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).