This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American
Should you happen to live in the United Kingdom, Matt Parker -- a.k.a. @StandUpMaths on Twitter -- probably needs no introduction. He's a former math teacher from Australia, relocated to London, who combines his love of math with stand-up comedy. Parker is regular on the hugely popular BBC Radio4's Infinite Monkey Cage (hosted by physicist Brian Cox and Robin Ince), for starters, and has sold out comedy shows at the Edinburgh Festival Fringe, in addition to appearing in countless other media venues over the years.
And now he's the author of a fantastic new book, Things To Make and Do in the Fourth Dimension, released this week in the US, wherein he guides the reader on a magical math-y tour, with tips on the fastest way to tie your shoes, how to make a working computer out of dominoes, and the fairest way to slice a pizza -- all useful life skills, I'm sure you'll agree. Jen-Luc Piquant chatted with Parker via Skype during his whirlwind stateside book tour about the challenges of communicating abstract mathematical concepts to popular audiences, what knitting patterns and sheet music have in common with math, and what he thought about that tesseract scene in Christopher Nolan's blockbuster sci-fi film, Interstellar.
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JLP: Your book is an intriguing mix of clear explication, hands-on activities, and fascinating historical nuggets. Out of curiosity -- since we're science history buffs here at the cocktail party -- who is your favorite historical mathematician?
MP: I wanted to find mathematicians who aren't as well known, but did amazing things. I talk a lot in the book about Edouard Lucas. On the one hand, he was an amazing mathematician, and on the other, he was an incredible communicator. Without him, we wouldn't know about the Fibonacci numbers. Fibonacci counted some rabbits and disappeared off the face of history, and Lucas noticed this later on, generalized them, and named them Fibonacci numbers. Encryption is based on prime numbers.
My favorite story I discovered while writing the book. I don't speak French and I couldn't find an English translation of Lucas's math book, but the diagrams are the same. If I see a diagram, I know what he's doing. I saw one, and even with my rudimentary French, I realized Lucas was talking about electronic computers. This was in the 1800s. I got that passage translated, and it was Lucas talking about what Charles Babbage and Ada Lovelace had done, theoretically coming up with how computers might work well before it was possible to build them. He even made the link to how it might be possible to build them with electronics. I had no idea anyone else had said, "Hey, we could make these things out of electrical circuits." It's a shame he never lived to see any of it happen.
JLP: For some of the hands-on activities in the book, you draw on your background in magic and card tricks, many of which you've used in your public appearances and standup routines. What was the thinking behind that?
MP: That comes out of having to teach teenagers math. One of the problems with math is that it can be very boring, very easily. Math teachers have to be creative people because you've got to find a way to bring a very dry subject to life. There's all sorts of fantastic things you can do with mathematics, but we're forced to learn the repetitive skills and then we never let them out onto the field to play. If you want to motivate an adolescent to learn something, you've got to give him/her a very good reason. Normally we threaten them: one day you'll have a mortgage or need to pass an exam. But if you can show them something immediately that they can use it for, they're much more enthusiastic. If a teenager thinks that by learning some math they can do a card trick to annoy their friends and family, they suddenly go, "Yeah, we're gonna learn this."
A huge number of magicians are closet math nerds. They deliberately hide their mathematical techniques because that's the whole culture in magic: you find a way to make it seem as non-mathematical as possible. I do the opposite, showing the math going on behind the scenes. But I was careful [not to reveal carefully guarded secrets]. Everything I talk about is already in the public domain.
JLP: It's interesting that you mention the repetition. I took piano lessons as a child, and the first thing I was taught was practicing scales. I practiced dutifully, but the music never really came alive for me until I heard the intricate chromatic scales in Chopin. It made me want to practice harder, so that I could one day attempt to play Chopin.
MP: People have compared mathematics to sheet music, because if you had never heard music, and all you'd ever seen was sheet music, and people writing down the notes, it looks like meaningless manipulation of symbols. People may say, "Oh, but look at the patterns, they're so beautiful," but if you hadn't heard the music, you'd think, "These guys are bonkers." It's the same thing in math. There's this amazing beauty behind it, but there's no equivalent of listening to it, so for non-mathematicians, it's just meaningless symbols. Part of what I wanted to do [in this book] was to give people a glimpse into why mathematicians get so excited about all these symbols. There is a logic and unexpected connections going on behind them.
My mom knits. She's knitted me a few mathematical objects, including a hat [pictured in the book]. I watched her work out the knitting pattern and I had a moment of insight into what it's like being a mathematician talking to a non-mathematician. I thought, "You could be writing anything, it's just random symbols, this doesn't make any sense!" Yet she was so engrossed and excited about how it all fit together; she could visualize it. When I finally saw the hat, I got some sense of what she was doing. So that's what I'm trying to do with math in this book. I'm trying to show people the hat.
JLP: I noted you used a few examples of hyperbolic crocheting in the book. That's another popular means of grasping abstract topological concepts, championed by the Institute for Figuring, for example.
MP: Crocheting is the best way of showing a hyperbolic surface, even better than 3D printing, which is expensive. With a hyperbolic surface, the further you travel along it, the more surface area you get. But any one local bit, if you flatten it out, you're just pushing the wrinkles somewhere else. With crocheting, you can do that [for real]. You can pick it up and flatten one bit and the rest clumps somewhere else.
JLP: You explore higher dimensional shapes in more depth in one of the later chapters. How did you approach the challenge of visualizing shaped in higher dimensions?
MP: Anything past five dimensions, you no longer have any intuitive grasp of higher spaces. Mathematicians are as lost as we are. It's not like they can visualize a 6D shape and then imagine plugging it into a 7D shape. What I wanted to get across was that mathematicians are people who enjoy the fact that it's difficult. They love the fact that you can't really visualize these objects, but if you're very careful and meticulous with the mathematics, you can still understand and manipulate them, investigate how they behave. They just delight in the difficulty.
I deliberately placed a chapter earlier in the book on graph theory and networks. When you can no longer visualize the shapes, all you are left with is the way the corners or the vertices are connected. You can draw that as a network. So even though you can't picture how it would pop out into higher dimensions, you can still see the network with all its corners stuck together, and you can marvel at the beauty of the structure and all the patterns. They never tell the full story, though. You're seeing a glimpse, one particular angle on these amazing shapes. We can put just a little bit on a flat page.
JLP: Finally, we have to talk about the tesseract in Interstellar. Sure, there were plenty of quibbles about the physics in the film, but I thought it was an interesting visual depiction of something that's very difficult for most of us to visualize.
MP: I'm married to a physicist and I think she got more annoyed by the physics than I did. I've seen worse. Their representation of time as a spatial dimension I thought was very nice. As always, you can imagine things in a dimension lower. If you had a cube and you wanted to show it to someone who is flat, one thing you could do is slice it up into squares and then put them next to each other -- kind of smear out the cube, like butter, into a very long rectangle, effectively making slices of the cube.
[The film] had cubes stacked one after each other to show that smearing out of time. It's actually a very nice way to visualize time as a dimension, although I prefer my fourth dimension to be spatial. Whether or not the fifth dimension is love is still very much an open area of mathematical research.
For more on how to visualize higher dimensions, check out the excerpt below!
Visualizing A Higher-Dimension Sphere
OK, so what is a higher-dimension sphere? On the surface, it might seem simple: a sphere is all the points which are a certain distance – the radius – from a fixed center point. A circle is a 2D sphere. A sphere is a 3D sphere. (OK, that one’s obvious.) A 4D sphere is all the points which are the same distance from a center, and so on. Unfortunately, these spheres are much more slippery than you would expect. Which is why, for safety reasons, we need to keep them locked in hypercubes so they cannot get away. We’ll pack the rest of the hypercube full of padding spheres so that the sphere at the focus of our attention cannot move around inside its box.
All these precautions to keep our hypersphere under control may seem disproportionate. Well, let’s find out if they are or not.
We’ll start in 2D and work our way up. Our 2D cube is a square, and I’m going to use one with sides which are 4 units long. This is because it neatly fits four circles of radius 1 unit into it. The center of each of these circles will sit a quarter of the way in from two of the edges.
We’re now going to carefully put our specimen circle (a 2D sphere) in the very middle of this box full of other circles. Unlike the padding circles, which remain a fixed radius of 1, we’ll let the middle circle expand to be as big as the space in the middle of the box allows.
We can now calculate exactly how big this middle circle can be. The box-length of 4 makes sense now, as it means that the distance from the middle of each padding circle to the center of the box can easily be calculated using Pythagorean theorem. For our 2D square, this diagonal length is 2 ≈ 1.414; the first 1 of this is used up by the padding circle, so our middle circle can only have a radius of up to 0.414 before it touches the padding. It’s not a very big circle, but at least we know exactly how big it is, where it is and what’s containing it.
We can fit eight padding spheres into our 4 × 4 × 4 3D box. The number of unit spheres is going to double as we go up each dimension, and they’ll always be centered one unit from all their nearby edges so that they’re just kissing them. The diagonal from the center of each padding sphere to the center of the cube is 3 ≈ 1.732, which allows our specimen sphere to expand to a radius of 0.732: not a big leap.
It makes sense that, as this sphere has more directions to move in, it can grow slightly bigger. However, as we go up each dimension, we’re adding more and more padding spheres to keep the specimen sphere/hypersphere/and so on, trapped, so surely it will not be able to escape. Its growth should tail off.
However, as you may have guessed, the center sphere is not going to behave itself: somehow, it will escape, even though, by definition, it will remain centered in the middle of the cube and be surrounded by other spheres holding it in place. Checking the numbers for 4D shows us that everything continues to line up for a 4 × 4 × 4 × 4 tesseract, with sixteen padding unit spheres. The distance from the center of each padding 4-sphere to the center of the 4D box is a very exact and tidy 2 (i.e. 4 ), giving our center specimen sphere a radius of 1. It seems that four dimensions have given it enough freedom to expand out to be the same size as the padding spheres around it.
Onwards to five dimensions, and things start to get a bit strange: our specimen sphere has continued growing and now has a radius of 1.236, bigger than the padding spheres around it (from now on, I’ll just call it a sphere, regardless of how many dimensions it’s in). What started as a small gap in the middle when we were in 2D is now a large chasm in 5D. Originally, we expected the middle specimen sphere to get a bit bigger as it gains more dimensions but that this 331 expansion would eventually, and rapidly, tail off. It doesn’t. Our center specimen sphere continues to get bigger and bigger at an alarming pace as more dimensions are added.
The big surprise is when the box enters ten dimensions and the inner sphere’s radius hits 2.162, which means that it is actually reaching outside the box. Not only has this sphere managed to expand despite the padding spheres around it, it has now escaped the box entirely. In nine dimensions, it manages just to touch the box, with a radius of 2, and then, for any dimensional space beyond that, it sticks out through the sides of the hyper-box (or hypercube). From twenty-six dimensions onwards, the sphere is more than twice as big as the box it’s inside. And it shows no sign of slowing down. The size of this center sphere does not converge as the dimensions get higher: it continues to diverge off into the distance.
The numbers do not lie, but we still need to explain how the sphere reaches out of the box. The box doesn’t change shape – it’s always 4 units long in all directions. Importantly, the other spheres are a fixed radius of 1. We do not let them expand as much as they could; we just arrange them all so they touch the outside wall of the box and the other spheres next to them. As the number of dimensions goes up, the gaps between these packing spheres get bigger and the center sphere somehow grows spikes which can reach through those gaps and out of the box. It was mathematician Colin Wright who first gave me this puzzle and, in his words, it’s best to think of higher-dimensional spheres as being spiky. It seems these spheres are covered in multi-dimensional bristles. Now that I didn’t see coming.
Excerpted from Things To Make and Do in the Fourth Dimension by Matt Parker, to be published in December 2014 by Farrar, Straus and Giroux, LLC. Copyright © 2014 by Matt Parker. All rights reserved. Used with permission.