Konstruksi Fraktal Rauzy Dengan Memodifikasi Substitusi Tribonacci

Abstract

The Rauzy fractal is a geometric object generated from substitution systems and has attracted considerable attention due to its self-similar structure and applications in symbolic dynamics, number theory, and tiling theory. Most previous studies have focused on the standard Tribonacci substitution, while the geometric properties of Rauzy fractals generated from modified substitutions remain relatively unexplored. Therefore, this study aims to construct Rauzy fractals using two modified Tribonacci substitutions and analyze the effects of these modifications on the geometric characteristics and fractal properties compared with the standard Tribonacci Rauzy fractal. The research employed a descriptive mathematical approach. The construction process consisted of forming the substitution matrix, determining the dominant eigenvalue and its corresponding eigenvector, generating the fixed point, computing Parikh vectors, constructing the canonical stepped line, and projecting the resulting points onto the contracting plane using orthogonal projection. The projected points were then visualized using MATLAB. The results show that both modified substitutions successfully produce Rauzy fractals through the same construction procedure as the standard Tribonacci Rauzy fractal. However, the resulting fractals exhibit different geometric appearances, particularly in branch orientation, point distribution, and overall shape. Despite these geometric differences, both modified Rauzy fractals preserve the self-similarity property that characterizes fractal structures. In addition, the fractal pattern generated by substitution σ1 becomes clearly visible at the 15th iteration, whereas the pattern generated by substitution σ2 becomes evident at the 30th iteration. These findings indicate that modifying the Tribonacci substitution influences the geometric development of the Rauzy fractal without altering its fundamental fractal properties.