Sandpiles - Numberphile
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- čas přidán 12. 01. 2017
- Luis David Garcia-Puente discusses sandpiles, and how they produce amazing "fractal zeroes".
Dr Garcia-Puente is an associate professor at Sam Houston State University and was interviewed while attending an MSRI-UP summer program.
We'd also like to thank David Perkinson and Cameron Fish for helping with sandpile visualisations. See more at people.reed.edu/~davidp/ and have a play at people.reed.edu/~davidp/web_sa...
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.
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I had no idea where this was going, but that was really beautiful in the end. Well worth the long view!
Peg Y do a video explaining how we express numbers like a bigg or a big boowa.
I agree, I really want to see an animation of a huge sandpile toppling.
Agreed!
I'm halfway and was like "this ain't going nowhere!". Taking your word for it and checking the end result :)
EDIT: It was worth it in the end. Cool!
my thoughts exactly
I've done an REU (research experience for undergraduates) and worked with Luis on this topic. He certainly has a knack for explaining concepts like these (in this case, implicitly illustrating the axioms of a group using sandpiles as an example) in way that even a general audience can grasp. Thanks Numberphile for showcasing Professor Luis' work and talent!
🎉
Group theory is such an underappreciated area in mathematics. Thank you for this great video!
Wait, what happend if the sandpile,
is the parkar square?
I'm on it...!
I had to create a program to do it, but i got:
1 3 1
3 1 3
1 3 1
Amazing! you're awesome!
Idan Zamir
Aw, thanks!
you have proven Parker's identitiy !
Would be cool if there was a video of the topple taking place.
It'd be difficult to choose colours which would show the detail in that range of numbers though. Probably not impossible, but difficult.
+Andy Brice
"{{0,0,0},{0,0,0},{0,0,0}} + {{0,0,0},{0,2^32,0},{0,0,0}} and topple until valid"
results in the same value as
"({{0,0,0},{0,0,0},{0,0,0}} + {{0,0,0},{0,1,0},{0,0,0}} and topple until valid) 2^32 times"
So this sketch would work:
while(true){
AddOne();
Draw();
while(!Valid){
Topple();
Draw();
}
}
No cell should ever have a higher value than 7. The highest valid value is 3 and then it could "get toppled into" 4 fom its neighbouring cells.
Check out the second link in the description, press DEL and then shift-left-click somewhere in the center of the grid to add a "source" cell. That simulates having a huge pile in the middle of the grid (essentially of infinite size), and you can watch the animation in real time.
@@quaternaryyy how does that thing work?
I try tapping and no sand appears
I set it to drop sang
@@Xnoob545 go into brush, click set clicked cells to n grains type in your number and then just click anywhere
Those 23 dislikes must've been sand grains that fell off the edge
L
best version of this meme I've ever seen.
yep, opinions are like grains of sand--everybody is one.
Well, I changed my like to dislike after realizing that he calls 0 a natural number.
@@TruthNerds I thought it is a difference in Russian and American versions of definitions, but, in hindsight, mathematics is not the place to have these kinds of inconsistency.
Natural numbers ARE starting from 1 for every nation, then, correct?
The best handwriting on Numberphile so far.
This dynamic reminds me of Conway's Game of Life work.
I thought this also. It's amazing to see such complexity arise from such simple rules and starting conditions. I'd be interested to see what happens for different starting conditions, like if randomly dumped a few large piles around instead of just 1.
It is, in fact, a form of cellular automata.
It's more than that: it's a set of cellular automata with an actual group structure to it, which is something I've never seen before
to be honest, I was not sure if that's cellular automata because I don't know the formal definition of it, so I can't really say, what it is...
I clicked out of my full screen playlist to suggest "isn't this similar to Conway's Life?" Glad I'm not the only one.
The word "mindblowing" is used incredibly liberally these days but this really did blow my mind. I'm still feeling numb from trying and failing to grasp the implications of this.
I'm excited for every Numberphile video that shows up in my subscriptions because they are interesting and entertaining but this stuff played in a whole different league.
+Penny Lane thanks. Lovely comment.
+
Numberphile
>> failing to grasp the implications of this.
This has something to do with complexity in biological ecosystems, and in immune systems, and in genetics.
@@otonanoC nice
By far, this is one of the most inspiring videos on this channel. I can't really explain why...
should we push this to the next dimension? A 3 dimensional grid?
Yes and infinite dimensions
6 topples, you might have 5 grids?
Prepare your 4d eyes to see it
@@ferencgazdag1406 The cells can have very small but different levels of opacity for each value.
@@NesrocksGamingVideos It would still be inconvenient.
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?
what do you mean by that?
That’s actually not a coincidence
@@Jared-ss3jx He means the zero-pile has a 2 in the corners, where 2 grains fall off the edge, a 1 along the edges, where one grain falls off, and 0 in the center.
@@Jared-ss3jx that has many definitions
I like how I fail all my Precalculus tests but still enjoy and understand all of Numberphile's videos.
How did normal/AP Calc work out for ya?
woah. started with such simple concept and ending was out of the park. great video
HOW U LOOK LIKE THE PERSON WHO MADE THIS VIDEO HOW???
@@cheeseburgermonkey7104 🐒😊
Dr. Garcia-Perente is probably my favorite interviewee/lecturer with only a single video with numberphile.
The end was really surprising and beautiful. Do watch it to the end
Join Brady's occasional email list (or Numberphile's Patreon, of course) for a chance to get occasional freebies, such as signed Numberphile postcards... eepurl.com/YdjL9
Numberphile Why should i donate you?
there is no obligation to donate john just do it if you want
One reason is if you appreciate the creator of the videos. Another would be for the benefits, you can read about the benefits to support on his patreon page.
veggiet2009 Nobody did ask you
john smith for a chance to get occasional freebies, such as signed Numberphile postcards!!!! don't you listen?
I was waiting for Brady's usual question... "Is this just a game that someone made up, or did it have some real practical reason for exploration?" "Can we learn something about other things because of research in this area?"
+Dave Brosius I don't think I ask that as often as you may think. I quite enjoy these things just for being awesome.
My instinct guess is that it is too beautiful to NOT have usefull applications for anything else
I actually was hoping for the question. The ending was really beautiful, but sometimes it seems that mathematicians are doing things just for the sake of doing things. It's interesting to see the reasoning behind this thing existing, even if it is "we just wanted to see what would happen".
Change "grain of sand" to atom, or proton and go back to the start of the universe and evolve the pile. :D
The most hilarious point about this is, that it mostly leads to some adoption down the road.
Just take a look at Surreal Numbers, Quaternion or Fractals.
Everybody agreed they are useless, until somebody found them useful and they started to pop off.
Surreal Numbers found their way into to Algebra, where they belong.
Quaternion are the basis for any fast approach to 3D Rotations.
Fractals are everywhere, from your mobile device, to decryption, to randomizing and even in modern medicine.
There is most likely always an adoption at some point in the future.
I love this host, he makes this seemingly trivial aspect of math not only engaging but extremely easy to follow, you love to see it
Perfect! Best numberphile video ever! Was a little Boring at first, but thank fucking God I watched it whole!
thanks for sticking with it!
Numberphile ,this sandpile algebra is insanely beautiful in its fractal form but are other arithmetical operations applicable in it and what would happen if we keep on increasing the no. of maximum sand grains in each cell of the infinite sandpile grid.
Yugandhar, if you're curious about these kinds of questions you should learn to program and have a play ;P that's half the fun of it.
8bitpineapple can you please teach me how?
+
I hope we get more Numberphile videos on sandpiles in the future! There are so many concepts to explore!
-How do you calculate the identity for any given sandpile?
-What if you changed the rules for collapsing in some way? (Maybe collapse the four by distributing one to each of the diagonal cells, rather than each of the adjacent ones?)
-What if you considered all numbers up to and including 4 as "stable" (don't need to be collapsed)? What about up to and including 5? 6?
-What kind of cool patterns are there when dealing with sandpiles that don't have symmetric patterns (randomly generated numbers for each cell)?
-What happens when you subtract sandpiles from each other, rather than just add? What about multiply? (Too bad you would run into issues with division)
And most obvious of all:
-Why do identity sandpiles and sandpiles collapsed from one center cell result in such beautiful fractals?
That final question has no answer, because beauty is necessarily subjective.
Did this guy go through your whole "brown paper" budget for 2017?
I would love to see that sand topple from the beginning for the 2^30. Many iterations per second of course
I doubt that the person who made that actually calculated it iteration by iteration, so it would likely take much more computation to do that
It went from "okay" to "interesting" to "fun!" to "cool" to "woooooow". Really well explained, really well put together, and what a payoff!
This was by far one of the most satisfying and interesting videos I've watched in a while. I hadn't realized how long it was till I paused to get a more detailed look at the fractals. Thank you for producing such unique and wonderful content
IT did not feel like a 24 min long video, but way shorter. Great video
On the contrary, it felt way too long. A lot of the tedious small number addition should have been cut.
Wow, ive been following this channel and others of yours for over 3 years by now and i honestly say this is my favourite video so far. I think the professor did a really good job at explaining this very abstract concept by giving so many examples. i found myself even skipping back to grasp the full concept so I was happy you made the video as long as needed. Good job and thanks to both of you!
That left me speechless. What a great finale.
Every video keeps amaze me... The end here is beautiful, ppl thinking video is "too long", stay until the end, it worths the effort.
+Flandre Scarlet ;)
I was tempted to stop watching a little past halfway through. It really took a turn towards 'woah' and all came together at the end.
it has its rewards!
Haha that's exactly when I went to the comments. But then when I saw your comment, I stopped reading and watched to the end!
At 5:06, the appearance of the zeroes are synchronized with him saying "zero", loving the little attention to detail
I'm a long time viewer of numberphile and I freaked out when I saw this video with Dr. Garcia! He was my calculus 2 teacher in college and that was the class that I discovered my love for math and made me change my major to math along with graduate studies in math.. and now i see him doing the same inspirational stuff on this channel. Crazy stuff man
Jon Woek Shoah that’s pretty epic
The long videos are always the best
Probably the best and most interesting video on this channel.
It is high up one the list, for sure.
Thank you very much for the sound correction right around the 20-minute mark, I saw it hiccup and then heard the switch to the different microphone and that was far preferable to an audio de-sync or loss of audio. Audio guys don't get a lot of recognition for the work they do and I want to say thank you to whoever caught and fixed that.
I really liked Luis' presentation, both the subject matter (which I am a huge nerd for), as well as the manner in which he presented it. It was very clear and easy to follow and I hope he shows up in future Numberphile videos.
I can always watch old numberphile videos and still be amazed
Those fractals are amazing
Could we get zero for 1920x1080 sandpile group, please.
And recursively fill the center rectangle with zero sandpiles of the rectangle's size.
You can download the program from the description and basically do everything they showed you in the video!
In the identity for the 1920 x 1080 group, the middle 392 columns are all 2's. You're on your own for the rest of it.
I'm amazed with the richness of Mathematics, and the hidden beauty lurking in the dark, waiting to be unveiled.
Keep it up Brady! Love your videos!
Wow, i was kind of skeptical about how abstract this was going in but it blew me away as it progressed. Talk about beauty in numbers. Im glad im subscribed to Numberphile.
The concept of zeros in sets like this are interesting. I wonder if they contain other properties of our normal set of numbers such as "even/odd" or "prime."
AlabasterJazz I wonder how you would define factorization in this system.
Primes can be more generally studied in "Rings", which are sets like in
the video but where you also can multiply (which you obviously need to
even make sense of "prime" and factorization) and the "normal" rules for
multiplication apply (like associativity/distributivity) and probably
also the existence of "1", meaning something like a zero just for
multiplication. If these thing would work for the sandpiles, then you could define "a divides b" via "there exists a sandpile c thus b=c*a", and start talking about primes and factorization.
The most obvious way to define multiplication is via just multiplying correpsonding entries and then toppling, but wether that has an identity I'm not sure (considering the all 1 grid is not in S)
For the 2x2 or 3x3 sandpile groups, defining A x B with cellwise multiplication will not make them rings. In a ring, if I is the additive identity, then A x I = I x A = I for all A. Let A =
0, 2, 2
2, 2, 1
2, 1, 2
A + I = A, so A is in S, but A x I =
0, 3, 0
3, 0, 3
0, 3, 0
For the 2x2 group, the sandpile
0, 3
3, 3
deals the death blow.
It's nice to see some algebra on numberphile.
Bravo. This is easily in the top 5 videos this channel has ever produced.
I thought it was too complicated for a person who is not advanced on maths, then I went "Oh, I got it that's working like a 0 because of the grains", then the graphs appeared and I thought "Wow, that looks amazing"... and at the end the zoom blew my mind.
Music and math are so beautiful.
Super fascinating and well explained!
+crashtextdummie thank you
Identity sandpile, like identity matrix?
Penguin Design Yep, it's the neutral element concerning addition in the set of sandpiles coming from All-3s.
Just like the identity matrix is the neutral element concerning multiplication in the set of matrices.
Now you talk about multiplication, and that was also my thought.
They introduce the +0. But what about the *1?
The neutral Element in Mulitiplication
Thiemo Krebsbach isn't +0 and *1 the same though? They are both identities.
0 is the additive identity, and 1 is the multiplicative identity. It depends on the operation. For exponentiation, the identity is 1. For matrix multiplication, the identity is the identity matrix. For matrix addition, it is the zero matrix.
Well, they didn't even define a multiplication operation between sandpiles.
Dr. Garcia-Puente is by far one of the best Math lecturers at Sam Houston State University. I had him for Discrete Math and Applied Algebra, and I can vouch for his unparalleled quality.
Beautiful how math does the unexpected and yet demonstrates an intricate pattern.
I need more
When I read this comment, it was at the bottom of the truncated comments. Right under it was the "Show More" button. I got a giggle out of that. :3
So there is an identity, an inverse for every element, commutativity and closure.
Is the operation associative though? The set S being an Abelian Group would be so cool
Thats the question
Googling a little bit, I believe found an answer saying indeed, it is
TG MrNacknime at first I thought the sandpiles would be a subgroup of (GL_3(Z/4Z),+) and then he threw in that odd decomposition and I was really wondering whether it would have any group structure, and it's really cool that it does!
Yes it is associative. To prove that, consider these grids as equivalence classes 'modulo', where modulo here means that you can substract 4 in any cell while adding 1 in the adjacent cells (or the opposite, add 4 in a cell and substracting 1 in adjacent cells), any time you need. In that new set, the addition is the usual one for matrices, cell-wise, and it is nicely commutative and associative. In addition (pardon the pun), that gives you a simple algorithm to find the inverse of a grid g. The identity [2,1,2,1,0,1,2,1,2] is equivalent to [4,5,4,5,4,5,4,5,4] in which every number is greater than 3. Therefore you just find the complement of g by a regular substraction (and if needed, you 'modulo' it).
In case I wasn't clear, here is an example.
You find to find the opposite (or inverse) of g:
g = [[1,1,1],[2,2,2],[3,3,3]]
identity is i = [[2,1,2],[1,0,1],[2,1,2]] ~= [[4,5,4],[5,4,5],[4,5,4]]
i - g = [[4,5,4],[5,4,5],[4,5,4]] - [[1,1,1],[2,2,2],[3,3,3]]
= [[3,4,3],[3,2,3],[1,2,1]] (regular matrix substraction)
~= [[1,3,1],[1,1,1],[2,3,2]] (equivalence by toppling the sand where needed)
This is similar to what I am studying right now, and I love it. This might be my favorite numberphile yet!
That ending is amazing. Definitely worth spending my 24 minutes watching this.
Pandemic players when they see 4 grains spreading to other cells
*war flashbacks
Wow, this is amazingly beatiful
One of THE best videos on Numberphile! Thanks!
That's pretty amazing. I've never seen these kinds of group structures with weird zeros before.
Wow this video showed how beautiful numbers can really be! ^^
So the maximum amount of sand for a cell is n-1 where n is the number of neighbors one cell has?
The triangular grid one had 6 stable states, which I find weird seeing as triangles only have 3 neighbors.
And if they count the triangles touching at each vertex then it would have 12 neighbors. And if they counted the opposite facing triangles it would be the correct number of 6 neighbors but it would make the square have 8 neighbors.
It is weird.
PerunaVallankumous yepp. that's the maximum untoppled pile
I am guessing that it is each intersection, or node in the triangular grid were you put the sand, which makes it more of a hexagonal grid but it is the only explanation I can think of.
I believe it has to do with the minimum length of cycles (6 for the triangular grid, 4 for the square grid, 3 for a hexagonal grid), but that doesn't seem to match the definition used on Wikipedia, which has it depend only on degree (3 for a triangular grid, 4 for a square grid, 6 for a hexagonal grid).
Yes, that is correct according to wikipedia. en.wikipedia.org/wiki/Abelian_sandpile_model
Super cool. The end definitely justifies the build up. The phrase, it blew my mind, gets some experiential understanding here...
I saw a book about this at JMM last year and wondered what was creating the beautiful images on the cover. It’s great to finally find out!
I love his shirt :)
How *do* you calculate the Identity?
A brute-force method I can think of is just creating a set of all possible sand-pile grids then adding them to a sand-pile grid full of 3's to get the special set of sand-piles. Then you pick a sand-pile from the special set and start adding other sand-piles from the set until you find the one that doesn't change anything.
Surely there has to be a way other than brute force.
Lordious
You don't need to add two different sand piles to each other from the set, can't you pick single sand piles to add to themselves and brute force until you find the right one then?
He said there was an algorithm that generated the identity for m x n grids and hinted that the run time grew exponentially.
Didn't think I was gonna like it at first, but now I am amazed :)
That may be my favorite Numberphile video of all time, just for the big WOW factor at the end!
It would be cool to do this on like a toroidal grid by which I mean the edges are connected. Then it would be cool to see if sand could topple indefinitely on one of those.
You'd probably need to modify the rules a bit.
In generalizations of the sandpile model, the sandpile is modeled with a group (a network of nodes connected by edges where grains are placed on the nodes and travel along the edges to neighboring nodes). In these generalizations, there is usually a "sink" node where the grains of sand go to disappear in order to prevent infinite toppling. In the case of the grid model, the "sink" is the edge of the "table" where the sand falls off.
One method is by brute force. Take any sandpile s in S (for example, the maximal sandpile) and add it to every other sandpile in S. Once you find the sandpile t such that s + t = s, then you know that t is the identity sandpile.
If you're reading this, have a Great day! 😄😄😄
That was a really great episode - thank you very much. It was very well explained as well!
Numberphile is my favorite channel to watch *really* early
C++ has just called me. It wants to do something cool tonight.
Okay, one question though: Do these numbers have rules like a-b-c = (a-b) - c = a - (b+c) or do they not behave that way?
The question of subtraction is a weird one when it comes to sandpiles - mostly because saying that a pile has a negative number of grains would create a kind of "sandsink" - then arises the questions of how a sink would topple, if at all, and how it might return to zero.
Better to restrict the operations to ones that don't necessitate negative elements, like addition and the extreme weirdness of sandpile multiplication
@@rovingfortune395 so why can't we have sand sink...we can define that whenever there is a sink it will gain 4 grains from its neighbours.....just an idea....also If we do consider this I think there might be some boxes which will keep on oscillating and would never reach a solution in finite steps.
Dr. Garcia!! I loved your Discrete Mathematics class and Linear Algebra class. I can't believe you're featured on Numberphile, that's awesome!
love the sandpile distribution graphic!! Great video, as always.
Parker Sandpile.
1 3 1
3 1 3
1 3 1
Se llama Luis David Garcia-Puente porque es una puente que dirige a sabiduría y conocimiento
I would have loved to attend MSRI-UP this past summer. Great video, thank you for sharing.
I smiled every time he said two and zero. Love the accent.
I wonder what would happen on a toroidal grid?
yes. or what will happen for objects with more then 1 hole. Like a double-donut. It might have application in topology.
Why did he only allow 0, 1, 2 and 3 in the square grid, but allowed 0, 1, 2, 3, 4 and 5 in the triangular grid?
Thanks, but I got that. Yet, why not allow more? Why not topple when 7 sand grains are in a cell of the square grid?
theFizzyNator can you expand upon "to make toppling make sense" please?
I think it's quite arbitrary but there's still a constraint: you need to topple with at least the number neighbors grains. If not, you wouldn't have enough grains to give to every neighbor.
But, as orochimarujes pointed out, it would be possible to select randomly which neighbors get a grain. I think that could give interesting results, still.
I'll explore myself toppling at an higher threshold
Not necessarily. I think random toppling would still have structure because the randomness would average out. It would be interesting to see this.
This is one of the best videos on the channel.
Totally blown away at the end!
Sandpiles? More like Sandphiles.
Fine, I'll go 😢
Is there one for a hexagonal grid?
James Moran probably.
There is a game called Hexplode that works on a hexagonal grid - the only difference comes from the fact that it is played on a finite grid - when the cell only has 2 neighbours, its maximum is 2, when it had 4 its maximum is 4 and so on. Makes for more interesting strategy, but loses something if the unity of the real sandpile group.
This absolutely blew my mind. It seemed pretty boring during all the computing, and it seems frivolous to ask that kind of question (which is a way to describe a lot of math, haha), but it really drew me in to want to experiment with it once they showed those fractals. That was beautiful and I want to try something like that myself
Does the order of sand-pile toppling effect the end result? What if you have two 4's next to eachother. Surely the result changes based on which 4 topples first.
EDIT: After some critical thinking, no the result does not change because a 5 leaves a 1 after it topples. Two 4's next to eachother always produce two 1's and the same surrounding numbers regardless of the order they are toppled.
Sandpile! Notice me!
That was awesome... I absolutely adore numberphile keep it up, proud of you.
Really liked this, it's probably the best video I've seen here in a while.
Awesome finale! But I think you should have made video a bit shorter, a lot of people will give up on watching.
So is this an example of a group?
Only if this operation is associative. In which case, since each of them is finite and also abeliam has some decomposition into a product of Z_ps which would be pretty frickin cool. So i hope that it is.
That went in a surprisingly beautiful direction
Great video! I've seen the abstract definition of a sandpile group before, but never really thought about what the zero in this group means... Very helpful to do calculations on small examples, and to see the color-coded picture towards the end!
Sounds like senpai amirite?
looked for this comment, found it finally :)
MDFlight I commented the same before it was cool
Can explain the joke here?
Bragtime Notice me sandpile!
Sandiles?
Salandits
Sundials?
senpais
Palossand
Shiny Swalot you're subbed to carl too, right?
love it!! math is really beautiful and with your videos you make us appreciate it, thank you!!!
I like the sound effect when the sand topples in the illustrations.
So, imagine the big bang was a *HUGE* number of particles (sand grains) in the center of the grid (the universe itself?) following mathematical rules of toppling. Now topple them.
Wait weren't Natural Numbers from 1,2,3...
and Whole Numbers from 0,1,2,3...?
That would be too easy gotta mix things up.
No. The "whole numbers" are the integers and include the negatives.
There is some inconsistency about what constitutes the natural numbers, though. Dr Garcia-Puente is using what I think is the best convention. The "counting numbers" are the positive whole numbers (since you can't count zero things) and the natural numbers are zero along with the counting numbers.
natural numbers include 0
Sauradeep Chakraborty There's different conventions. It seems that excluding 0 from the natural numbers is more common in school curriculums, but including 0 is more common in professional mathematical writing. If you want to be unambiguous, you can say non-negative integers (i.e. 0, 1, 2...) and positive integers (i.e. 1, 2, 3...).
Benjamin Przybocki Yeah. That's what I do. Its just that through 5th grade we've been asked this question in numerous tests: do natural numbers include 0 and always the correct answer was no. So I thought after watching this that maybe my life is a lie.
Many thanks for not spoiling the ending in the thumbnail! The gradual buildup of “holy crap” was most enjoyable :-D