The Optimal Ratio of a Generalized Chaos Game in Regular Polytopes
Christoffer Tarmet
TL;DR
The paper addresses how to choose an optimal scaling ratio $r_{\text{opt}}$ in the Generalized Chaos Game for regular polytopes in $n$ dimensions, showing that this ratio depends on polytope geometry rather than being universal. It develops a geometrical framework using contraction mappings and the Hausdorff metric to derive a general formula $r_{\text{opt}}=\dfrac{\delta_{\parallel}}{\delta_{\parallel}+\ell}$, with a special case yielding $r_{\text{opt}}=\tfrac{1}{2}$ when $\delta_{\parallel}=\ell$. The authors provide practical methods to compute $\delta_{\parallel}$, compile comprehensive tables of $r_{\text{opt}}$ across dimensions up to 5, and validate the results via Python-based overlap tests. They demonstrate that applying the optimal ratio enhances fractal structure in high-dimensional Chaos Games and discuss implications for applications, notably in biology where structure can emerge in sequence representations at $r_{\text{opt}}$ rather than at conventional choices like $r=0.5$.
Abstract
This paper investigates the concept of an optimal ratio for regular polytopes in $n$-dimensional space within the framework of the Generalized Chaos Game. The optimal ratio, $r_{\text{opt}}$, is defined as the value at which the self-similar regions of the resulting fractal touch but do not overlap. Using a series of Python simulations, we explore how the optimal ratio varies across different polytopes, from two-dimensional polygons to three-dimensional polyhedra and beyond. The results, visualized through plots generated for various polytopes and values of the scaling factor $r$, demonstrate that the optimal ratio is not universal but rather depends on each polytope's specific properties. A formula is then derived for determining the optimal ratio for any regular polytope in any dimension. The formula is then experimentally verified using multiple Python programs designed to search and find the optimal ratio iteratively.