Exploring the Method of Colour Stealing for Contractive Iterated Function Systems

Plotting fractals generated by an Iterated Function System (IFS) can be challenging and computationally intensive, so an algorithm referred to as the chaos game is employed. Here, given a seed point, IFS mappings are chosen at random in sequence, with each subsequent point mapped from the one before it through the new mapping. Utilizing this approach, we may plot attractors accurately and quickly. Attractors may be coloured in many ways, but of interest is the method of colour stealing (Barnsley, Superfractals. Academic Press, London [1]; Barnsley, Theory and Application of Fractal Tops, Fractals in Engineering, Tours. Springer, France [2]; Kunze et al., Maple Conference 2006 Proceedings [3]). Complications to the existing scheme arise in implementation, particularly when considering assigning colour values to pixels. These lead us to explore some slight modifications of the original framework, making use of the notions of finite code space and a metric for use in practical computation. Further, we explore an extension of the notion of the fractal top by defining a general projection function and showcase some resulting attractors.