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🔺 Sierpiński Tetrahedron — 3D Fractal

Explore the Sierpiński tetrahedron, a 3D fractal built by recursively replacing a tetrahedron with four half-scale copies. Rotate it in real time, switch to chaos-game generation, and watch the fractal dimension stay exactly 2.

Geometry3DEasy60 FPS

About this simulation

This is the 3D analogue of the Sierpiński triangle: a tetrahedron with a smaller tetrahedron removed from its centre at every recursion level, repeated up to 6 times. The same fractal can also be generated by the "chaos game" — repeatedly jumping halfway toward one of four randomly chosen vertices — which converges to an identical shape from pure randomness instead of explicit subdivision.

🔬 What it shows

A self-similar 3D fractal built either by recursively removing central tetrahedra (Recursive mode) or by plotting millions of random midpoint jumps toward four vertices (Chaos game mode), both converging to the same Sierpiński tetrahedron.

🎮 How to use

Switch between Recursive and Chaos game Mode, adjust Recursion depth and Zoom, toggle Auto-rotate and its Speed, pick a Color scheme (By depth, Ember, Ice), and use Wireframe, Reset or Save PNG.

💡 Did you know?

The Sierpiński tetrahedron has a fractal (Hausdorff) dimension of exactly 2 — meaning it behaves mathematically like a 2D surface despite living in 3D space, since it's built from 4 self-similar copies each scaled by 1/2 (log 4 / log 2 = 2).

Frequently asked questions

What exactly happens at each step of Recursive mode?

At each recursion level, every solid tetrahedron is replaced by 4 smaller copies of itself (at half scale) placed at its corners, with the central tetrahedron between them removed — this is repeated up to the chosen Recursion depth, so depth 6 has undergone this replacement six times.

How can the "Chaos game" produce the same shape from pure randomness?

Starting from any point, repeatedly jumping halfway toward a randomly chosen one of the four base vertices causes points to statistically avoid the "forbidden" central regions over many iterations, so the accumulated dots trace out the fractal structure purely through probability, with no explicit subdivision rule at all.

Why does increasing Recursion depth make the structure look "hollower" rather than more solid?

Each recursion step removes volume from the interior (the central tetrahedron at every scale), so as depth increases, the fraction of the original solid tetrahedron's volume that remains shrinks toward zero, even though the outer silhouette stays roughly the same size.

What does "fractal dimension of 2" actually mean for a 3D object?

Fractal dimension measures how detail scales with magnification: an ordinary solid has dimension 3, but this structure is built from 4 copies at half scale, giving dimension log(4)/log(2)=2 — it fills space more like a folded 2D sheet than a genuine 3D solid, despite occupying 3D space visually.

Why does switching Color scheme to "By depth" help you understand the structure?

Coloring sub-tetrahedra by their recursion level visually separates each generation of the subdivision process, making it far easier to see how each large piece is built from smaller identical copies — the core idea of self-similarity that defines any fractal.

⚙ Under the hood

Explore the Sierpiński tetrahedron — a 3D fractal built by replacing a tetrahedron with four half-scale copies (4^d at depth d). Drag to rotate, switch to chaos-game mode; dimension stays exactly log4/log2 = 2.

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3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install