Fractals - Chaos Game 3D

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Same rules as the 2D chaos game, just with the vertices of a polyhedron instead of a polygon: drop a point, then repeatedly move it some fraction (ratio) of the way toward a random vertex. At ratio = 0.5 a tetrahedron's four vertices produce the 3D analog of the Sierpinski triangle, the Sierpinski tetrahedron — four shrunken copies of the whole shape, one pinned to each corner, hollowing out the middle at every scale.

Unlike the 2D page, this one doesn't jump ratio to the value where neighboring copies touch corner-to-corner — working that out in 3D depends on the specific polyhedron's geometry (how its faces, not just its vertices, meet), not a single formula shared by every shape the way the regular-polygon case was. Nudge ratio by hand instead and watch the structure open up or overlap.

The five Platonic solids (Tetrahedron through Icosahedron) are computed directly from their well-known coordinate formulas — simple sign combinations of one or two numbers, optionally involving the golden ratio φ. The thirteen Archimedean solids use the same vertex coordinates as the Polyhedra page's existing .wrl files instead: several of them (the snub forms especially) don't have a simple closed-form formula — their coordinates come from numerically solving a cubic equation — so reusing already-verified data beat re-deriving them by hand.

Rendered by the same point-cloud renderer as the 3D IFS and Lorenz Attractor pages, including its "color by vertex" categorical coloring — the exact 3D counterpart of the 2D chaos game's own color by vertex checkbox.

The whole page was created with Claude Code. These were the most significant prompts that shaped this page:

1. Extend the chaos game to 3D: a dropdown of polyhedra, reusing the 3D point-cloud renderer built for IFS 3D. No automatic touching-ratio calculation this time — that's polyhedron-specific geometry, out of scope here.
2. Only fall back to the Polyhedra page's .wrl coordinate files where the vertex coordinates can't be derived directly; compute the five Platonic solids from their known formulas instead.
3. The default density shading (tuned for IFS 3D's dense, self-overlapping fern) left a chaos-game point cloud looking washed out, since it's a much sparser attractor — lower the shading's level count to match.
4. Raise the Points slider's max from 200,000 to 1,000,000 — the many-vertex Archimedean solids (the snub forms especially) can look sparse at the old ceiling. Scale the shading level count with the slider so raising it doesn't wash the shape back out.