to untwist them and then reattach them, you get the left-hand picture. This proves that the complements are equivalent. But the left-hand link falls apart completely if you cut the cylindrical tube, whereas the right-hand link does not, so the two links are inequivalent.

"The argument seems to involve cutting things," Boris protested. "I thought you weren't allowed to cut the space."
"Okay, so I lied. You can cut it, provided you glue it back together later. Just as it was."
"But then you can untie a knot," Boris protested. "Cut the tube, undo the knot, then glue it back."
"Which is why, Boris, dear, you have to deform the space around the knot and not just the knot," said Charlotte, who was finding the video more to her taste than she'd expected. "Which in turn is why it's called Not Knot"
"Thanks, Charlotte," Boris said. "Hmm, it's getting more complicated now."
"Yes. It's leading up to the recent discovery that knot complements have a natural geometric structure, which you can use to tell the difference between inequivalent knots. The interesting thing is that it's non-Euclidean geometry that shows up," Alison explained.
"You're telling me there's more than one kind of geometry?" Charlotte asked.

"There are lots of different kinds of geometry-ordinary Euclidean geometry is just one of them. The main difference in non- Euclidean geometry is that parallel lines behave in funny ways and may not exist at all. You can visualize two-dimensional non-Euclidean geometry by replacing the plane with curved surfaces, like spheres or saddle shapes, and drawing the lines and stuff on those. But for knot complements, you need to think about three-dimensional curved spaces, and that's hard. So what the video does is fly you around inside such a space and show you what it would look like."
Boris looked at the screen, where tiny cars were chasing one another around a cone. "Flies? With a car?"
"The flying bit comes later. Here they're showing you how to make non- Euclidean geometries by kind of cutting a slice out of ordinary space and gluing the edges together. Just as you can make a cone from a circular piece of paper with a pie-shaped slice cut out. Only you just have to imagine what the result of such a gluing process would be. The idea is that whenever you draw a line that hits an edge of the slice, you immediately transfer to the corresponding place on the other edge of the slice and carry on drawing. That kind of bends the lines, even though each bit of them is straight-so not only do you get funny effects with parallel lines but also a 'straight' line that can bend around and cross itself."
Boris looked puzzled. "Alison, how can a straight line bend?"
"When I say 'straight' I mean the line that covers the shortest distance. When the space has a non-Euclidean geometry, straight lines don't always look straight to us-looking in from the outside. But if there were light rays that followed the shortest paths, then to a creature living in the space the lines would look straight".
"Another way to create non-Euclidean geometries is to use mirrors," Alison went on.
"Like a kaleidoscope. The mirrors have the same effect, changing the direction of light rays. For instance, imagine you are inside a cubical room whose floor, walls and ceilings are all mirrors. What do you see?"

 Reflective Musician Musician Mike Oldfield borrowed the ideas of the geometry of "a cubicle room of mirrors" in his "Guilty" single cover.In other stage performances he has used the art of Escher , which also exploits mathematical geometries to produce fascinating pictures.

Charlotte thought for a moment. "Lots of copies of me."
"Yes. The images of the cube in the mirrors would tile space, and in whatever direction you looked, you'd see yourself. Well, a reflection of yourself. But mathematically we can pretend that each reflection really is you, the same you. That has the effect of 'gluing' opposite mirrors together, just as we glued the edges of the cone. But now you get a three-dimensional space with weird geometry. For instance, the straight line starting at your forehead and traveling horizontally forward eventually runs back into your forehead."
"So straight lines can make U-turns?"
"Yeah. But they stay straight, which is perfectly natural if you happen to be a creature that lives in a knot complement. Now, you can set up a different kind of mirror-mathematical, not physical-that turns things upside down as well as reflecting them. Call them inverting mirrors, okay? If the walls of the cube were inverting mirrors, you'd see lots of copies of yourself, but some of them would be upside down [see top illustration on next page].
This particular geometry, a cube with inverting mirrors for faces, is one possible geometry for the complement of the Borromean rings. The video explains why in detail, but I'll try to summarize. An inverting mirror-in the sense we've just been talking about--is a place where space wraps around it self in strange ways. A kind of space warp. With me so far?"
"Hanging on by my toenails."
"Now, take the three separate tubes that make up the Borromean rings and stretch each tube until it gets very long and thin, with its edges straight for most of their length. Like an athletics stadium, only with the straight parts of the track being much longer. If you do it right, you can get a pair of parallel tubes that runs north-south, say; another pair that runs east-west; and a third pair that runs up-down. Plus extra U- shaped pieces joining them at each end, which we'll push away to infinity so they don't matter."
"Gotcha."
"You can find a similar arrangement of lines on a cube. On the floor and ceiling, run a line down the middle, going north-south. On two of the walls, draw horizontal lines running east- west. On the remaining walls, run vertical lines up the middle. The important thing is that those lines stay fixed when you 'reflect' them in inverting mirrors on the cube's faces. That lets you relate the corresponding geometry to the complement of the Borromean rings. It means that when you perform all the inverting reflections, the images of the lines on the cubes fit together so that they stretch away to infinity, just as the Borromean rings do after we push the U-bends off to infinity. And that means that the space around them-the funny geometry with lots of copies, some upside down-is just like the space around the Borromean rings.
"I'll believe you," Charlotte said.
"Okay-or else watch the video again, carefully."
"The night is young yet," Boris said. He stopped, spellbound. "Hey, this bit is really neat. It's like being inside some kind of cage and moving around through the bars."
"This is the flying part," Alison explained. "The bars are the edges of dodecahedrons. Well, really they're all the same dodecahedron, because you have to imagine everything glued together as the faces of the cube were. But if you lived inside such a space, you'd see multiple copies of everything, like the picture with the inverting mirrors. It's another geometry you can get from the complement of the Borromean rings by using a different kind of 'mirror.' Regular mirrors on the faces of a cube produce cubic images that tile space, and similarly non-Euclidean mirrors on the faces of a dodecahedron produce dodecahedral images that tile non-Euclidean space. The edges are at right angles, so you can fit four of them together, and that's why they tile."

 Inverted reflections in all directions: one possible geometry (or the compliment of the Borromean rings A non-Euclidean tiling of space by regular dodecahedrons

"Hold it," Charlotte interjected. "A dodecahedron I understand-it's a solid with 12 pentagonal faces."
"Right."
"But they don't meet at right angles."
"Not in Euclidean space, no. But this space is non-Euclidean-curved, if you wish. Bent just enough to make them into right angles."
"I guess I see that."
"The video really does let you see it. You fly through it; you can see what it looks like, really feel the weird curvature effects that come from non-Euclidean geometry [see bottom illustration on this page]. It's quite strange and beautiful."
"Yeah," Boris said. "It kinda gets to you after a while. I could imagine living in a space like this. It's roomy."
"How do you mean?"
"Well, flying around it like this you can see that surrounding each dodecahedral tile there are a lot more other tiles than you could fit into ordinary Euclidean space. The amount of space gets bigger than you'd expect as you move outward."
"That's called negative curvature. It shows you're in what's called hyperbolic space, one particular kind of non-Euclidean geometry. And that's the central point of the video. According to a recent discovery made by William P. Thurston of the Mathematical Sciences Research Institute, nearly all knot and link complements have a natural hyperbolic geometry. There are a few exceptions, but they're all known. And you can use the geometric structure to tell all of the others apart. That is, inequivalent knots or links have different geometric structures. It's an amazing connection-I dare not say 'link'-between the flexible geometry of topology and the rigid geometry of non-Euclidean spaces. So now a very old-fashioned branch of math-non- Euclidean geometry-is back in vogue."
"Great," said Charlotte, rewinding the tape. "But what would be really great is if they hired Ernie Scrambledegger to star in the sequel."
"The Geometry Center got a big grant," Alison said. "But not that big."