These sets of transforms are known as iterated function systems (IFS), and the algorithm dubbed "the chaos game" by Barnsley. If I recall correctly the algorit

These sets of transforms are known as iterated function systems (IFS), and the algorithm dubbed "the chaos game" by Barnsley. If I recall correctly the algorithm actually distributes uniformly over a probability measure supported by the set, so for picture-making purposes it is typically done with a probability associated with each transforms chosen to even out the visitation (otherwise some details will take forever to be seen). The whole area is a consequence of Banach's fixed point principle (a very fundamental result), with Hutchinson I think extending it to unions of contractive maps. Math has many beautiful corners.