[pacaro]

There are so many ways to produce sierpinski gaskets.

It you specify n points and the pick a new point at random, then iteratively randomly select (uniformly) one of the original n points and move the next point to the mid point of the current point and the selected point. Coloring those points generates a sierpinski triangle or tetrahedron or whatever the n-1 dimensional triangle is called

[linschn]

That's called a simplex :)

The same as in the simplex algorithm to solve linear programming problems.

[CrazyStat]

I programmed this on my TI-83 back in the day and spent many hours watching it generate triangles during boring classes.

You can generate many other fractals (e.g. fern shapes) in a similar way, though the transformations are more complicated than “move halfway to selected point”.

[deadfoxygrandpa]

yes, those are called iterated function systems (IFS) fractals