Labyrinth Tiling

Visualization

@
$dim=4
$subspace=[s,0]
g=1+s-s^3
s=$companion([1,0,0,0])
r=$exchange()
q0=1*[0,1,0,0]
q1=1
q2=s^2*[0,-1,-1,-1]
q3=s^4*[-1,-1,0,0]
q4=s^6*[-1,-1,0,1]
q5=s^2*[0,0,0,-1]
q6=s^4*[-1,-2,-1,0]
q7=1*[0,1,1,0]
q8=s^6*[-1,0,1,1]
h0=q0
h1=q4
h2=q1
h3=q2
h4=q6
h5=q5
h6=q7
h7=q8
h8=q3
h9=q0
h10=q1
h11=q2
h12=q4
h13=q5
h14=q3
h15=q0
h16=q1
h17=q2
h18=q3
$root=A0
A0=g^-1*h0*A0|g^-1*h1*A1|g^-1*h2*A1|g^-1*h3*A1|g^-1*h4*A1|g^-1*h5*A2|g^-1*h6*A2|g^-1*h7*A2|g^-1*h8*A2
A1=g^-1*h9*A0|g^-1*h10*A1|g^-1*h11*A1|g^-1*h12*A1|g^-1*h13*A2|g^-1*h14*A2
A2=g^-1*h15*A0|g^-1*h16*A1|g^-1*h17*A1|g^-1*h18*A2

Overview

The Labyrinth tiling is an aperiodic tiling with 8-fold rotational symmetry, related algebraically to the Ammann–Beenker tiling. Both use the same 4D rational space with $s^4 = -1$ and the same expansion factor $g = 1+s-s^3$ (the silver ratio $1+\sqrt{2}$ on the rendering eigenplane).

The tiling has three prototile types $A_0, A_1, A_2$ with mutual dependencies, forming a connected GIFS system. The tile shapes have fractal boundaries arising from the algebraic cut-and-project structure.

Algebraic Structure

The companion matrix $s$ (AIFS: $companion([1,0,0,0])) satisfies $s^4 + 1 = 0$, so $s$ represents $45°$ rotation. The expansion at eigenvalue $e^{i\pi/4}$ is:

The 9 unique isometry maps $q_0, \ldots, q_8$ are reassigned to $h_0, \ldots, h_{18}$ to define the three-attractor GIFS:

Boundary Dimension

The tile boundaries have Hausdorff dimension $1$, but the boundary curves are not straight — they are infinite-sided polygons. Each boundary piece between two neighbouring tiles is a self-similar curve that spans the full 2D plane (it cannot be contained in any line), yet its dimension equals $1$ exactly.

This distinguishes the Labyrinth from polyhedral tilings such as the Ammann–Beenker, where tile boundaries are finite unions of straight line segments. Both have $\dim_H(\partial A) = 1$, but the Labyrinth boundaries are curved.

References

• Labyrinth — Tilings Encyclopedia

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