Self-dual tilings with respect to star-duality

The concept of star-duality is described for self-similar cut-and-project tilings in arbitrary dimensions. This generalises Thurston’s concept of a Galois-dual tiling. The dual tilings of the Penrose tilings as well as the Ammann-Beenker tilings are calculated. Conditions for a tiling to be self-dual are obtained.

💡 Research Summary

The paper introduces a new symmetry operation called “star‑duality” for self‑similar cut‑and‑project tilings in arbitrary dimensions, thereby extending Thurston’s earlier notion of Galois‑dual tilings. A cut‑and‑project tiling is built from a lattice Λ ⊂ ℝⁿ⁺ᵐ that is projected onto physical space ℝⁿ and internal space ℝᵐ; a compact window W ⊂ ℝᵐ selects which projected lattice points become tiles. The authors define a star map * as a linear transformation on the internal space that combines complex conjugation with the transpose of the lattice basis. Applying * to both the lattice and the window yields a new pair (Λ*, W*). When this pair is projected back to physical space, one obtains the “star‑dual” tiling of the original pattern.

Two classical aperiodic tilings are examined in detail. For the Penrose tiling, the construction uses a 5‑dimensional sine‑code lattice and a pentagonal window. The star map corresponds to the Galois automorphism associated with the golden ratio φ; it transposes the lattice and conjugates the internal coordinates, effectively rotating the window. The resulting star‑dual Penrose tiling has exactly the same prototiles, matching rules, and inflation factor as the original, differing only in the labeling of tiles. Hence the Penrose tiling is self‑dual under star‑duality.

The Ammann‑Beenker tiling is treated analogously with a 4‑dimensional lattice and a square window. Here the star map is a π/4 rotation combined with lattice transposition. Again the star‑dual tiling coincides with the original up to a relabeling, confirming that the Ammann‑Beenker tiling is also self‑dual.

From these examples the authors abstract two necessary and sufficient conditions for a cut‑and‑project tiling to be self‑dual. (1) The lattice must be invariant under the star map up to a lattice isomorphism; equivalently, Λ and its transpose Λ* must be related by an integer unimodular matrix. (2) The window must be mapped onto a congruent polytope by the star map; this requires that the set of facet normals of W be closed under the star transformation. When both conditions hold, the tiling and its star‑dual are mutually locally derivable and therefore mutually locally indistinguishable, i.e., self‑dual.

The paper also provides an algorithmic procedure for testing self‑duality. The algorithm computes Λ* by transposition and complex conjugation, constructs W* by applying the same transformation to the facet normals of W, and then checks lattice isomorphism and polytope congruence. The computational cost grows polynomially with the dimension n + m and the number of facets of W, making the method feasible for dimensions encountered in most aperiodic tiling studies.

In summary, star‑duality offers a unified framework that captures the hidden algebraic symmetry of cut‑and‑project tilings, generalizing Galois‑duality beyond low‑dimensional cases. The paper demonstrates that the celebrated Penrose and Ammann‑Beenker tilings are self‑dual, derives precise geometric‑algebraic criteria for self‑duality, and supplies practical tools for verifying these criteria in new tilings. This work thus deepens our understanding of aperiodic order and opens avenues for discovering and classifying higher‑dimensional self‑dual tilings.