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  • So a cube has perfect symmetry, right? You know what I mean. Every face is the same.

  • And a tetrahedron has perfect symmetry. Okay, there are four faces,

  • and they're all the same.

  • You can ask, and Euclid did, what are all of the symmetric

  • three-dimensional objects? And it turns out there are only five of them. Most people know what a cube is, and a tetrahedron.

  • But there are also an octahedron, which has eight faces, you know, four on top and four on the bottom. And it's again, perfectly

  • symmetrical. Every way you twist and turn it, it looks the same, just like a cube. And then there are two others.

  • There's the icosahedron and the dodecahedron.

  • But that's it. So first you might think, well,

  • if there are so many funny ones, why aren't there lots of different ones? So I proved that in class,

  • I mean, it's a real theorem, and I said, therefore, there are only five fair dice.

  • Because each of those dice would be fair, that is, if I wanted to generate a number between one and eight, if I rolled the

  • octahedron, each of these faces is equally likely to come out. Somehow, intuitively,

  • we know that. You know, that's a fair dice

  • which has eight sides. And a kid raises his hand in class and said, I have a thirty sided die.

  • And I said, no you don't.

  • And he said, yeah I do.

  • When you first meet it, it's just some lumpy roundish thing. But in fact,

  • it's called a rhombic triacontahedron, and it has thirty faces, and all of the faces are the same size.

  • They're all little rhombuses.

  • That's why it's called a rhombic thing. Now, this die isn't as

  • symmetric as the other ones. This vertex has five faces that meet at this vertex.

  • But this vertex has three faces that meet at this vertex.

  • So it's not as symmetrical, for example, a cube has three faces at each vertex,

  • you know, and every edge has two sides. And so there's a question, is it fair, and what does that mean?

  • Well, of course we know what it means in practice. We mean if we throw it

  • you know, it should land on each of the thirty faces a thirtieth of the time.

  • But let me talk about that for a second. When I was a graduate student,

  • we had a guy who was a retired executive

  • who wanted to test the laws of chance. And he came into the department,

  • and he said, I like throwing dice when I watch television. And actually, Brady,

  • it might be that some of your viewers might want to participate in this. And in the end, he wound up rolling a die

  • three and a half million times,

  • and recording how many times did each face come out. And here are some things you learn. The first thing is that

  • dice, if you roll them a lot, get round. Of course they do. If you roll a die

  • 20,000 times it, you know, bounces around, it changes. So we had to give him new dice. The notion of long-term frequency,

  • it's actually pretty fictional. The second thing was, dice the way they're made, the dice

  • he was using, have pips drilled in them and filled with paint. So there's a six has six spots in it.

  • Well, those drill holes are lighter.

  • So the six face actually has less mass, and the one face, which is opposite, has more.

  • You know, they weren't fair.

  • Now, casino dice are much more carefully made. The holes are filled with paint of the same specific gravity as the

  • surrounding material, so they really are pretty close to fair. When you start thinking about anything,

  • even something as silly as, as, what does it mean to say a dice, die is fair, there are things to think about, of course.

  • Okay, so I wanted to think about, what does it mean

  • to say this die is fair?

  • There's the notion of the symmetry group of a die. A symmetry of an object, any object, is the set of all

  • transformations I can make, like turning it a quarter, or

  • turning it this way, or turning it over, which bring it, atom for atom, where it was, okay?

  • So that's called a symmetry group. And of course, if I have one symmetry, like turning at a quarter turn, I can do it twice.

  • I can go once, twice, right? So I can combine the symmetry of flip that way with the symmetry of a flip that way.

  • So, we say that the symmetries of an object form a group. All that means is, you can

  • take one symmetry, and then take another symmetry, and compose them, do them twice. You have to be able to undo it.

  • If I just did this

  • I can undo it.

  • Brady: "What's a, what's a function you could perform on a dice, for example, that you couldn't undo just by doing it backwards?"

  • Suppose that the die was made out of Jell-O and I flattened it.

  • I couldn't undo that, probably. That wouldn't be a symmetry.

  • Brady: "Okay. That is the definition."

  • Right, right. Exactly, exactly.

  • Takes it atom for atom to where it is, and, and, you know, of course you can reverse it. Different objects have different

  • notions of symmetry. So this tetrahedron, it has less symmetry, somehow. The number of different things I can do to it,

  • it's less than what I can do to this. There are twelve

  • symmetries that I can make of this if you start fiddling,

  • and there twenty-four symmetries that I can make with this die. And obviously some giant thing like this,

  • you know, there are a lot of different things I could do to it.

  • Joe Keller and I defined an object to be fair by symmetry if its group of transformations

  • was transitive on the faces. And all that means is, I can take the die

  • and leave it atom for atom the same, but have any face where this face is.

  • You know, right now, the two is up in the rhombus.

  • Well, I should be able to make the ten come up by a symmetry, and have the die

  • just exactly where it was, but have the ten up. We define the die to be fair by symmetry if its symmetries

  • act transitively on the faces, meaning I can take any face to any other face. This die, its symmetry group isn't

  • transitive on the vertices. I can't take this vertex,

  • which has five things around, it into this vertex, which has three things around it. I could move this vertex to there,

  • but the surrounding territory will be different. We did prove that this die has a

  • transitive symmetry group.

  • Brady: "But not by vertex, but yes by face."

  • Yes by face. And not by edges, either, actually. So we did show that this die has a transitive symmetry group.

  • Brady: "That's fair."

  • So it's fair. It sounds fair now. If every face is the same, and since the bets are gonna be on the faces,

  • it sounds fair.

  • Then we had the idea, what are all the fair dice? Euclid found the

  • completely fair dice. That is, Euclid's Theorem is

  • what are all the

  • objects whose groups are transitive on their faces, on their edges and on their vertices? And the cube, of course, you could see, every vertex

  • looks the same, right? And similarly, every edge looks the same. Not, not this guy.

  • And with Joe, we classified all of the fair dice.

  • Suppose I had a five-gon.

  • And then let's put a point up here and down here, and

  • connect that point. Fill in the solid so it's got five edges.

  • And then it'll have five triangles going up, and five triangles going down.

  • That's a fair object, because these five

  • triangles all are the same, and they're obviously the same as these five triangles,

  • and so every triangle is the same. But I could put that point wherever I want, and you'd get very different looking dice, right?

  • You'd get a kind of little tiny thing or a very long thing.

  • But, so that's a one parameter family of ten-sided dice. What we found is that there are thirty families of fair dice.

  • We classified them, and then we found out that Archimedes had been there before us, about

  • twenty-five hundred years ago.

  • Brady: "How many members does each family have? Or is it infinite?"

  • Infinite, infinite, because there's a continuous parameter.

  • But, it's interesting that there were, you know, two continuous parameters.

  • We have a list. Some of the things on the list don't have a parameter in them. Some of them do.

  • We found all the possible fair dice.

  • Brady: "You told me you could do that with a pentagon, and I follow, that made sense,

  • "and I could imagine you could do that with a hexagon and a seven-gon,

  • "why couldn't you do it with a 200,000-gon?"

  • It, it, it's true.

  • Now, it's true what you say, and you've caught me, Brady, because that,

  • exactly, that's exactly correct. So why isn't that infinitely many, um,

  • infinitely many fair dice? It is. And so, let me amend the statement of my theorem.

  • That's an infinite family. That is an infinite family.

  • And it has what we call

  • Dihedral symmetry, that is, the only symmetry that those dice have is,

  • these sides are all the same, and these sides are all the same, and these sides are all the same as these sides, okay?

  • So dice with more than dihedral symmetry, that is we call them interesting dice. They have extra symmetry.

  • Brady: "So the example I just gave is trivial to you almost."

  • Well, it's an important example. If I took this tetrahedron, and if I had another one,

  • and I stuck them together, that would be a triangle, a point above and a point below, right? So here it is. I made one.

  • And obviously, you know, this is a six-sided thing.

  • Here, here are three sides,

  • here are the other three sides. These three are all the same, these three are all the same.

  • So this is a fair thing. Now, if you flipped it, of course you'd have to roll it, and you'd have to talk about the

  • side that lands down. Here's another six-sided die. Is there any sense in which we can say that this die is

  • more or less fair than this die?

  • That's a philosophical question, it's a math question.

  • The symmetry group of this die, there are just six symmetries, you can twist it around

  • three things, or you could turn it over and twist it around three things. The size of the symmetry group is as small as it could be.

  • Whereas this die has twenty-four symmetries. What does that mean?

  • I know, I once did a numberphile video about flipping a coin,

  • so I got a friend with a stopwatch,

  • and I went one two three flip! one two three flip!

  • and when we were talking about that,

  • we said, well of course, physics should come into this

  • description of fairness someplace. Of course it should come in with dice, and it's the same

  • physics. It's mechanics. If I roll the die very carefully, so it just goes around these four faces, on a blanket.

  • Well, that's not very fair, because these faces could not come up.

  • When you release the die from your hand, if you're actually rolling it, it has

  • velocity, and it has angular

  • velocities, and there's a phase space.

  • What direction is it going in, and how fast, and then how fast is it turning in each of various directions?

  • And there are actually twelve dimensions of parameters needed to describe the initial conditions. What we can show, is that,

  • that twelve-dimensional space of initial conditions is partitioned up into six regions,

  • regions where, if the die leaves your hand when the initial

  • positions are in this part of the phase space,

  • it comes up at side one. All of the initial conditions where the die comes up at side two. It can only come up in one

  • of six faces. So that partitions the, this twelve-dimensional space into

  • regions, six different regions. The partitioning up of phase space is much finer for this die

  • than for this die. This partitioning is cruder. For any way of rolling,

  • small changes in the initial conditions, the difference

  • between your hand and your brain, make for a big difference in what side it comes up, because the

  • partitioning of the phase space is finer.

  • So there is a sense in which symmetry isn't the only determining factor, physics and symmetry combine to allow a

  • reasonably satisfactory analysis of just how fair dice are.

  • Brady: "So the cube is fairer."

  • The cube is fairer.

  • Tiny little changes in just how you release it will make for the difference between side one and side two,

  • whereas the basins of attraction for side, you know, one up to six, with this

  • two tetrahedral die, are cruder. And so, it would be easier to control, for example. You know a coin is the

  • simplest kind of dice, the simplest kind of die, it just has two faces, right?

  • Obviously, heads and tails are symmetric, but of course when you really flip a coin,

  • you know, the flips matter.

  • And I once had a colleague who was at a junior high school, and he called up and said, I gave my kids

  • the problem of flipping a coin, and when they flipped coins, it was very very patterned.

  • You know, head tail head tail head tail head tail.

  • And he couldn't understand it. And I went in and watched the kids flip, and they were bored out of their minds, and they were

  • just doing wimpy little flips.

  • You know, so that it would flip once. Now of course it's going to come up head tail head tail.

  • So, you know, how you flip can make a difference.

  • How you roll a die can make a difference.

  • This isn't the end. In fact, our next video will be the second part of this

  • interview with Professor Diaconis,

  • talking more about fair dice and casino dice. If you can't wait until then, there are links on the screen and in the video description.

  • You can watch it right now.

  • You know talking to me about dice and fairness is like talking to a California wine person about wine.

  • It can go on forever right?

  • So let me go back...

  • And as some of you know, there are plenty more dice videos on numberphile.

  • You can look at our back catalogue. And look at this one,

  • it's coming very soon from Tadashi.

  • Four four four four and zero zero.

  • That's something to get excited about.

  • Links on the screen and in the video description. Thank you so much, everyone, for watching our videos. We really appreciate it.

So a cube has perfect symmetry, right? You know what I mean. Every face is the same.

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フェアダイス(前編) - Numberphile (Fair Dice (Part 1) - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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