🔺 Sierpiński Tetrahedron
A 3D fractal pyramid
Dimension = log4/log2 = 2.000
Tetrahedra: 1
Mode
Geometry
Rotation
Color
Controls
Stats
Depth
2
Count 4^d
16
Info & Theory

The Sierpiński tetrahedron (also called the Sierpiński pyramid or tetrix) is the 3D analogue of the Sierpiński triangle. Start with a regular tetrahedron — four vertices, four triangular faces.

Recursive construction

Shrink the tetrahedron to half size and place one copy at each of the four corners. Repeat on every copy. At depth d there are 4^d small tetrahedra, and the empty octahedral hole in the middle grows at every level.

Self-similarity

The limit shape is exactly self-similar: it is made of N = 4 copies of itself, each scaled by a factor s = 1/2.

Fractal dimension exactly 2

The Hausdorff (similarity) dimension is D = log N / log(1/s) = log 4 / log 2 = 2. Although it lives in 3D space, its dimension is the integer 2 — the same as a flat surface. Its total surface area stays constant while the enclosed volume tends to zero.

Chaos game in 3D

A second way to build it: pick a random start point, then repeatedly jump halfway toward one of the four tetrahedron vertices chosen at random, plotting each landing point. The cloud of points converges to the same fractal — no recursion needed.

  • Drag the canvas to rotate; scroll or the slider to zoom.
  • Painter's algorithm sorts faces back-to-front per frame.
  • Cap is depth 6 (4^6 = 4096 tetrahedra).

About this simulation

Written by MySimulator Team · Reviewed by MySimulator Editorial Review

Last updated: 5 July 2026

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.