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.