Shield Tiling

Visualization

@
$dim=4
$subspace=[s,0]
g=1+s
s=$companion([1,0,-1,0])
r=$exchange()
h0=s^11*[-1,0,1,1]
h1=s^9*[-1,0,0,0]
h2=s^4*[0,0,-1,-1]
h3=s^11*[-1,0,1,0]
h4=s^5*[0,0,-1,0]
h5=s^4*[0,0,-1,-1]
h6=s^11*[-1,0,1,1]
h7=s^2*[0,1,0,0]
h8=s^5*[0,0,-1,-1]
h9=s^2*[0,0,-1,0]
h10=s^11*[0,1,0,0]
h11=s^10*[0,1,1,1]
h12=s^4*[-1,-1,-1,0]
h13=s^3*[-1,-1,-1,0]
h14=s^7*[-2,-1,1,2]
h15=s^2*[0,-1,-2,-1]
h16=s^9*[-1,0,1,1]
h17=s*[0,0,-1,-1]
h18=s^5*[-2,-2,0,1]
h19=1*[1,1,-1,-2]
h20=s^5*[-1,-1,0,0]
h21=s*[1,1,-1,-2]
$root=A3
A3=g^-1*h9*A0|g^-1*h10*A0|g^-1*h11*A0|g^-1*h12*A0|g^-1*h13*A0|g^-1*h14*A0|g^-1*h15*A0|g^-1*h16*A1|g^-1*h17*A1|g^-1*h18*A1|g^-1*h19*A2|g^-1*h20*A2|g^-1*h21*A2
A2=g^-1*h4*A0|g^-1*h5*A0|g^-1*h6*A0|g^-1*h7*A0|g^-1*h8*A2
A1=g^-1*h1*A1|g^-1*h2*A1|g^-1*h3*A2
A0=g^-1*h0*A3

Overview

The Shield tiling is a classic aperiodic tiling with 12-fold (dodecagonal) rotational symmetry, discovered by Franz Gähler in 1988. It is mutually locally derivable (MLD) from the Socolar tiling and the Wheel tiling, all three belonging to the same MLD class.

The tiling admits local matching rules: tiles must fit edge-to-edge with matching orientation arrows and the tile decorations must form crosses at interior vertices. The window (acceptance domain) of the cut-and-project scheme is a regular dodecagon.

Algebraic Structure

The 12-fold structure lives in the cyclotomic field $\mathbb{Q}(e^{i\pi/6})$. The companion matrix $s$ satisfies $s^4 - s^2 + 1 = 0$ (12th cyclotomic polynomial), so $s$ represents a $30°$ rotation.

The expansion is

which at the eigenvalue $s_0 = e^{i\pi/6}$ evaluates to $1 + e^{i\pi/6}$, giving area inflation factor $|g|^2 = 2 + \sqrt{3}$.

The GIFS has four tile types $A_0, A_1, A_2, A_3$ with the cyclic dependency $A_0 \to A_3 \to (A_0, A_1, A_2)$:

References

• Shield — Tilings Encyclopedia
• Socolar — Tilings Encyclopedia
• Gähler, F. (1988). Crystallography of Dodecagonal Quasicrystals. Springer.
• Socolar, J. E. S. (1989). Simple octagonal and dodecagonal quasicrystals. Phys. Rev. B, 39, 10519–10551.

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