Rauzy Fractal

Visualization

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

Overview

The Rauzy fractal (also called the Rauzy gasket or tribonacci fractal) was introduced by Gérard Rauzy in 1982 in connection with the tribonacci substitution

$\sigma: 1 \mapsto 12,\quad 2 \mapsto 13,\quad 3 \mapsto 1.$

It is a compact subset of $\mathbb{R}^2$ with non-empty interior that tiles the plane by integer translates. Its boundary is a fractal curve with Hausdorff dimension between 1 and 2.

Tribonacci Constant

The tribonacci constant $\beta \approx 1.3247$ is the real root of

$\beta^3 = \beta^2 + \beta + 1,$

i.e. the dominant eigenvalue of the substitution matrix. The companion matrix for the polynomial $x^3 + x^2 + x - 1$ (the $companion([-1,1,1]) constructor in AIFS) encodes this algebraic structure in a $3 \times 3$ integer matrix.

Rational Form

The Rauzy fractal is the attractor of three affine maps in $\mathbb{Q}^3$:

$g^3 A = g^2 h_1(A) \cup g\, h_2(A) \cup h_3(A)$

where $h_1(\mathbf{x}) = \mathbf{x},\quad h_2(\mathbf{x}) = \mathbf{x}+[0,1,0],\quad h_3(\mathbf{x}) = \mathbf{x}+[0,1,1],$

and $g$ is the companion matrix for $x^3+x^2+x-1$. Equivalently:

$A = (g^{-1} \,|\, g^{-2}*[0,1,0] \,|\, g^{-3}*[0,1,1]) * A$

The rendering plane is the 2D eigenspace of $g$ corresponding to its complex conjugate pair of eigenvalues $\lambda, \bar\lambda$ with $|\lambda| = 1/\sqrt{\beta} \cdot \beta \approx 1.356$. The directive $subspace=[g,0] selects this eigenspace (index 0 in the sorted eigenvalue list).

Contraction Factors

On the projected 2D plane the three maps contract by factors $|\lambda|^{-1} \approx 0.738$, $|\lambda|^{-2} \approx 0.545$, and $|\lambda|^{-3} \approx 0.402$.

Properties

• Dimension: 2 (positive area; the Rauzy fractal is a 2D tile)
• Tiling: tiles $\mathbb{R}^2$ by translates of the integer lattice $\mathbb{Z}^2$ projected to the plane
• Boundary dimension: $\approx 1.093$; satisfies $\dim_H(\partial A) = 2\log\lambda/\log\tau$, where $\tau$ is the tribonacci constant and $\lambda$ is the real root of $\lambda^4-2\lambda-1=0$
• Symmetry: 3-fold symmetry related to the three letters of the tribonacci alphabet

Boundary Dimension

The boundary $\partial A$ has Hausdorff dimension

where $\tau \approx 1.8393$ is the tribonacci constant (real root of $\tau^3-\tau^2-\tau-1=0$) and $\lambda \approx 1.3953$ is the real root of

Here $\tau = p$ is the squared norm of the expansion $g$ acting on the 2D projection plane.

References

• Rauzy, G. (1982). Nombres algébriques et substitutions. Bulletin de la Société Mathématique de France, 110, 147–178.
• Mekhontsev, D. (2019). An algebraic framework for finding and analyzing self-affine tiles and fractals. §9.5.
• Rauzy fractal — Wikipedia
• Hinsley, S. R. Self-Similar Tiles — Rauzy Tiles.

Similar