CAP (Hat Monotile)

Fractal dimension: 2

Visualization

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AIFS program

@
$dim=8
$subspace=[s,0]
g=1+s^2+s^3-s^7
s=$companion([1,1,0,-1,-1,-1,0,1])
r=$exchange()
O=[1,s,s^2,s^3,s^4,s^5,s^6,s^7,s^8,s^9,s^10,s^11,s^12,s^13,s^14,s^15,s^16,s^17,s^18,s^19,s^20,s^21,s^22,s^23,s^24,s^25,s^26,s^27,s^28,s^29,(s*r),(s^2*r),(s^3*r),(s^4*r),(s^5*r),(s^6*r),(s^7*r),(s^8*r),(s^9*r),(s^10*r),(s^11*r),(s^12*r),(s^13*r),(s^14*r),(s^15*r),(s^16*r),(s^17*r),(s^18*r),(s^19*r),(s^20*r),(s^21*r),(s^22*r),(s^23*r),(s^24*r),(s^25*r),(s^26*r),(s^27*r),(s^28*r),(s^29*r),r]
h0=O[0]*[-2,-2,2,0,0,0,0,-2]
h1=O[0]*[2,2,0,0,0,-2,-2,2]
h2=O[0]*[0,0,-2,0,0,2,2,0]
h3=O[15]
h4=O[0]*[-2,0,3,0,0,-1,-3,-2]
h5=O[10]*[-2,0,3,0,0,-1,-3,-2]
h6=O[20]*[-2,0,3,0,0,-1,-3,-2]
h7=O[0]*[2,4,1,0,0,-3,-5,2]
h8=O[10]*[2,4,1,0,0,-3,-5,2]
h9=O[20]*[2,4,1,0,0,-3,-5,2]
h10=O[0]
h11=O[15]*[1,3,0,0,0,-1,-1,1]
h12=O[0]*[1,3,0,0,0,-1,-1,1]
h13=O[5]
h14=O[25]*[-1,1,2,0,0,-1,-5,-1]
h15=O[10]*[-1,1,2,0,0,-1,-5,-1]
h16=O[15]*[1,3,0,0,0,-1,-1,1]
h17=O[0]*[1,3,0,0,0,-1,-1,1]
h18=O[5]
h19=O[25]*[-1,1,2,0,0,-1,-5,-1]
h20=O[10]*[-1,1,2,0,0,-1,-5,-1]
h21=O[0]*[1,5,1,0,0,-2,-4,1]
$root=H
H=g^-1*h0*H|g^-1*h1*H|g^-1*h2*H|g^-1*h3*T|g^-1*h4*P|g^-1*h5*P|g^-1*h6*P|g^-1*h7*F|g^-1*h8*F|g^-1*h9*F
T=g^-1*h10*H
P=g^-1*h11*H|g^-1*h12*H|g^-1*h13*P|g^-1*h14*F|g^-1*h15*F
F=g^-1*h16*H|g^-1*h17*H|g^-1*h18*P|g^-1*h19*F|g^-1*h20*F|g^-1*h21*F

Overview

The CAP (Canonical Aperiodic Prototile) tiling is a self-similar substitution tiling underlying the hat monotile — the first known aperiodic monotile (a single tile shape that can tile the plane only aperiodically), discovered in 2022 by David Smith and published in 2024.

The substitution uses four metatile types (H, T, P, F), which are “super-tiles” assembled from the original hat-shaped tiles. The H-metatile (hat) is the most complex (10 maps); T (thin) is the simplest (1 map); P (parallelogram) and F (fat) are intermediate.

The CAP substitution was used by Baake, Gähler, and Sadun to derive the cut-and-project description of the hat tiling, establishing it as a proper quasicrystal.

Algebraic Structure

The underlying algebraic structure has 30-fold symmetry. The 8D rational space uses $s$ (AIFS: $companion([1,1,0,-1,-1,-1,0,1])), which satisfies $x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 = 0$ (related to the 15th cyclotomic polynomial).