Sandpiles - Numberphile

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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.
NUMBERPHILE
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Videos by Brady Haran
Patreon: / numberphile
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Brady's latest videos across all channels: www.bradyharanblog.com/
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12 янв 2017

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Комментарии : 1 тыс.

@pegy6384 7 лет назад

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

@p.mil.1147 7 лет назад

Peg Y do a video explaining how we express numbers like a bigg or a big boowa.

@PeguinDesign 7 лет назад

I agree, I really want to see an animation of a huge sandpile toppling.

@fossilfighters101 7 лет назад

Agreed!

@devling6606 7 лет назад

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!

@marksmod 7 лет назад

my thoughts exactly

@reuben2011 7 лет назад

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!

@DrKaii Год назад

🎉

@xStrongHD 6 лет назад

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

@idanzamir7540 7 лет назад

Wait, what happend if the sandpile, is the parkar square?

@robinsparrow1618 7 лет назад

I'm on it...!

@robinsparrow1618 7 лет назад

I had to create a program to do it, but i got: 1 3 1 3 1 3 1 3 1

@idanzamir7540 7 лет назад

Amazing! you're awesome!

@robinsparrow1618 7 лет назад

Idan Zamir Aw, thanks!

@TheGamblermusic 7 лет назад

you have proven Parker's identitiy !

@inquaanate2393 7 лет назад

Would be cool if there was a video of the topple taking place.

@iAmTheSquidThing 7 лет назад

It'd be difficult to choose colours which would show the detail in that range of numbers though. Probably not impossible, but difficult.

@ihrbekommtmeinenrichtigennamen 7 лет назад

+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.

@quaternaryyy 7 лет назад

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.

@Xnoob545 4 года назад

@@quaternaryyy how does that thing work? I try tapping and no sand appears I set it to drop sang

@draganjonceski2639 4 года назад

@@Xnoob545 go into brush, click set clicked cells to n grains type in your number and then just click anywhere

@nowymail 7 лет назад

The best handwriting on Numberphile so far.

@shield543 7 лет назад

Those 23 dislikes must've been sand grains that fell off the edge

@Someone-cr8cj 6 лет назад

@daniellebarker7205 5 лет назад

best version of this meme I've ever seen.

@jasonstone1833 5 лет назад

yep, opinions are like grains of sand--everybody is one.

@TruthNerds 5 лет назад

Well, I changed my like to dislike after realizing that he calls 0 a natural number.

@sirhasslich536 5 лет назад

@@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?

@unvergebeneid 7 лет назад

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.

@numberphile 7 лет назад

+Penny Lane thanks. Lovely comment.

@KrupaHebbar15 7 лет назад

@bernardweisblum2060 7 лет назад

Numberphile

@otonanoC 5 лет назад

>> failing to grasp the implications of this. This has something to do with complexity in biological ecosystems, and in immune systems, and in genetics.

@fasligand7034 4 года назад

@@otonanoC nice

@peppybocan 7 лет назад

This dynamic reminds me of Conway's Game of Life work.

@St3venAU 7 лет назад

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.

@alexanderf8451 7 лет назад

It is, in fact, a form of cellular automata.

@maxkolbl1527 7 лет назад

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

@peppybocan 7 лет назад

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...

@drskelebone 7 лет назад

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.

@jonathanc8845 5 лет назад

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?

@Jared-ss3jx 3 года назад

what do you mean by that?

@debblez 3 года назад

That’s actually not a coincidence

@cheshire1 2 года назад

@@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.

@DrKaii Год назад

@@Jared-ss3jx that has many definitions

@TheRubixPro 7 лет назад

I like how I fail all my Precalculus tests but still enjoy and understand all of Numberphile's videos.

@trickytreyperfected1482 4 года назад

How did normal/AP Calc work out for ya?

@utl94 7 лет назад

By far, this is one of the most inspiring videos on this channel. I can't really explain why...

@guanche011 7 лет назад

The end was really surprising and beautiful. Do watch it to the end

@chasetuttle2121 7 лет назад

should we push this to the next dimension? A 3 dimensional grid?

@Nulley0 4 года назад

Yes and infinite dimensions

@whatisthis2809 4 года назад

6 topples, you might have 5 grids?

@ferencgazdag1406 4 года назад

Prepare your 4d eyes to see it

@NesrocksGamingVideos 4 года назад

@@ferencgazdag1406 The cells can have very small but different levels of opacity for each value.

@ferencgazdag1406 4 года назад

@@NesrocksGamingVideos It would still be inconvenient.

@DrTacoPHD665 2 года назад

Dr. Garcia-Perente is probably my favorite interviewee/lecturer with only a single video with numberphile.

@GaneshNayak 7 лет назад

woah. started with such simple concept and ending was out of the park. great video

@cheeseburgermonkey7104 4 года назад

HOW U LOOK LIKE THE PERSON WHO MADE THIS VIDEO HOW???

@DrKaii Год назад

@@cheeseburgermonkey7104 🐒😊

@MrMebigfatguy 7 лет назад

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?"

@numberphile 7 лет назад

+Dave Brosius I don't think I ask that as often as you may think. I quite enjoy these things just for being awesome.

@TheGamblermusic 7 лет назад

My instinct guess is that it is too beautiful to NOT have usefull applications for anything else

@SlackwareNVM 7 лет назад

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".

@gerstensaft2936 7 лет назад

Change "grain of sand" to atom, or proton and go back to the start of the universe and evolve the pile. :D

@tpat90 7 лет назад

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.

@jackofallspades98 7 лет назад

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?

@angelmendez-rivera351 10 месяцев назад

That final question has no answer, because beauty is necessarily subjective.

@alexbontempo2438 2 года назад

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

@radix4801 7 лет назад

Did this guy go through your whole "brown paper" budget for 2017?

@stevelast3686 7 лет назад

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

@bestnocture 7 лет назад

Perfect! Best numberphile video ever! Was a little Boring at first, but thank fucking God I watched it whole!

@numberphile 7 лет назад

thanks for sticking with it!

@yugandhardesai8493 7 лет назад

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.

@8bit_pineapple 7 лет назад

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.

@bestnocture 7 лет назад

8bitpineapple can you please teach me how?

@Endoterrestrials 7 лет назад

@adamplace1414 2 года назад

It went from "okay" to "interesting" to "fun!" to "cool" to "woooooow". Really well explained, really well put together, and what a payoff!

@Fallub 2 года назад

That left me speechless. What a great finale.

@Snakeyes244 7 лет назад

I would love to see that sand topple from the beginning for the 2^30. Many iterations per second of course

@debblez 3 года назад

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

@MrDaanjanssen 7 лет назад

IT did not feel like a 24 min long video, but way shorter. Great video

@jursamaj 4 года назад

On the contrary, it felt way too long. A lot of the tedious small number addition should have been cut.

@BelialsRevenge 7 лет назад

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!

@thexavier666 3 года назад

I can always watch old numberphile videos and still be amazed

@sophieward7225 7 лет назад

The long videos are always the best

@gui1521 7 лет назад

Every video keeps amaze me... The end here is beautiful, ppl thinking video is "too long", stay until the end, it worths the effort.

@numberphile 7 лет назад

+Flandre Scarlet ;)

@jordantistetube 7 лет назад

At 5:06, the appearance of the zeroes are synchronized with him saying "zero", loving the little attention to detail

@vipermagi5499 7 лет назад

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.

@ozboltmenegatti 7 лет назад

Could we get zero for 1920x1080 sandpile group, please.

@wesofx8148 7 лет назад

And recursively fill the center rectangle with zero sandpiles of the rectangle's size.

@FLooper 7 лет назад

You can download the program from the description and basically do everything they showed you in the video!

@DrGerbils 7 лет назад

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.

@zacontraption 7 лет назад

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.

@numberphile 7 лет назад

it has its rewards!

@nowonmetube 4 года назад

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!

@OnixFilms 7 лет назад

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.

@SnoutyPig 7 лет назад

Beautiful how math does the unexpected and yet demonstrates an intricate pattern.

@xystem4701 7 лет назад

Those fractals are amazing

@IcepickL 7 лет назад

It's nice to see some algebra on numberphile.

@DoctorAsshole1 7 лет назад

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.

@luiservela 7 лет назад

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!

@zairaner1489 7 лет назад

Probably the best and most interesting video on this channel.

@utl94 7 лет назад

It is high up one the list, for sure.

@AlabasterJazz 7 лет назад

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."

@Aodhan2717 7 лет назад

AlabasterJazz I wonder how you would define factorization in this system.

@zairaner1489 7 лет назад

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)

@DrGerbils 7 лет назад

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.

@ljfaag 7 лет назад

That's pretty amazing. I've never seen these kinds of group structures with weird zeros before.

@G.Aaron.Fisher 7 лет назад

Bravo. This is easily in the top 5 videos this channel has ever produced.

@crashtextdummie 7 лет назад

Super fascinating and well explained!

@numberphile 7 лет назад

+crashtextdummie thank you

@numberphile 7 лет назад

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

@johnsmith-ke3nb 7 лет назад

Numberphile Why should i donate you?

@nickyboy909 7 лет назад

there is no obligation to donate john just do it if you want

@veggiet2009 7 лет назад

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.

@johnsmith-ke3nb 7 лет назад

veggiet2009 Nobody did ask you

@SachielxLAEx 7 лет назад

john smith for a chance to get occasional freebies, such as signed Numberphile postcards!!!! don't you listen?

@SaveSoilSaveSoil 3 года назад

Totally blown away at the end!

@jonwoek5750 7 лет назад

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

@thewarlord6529 5 лет назад

Jon Woek Shoah that’s pretty epic

@dakoitwuther7181 7 лет назад

I need more

@N3bu14Gr4y 7 лет назад

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

@ah-ray 7 лет назад

Wow, this is amazingly beatiful

@paroxyzm21 6 лет назад

One of THE best videos on Numberphile! Thanks!

@madelinescyphers5413 7 лет назад

This is similar to what I am studying right now, and I love it. This might be my favorite numberphile yet!

@MrNacknime 7 лет назад

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

@zairaner1489 7 лет назад

Thats the question

@zairaner1489 7 лет назад

Googling a little bit, I believe found an answer saying indeed, it is

@sebster100 7 лет назад

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!

@Risu0chan 7 лет назад

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).

@Risu0chan 7 лет назад

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)

@j0shmyg0sh90 Год назад

Pandemic players when they see 4 grains spreading to other cells *war flashbacks

@DoctorSinister1987 7 лет назад

That was a really great episode - thank you very much. It was very well explained as well!

@faktablad 4 года назад

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!

@Sopel997 7 лет назад

C++ has just called me. It wants to do something cool tonight.

@PeguinDesign 7 лет назад

Identity sandpile, like identity matrix?

@schmuelinsky 7 лет назад

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.

@TiKayStyle 7 лет назад

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

@-yake- 7 лет назад

Thiemo Krebsbach isn't +0 and *1 the same though? They are both identities.

@siddharth_desai 7 лет назад

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.

@dermaniac5205 7 лет назад

Well, they didn't even define a multiplication operation between sandpiles.

@thesonluong3982 5 лет назад

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

@Jose-pq4ow 7 лет назад

Those images look awesome!

@jenniferneumann706 7 лет назад

Wow this video showed how beautiful numbers can really be! ^^

@mighty8357 7 лет назад

I love his shirt :)

@americalost5100 4 года назад

Super cool. The end definitely justifies the build up. The phrase, it blew my mind, gets some experiential understanding here...

@Zahlenteufel1 7 лет назад

Didn't think I was gonna like it at first, but now I am amazed :)

@kauhanen44 7 лет назад

So the maximum amount of sand for a cell is n-1 where n is the number of neighbors one cell has?

@livedandletdie 7 лет назад

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.

@Keithfert490 7 лет назад

PerunaVallankumous yepp. that's the maximum untoppled pile

@mangomalarkey 7 лет назад

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.

@EebstertheGreat 7 лет назад

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).

@JohnSmith-zq9mo 7 лет назад

Yes, that is correct according to wikipedia. en.wikipedia.org/wiki/Abelian_sandpile_model

@mechanicalsnail4703 7 лет назад

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.

@reuben2011 7 лет назад

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.

@reuben2011 7 лет назад

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.

@robertbauer499 7 лет назад

I would have loved to attend MSRI-UP this past summer. Great video, thank you for sharing.

@johnchancey3941 7 лет назад

That may be my favorite Numberphile video of all time, just for the big WOW factor at the end!

7 лет назад

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?

@rovingfortune395 5 лет назад

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

@sahilnaik3079 5 лет назад

@@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.

@lucashoffses9019 7 лет назад

How *do* you calculate the Identity?

@wesofx8148 7 лет назад

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.

@lucashoffses9019 7 лет назад

Surely there has to be a way other than brute force.

@aashishkariya8328 7 лет назад

Lordious

@katzen3314 7 лет назад

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?

@DrGerbils 7 лет назад

He said there was an algorithm that generated the identity for m x n grids and hinted that the run time grew exponentially.

@jaysun4069 7 лет назад

That went in a surprisingly beautiful direction

@tillybillyboyboy 7 лет назад

love the sandpile distribution graphic!! Great video, as always.

@enderwiggins8248 5 лет назад

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

@MrMakae90 7 лет назад

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?

@MrMakae90 7 лет назад

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?

@christrengove7551 6 лет назад

theFizzyNator can you expand upon "to make toppling make sense" please?

@yaeldillies 6 лет назад

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

@Euquila 6 лет назад

Not necessarily. I think random toppling would still have structure because the randomness would average out. It would be interesting to see this.

@jayashrishobna 6 лет назад

more of this guy please!

@Stemma3 4 года назад

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.

@Bismarck_Games 7 лет назад

I wonder what would happen on a toroidal grid?

@aion2177 3 года назад

yes. or what will happen for objects with more then 1 hole. Like a double-donut. It might have application in topology.

@wesofx8148 7 лет назад

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.

@lizapiashko9105 4 года назад

Numberphile is my favorite channel to watch *really* early

@JansenPrice 7 лет назад

I like the sound effect when the sand topples in the illustrations.

@Mizziri 7 лет назад

Is there one for a hexagonal grid?

@CraftQueenJr 5 лет назад

James Moran probably.

@rovingfortune395 5 лет назад

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.

@TheScabbage 7 лет назад

Parker Sandpile. 1 3 1 3 1 3 1 3 1

@rafaellisboa8493 7 лет назад

That was awesome... I absolutely adore numberphile keep it up, proud of you.

@theoractliffe4878 7 лет назад

I smiled every time he said two and zero. Love the accent.

@Samboy_Chips 7 лет назад

Sandpiles? More like Sandphiles. Fine, I'll go 😢

@MrCyanGaming 7 лет назад

If you're reading this, have a Great day! 😄😄😄

@jweezy101491 7 лет назад

This is one of the best videos on the channel.

@CasualGraph 7 лет назад

Really liked this, it's probably the best video I've seen here in a while.

@crazydrummer4827 7 лет назад

Awesome finale! But I think you should have made video a bit shorter, a lot of people will give up on watching.

@MrMakae90 7 лет назад

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.

@christophersmith2890 5 лет назад

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

@zhangsc91 7 лет назад

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!

@Milehupen 7 лет назад

Sandpile! Notice me!

@Adraria8 7 лет назад

So is this an example of a group?

@forbesmccann5063 5 лет назад

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.