Chaos Game pada Segiempat dengan Rasio Kompresi Bertambah Secara Linier

Abstract

The chaos game is an iterative method for generating fractal patterns by selecting an initial point at random and moving it toward one of the vertices of a geometric shape by a certain distance determined by a compression ratio. This study aims to analyze the effect of a linearly increasing compression ratio on the chaos game pattern generated on a quadrilateral using a random selection rule along with three modified point-selection constraints. The compression ratio is defined by the equation 𝑟_𝑛 = 𝑟0 + 𝑘_𝑛, with 𝑟_0 = 2, 𝑛 = 5000, and the increment value 𝑘 taken from a specified range that includes negative, zero, and positive values. The results show that a linearly increasing compression ratio at each iteration, under various values of 𝑘, significantly affects the stability of the resulting chaos game pattern. When 𝑘 approaches zero, either from the positive or negative direction, the changes in 𝑟_𝑛 and 𝛼_𝑛 occur slowly and remain stable. Under these conditions, the iterated points spread evenly across the quadrilateral, and the resulting pattern preserves the essential fractal characteristic of self-similarity. Conversely, when 𝑘 becomes increasingly positive, 𝑟_𝑛 grows rapidly and 𝛼𝑛 becomes smaller, causing the movements toward the reference vertices to be extremely short. As a result, points accumulate around the vertices while the central region becomes empty, leading to the loss of fractal properties. When 𝑘 becomes increasingly negative, 𝑟_𝑛 decreases rapidly toward zero or even becomes negative, causing 𝛼𝑛 to become unstable and the points to move outward or out of the quadrilateral, producing irregular patterns that are no longer fractal. Therefore, it can be concluded that the stability of the chaos game pattern is strongly influenced by the rate of change in the compression ratio, and a fractal pattern forms only when 𝑘 is very close to zero, allowing 𝑟_𝑛 to change gradually and remain stable throughout the iterations.