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        riddles >> medium >> A Chaos Game
        
(Message started by: BMAD on May 23^rd, 2014, 2:59pm)

Title: A Chaos Game
Post by BMAD on May 23^rd, 2014, 2:59pm

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?
Does similar designs occur in other shapes?

Title: Re: A Chaos Game
Post by towr on May 24^th, 2014, 4:24am

I know this one, [hide]you get a Sierpinski triangle; you also get it if you don't choose one vertex randomly but choose all three (so tripling the points each iterations)

I think it should work for squares if you pick the next point not halfway, but 2/3rds of the way to the chosen vertex.

I remember trying some variations at the time, but that was long ago.[/hide]

Title: Re: A Chaos Game
Post by rmsgrey on May 26^th, 2014, 7:07am

The generic version is to take a self-similar fractal - one that can be broken up into multiple smaller disjoint copies of itself - and, for one such decomposition, find the transformations that convert the full fractal into each of those smaller copies. Starting with a point that's part of the fractal, pick one of those transformations at random and apply it. Iterate until you're happy with the image you've built up.

An automated technique based on this idea was used to provide (lossy) image compression for Microsoft Encarta.