Ammann–Beenker Tiling

Visualization

@
$dim=4
$subspace=[s,0]
s=$companion([1,0,0,0])
r=$exchange()
g=s-s^3+1
h0=s^4*r*[-1,-1,-1,-1]
h1=s^5*[-1,-1,-1,-1]
h2=s^3*[0,0,-2,-2]
h3=s^5*[0,-2,0,0]
h4=s^3*[1,-1,-1,-1]
h5=s^4*[-1,-1,-1,-1]
h6=s^3*r*[-1,-1,-1,-1]
h7=[1,1,1,-1]
h8=r*s*[1,1,1,-1]
h9=s^4*[0,-2,0,0]
h10=s*r*[-1,-1,1,1]
h11=r*s*[2,0,2,0]
A=g^-1*(h0*A|h1*A|h2*A|h3*B|h4*B)
B=g^-1*(h5*A|h6*A|h7*A|h8*A|h9*B|h10*B|h11*B)

Overview

The Ammann–Beenker tiling is an aperiodic tiling of the plane with exact 8-fold (octagonal) rotational symmetry, first described by Robert Ammann around 1977 and independently studied by F. P. M. Beenker in 1982. It is one of the canonical examples of a quasicrystal tiling.

The tiling uses two prototiles — a rhombus (45°/135° angles) and a square — in a ratio of $1 : \sqrt{2}$. They are assembled by a self-similar substitution rule, captured here as a Generalized IFS (GIFS) with two interleaved attractors $A$ (rhombus) and $B$ (square).

The Two Tile Types

Substitution Rules

Each tile decomposes into smaller copies of both tile types under the substitution:

$g A = h_0(A) \cup h_1(A) \cup h_2(A) \cup h_3(B) \cup h_4(B)$

$g B = h_5(A) \cup h_6(A) \cup h_7(A) \cup h_8(A) \cup h_9(B) \cup h_{10}(B) \cup h_{11}(B)$

The rhombus $A$ decomposes into 3 rhombuses and 2 squares; the square $B$ decomposes into 4 rhombuses and 3 squares. The total inflation factor is $g = s - s^3 + 1$, corresponding to scaling by $\delta = 1 + \sqrt{2}$ (the silver ratio).

Algebraic Structure

The tiling lives in $\mathbb{Q}(\zeta_8)$ where $\zeta_8 = e^{i\pi/4}$ is a primitive 8th root of unity. The companion matrix $s$ (defined as $companion([1,0,0,0]) in AIFS) is the companion for $x^4 + 1$ — the 8th cyclotomic polynomial — giving $s$ eigenvalues $e^{i k\pi/4}$ for $k = 1, 3, 5, 7$.

The directive $subspace=[s,0] projects onto the dominant complex conjugate pair $\{e^{i\pi/4}, e^{-i\pi/4}\}$ (eigenvalue index 0), yielding the 2D 8-fold symmetric projection.

The exchange matrix $r$ (defined as $exchange() in AIFS) represents complex conjugation in the rational space, needed for the orientation-reversing maps.

8-Fold Symmetry

The eight-fold rotational symmetry arises from the 8th cyclotomic field: multiplying by $s$ in the rational space corresponds to rotation by $45° = \pi/4$ in the projected plane. The 12 maps $h_0, \ldots, h_{11}$ include all eight rotational orientations of the prototiles, ensuring the global 8-fold symmetry of the tiling.

Connection to Quasicrystals

Ammann–Beenker tilings serve as 2D cross-sections of a 4D hypercubic lattice (the “cut-and-project” method). The 4D rational space $\mathbb{Q}^4$ used here is precisely this higher-dimensional lattice, and the $subspace projection corresponds to the physical-space projection in the cut-and-project construction. The inflation factor $\delta = 1 + \sqrt{2}$ (the silver ratio) plays the same role as the golden ratio $\phi$ does in Penrose tilings.

References

• Ammann, R., Grünbaum, B. & Shephard, G. C. (1992). Aperiodic tiles. Discrete & Computational Geometry, 8(1).
• Beenker, F. P. M. (1982). Algebraic theory of non-periodic tilings of the plane by two simple building blocks. Eindhoven TH report.
• Ammann–Beenker tiling — Wikipedia
• Ammann–Beenker tiling — Tilings Encyclopedia

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