Octagonal Tiling (1225)

Visualization

@
$dim=4
$subspace=[s,0]
g=1+s-s^3
s=$companion([1,0,0,0])
r=$exchange()
h0=1*[0,2,0,0]
h1=1*[0,0,0,0]
h2=r*s*[0,0,0,0]
h3=1*[2,2,0,0]
h4=1*[2,2,1,0]
h5=r*s^2*[0,0,0,0]
h6=s^5*[-2,-2,0,2]
h7=1*[2,0,0,0]
h8=s^6*[-1,0,3,2]
h9=s^2*[1,-2,-3,-2]
$root=A0
A0=g^-1*h3*A1|g^-1*h4*A1|g^-1*h5*A0|g^-1*h6*A0|g^-1*h7*A0|g^-1*h8*A0|g^-1*h9*A0
A1=g^-1*h0*A1|g^-1*h1*A0|g^-1*h2*A0

Overview

The Octagonal 1225 tiling is an aperiodic 2-prototile tiling with 8-fold rotational symmetry. It shares the same algebraic framework as the Ammann–Beenker and Labyrinth tilings — all three use the 4D rational space with $s^4 = -1$ and expansion by the silver ratio $g = 1+s-s^3$.

With only two prototile types ($A_0$ and $A_1$) and 10 affine maps, it is the most compact member of the 8-fold family in this file.

Algebraic Structure

The companion $s$ (AIFS: $companion([1,0,0,0])) satisfies $s^4+1=0$ (rotation by $45°$). The silver-ratio expansion $g = 1+s-s^3$ contracts by $1/(1+\sqrt{2})$ on the eigenplane of $e^{i\pi/4}$.

The GIFS has two tile types: $A_0$ (the larger tile, 7 maps) and $A_1$ (the smaller tile, 3 maps):

References

• Ammann-Beenker — Tilings Encyclopedia

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