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Chaos Game


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24 replies to this topic

#1 BMAD

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Posted 22 July 2013 - 03:05 PM

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?

 


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#2 Anza Power

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Posted 22 July 2013 - 04:38 PM

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...

Attached Files


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#3 BMAD

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Posted 22 July 2013 - 05:56 PM

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

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Posted 22 July 2013 - 09:09 PM

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

That's cool, now for explanation...

Attached Files


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#5 BMAD

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Posted 22 July 2013 - 11:21 PM

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 by BMAD, 22 July 2013 - 11:22 PM.

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#6 bonanova

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Posted 22 July 2013 - 11:28 PM

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.
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#7 BMAD

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Posted 22 July 2013 - 11:34 PM

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...


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#8 Rainman

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Posted 22 July 2013 - 11:41 PM

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.


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#9 BMAD

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Posted 22 July 2013 - 11:45 PM

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?


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#10 Rainman

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Posted 23 July 2013 - 12:37 AM

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 :P)


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