Play the chaos game with polygons and ratio rules. Watch Sierpinski triangle, fern, and custom attractors emerge from random iterations.
Random choices can settle into a sharply structured set. The Chaos Game Visualizer starts from a point, repeatedly selects among polygon targets or preset rules, moves the point by the chosen ratio, and plots the resulting orbit. It supports experiments with the Sierpinski triangle, fern, and custom attractors as iterations accumulate. The usual surprise is that randomness controls the route rather than the overall limiting geometry: separate runs need not contain identical points, yet they can reveal the same recognizable attractor.
For a polygon rule, each iteration randomly chooses an eligible vertex and places the next point on the segment from the current point toward that vertex. A ratio of 0.5 places it halfway. The new location becomes the current point for the following iteration, so the plotted dots form one dependent orbit, not a collection of independent samples.
At the start, the orbit can retain a visible trace of its initial point. Under contracting rules, that influence shrinks and later points gather on the rule set's attractor. For the three-vertex triangle rule with equal random choice and a 0.5 move, repeated points fill out the Sierpinski gasket. The fern preset and custom configurations use different rule sets, so their limiting shapes and densities can differ.
With a 0.5 move, each triangle vertex maps points into the smaller corner triangle nearest that vertex. None of those three mapped regions covers the central inverted triangle. Repeating the same choice-and-move process creates corresponding uncovered regions at progressively smaller scales, while the random choices determine which surviving regions the orbit visits and in what order.
Generally, no. Random vertex or rule choices change the exact point sequence between runs. For a contracting rule set, sufficiently long runs should reveal the same large-scale attractor and broadly similar density, but individual coordinates, local clusters, and temporary gaps can differ. The visualization therefore supports structural comparison, not exact point-for-point reproduction.
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