The well-known “chaos game” algorithm for generating the Sierpinski gasket stochastically is generalized to any order of regular polygon. A simple formula is derived that gives a value, referred to as the “kissing number,” to produce a crisp analog to the Sierpinski triangle for any order of regular polygon. In addition, a second kissing number is revealed that produces another fractal for each polygon. Finally, some surprising aspects of the algorithm’s behavior are revealed, including that the resulting figures contain a geometrically encoded history of the order of vertices targeted by the algorithm. This work was inspired by a desire to generalize the usual algorithm in a way that could be useful towards artistic ends and of interest for instructional ends at a variety of levels.