This video from the excellent Numberphile YouTube channel shows a very simple game that has the most unexpected outcome. My only criticism of this video is that I’d like to be shown the explanation, even if it’s complicated! Maybe readers can help me.

Well not so obvious. By 45 seconds, I had confidently expected to end up with a cluster in the middle of the triangle. I had not expected it to end up with a Sierpinski gasket. I can’t figure out why that is, in particular why there are areas of avoidance.

If you think about the fact that there are only ever three directions you are going to go towards fixed points, and the distance is limited, and once you’re inside the confines of the triangle you’ll stay there, you’re always going to get triangles with gaps like that. In a square, squares, in a pentagon, pentagons. It does make sense.

I also don’t think it is obvious – especially why there are areas that the point cannot reach (or can it and it is just very unlikely?). I was expecting that everything would average out and converge to a spot in the middle of the triangle.

Out of curiosity, what was your expectation? It looks like just a random walk among points in the Sierpinski triangle. Each move stays in the set, and the walk doesn’t collapse toward a fixed point or neighborhood so you get a broad sampling of the set.

The explanation is much like the explanation of evolution. You have a random element (select the vertex) combined with a non-random element (rule says move halfway to selected vertex). The non-random rule makes certain outcomes more likely and/or even guarantees certain outcomes.

Consider points outside the triangle. If you start outside the triangle you will always move closer until you are inside. Once inside, you can never get back outside. So the outside remains empty. Now look at the big “hole” in the middle. How can you get inside that hole if you don’t start there? Only by being outside the triangle! Once inside the triangle, you can never land in the hole again (work it out). The three 2nd largest “holes” can only be reached from outside or from the large hole. And so on. The tertiary holes can only be reached from the outside, from the central hole, or from the secondary holes.

The structure basically reveals a hierarchy of “areas you can never get back to once you’ve left them”. So the structure we see has more to do with the empty areas than the (seemingly) filled in areas.

It can also be shown that if you start *exactly* on an edge of the triangle, or on an edge of any inner triangle, then you will always land *exactly* on an edge of some other, smaller, inner triangle. When you do not start on an edge, you are guaranteed to get get closer and closer to being on the edges.

Other sets of points and rules reveal other structures. The fern pattern being one such.

Yours and other comments help me to understand that these patterns must emerge because they are simply outlines of the constraints placed on where dots are placed.
But although I now understand it better I can still say: Mind blown.

Work it backwards: pick any point, then double its distance to the corners to see if that point could have been chosen as a halfway point from a preceding move.

This is related to the response of Ann German #2 and S.K. Graham #5, but I think it gives you a procedure to follow which may provide you some insight.

I would venture that every reader of this website will find something to fascinate themselves in one of Brady Haran’s many YouTube channels. I just copied and pasted the following titles from his BradyStuff channel. Coincidentally perhaps, my favourites are the first four on this list.

If you want something deceptively simple and equally mindblowing, consider the Koch snowflake. A pattern with an (obviously) finite area but an infinitely long edge.

It’s easy to see how the edge gets infinitely long, as each iteration increases its length by 4/3 (and you can iterate forever).

Related problem: How long is the coastline of Britain.

However, I propose that what makes this light up everyone’s brain is nothing more than the human tendency to see patterns in anything, whether they “mean” anything or not. In this case, there’s some simple rules that produce an amazing, unexpected result.

And I say that’s a good thing! But we know how that can go bad, don’t we!…

Detailed explanations are all good and well, but to my mind the general explanation is more informative on dynamical systems. The difference (or differential) equations (without or with randomness) that ends up in attractors (whether strange or not) are a specific group.

Those equation systems are models of dissipative systems with a forcing. Some of them exist in nature. Driven systems that lose energy include homeostatic cells. There have been attempts to model life emergence as attractors, but to my knowledge nothing has come out of it. (Perhaps because homeostatic cells are mature systems.)

Similar simple but informative observations abound around models of dynamics. For example how mathematically convex models admit just one global extremum, which is not prior obvious I think.

## 32 Comments

Try it out! I am sure there’s an online version.

(I wrote a program to do this years ago which I no longer have.)

Because you only ever go 1/2 of the way, those lines define the length of the internal triangles which mimic the angles of the outer triangle.

should be “define the length of the side” of the internal triangles, etc.

This is really cool. It’s obvious what will happen if you think about it, but that doesn’t make it any less cool. I love the fern!

I’d really like to see and learn more about some of the patterns/rules people have come up with.

Well not so obvious. By 45 seconds, I had confidently expected to end up with a cluster in the middle of the triangle. I had not expected it to end up with a Sierpinski gasket. I can’t figure out why that is, in particular why there are areas of avoidance.

cr

If you think about the fact that there are only ever three directions you are going to go towards fixed points, and the distance is limited, and once you’re inside the confines of the triangle you’ll stay there, you’re always going to get triangles with gaps like that. In a square, squares, in a pentagon, pentagons. It does make sense.

I also don’t think it is obvious – especially why there are areas that the point cannot reach (or can it and it is just very unlikely?). I was expecting that everything would average out and converge to a spot in the middle of the triangle.

Well, not exactly converging. But I was expecting it to jump around the middle.

Out of curiosity, what was your expectation? It looks like just a random walk among points in the Sierpinski triangle. Each move stays in the set, and the walk doesn’t collapse toward a fixed point or neighborhood so you get a broad sampling of the set.

The explanation is much like the explanation of evolution. You have a random element (select the vertex) combined with a non-random element (rule says move halfway to selected vertex). The non-random rule makes certain outcomes more likely and/or even guarantees certain outcomes.

Consider points outside the triangle. If you start outside the triangle you will always move closer until you are inside. Once inside, you can never get back outside. So the outside remains empty. Now look at the big “hole” in the middle. How can you get inside that hole if you don’t start there? Only by being outside the triangle! Once inside the triangle, you can never land in the hole again (work it out). The three 2nd largest “holes” can only be reached from outside or from the large hole. And so on. The tertiary holes can only be reached from the outside, from the central hole, or from the secondary holes.

The structure basically reveals a hierarchy of “areas you can never get back to once you’ve left them”. So the structure we see has more to do with the empty areas than the (seemingly) filled in areas.

It can also be shown that if you start *exactly* on an edge of the triangle, or on an edge of any inner triangle, then you will always land *exactly* on an edge of some other, smaller, inner triangle. When you do not start on an edge, you are guaranteed to get get closer and closer to being on the edges.

Other sets of points and rules reveal other structures. The fern pattern being one such.

Yours and other comments help me to understand that these patterns must emerge because they are simply outlines of the constraints placed on where dots are placed.

But although I now understand it better I can still say: Mind blown.

Elon Musk is right!

Stephen Wolfram is right!

But chaos is (cellular automata, fractals) is classical as far I can tell. So if Stephen is right, he’s only classically right.

This is pretty cool. http://www.bowdoin.edu/news/events/archives/002489.shtml

Basically very difficult-to-manually-model parts of the heart can be modeled accurately using fractal math.

That is interesting.

To me it shows that the universe is pregnant with natural constructions which is another reason evolution is true.

Yes…after all (and I got this from Dennett) evolution can be understood as an algorithm.

Anyone know of a program where I can create my own patterns?

Very, very cool.

I’ve got a trick where I make a pile of money disappear with dice, but it involves a croupier.

Sub

Work it backwards: pick any point, then double its distance to the corners to see if that point could have been chosen as a halfway point from a preceding move.

This is related to the response of Ann German #2 and S.K. Graham #5, but I think it gives you a procedure to follow which may provide you some insight.

I would venture that every reader of this website will find something to fascinate themselves in one of Brady Haran’s many YouTube channels. I just copied and pasted the following titles from his BradyStuff channel. Coincidentally perhaps, my favourites are the first four on this list.

ComputerphilePeriodic Videos

Numberphile

DeepSkyVideos

nottinghamscience

foodskey

BackstageScience

FavScientist

BradyStuff

Another one he put in a different list is

Objectivity. Intriguing things to be found in the Royal Museum in Great Britain.I wonder if the result would be much different if the sequence one through six was chosen instead of rolling dice?

So how much of it is chaos?

Reblogged this on The Logical Place.

If you want something deceptively simple and equally mindblowing, consider the Koch snowflake. A pattern with an (obviously) finite area but an infinitely long edge.

It’s easy to see how the edge gets infinitely long, as each iteration increases its length by 4/3 (and you can iterate forever).

Related problem: How long is the coastline of Britain.

cr

(People talk about quantum weirdness. I think fractals have a weirdness coefficient right up there with quantum).

cr

That proves it! we are definitely living in a Computer.

I love this and all the comments are good.

However, I propose that what makes this light up everyone’s brain is nothing more than the human tendency to see patterns in anything, whether they “mean” anything or not. In this case, there’s some simple rules that produce an amazing, unexpected result.

And I say that’s a good thing! But we know how that can go bad, don’t we!…

Detailed explanations are all good and well, but to my mind the general explanation is more informative on dynamical systems. The difference (or differential) equations (without or with randomness) that ends up in attractors (whether strange or not) are a specific group.

Those equation systems are models of dissipative systems with a forcing. Some of them exist in nature. Driven systems that lose energy include homeostatic cells. There have been attempts to model life emergence as attractors, but to my knowledge nothing has come out of it. (Perhaps because homeostatic cells are mature systems.)

Similar simple but informative observations abound around models of dynamics. For example how mathematically convex models admit just one global extremum, which is not prior obvious I think.

I’m impressed and not impressed at the same time. It is impressive, but not

reallyunexpected. Love it.