Vicsek Fractal

Visualization

@
$dim=2
f1=[0,0]*[0.333,0,0,0.333]
f2=[0.667,0]*[0.333,0,0,0.333]
f3=[0.333,0.333]*[0.333,0,0,0.333]
f4=[0,0.667]*[0.333,0,0,0.333]
f5=[0.667,0.667]*[0.333,0,0,0.333]
S=(f1|f2|f3|f4|f5)*S

Overview

The Vicsek fractal (also called the box fractal) was introduced by Tamás Vicsek in 1983 in the context of diffusion-limited aggregation. Starting from a 3×3 grid, keep the four corner squares and the central square, then recurse. The resulting pattern resembles a cross at every scale.

Construction

At each step, subdivide the square into a 3×3 grid and keep only the five marked cells:

X . X
. X .
X . X

Repeat for each retained sub-square. The attractor is the intersection of all iterations.

Definition

Five affine maps, each scaling by $\frac{1}{3}$ and translating to one of the five retained positions:

$f_k(x, y) = \frac{1}{3}\begin{pmatrix}x \\ y\end{pmatrix} + \begin{pmatrix}t_x^{(k)} \\ t_y^{(k)}\end{pmatrix}$

with translations $(0,0)$, $(\frac{2}{3}, 0)$, $(\frac{1}{3}, \frac{1}{3})$, $(0, \frac{2}{3})$, $(\frac{2}{3}, \frac{2}{3})$.

Properties

• Fractal dimension: $\frac{\log 5}{\log 3} \approx 1.465$
• Lebesgue measure: 0
• Number of transforms: 5, all contraction ratio $\frac{1}{3}$
• Lattice symmetry: The fractal has the same 4-fold symmetry as the square lattice

References

• Vicsek, T. (1983). Fractal models for diffusion controlled aggregation. J. Phys. A: Math. Gen.
• Vicsek fractal — Wikipedia

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