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0:00
Okay, so today I want to show you a fractal zero. Okay, so you may ask, you know, what is a zero? Well, zero is something that solves this equation, a plus x equals a. But then you say, well, what is a? If we are talking about, say, the counting numbers, you know, so we say 1, 2, 3, and so on. If I am only allowed to use the counting numbers, then I, you know, x cannot be, there is no counting number that will solve this equation. So in the couning, the counting numbers do not have a zero.
0:39
Well, the, the natural numbers do, right? So we put zero, and then we call these the natural numbers, 0, 1, 2, 3, 4 They do have a zero it's x equals zero. But this zero, okay, so in the natural numbers, if you were going to solve a different equation, for example, a plus x equals 0, in the natural numbers, then this only has one solution if a is zero, if a is not zero, then this has no solutions, right? So then how do you solve this? Well, you solve this by then,
1:16
you know, adding even more numbers, and then talk about what we call the integers. Dot dot dot, minus 3, minus 2, minus 1, 0, 1, 2, and so on. In this set, these two equations, you know, have solutions. So there is a zero, and, and you can always solve a plus x equals zero. Today, I will show you a different type of numbers, okay, that have this very special zero that looks like a fractal. Okay, and these new numbers that I will talk about, they are called sandpiles. So what is a sandpile?
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So this is a, you know, it's going to be a three by three grid, okay? And on this three by three grid I am going to put numbers that are either 0, 1, 2 or 3. So in every one of these nine cells. I'm going to put either a zero, a one, a two or a three, and that's going to be my sandpile. So these numbers represent numbers of grains of sand that you put on one cell in this grid. And so these are my numbers, and then I need to tell you about, well,
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what's the addition. How am I going to add them? So, for example, 120, 211, 013. Let's add. Let's say 213 101, 010 Ok, so I have it here two sandpiles. How do I add them? Well, I add them by adding every number in each cell. So here we have 1 plus 2, that's going to be 3, 2 plus 1 that's also 3, 0 plus 3 is 3, 2 plus 1 is 3, 1 plus 0 is 1, 1 plus 1 is 2, 0 plus 0 is 0, 1 plus 1 is 2, 3 plus 0 is 3. Okay? So that's, these are my new numbers, and how to add them. Now, I can run into trouble,
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yes, for example, if I change one of these numbers, for example, this 1 for a 2. Then I'm going to have 2 plus 2 which is 4 and I am not allowing 4. So how do I fix that problem? Okay, and so this is why these are called sandpiles. So there is some dynamic thing that happens in here. And so when I when an entry reaches 4, it is unstable, and it's going to topple and send the grains of sand to their neighbors, okay? Brady: "Like a house of cards falling over." Like a house of cards, that's right.
4:01
So in here, because this 4 is in a corner, then it has two neighboring cells, okay? And then two other, you know, imagine that this is sitting on top of a table, and then, so it's going to send one grain of sand to this 3, and then one grain of sand to this other 3. Okay, everything else remains the same. And then it's going to lose one grain of sand to this edge, and one grain of sand to this edge, and now it's going to have 0. Now this process created two new 4s,
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so then I have to repeat it again, because remember my rules is that the entries have to be between 0 and 3. And you do this process until you have no more 4s. And here as you see, we created one, yet one more 4, so we need to repeat this process, and that's the addition of the two sandpiles that we have at the beginning, okay? So then the question is, do we have a zero in this, in this set of sandpiles?
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And, well, it turns out that it's, you know, from the definition of, of this addition, the zero is, unsurprisingly, 000, 000, 000. If I add anything to this, I am not going to do anything. Well, that was sort like a boring zero, and it is boring because if I want to solve an equation like a plus x equals 0 where this is one, you know, every single thing is a sandpile, then this equation has no solutions.
5:32
So if I put two sandpiles together, you know, if I have a sandpile a, and a sandpile x, and I put them together, I will never be able to topple everything until I get a zero. There is no minus a sandpile. It doesn't exist. And, the reason, okay, so if we, if we put numbers into our sandpile, so we say, a1, a2, a3 ... a8, a9, so we put nine numbers, you know, and this is like the result of some addition,
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then there will there will be topplings, okay, there will be topplings. But at the end of the day, You lose sand only if you are on the edge of this grid. But, however, every single time you are on an edge, sand gets sent to another neighboring cell. So, you will never be able to get rid of the entire sand, so there is no minus a sandpile. So, how do we fix that? In the natural numbers, okay, we fix that by making them bigger, into the integers. So we threw, essentially,
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all the negative numbers. And here we're going to do something different. We are actually going to shrink our set. So what is the set that we are considering? We are considering every single sandpile with this, so, our, the set that we're considering, let me call it M, and this set M is the set of all sandpiles. All sandpiles on the three by three grid. Yes, where a sandpile is just a collection of nine numbers where every number is between 0, 1, 2 or 3, and in fact,
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how many such sandpiles are there? We have nine cells, This cell can be a 0, a 1, a 2, or a 3, so it has four possibilities. This cell has also four possibilities. This cell has four possibilities as well. So a basic counting says that we have four to the nine total sandpiles. And this number happens to be 262,144. So this is the set of all sandpiles. But now we're going to shrink it. Okay, so we are not going to consider all the sandpiles.
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We are just going to consider a smaller collection of them, and that smaller collection is the collection of sandpiles that I can obtain by starting with the all 3s Yes, the all 3s is the sandpile that has the most sand. Okay, and then I'm going to add whatever I want to add to it, and when I'm done, you know, the resulting sandpile is going to be in my new collection, okay? So the new collection, the smaller collections, is
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all the sandpiles that I can obtain from the all 3s by adding extra sand, and then toppling. So, for example, if I add just one grain of sand in the center cell, then, well, I have everywhere 3s, except a 4 in here. Brady: "Oh, you're going to start some big toppling now." Yes. Yes, so I start toppling. And so this 4 is going to send one grain of sand to each of its four neighbors. So 343, 404, 343. I have now four unstable vertices. I can topple them all simultaneously.
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So each of them is going to lose four grains of sand, so now they are going to be zeroes.
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They are going to send, each is going to send one grain of sand to the center. So now the center has four grains of sand again. And the corners will receive one grain of sand from one side and one grain of sand from the other, so now the corners have five.
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Okay. And we need to repeat this process. This 4 is going to send one grain of sand to each of its neighbors. Now it has zero. The corners are still 5. We topple the corners and we get,
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you know, they are going to lose four grains of sand, so each corner will have 1 at the end,
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and then this number is going to get one grain of sand from here, one grain of sand from these other 5s, so now they are going to be 3s.
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Brady: "So that is part of my new set." This is part of my new set. Correct. Yes. So this new set, okay, has many many sandpiles, you know. The original set had 262,144. Our new set, so this set that I'm going to call S, which is the sandpiles arising from the all 3s, it's much smaller. It's about a half, it's half as big. It has more or less a hundred thousand. Now, one, we have one big problem here. And the big problem is that the 000, 000, 000 is not in S.
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Because, you know, we start from max, we add sand to it, we are never going to get rid of all the sand. So this zero is not in our sub collection So we lost our zero. We wanted to find sort of the minus sandpile, and now we have even a bigger problem, which is, we don't have a zero. However, let me show you just a couple. These computations, as you can see, they take a little bit of time. So let me show you just a couple more computations, okay? So it turns out, okay, that
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the all 2s is in S Brady: "It is. You can get all 2s." You can get the all 2s. So this all 2s is the all 3s
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plus 131, 333, 131.
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And, also, let me show you another one that, another element that is in S. So this element 212, 101, 212 is also in S. So now I'm going to add these two new
11:57
sandpiles that are in S, okay? So I'm going to add the 222, 222, 222. So the all 2s plus this new element that we got, 212, 101, 212. If I add them, then I get 434, 323, 434. Yes, and I have 4s in the corners, so I'm going to topple them, And, so each of the 4s is going to lose four grains of sand, so now I'm going to have zeros in the corners,
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and then this 3 is going to receive one grain of sand from this 4, another grain of sand from this other 4, so now they're going to be 5s. 5s, and I still have a 2 in the center. Okay, so now all the 5s
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topple, and they are going to lose four grains of sand, so all the 5s are going to be now 1s.
13:03
Okay, they are going to send, each 5 is going to send one grain of sand to the center. So now the center cell has 2 plus 4 meaning 6 grains of sand, and then each of the corners is going to get one grain of sand from each side, so now they are going to have 2s.
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And this 6 will topple. It's going to lose four grains of sand,
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and we get the all 2s back.
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This is something that is a little bit surprising, okay? So we have to sandpiles. We have the all 2s, and then we have this other sandpile, we put it on top of each other, and we get the all 2s back. So it's acting like a zero. And in fact, if we were to do this computation again on the all 3s, or on any other sandpile in S, we will get the same sandpile. So, the zero in our new set S, our zero is this 212, 101, 212. Brady: "So that will have that effect on any other sandpile within S."
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Correct. Correct. Brady: "It's like a magic sandpile!" It's like a magic sandpile. And in fact, it can be used to separate these two sets, like the things that are in S, and the things that are not. If you add a sandpile that it doesn't come from the all 3s, okay, then you're not going to get the same sandpile after you add- Brady: "It's like a test you can apply." It's a test, yes, it's a test. Brady: "What happens if you add it to itself?"
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Oh, that's a good question. So, what do you think that should happen? Brady: "I think it should ... result in itself." We can check that, yes? So, to each cell, we could have just multiply every entry by two, but, okay, so let's do it, so 424, 202, 424. Again we have 4s in the corners, they are going to topple, 0 in the center, and when the 4s topple,
15:09
you know, they are going to be going to 0, they lose four grains of sand. The corners are going to receive two grains of sand each, and the center is going to get one grain of sand from each of the 4s. I need to topple the center, and I get exactly 212, 101, 212. Yes, that's, you know, the addition of this element plus itself is equal to 212, 101, 212. Brady: "What do we call that special pile? I called it a magic pile, but what's its proper name?" Well, we call it the identity sandpile.
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But, you know, it's also the zero of this set. So we indeed call it a zero. Let me show you yet one more beautiful property of this zero. Remember, at the beginning we had these equations, a plus x equals 0. Now our zero is this 212, 101, 212. It turns out that in S we can always solve these equations. So there is always a minus sandpile on S. We can always solve this equation. So this set S has, you know, it behaves sort of like the integers. They, it has a zero, but it also has
16:23
you know, these additive inverses, this minus sandpile. You know, if we start from the all 3s, right? So, what could I add here to the all 3s to get to this zero sandpile? So, is there such a thing as minus a sandpile? Something that I add to the 3s to get to zero? Well, let's see. So if I add 333, 313, 333, I get the zero sandpile. Brady: "So in this very specific set we created, this is a negative sandpile." That's right. This is minus the all 3s.
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This is what I add to the all 3s to get to the zero. Brady: "But, Luis, if I had something else there," So for example, like the all 2s, Brady: "yes, that wouldn't have a negative effect." Correct. Correct. This minus sandpile is unique. For the all 2s, I have 232, 323, 232. This is also the zero sandpile. And, actually this works for any sandpile that comes from max, from the all 3s. I can always find a minus sandpile. So this zero, it's, you know, it's it's very special. I told you about,
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you know, that at the beginning, I was going to talk about fractal zeroes. This doesn't look like a fractal. I mean, it's very pretty, but it doesn't look like a fractal. So let me now tell you about fractals, okay? We can play the same game that we have played on a three by three grid. We can play it on any grid. For example, on the two by two grid. And if we play it on a two by two grid, the identity there is also an identity, and the identity is very simple. It's the all 2s.
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Brady: "Is that the identity for what set though, for the max set?" Yes. We have a two by two grid, and we look at the sandpiles on the two by two grid that come from max. Max in this case is 33, 33. So this is the zero sandpile on a two by two grid. On a three by three grid, well, it's, you know, the one we have computed, 212, 101, 212. Okay, we could do this on a three by three grid, on a four by four, or even on rectangular grids, like, you know, five by ten, or something like this.
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So let me show you a few more zeros, okay? So this is the three by three, what about a four by four? How does it look like? 2332, 3223, 3223, 2332. They, we always have to have some symmetry, because, you know, we are, you know, the grid itself, you know, it's very symmetric. So we can continue doing this. We can, let me show you, maybe one more, or a few more? Let me show you the five by five. Okay. So, the five by five is a 23232, 32123, 21012, 32123, 23232. So, they have some structure, okay?
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As you saw, they are symmetric, and we can, there is actually an algorithm to compute every single one of these. zeroes for any rectangular grid. So, but, this algorithm is, you know, not an algorithm, you know, for small cases it produces them very fast, but in very large grids, it will take, you know, maybe a lifetime to compute this. But we can compute, let's say, for example, a 400 by 400, okay? So, let me show you the picture. So this is the zero sandpile on a
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400 by 400 grid. And it's full of colors. So let me explain to you the colors. The black means zero grains of sand, the yellow means one grain of sand, blue means two, and red means three. And as you can see in this picture, there is, yes, a lot of symmetry, but then there is also this fractal structure that they have. And in fact, so this is something that is very interesting to people, you know, to try to describe exactly what type of fractal structure do we have.
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Brady: "So if I had a one million by one million grid, I would find this somewhere within it?" Correct. Correct. Yes. And, and, you know, so we know a few things about this, this type of fractals, but not a lot, okay? So, roughly, you know, the, you know, so you may ask, for example, what is the density of this, this picture? So how many grains of sand, you know, in average, are on a cell? And, well, you know, it has to be less than three. It turns out that it has to be bigger than two, okay?
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And it's more or less 2.1, okay? It's close to the point, so it's really near 2. In fact, you know, we can you know generalize this. Say, we, I showed you how to do this with grids. But we can do this with other type of grids, like hexagonal grids, and triangular grids, and get different zeroes, different sets. We could even play this game with an infinite grid. And very interesting things happen when you play on infinite grids. Would you like to see? So what I'm going to show you is
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an image created, created by Wesley Pegden, where you have an infinite grid, and then you put lots and lots of sand to one cell, and then you topple and see what happens. So in this case, we're going to put two to the thirty number of grains of sand in one cell, and then we're going to topple and see what happens. And the picture that we get is this type of picture. It's also, you know, fractal. Blue is zero.
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This is played on an infinite grid. So when you topple you're going to reach your, there is going to be some perimeter, and, and then after that you're not going to have grains of sand. So zero is blue, then a lighter blue is 1, yellow is 2, and red is 3. It's very interesting that, you know, the picture that you get looks sort of like a circle. So this avalanche doesn't just send grains of sand everywhere,
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but it sort of like preserves this circular structure. And we can even zoom in this picture and see what happens. So we can zoom in, and you see that there is this sort of fractal type structure. And, you know, things that, you know, only by looking very very close, we see, you know, that very intricate things are happening, right? So we can play this with different types of grids. This is, you know, a triangular grid, and this is, again, putting lots and lots of grains of sand in one cell,
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and then toppling. And in here, you know we have more neighbors per cell. So our max, you know, before we topple, we, you can put larger piles of sand. So in here we can put all the way up to 5. And that is stable. So we have more colors, and then it produces this type of also fractal looking picture.
23:59
...point goes back to itself. So we know that this point is on the straight line, because it intersects the circle inversion
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