The Optimal Ratio of a Generalized Chaos Game in Regular Polytopes

TL;DR

This study explores the optimal scaling ratio in the Generalized Chaos Game for regular polytopes across dimensions, deriving a formula and verifying it through simulations.

Contribution

It introduces a formula for the optimal ratio in the Generalized Chaos Game for regular polytopes in any dimension, supported by computational verification.

Findings

Optimal ratio varies with polytope properties

Derived a universal formula for the optimal ratio

Validated the formula through Python simulations

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.