Sphinx

Visualization

@
$dim=2
$subspace=[s,0]
s=$companion([1,-1])
r=$companion([-1,0])
h0=r*s^5
h1=r*s^2*[0,-1]
h2=r*s^5*[2,-1]
h3=s^4*[0,-2]
A=2^-1*(h0|h1|h2|h3)*A

What is the Sphinx?

The sphinx is a hexiamond — a shape made from six equilateral triangles joined edge-to-edge. Among the twelve distinct hexiamonds it is the only one whose outline resembles the profile of a crouching sphinx, giving the tile its name. The sphinx is a concave pentagon with five sides of equal length (two triangle-edge lengths each) and internal angles of 60°, 60°, 240°, 60°, 60°, 240°.

The sphinx is a rep-4 tile: four half-size copies tile a full sphinx exactly. Three of the copies are mirror images of the original (reflected), and one is a direct copy rotated by 180°.

Because the sphinx is chiral (it is not mirror-symmetric), two versions occur in every sphinx tiling: a right-handed sphinx (R) and its left-handed mirror image (L). A right-handed sphinx decomposes into three L-copies and one R-copy. A left-handed sphinx decomposes into three R-copies and one L-copy. This is encoded in the IFS by the three maps $h_0, h_1, h_2$ (which include the reflection $r$) and the single direct-rotation map $h_3$.

The sphinx is also rep-$n^2$ for every positive integer $n$: $n^2$ half-scale copies tile the original. In particular it is rep-9 (nine copies at scale 1/3), and this self-similar subdivision uses only direct copies (no mirror images required).

IFS on the Hexagonal Lattice

The sphinx lives on the triangular (Eisenstein) lattice $\mathbb{Z}[\omega]$, where $\omega = e^{i\pi/3}$ is a primitive 6th root of unity. In the AIFS program:

• s = $\omega$ (companion matrix of $x^2 - x + 1 = 0$, 60° rotation)
• r = reflection (companion matrix of $x^2 - 1 = 0$, i.e. the exchange matrix)
• s^3 = $\omega^3 = -I$ (180° rotation)
• s^4 = $\omega^4$ (240° rotation)

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

The attractor equation is:

Sphinx Tilings: Limit-Periodicity

The sphinx rep-tile generates a family of nonperiodic sphinx tilings of the plane by iterating the substitution rule. These tilings possess a remarkable structural property first described by Godrèche (1989) and rigorously established by Lee & Moody (2001):

The sphinx tiling is limit-periodic. The tile set can be partitioned into a countably infinite collection of periodic sublattices $L_1, L_2, L_3, \ldots$, where $L_i$ has period vectors of length $2 \times 2^i$. This is analogous to the chair tiling on the square lattice.

Because it is limit-periodic, the sphinx tiling can be obtained by a cut-and-project method with a p-adic internal space, and its diffraction spectrum is pure point (all Bragg peaks, no diffuse background) — a property shared with quasicrystals.

Frame Tilings and Entropy

A sphinx-shaped frame of order $n$ (defined by the number of unit triangles at the tail) can also be tiled by sphinx tiles in non-recursive ways. The number of distinct tilings grows exponentially:

This exponential growth (Huber et al. 2024) reflects the entropy of the tiling ensemble and is related to the chirality structure: each sphinx placed inside a frame may be either left-handed or right-handed, subject to the constraint that the pieces fit together.

References

• Godrèche, C. (1989). The sphinx: a limit-periodic tiling of the plane. Journal of Physics A, 22(24): L1163–L1166.
• Lee, J.-Y. & Moody, R. V. (2001). Lattice substitution systems and model sets. Discrete & Computational Geometry, 25(2): 173–201.
• Niţică, V. (2003). Rep-tiles revisited. MASS selecta, AMS, pp. 205–217.
• Huber, G. et al. (2024). Entropy and chirality in sphinx tilings. Physical Review Research, 6: 013227.
• Sphinx tiling — Wikipedia
• Sphinx — Tilings Encyclopedia
• Sphinx — MathWorld

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