There are many ways to generate fractals, such as nonlinear dynamical systems in continuous time (at least 3 rd order) or its discrete analogue known as an iterative map. The latter one is coined as the chaos game, where a sequence of points is created by using a polygon and an initial random point inside the polygon to recursively generate a new point that is sitting at a fixed proportional distance from the previous point and a randomly selected vertex of the polygon. With an appropriate proportional parameter and many iterations, this process generates a fractal shape. In this work, we present algorithms for generating fractals inside an arbitrary regular polygon, inside a polyhedron, and even a polychoron based on restricted vertex replacement for fast fractal generation. In the case of polychoron fractals, we project the fractals onto the 3D space to visualize geometric features. We further extend this idea to projections of fractals onto a surface such as a sphere.
