Square Family

Visualization

@
$dim=2
$subspace=[s,0]
s=$companion([1,0])
r=$companion([-1,0])
h0=1
h1=1*[1,0]
h2=1*[0,1]
h3=1*[1,1]
A=2^-1*(h0|h1|h2|h3)*A

Overview

The square family is a collection of rep-4 tiles: each tile is the attractor of an IFS of exactly four maps, each contracting by $\tfrac{1}{2}$. Because $4 \times \bigl(\tfrac{1}{2}\bigr)^2 = 1$, every tile in this family has positive Lebesgue measure (Hausdorff dimension exactly 2) and tiles the plane.

The family splits into two lattice types:

• G4 tiles — built on the square lattice $\mathbb{Z}[i]$ (Gaussian integers), using 90° rotations. The rotation matrix $s$ satisfies $s^2 = -I$ (companion of $x^2 + 1$); $r$ with $r^2 = I$ is a reflection.
• G6 tiles — built on the triangular lattice $\mathbb{Z}[\omega]$ (Eisenstein integers, $\omega = e^{i\pi/3}$), using 60° rotations. Here $s$ satisfies $s^2 - s + 1 = 0$ (companion of $x^2 - x + 1$).

In both cases the expansion factor is $g = 2$ (scaling by 2 each step).

Tile Gallery

IFS Structure

Each tile satisfies the fixed-point equation

where the four maps $h_k$ are affine contractions of the form $h_k(x) = R_k \cdot x + t_k$, with $R_k \in \{I,\, s,\, s^2,\, s^3,\, r,\, sr,\, s^2r,\, s^3r\}$ a symmetry of the lattice and $t_k$ a lattice vector.

Notable Members

The Chair and Sphinx have their own dedicated pages with detailed mathematical background:

• Chair (G4) — L-tromino rep-tile; limit-periodic tiling with a 2-adic cut-and-project structure ($\mathbb{Q}_2 \times \mathbb{Q}_2$ internal space); pure-point diffraction spectrum.
• Sphinx (G6) — hexiamond (6 equilateral triangles) rep-tile; chiral tile appearing in left- and right-handed versions; limit-periodic and pure-point diffractive; the number of sphinx-frame tilings of order $n$ grows as $e^{0.425\, n^2}$.

References

• Mekhontsev, D. (2019). IFStile — software for self-affine tilings. Github: mekhontsev/ifstile.
• Hinsley, S. R. Self-Similar Tiles.

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