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SIERPINSKI 3D

Procedural Iterated Function Systems & Chaos Attractors

FRACTAL MODE

FPS 60 FPS

Instances / Pts 1,024

Triangles 4,096

Vertices 4,096

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MATHEMATICAL THEORY

The Mathematics of Sierpinski Fractals

1. Sierpinski Tetrahedron (Tetrix)

Hausdorff Dimension: D = 2.0

The 3D Sierpinski Gasket (Tetrix) is formed by starting with a solid regular tetrahedron, scaling it by a factor of r = 1/2, and placing 4 identical copies at the 4 vertices of the original.

Hausdorff Dimension Formula:

$$D = \frac{\ln(N)}{\ln(1/r)} = \frac{\ln(4)}{\ln(2)} = 2.0$$

Remarkably, even though the Sierpinski Tetrahedron exists in 3-dimensional space, its fractional dimension is exactly 2.0! As recursion depth $n \to \infty$, its volume approaches zero ($V_n = (1/2)^n V_0 \to 0$) while its surface area remains constant.

2. Menger Sponge & Sierpinski Carpet

Hausdorff Dimension: D ≈ 2.7268

The 3D Menger Sponge divides a cube into 27 sub-cubes ($3 \times 3 \times 3$) and removes the 6 center face cubes and 1 central core cube, leaving $N = 20$ cubes of size $r = 1/3$.

$$D = \frac{\ln(20)}{\ln(3)} \approx 2.726833$$

In the infinite limit, the Menger Sponge has zero volume but infinite surface area!