Danzer's 7-Fold Tiling

Visualization

@
$dim=6
$subspace=[s,0]
g=1+s^2-s^5
s=$companion([1,-1,1,-1,1,-1])
r=$exchange()
h0=1*[0,0,0,0,0,0]
h1=1*[1,0,1,0,0,-1]
h2=r*s^4*[1,0,0,0,0,0]
h3=s^6*[-1,0,-2,0,-1,1]
h4=s^10*[-1,0,0,1,0,1]
h5=1*[1,0,0,0,0,0]
h6=s^5*[0,-1,-1,-1,0,0]
h7=s^9*[0,0,-1,0,0,1]
h8=s^13*[0,1,0,0,0,1]
h9=s^6*[0,-1,-1,0,0,0]
h10=s*[1,-1,1,0,1,-1]
h11=r*s^4*[0,0,0,0,0,0]
h12=1*[1,0,0,0,0,0]
h13=s^4*[0,-1,0,-1,0,-1]
$root=A0
A0=g^-1*h0*A0|g^-1*h1*A0|g^-1*h2*A1|g^-1*h3*A1|g^-1*h4*A2
A1=g^-1*h5*A0|g^-1*h6*A0|g^-1*h7*A1|g^-1*h8*A1|g^-1*h9*A1|g^-1*h10*A2
A2=g^-1*h11*A0|g^-1*h12*A1|g^-1*h13*A2

Overview

Danzer’s 7-fold tiling is an aperiodic substitution tiling with 7-fold rotational symmetry, discovered by Ludwig Danzer. It uses three triangular prototile types with angles that are multiples of $\pi/7$:

• $A_0$: triangle with angles $\pi/7$, $2\pi/7$, $4\pi/7$
• $A_1$: triangle with angles $2\pi/7$, $2\pi/7$, $3\pi/7$
• $A_2$: triangle with angles $\pi/7$, $3\pi/7$, $3\pi/7$

The four edge lengths are $\sin(\pi/7)$, $\sin(2\pi/7)$, $\sin(3\pi/7)$, and $\sin(2\pi/7)+\sin(3\pi/7)$. Simple local matching rules exist for the tiling.

Algebraic Structure

The 7-fold structure lives in the 14th cyclotomic field $\mathbb{Q}(e^{2\pi i/7})$. The companion $s$ (AIFS: $companion([1,-1,1,-1,1,-1])) satisfies $x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^7+1)/(x+1) = 0$, so $s = e^{2\pi i/7}$ (primitive 7th root of unity). The $\dim=6$ rational space reflects the degree-6 cyclotomic polynomial.

The expansion is $g = 1+s^2-s^5$, which on the eigenplane of $e^{2\pi i/7}$ gives the 7-fold inflation factor.

The GIFS equations:

Variant 1

A second variant uses slightly different maps $h_0, h_2, h_4, h_{11}$ while keeping the same GIFS dependency structure.

References

• Danzer's 7-fold — Tilings Encyclopedia
• Nischke, K.-P. & Danzer, L. (1996). A construction of inflation rules based on n-fold symmetry. Discrete and Computational Geometry, 15(2), 221–236.

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