Create an equilateral triangle and choose a random point inside it, then repeatedly place a new point midway between the current point and one of the triangle's corners (chosen randomly), and eventually (after throwing the first few points away) the points will look more and more like a Sierpinski triangle.
The following array shows progressively greater numbers of iterations of the preceding process, called the chaos game.
Interesting (to me, anyway) shapes also appear when we start with a shape other than a triangle.
As it happens, starting with a tetrahedron instead of a triangle gives a similar result in three dimensions. (In the following graph, each point is colored according to its distance from the observer's viewpoint.)
Designed and rendered using Mathematica versions 2.2 and 3.0 for the Apple Macintosh.
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