Golden Trapezoid

Visualization

@
$dim=4
$subspace=[g,0]
g=$companion([-1,0,1,0])
r=$companion([-1],[1],[-1],[1])
h0=-1*[-1,1,0,1]
h1=r*-1*[1,1,0,-1]
h2=r*[1,-1,0,0]
h3=1*[1,-1,1,0]
A=(g^-2*h0|g^-3*h1|g^-3*h2|g^-4*h3)*A

Overview

The Golden Trapezoid is a self-affine tile whose shape is determined by the roots of $x^4 + x^2 - 1 = 0$. Like the Penrose tilings, it is linked to the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$, but arises from a different algebraic polynomial.

The attractor fills a solid trapezoidal region of the plane with Hausdorff dimension $2$.

Algebraic Structure

The IFS is defined in $\mathbb{R}^4$ (the rational space $\mathbb{Q}[x]/(x^4+x^2-1)$) and projected to $\mathbb{R}^2$ via the dominant complex eigenspace of $g$.

The expansion matrix $g$ (defined as $companion([-1,0,1,0]) in AIFS) is the companion matrix for $x^4 + x^2 - 1 = 0$. Its eigenvalues come in two complex conjugate pairs:

• Dominant pair (index 0): modulus $|\lambda|^2 = \sqrt{\phi} \approx 1.272$
• Subdominant pair: modulus $< 1$

The directive $subspace=[g,0] projects onto the Jordan cell of index 0 — the dominant complex pair — yielding the 2D rendered attractor.

The matrix $r$ (defined as $companion([-1],[1],[-1],[1]) in AIFS) is the companion for $x^4 - x^3 + x^2 - x + 1 = \Phi_{10}(x)$, acting as a rotation/reflection in the rational space.

The Four Maps

The attractor satisfies $A = g^{-2}h_0(A) \cup g^{-3}h_1(A) \cup g^{-3}h_2(A) \cup g^{-4}h_3(A)$ with four maps at different scales:

The different scales break the strict self-similarity but the attractor remains a well-defined compact set whose Hausdorff dimension is 2.

Connection to the Golden Ratio

Both $x^4 + x^2 - 1 = 0$ and the Penrose case $\Phi_{10}(x)$ are degree-4 polynomials over $\mathbb{Q}$ with roots on the unit circle or at modulus $\neq 1$. The dominant eigenvalue magnitude $|\lambda| = \phi^{1/4}$ where $\phi$ is the golden ratio links this fractal structurally to Penrose-type aperiodic tilings.

References

• Hinsley, S. R. Self-Similar Tiles — Metallic Tiles (Golden bee and derivatives).

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