Chaos Game

[BMAD]

This is an excellent program for my programmers out there but anyone can successfully examine this:

The Chaos Game:

1. Label an isosceles triangles vertices 1,2, and 3.

2. Select some form of a randomizer that will pick the numbers 1,2, and 3.

3. Pick a point inside the triangle and put a dot there.

4. Use your randomizer to select one of the numbers from 1,2, and 3.

5. Place a dot midway between the vertex with that number and the current dot.

6. Now, using that new dot as a reference point, repeat steps 4, and 5.

7. Continue these trials until you notice something magical (hopefully)

Why is this happening?

[Anza Power]

I don't see what's happening...

Attached are two HTML files that do this, one that has a point moving around and one that draws a line so you can see the entire path.

You'll need an updated browser to view these because they're HTML5...

triangle.html

triangle_history.html

[BMAD]

Don't erase your prior points, show them all as each new point is found

[Anza Power]

Ah ok I see, here's a better one.

That's cool, now for explanation...

triangle_narrow.html

[BMAD]

Nicely done Anza! For other people to see the "pattern" forming, you need to let the above html run for about five minutes to clearly see it.

The longer it runs, the better.

Edited July 22, 2013 by BMAD

[bonanova]

Explanation:

The prohibited area comprises a series of triangles of doubled linear scale proceeding from the vertices.

That means if you start outside this area you can't get inside it by going half-distances toward a vertex.

But I'm not clear what happens if your initial point were inside.

[BMAD]

But I'm not clear what happens if your initial point were inside.

Look backwards. If the second point were in the middle empty area, where does that put the first point?

Maybe this logic can be extended to the other smaller empty areas as well...

[Rainman]

The almost empty area consists of triangles of different sizes. You can only get to one of these almost empty triangles from a larger almost empty triangle. So no almost empty triangle can contain more than one dot. You can get out of the almost empty area if you started inside it, but you can't get back inside it once you've left.

[BMAD]

Does this patterning of similar shapes occur in other figures as well (like a square, rectangle, octogon, etc.)? Or is it relegated to only triangles?

[Rainman]

The central almost empty area of a polygon is the set of internal points which are more than halfway out from every corner. At most one such point exists except in triangles, and that is the midpoint of the polygon. The midpoint can be almost empty and give rise to infinitely many other almost empty points. However, since they are discrete points, they don't form any noticable pattern. So triangles are the only polygons with these patterns. (Proofs omitted because I'm just going by intuition here )

[BMAD]

Here is the effect of applying the 'game rules' to the pentagon in the shape of a home plate. See that the empty space pattern in the pattern of the starting figure occurs again.

Also notice that this shape is irregular and not triangular.

[Rainman]

And that's why you should never go by intuition in mathematics (or at least have better intuition than I do)

[BMAD]

octogons anyone?

[bonanova]

But I'm not clear what happens if your initial point were inside.

Look backwards. If the second point were in the middle empty area, where does that put the first point?

Maybe this logic can be extended to the other smaller empty areas as well...

I can't look backwards from the initial point.

If the initial point were in the central area, the second point would not also be there.

[BMAD]

But I'm not clear what happens if your initial point were inside.

Look backwards. If the second point were in the middle empty area, where does that put the first point?

Maybe this logic can be extended to the other smaller empty areas as well...

I can't look backwards from the initial point.

If the initial point were in the central area, the second point would not also be there.

I asked you to look backwards from the second point. if the second point is in the center, then there could not have been an initial point as any where twice the distance from the center is outside of the shape, therefore we could not under the rules and point system have points in the center.

[BMAD]

So the only point that could ever end up in the center of the shape is the initial point but the resulting image would be the same as no other dot could go in that empty space.

Edited July 23, 2013 by BMAD

[Pickett]

But when I ran my own program, the only shape that I get a similar pattern on is a triangle:

I see a pattern forming with a pentagon, but not quite as well defined:

Hexagon is pretty nifty, but still not very well defined:

With the one, you can START to see the pattern mentioned above...but it's not EXCLUSIVE:

I also tried square and octagon...but they didn't give me anything really neat...

[Pickett]

Is also pretty neat:

[Pickett]

Gives cool perspective...

Alright, I'm done now...this was just fun to do!

[BMAD]

But when I ran my own program, the only shape that I get a similar pattern on is a triangle:

triangle.png

I see a pattern forming with a pentagon, but not quite as well defined:

pentagon.png

Hexagon is pretty nifty, but still not very well defined:

hexagon.png

With the one, you can START to see the pattern mentioned above...but it's not EXCLUSIVE:

homeplate.png

I also tried square and octagon...but they didn't give me anything really neat...

square.png octagon.png

Great point. My program allows me to tweak the ratio (or pull) to make the 'hard to see' patterns in the above drawings easier to see. I assure you the image is there yours just has more noise. But if you examined frequency by location (or clustering) the images wouuld clean up like mine. But you are right, the empty spaces in my plate examples aren't completely empty. Edited July 23, 2013 by BMAD

[Pickett]

If any of you would like to play around with the little app I created for this (hats off to Anza Power for giving me the idea to use HTML5 for visualizing this), here it is:

TestCanvasDraw.html

Viewing the source (if you're at all a developer) should show you how to create your own shapes and try it out.

[bonanova]

What fraction of the area is white (prohibited)

1. For the triangle? (straightforward)
2. For the trapezoid? (more thought-provoking)

[BMAD]

The biggest white area or all of the white areas. The sum, I am almost certain they approach a limit.

By the way if anyone was wondering about this problems category. It belongs to set of mathematics known as fractals. Specifically iterated fractals of self similars.

[bonanova]

The total area that is white.

[gavinksong]

What fraction of the area is white (prohibited)

1. For the triangle? (straightforward)
2. For the trapezoid? (more thought-provoking)

Well, for a triangle, if we add up the white space, it is just the infinite sum of a geometric series with a factor of 3/4 (the big empty triangle in the middle is one-fourth of the total area, the three smaller triangles are one-fourth of the remaining area, and so forth). Strangely, it converges at the total area of the triangle. It we calculate the black space, it is three-fourths raised to the infinite power, which is zero. So apparently, the entire triangle is supposed to be prohibited space.

I think here bonanova's musings becomes relevant:

But I'm not clear what happens if your initial point were inside.

Instead of looking backwards, as we did to solve the problem initially, we should look forwards from the initial point. This is all intuition, but you will notice, that if the initial point is in the big empty triangle in the middle, then in the next step, it will always move to one of the three smaller white triangles. And from there, it will always move into one of its three smaller "shadows" (I think that is an appropriate term, and hopefully you understand what I mean), and so on. The important point here is that we always move into smaller and smaller "shadows".

When the initial point is thrown into the triangle at random, the probability that it lies into a white triangle is 100%. However, as the simulation progresses, the points are eventually forced into smaller and smaller white triangles. In other words, the amount of "prohibited space" is not actually an absolute value, but simply increases with step number (in a sense). As in, there is no prohibited space when you throw in the first point, but the second point cannot be in the large white triangle, the third point cannot be in the large white triangle or any of its shadows, the fourth cannot be in the large white triangle or its shadows or any of its shadows' shadows, and so on.

So to us, it seems as though there is some sort of absolute law because we see the larger empty white triangles, but in reality, it is only because we do not see the infinitesimally small white triangles with the dots in them. The entire process is just endless error-correction, moving toward an impossible figure, trailing behind an interesting asymptotic pattern in the process.