Chaos Game and the Serpinski Triangle

Monday morning Simon showed me the Chaos Game: he created three random dots on a sheet of paper (the corners of a triangle) and was throwing dice to determine where all additional dots would appear, always half-way between the previous dot and one of the corners of the triangle.

Very soon, he found it too much work to continue and I though he gave up. Later the same day, however, he suddenly produced the same game in Codea, the points filling in much faster than when he did it manually, yet following exactly the same algorithm. To my surprise, what resulted from this seemingly random scattering of dots was a beautiful Serpinski triangle.

How come a dot never happens inside one of the black triangles in the middle? – I asked.
Sometimes you start there, but the next dot (half-way towards one of the corners) is already outside the black triangle, Simon showed. (The screenshot above is of such an occurance. If you look carefully, you will see a dot in the middle).

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