Chair Tiling

Visualization

@
$dim=2
$subspace=[s,0]
s=$companion([1,0])
r=$companion([-1,0])
h0=1
h1=1*[1,-1]
h2=s^3*[1,-1]
h3=s*[1,-1]
A=2^-1*(h0|h1|h2|h3)*A

The Chair (L-Tromino)

The chair tile — also called the L-tromino or L-tile — is an L-shaped polyomino consisting of three unit squares, obtained by removing one corner square from a 2×2 block. It is one of the two distinct trominoes.

The chair is a rep-4 tile: four half-size copies tile a full chair exactly, all without mirror reflections. The four copies are arranged as one copy in the original orientation at the top-left corner, and three copies (rotated 0°, 90°, and 270°) fitted around the inner corner of the L.

IFS on the Square Lattice

The chair lives on the Gaussian integer lattice $\mathbb{Z}[i]$. In the AIFS program:

• s = $i$ (companion matrix of $x^2 + 1 = 0$, 90° rotation)
• s^3 = $-i$ (270° rotation)

The four contraction maps (each scaling by $\tfrac{1}{2}$) are:

The attractor equation is:

Chair Tilings: Nonperiodic but Not Aperiodic

Iterating the substitution rule produces chair tilings of the plane. These tilings are nonperiodic (no translational symmetry), yet the chair tile itself is not aperiodic: it is possible to tile the plane periodically with L-trominoes as well, so the chair tile does not force nonperiodicity on its own.

The tilings exhibit a striking near-periodicity: the vertex set of any chair tiling spans a perfect square lattice. Furthermore, the tiling can be decomposed into a countable family of periodic sublattices $L_1, L_2, L_3, \ldots$, where $L_i$ is fully periodic with period vectors of length $2 \times 2^i$. This property is called limit-periodicity (Baake, Moody & Schlottmann 1998).

p-adic Cut-and-Project Structure

The limit-periodic structure of the chair tiling has a precise algebraic interpretation. The tiling can be obtained by a cut-and-project scheme with a p-adic internal space:

where $\mathbb{Q}_2$ is the field of 2-adic numbers. This connects the chair tiling to the broader theory of model sets (mathematical quasicrystals). As a consequence, the diffraction spectrum of the chair tiling is pure point — all peaks are Bragg peaks.

The vertex point set of the chair tiling can be described as

making precise the sense in which the tiling is “2-adic periodic.”

Matching Rules and Aperiodic Enforcements

Although the undecorated chair tile does not force nonperiodicity, decorated versions do. Goodman-Strauss (1999) showed that the trilobite and cross tiles — obtained by adding simple geometric decorations to chair-related shapes — form a small aperiodic set that forces exactly the chair-tiling substitution structure. This is one of the simplest known aperiodic tile sets.

Cohomology

The topological invariants of the chair tiling space have been computed: Barge et al. (2010) calculated the Čech cohomology of the tiling space, confirming the limit-periodic structure through algebraic topology.

References

• Robinson Jr., E. A. (1999). On the table and the chair. Indagationes Mathematicae, 10(4): 581–599.
• Baake, M., Moody, R. V. & Schlottmann, M. (1998). Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces. Journal of Physics A, 31(27): 5755–5766.
• Goodman-Strauss, C. (1999). A small aperiodic set of planar tiles. European J. Combinatorics, 20(5): 375–384.
• Barge, M., Diamond, B., Hunton, J. & Sadun, L. (2010). Cohomology of substitution tiling spaces. Ergodic Theory and Dynamical Systems, 30(6): 1607–1627.
• Chair tiling — Wikipedia
• Chair — Tilings Encyclopedia

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