Cantor Dust

Visualization

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

Overview

Cantor dust is the two-dimensional generalisation of the classical Cantor set. It is constructed as the Cartesian product $C \times C$ of the Cantor set with itself — equivalently, by taking a unit square and at each iteration removing the central cross $(\frac{1}{3}, \frac{2}{3}) \times [0,1]$ together with $[0,1] \times (\frac{1}{3}, \frac{2}{3})$.

The result is a totally disconnected perfect set: every point is a limit point, yet no two points are connected.

Construction

At step $n$ retain only the $4^n$ squares of side length $3^{-n}$ located in the four corners of the $3^n \times 3^n$ grid. In the limit a self-similar dust remains.

Each of the four IFS maps scales by $\frac{1}{3}$ and translates to a corner of $[0,1]^2$. In the algebraic form the expansion $g = 3$ acts on integer vectors, and the four translations $[0,0]$, $[2,0]$, $[0,2]$, $[2,2]$ live in $\mathbb{Z}^2$:

$f_k(\mathbf{x}) = g^{-1}(\mathbf{x} + \mathbf{t}_k) = \frac{\mathbf{x} + \mathbf{t}_k}{3}$

with $\mathbf{t}_k \in \{(0,0),\,(2,0),\,(0,2),\,(2,2)\}$, giving outputs $\frac{2}{3}$ exactly (no floating-point approximation).

Properties

• Fractal dimension: $\frac{\log 4}{\log 3} \approx 1.261$
• Lebesgue measure: 0 (the dust has zero area)
• Topological dimension: 0 (totally disconnected)
• Cardinality: Uncountably infinite
• Self-similarity: Each quadrant of the dust is an exact $\frac{1}{3}$-scaled copy of the whole

Relation to the Cantor Set

The 1D Cantor set $C \subset [0,1]$ is the attractor of the two maps $g_1(x) = \frac{x}{3}$ and $g_2(x) = \frac{x}{3} + \frac{2}{3}$. Cantor dust is $C \times C$, and the four 2D maps are exactly all combinations of $g_1, g_2$ applied independently to each coordinate.

Its Hausdorff dimension equals $2 \times \dim_H(C) = 2 \times \frac{\log 2}{\log 3} = \frac{\log 4}{\log 3}$.

References

• Cantor, G. (1883). Über unendliche, lineare Punktmannichfaltigkeiten. Math. Annalen.
• Cantor set — Wikipedia

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