Octahedron Fractal

Visualization

@
$dim=3
A=2^-1*([1,0,0]|[0,1,0]|[0,0,1]|[-1,0,0]|[0,-1,0]|[0,0,-1])*A

Overview

The octahedron fractal is a 3D self-similar set constructed by placing six half-scale copies of itself at the six vertices of a regular octahedron. It is analogous to the Cantor dust and Vicsek fractal but in three dimensions with octahedral symmetry.

Algebraic Structure

The six vertices of the regular octahedron centered at the origin with circumradius $1$ are:

The six maps are:

The Hausdorff dimension is:

Properties

• The attractor has full octahedral symmetry ($O_h$ point group, order 48).
• Despite the high symmetry, the attractor is a fractal with non-integer dimension — not a surface or solid.
• It can be seen as the 3D analog of the six-branch Vicsek fractal.

References

• Octahedron — Wikipedia

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