Ammann A3 Tiling

Visualization

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

Overview

The Ammann A3 tiling is an aperiodic tiling of the plane discovered by Robert Ammann around 1977 and first published in Grünbaum & Shephard (1987), where it was named A3 in the series Ammann A1–A5. The tiling uses three prototile types and a self-similar substitution rule governed by the golden ratio $\varphi = \tfrac{1+\sqrt{5}}{2}$ as the inflation factor.

The tiling is a Euclidean windowed (cut-and-project) tiling with finite rotational symmetry, and falls in the same golden-ratio family as the Penrose tilings.

The Three Tile Types

Substitution Rules

Each prototile decomposes into smaller copies of the three types under inflation by $g$ (scaling by $\varphi$ in the rendering plane):

The substitution matrix (column $j$ = counts of each tile type in the decomposition of tile $j$) is:

The Perron–Frobenius eigenvalue of $M$ is $\varphi^2$, consistent with areas scaling by $\varphi^2$ under the inflation.

Algebraic Structure

The tiling lives in the ring $\mathbb{Z}[\varphi]^2 \cong \mathbb{Z}[x]/(x^4 + 3x^2 + 1)$. The companion matrix $x$ (defined as $companion([1,0,3,0]) in AIFS) satisfies

This polynomial factors over $\mathbb{Q}(\sqrt{5})$ as $(x^2 + \varphi^2)(x^2 + \varphi^{-2})$, giving eigenvalues $\pm i\varphi$ (modulus $\varphi$) and $\pm i/\varphi$ (modulus $1/\varphi$). The directive $subspace=[x,0] selects the eigenplane of $\pm i\varphi$ as the 2D rendering space.

The auxiliary variable $s = x^3 + 2x$ evaluates to $-i$ on the eigenvalue $i\varphi$, so $s$ acts as a 90° rotation in the rendering plane. The expansion matrix

has eigenvalue $g(i\varphi) = (-i)(i\varphi) = \varphi$, confirming that each inflation step scales the tiles by $\varphi$.

References

• Grünbaum, B. & Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman.
• Ammann A3 — Tilings Encyclopedia

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