Sierpiński

Fractal Structures  ·  Infinite Depth  ·  Self-Similarity

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01 / The Gasket

A triangle within
every triangle

Remove the central inverted triangle from an equilateral triangle. Repeat on each of the three remaining triangles. Repeat forever.

You will see a process emerge. An infinitely detailed edge between filled and void, where every fragment is an exact mirror of the whole.

The chaos game reveals something stranger still: make millions of random jumps toward corners and the same structure crystallises from pure disorder. It is an attractor, a fixed point of infinite recursion.

dim = log(3) / log(2) ≈ 1.585
More than a line, less than a plane.
Area converges to zero. Perimeter diverges to infinity.
Points 30,000

02 / The Tetrahedron

The fractal lifted
into three dimensions

Place four half-size tetrahedra at each vertex of the original. Remove the central octahedral void between them. The fractal dimension is exactly 2.0, the dimension of a surface, despite living in three-dimensional space.

Viewed from the right angle, it projects cleanly onto the 2D gasket. Two fractals, one idea.

dim = log(4) / log(2) = 2.000
Volume → 0. Surface area → ∞.
The removed void at each step is a regular octahedron.
Step 01
Solid tetrahedron, the seed form.
Step 02
4 half-size copies placed at each vertex. Central octahedral void opens.
Step 03
Same rule applied to all 4 copies → 16 sub-tetrahedra.
Step 04 → ∞
16 → 64 → 256 → ∞. Volume → 0, detail → ∞.
Sierpiński Gasket
1.585
Fractal Dimension
log(3) / log(2)
Area = 0 · Perimeter = ∞
Fractal boundary between 1D and 2D
Sierpiński Tetrahedron
2.000
Fractal Dimension
log(4) / log(2)
Volume = 0 · Surface area = ∞
A surface living inside 3D space
Both Structures
∞
Self-Similar Scales
Every fragment is the whole
No characteristic length exists
Scale invariance at every zoom level