Sandpiles – Numberphile

Sandpiles – Numberphile

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. 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, 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? 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, 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, 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. 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, 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? 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. 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, 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, 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, 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. 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 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. So each of them is going to lose four grains of sand, so now they are going to be zeroes. 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. 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, you know, they are going to lose four grains of sand, so each corner will have 1 at the end, 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. 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. 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 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 plus 131, 333, 131. 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 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, 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 topple, and they are going to lose four grains of sand, so all the 5s are going to be now 1s. 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. And this 6 will topple. It’s going to lose four grains of sand, and we get the all 2s back. 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.” 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?” 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, 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. 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 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. 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, 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. 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. 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? 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 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. 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? 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 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. 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, 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, 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. …point goes back to itself. So we know that this point is on the straight line, because it intersects the circle inversion

Comments (100)

  1. I had no idea where this was going, but that was really beautiful in the end. Well worth the long view!

  2. I wonder if one could use large sandpiles as a basis for a discrete log style cryptographic algorithm…

    We know the max sandpile is a generator G. Then, you can do scalar multiplication nG fairly easily using something like a square-multiply technique. Is it easy to recover n? Is there a Sand Pile "Discrete Log" Problem? Elliptic curves prove extremely useful in Secure Multiparty Computations largely due to their speed and their prime orders, allowing fractions in the calculations. I wonder what properties these groups have. Are they fast? How big would they have to be? What kind of order would these groups have?

  3. Reminds me of simulating the wave equation on a grid

  4. Срань господня, советский ковёр в конце. Так и знал.

  5. notice me, sandpile!

  6. This video was like the novel Father Goriot. first half was a bit boring and took too long, and then bam!
    But I miss one thing. How does the substraction work?
    I mean, we have the zero, so if I get some sandpile A, how to compute the -A sandpile? 0-A=??

  7. more of this guy please!

  8. Take an infinite grid, already filled with 3 in all positions. Add one grain of sand. What happens?

  9. After 15 minutes it sounds like he says senpai and not sandpile

  10. Surely the bit at the end is a hexagonal grid not a triangular one since triangles only have 3 neighbors.

  11. Am I the only one who thought the "2^30" grid looked like the Galactic Senate from Star Wars?

  12. 2 + 2 is 4

  13. Is this work published somewhere? I'm very interested in reading more about this.

  14. No, natural numbers begin with 1…the whole numbers begin with 0.

  15. Here's something to imagine.
    Imagine placing a grain of sand on a three sandpile plane that follows the rules of sandpiles (apart from losing sand off the edge. If you did correctly then it will turn to a checkerbord like the below one:


  16. Try adding the all 3's to itself. It takes FOREVER to topple.

  17. Group theory is such an underappreciated area in mathematics. Thank you for this great video!

  18. I really enjoyed this; a surprising and beautiful emergence of fractals.
    Could you do this with each cell in the grid being a sandpile from some other set S', rather than a number, somehow?
    Then you could take the smaller sandpiles as a 3×1 sandpile of RGB values, and really invert an image..

  19. How about
    212 000 212
    212 000 212
    in this case
    is not 0

  20. Odd how 1 = 2 = 4 work in sandpiles. Seems like broken math.
    Missing context – how does this provide value to real world applications.

  21. what's his shirt? prada? love it.

  22. Is it my imagination, or do the 'zero' sandpiles for 2×2 and 3×3 have values that are just as many as they lose to the 'edge', possibly reflecting that each box on the edge for those dimension only topples once?

  23. What is the smallest 8-digit number possible? Make sure your answer is in base 10!

  24. So does this mean that the set S is a group under sandpile addition?

  25. very interesting… I would like to see a follow up video on this

  26. One of THE best videos on Numberphile! Thanks!

  27. So what is the identity for the infinite grid?

  28. I would like to see the ancient mathmaticians and artists shown a computer and document their emotions.

  29. i've found some interesting patterns that arise if you repeatedly enter two into each cell. there is so much here. i want to measure this and be able to explain it mathematically but i'm not at that level yet 🙁 this is amazing.

  30. Where can I find a list of the zeroes for 6×6, 7×7, … grids?

  31. For those wondering, here are proofs of most of the claims in the video.

    I am assuming that you know what an Abelian group is.

    I will start off with a very important lemma. I will refer to this as the group construction lemma.

    Let A be a set with a commutative associative binary operator +:A×A->A. Assume there exist elements a and b in A, and a function f:A->A, such that a+a+b=a, and such that x+f(x)=a holds for all x in A. Then for G:={a+x|x in A} and 0:=a+b we get that (G,+,0) is an Abelian group.

    Proof of the group construction lemma:

    First note that + is well-defined on G, as (a+x)+(a+y)=a+(x+a+y). Hence it is also commutative and associative on G. Also note that 0 is obviously an element of G.
    Furthermore, for all a+x in G, we have (a+x)+0=a+x+a+b=a+a+b+x=a+x, so 0 is indeed an identity of (G,+).
    Finally, for all a+x in G, we have (a+x)+(f(x)+b+b)=a+(x+f(x))+b+b=a+a+b+b=a+b=0, so indeed every element of G has an inverse.

    I will now prove that every finite sandpile-like set M with the binary operator + defined as in the video has this property that the set S:={m+x|x in M} forms an Abelian group, where m is the maximal sandpile.

    First note that + is commutative, since addition of numbers is commutative.
    That + is also associative is less trivial, but it comes down to the fact that the order of the toppling does not matter. So to calculate a+b+c, you can first add together the cells and then do the toppling, which makes it clear that + is associative. I can elaborate more on this if someone requests so in a reply.
    We will now look for a and b that satisfy the conditions of the group construction lemma. For this, define the sequence x(i)=i*m=m+m+…+m (i>0 times). Since M is finite, x can not be injective, so there exist i<j such that i*m=j*m. We find i*m+(j-i)*m=i*m, so by induction, i*m+k(j-i)*m=i*m for all k. For k>i/(j-i) we get i*m+i*m+(k(j-i)-i)*m=i*m, so for a:=i*m and b:=(k(j-i)-i)*m we get a+a+b=a.
    Finally, we find a function f that satisfies the condition of the group construction lemma. We take f(x)=(i-1)*m+(m-x). Here m-x just means subtracting in each cell separately. This is well-defined, because m is the maximal sandpile. It should be obvious that x+f(x)=a holds for all x in M.
    By the group construction lemma, we find for G:={a+x|x in M} and 0:=a+b that (G,+,0) is an Abelian group. The only thing left to show is that G=S.
    That S is a subset of G follows, because a+x=i*m+x=m+((i-1)*m+x). That G is a subset of S follows, because m+x=a+((m-a)+x). Again, m-a means subtracting in each cell separately.

  32. math is not only a language, it is art

  33. The video is super awesome but the camera blur(not focused) is kinda irritating.

  34. What are the sequence. . 1+1/2+1/3+1/4… 1+1/2^2+1/3^2…. 1/1/2^3+1/3^3… 1/n^n upto 9… then 1-1/2^2+1/3^2…. and so many fractals.

  35. Has anyone played the game Chain Reaction on the Android appstore??

  36. The name should be atomic nucleus

  37. They payoff at the end was worth watching 24 minutes for sure

  38. Today i will show you a different type of numbers, that have this very special zero that looks like a fractal,
    What is a zero?
    *music starts

  39. drink every time he says zero

  40. please have luis on again some time!

  41. but… 0 in N ?? Haven't heard anyone going by that convention, can anyone tell me which litterature has that?

  42. Ever tried hexagonal sandpiles?

  43. 1100111 1100001 1101101 1100101 100000 1101111 1100110 100000 1101100 1101001 1100110 1100101

  44. I learned in school that Natural numbers begin with 1(eg 1,2,3,4) and whole numbers begin with 0 ( eg 0,1,2,3,4)????

  45. I like the symmetry of the zeros, magic I tell you.

  46. 222 333 131
    222 – 333 = 333
    222 333 131

  47. This is my favorite numberphile video

  48. Dr. Garcia!! I loved your Discrete Mathematics class and Linear Algebra class. I can't believe you're featured on Numberphile, that's awesome!

  49. Did I miss it? Surely if you add the identity pile to a pile of all ones, you get a different answer to all ones

  50. Why do acid hallucinations look like math?

  51. I wonder what this would look like if you did it on a penrose tiling.

  52. Are there any magic square sandpiles? Cough parker square cough

  53. this is the single slowest and most boring maths video i've seen. what's next? solving 2*x+1=2 in 2 hours?? stop slowing everything down so much, I want the maths not a private teacher feeding me the maths with a spoon half a bite per minute

  54. Natural numbers are 1,2,3… right? Not 0,1,2,3…

  55. Awesome to find a Mexican guy on numberphile.

  56. 19:34 this is kraftwerk

  57. That ending is amazing. Definitely worth spending my 24 minutes watching this.

  58. notice me, sandpile!

  59. 23:07 we can even Z O O M

  60. he says 'zero' around 89 times, and 'topple'/'toppled'/'toppling' around 22.

  61. Experienced camera operators are hard to find.

  62. Just wondering, so if M is the set of all 3 by 3 sandpiles…

    If you have A + X = A. A is any sandpile from set S. X is any sandpile from set M (not set S). Well, X could be the zero sandpile, but it could also be the magic identity sandpile you discussed. How many things could be X?

  63. That's what I'm gonna do next summer with the sand at the beach.

  64. What do you do when you get a 5,6 or 7 ??
    which BTW can happen

  65. They r whole nubers

  66. Se llama Luis David Garcia-Puente porque es una puente que dirige a sabiduría y conocimiento

  67. This is some serious mindf*ck.

  68. anyone else notice that the magic sandpile for S had values in each square representing the number of grains of sand that are lost to the grid when the pile topples?

  69. It's sort of like a game of Pandemic.

  70. Its a 'zero' because the values in each position is equal to the number of invalid neighbors it has

  71. this mathematician makes better use of the brown paper than just about anyone

  72. I love the way this guy says zero (lol).

  73. Very underrated video

  74. Big Bangs in the end. 🙂

  75. Yo a heads-up
    The sandpile simulator in the description doesn't function anymore.

    The website loads and all the tools work but the sandpile itself never loads, making it defunct.

  76. 1,2,3… is both the counting numbers and natural numbers? Whole Numbers are 0,1,2,3… Correct?

  77. I know a way of solving a+x=0

  78. how do you find "-sandpile"? its obviously not "just" a substraction

  79. Sandman wants to know your location.

  80. what about toppling a 7? Or is this possible because 3+3=6 is the highest combination made using these numbers?

  81. Is there any real life situation, perhaps something of the like of fundamental particles/forces interactions, 3 or more planet solar systems or some other intractable problem that these techniques can be employed to model or is this currently purely of interest only to mathematicians ?

  82. In the zero sandpile, every number is how much edges it's touching.

  83. There are so many variations which can be played with. There can be walls between adjacent cells of various heights (possibly infinite), which would change the number at which they could topple into one another – else they would have to flow around the walls. The edges can loop around (like Pac-Man/a torus/a cylinder, or as a Möbius strip, along any given axis). The toppling number can be changes (the minimum is the number of cells which count as adjacent, I think). Diagonals could count as adjacent (or only diagonals in one direction). The dimensionality can be varied (these were all two-dimensional) and the number of cells in each axis can be varied; it need not even be uniform (the edge can be jagged).

    The possibilities really seem endless.

  84. Until see this I didn't realise that I was not the only nut doing such weird number experiments. VERY INTERESTING! THANX!

  85. did i miss where he explains how the 5 topples, where does that extra 1 go?

  86. 10:46 …we are never able to get rid of all the sand…

    yeah, i know that problem for sure XD

  87. Correct me if I'm wrong, but natural numbers don't include a zero, right? Why does he refer to natural numbers with zero in them?

  88. I can smell the purple sharpie.

  89. 232 232 212
    232 232 212

  90. Anyone have a reference for the addition, zero, inverse stuff? (first part of the video)

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