Let’s play the following “chaos game”. Consider a regular triangle and take some initial point in it. Now pick one of the three vertices at random, connect it with the current point, and mark this line’s midpoint as the new point. If you repeat this process many times, what will the resulting shape look like? You might expect to find the points simply shattered everywhere, without much structure, but surprisingly, the points will in fact create a Sierpinski fractal.
There are lots of variations on this theme, some of which can be found on this Wikipedia page.
Animation made in Mathematica.