Sierpiński Tetrahedron

Visualization

@
$dim=3
A=2^-1*([1,1,1]|[-1,1,-1]|[1,-1,-1]|[-1,-1,1])*A

Overview

The Sierpiński tetrahedron (also called the tetrix) is the 3D analog of the Sierpiński triangle. Four self-similar copies of scale $1/2$ are placed at the four vertices of a regular tetrahedron inscribed in the unit cube. Repeating this substitution indefinitely yields a fractal whose Hausdorff dimension equals exactly 2 — it is a 2D surface folded into 3D space.

Algebraic Structure

The four vertices of the inscribed tetrahedron are:

These are the four vertices of the cube $\{-1,1\}^3$ with an even number of $-1$ coordinates. The four maps are:

The Hausdorff dimension equals:

Despite the dimension of 2, the attractor is not a surface — it has no interior and is topologically very different from a 2D manifold.

Properties

• The total volume is zero: at each step the fraction of space occupied decreases by a factor of $4/8 = 1/2$, converging to zero.
• Its Hausdorff dimension of exactly 2 means it is a measure-zero subset of $\mathbb{R}^3$ with the same metric complexity as a surface.
• Each of the four self-similar sub-copies is a scaled, translated copy of the whole attractor.

References

• Sierpiński tetrahedron — Wikipedia

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