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Explore the generative art of self-similar fractals using chaos game and deterministic iteration methods
Start from any point and randomly select a transformation rule to apply each time. After thousands of repetitions, the point cloud gradually forms the fractal attractor. This method demonstrates how randomness produces deterministic patterns.
Start from an initial geometric shape and apply all transformation rules to all points simultaneously. Each iteration generates a new set of points, showing the hierarchical construction and self-similarity of the fractal.
Each transform is a distance contraction: d(T(x), T(y)) ≤ r·d(x, y) where r < 1. According to the collage theorem, the unique attractor satisfies self-similarity.
For self-similar fractals: D = log(N) / log(1/r), where N is the number of similar copies and r is the scaling factor. For example, the Sierpinski triangle has a dimension of approximately 1.585.