Spacetime Quasicrystals

Self-similar quasicrystals (like the famous Penrose and Ammann-Beenker tilings) are exceptional geometric structures in which long-range order, quasiperiodicity, non-crystallographic orientational symmetry, and discrete scale invariance are tightly interwoven in a beautiful way. In this paper, we show how such structures may be generalized from Euclidean space to Minkowski spacetime. We construct the first examples of such Lorentzian quasicrystals (the spacetime analogues of the Penrose or Ammann-Beenker tilings), and point out key novel features of these structures (compared to their Euclidean cousins). We end with some (speculative) ideas about how such spacetime quasicrystals might relate to reality. This includes an intriguing scenario in which our infinite $(3+1)$D universe is embedded (like one of our spacetime quasicrystal examples) in a particularly symmetric $(9+1)$D torus $T^{9,1}$ (which was previously found to yield the most symmetric toroidal compactification of the superstring). We suggest how this picture might help explain the mysterious seesaw relationship $M_{\rm Pl}M_{\rm vac}\approx M_{\rm EW}^{2}$ between the Planck, vacuum energy, and electroweak scales ($M_{\rm Pl}$, $M_{\rm vac}$, $M_{\rm EW}$).

💡 Research Summary

The paper “Spacetime Quasicrystals” extends the well‑known mathematical structure of Euclidean quasicrystals—exemplified by the Penrose and Ammann‑Beenker tilings—to Minkowski spacetime. The authors begin by reviewing the essential ingredients of Euclidean quasicrystals: non‑periodicity combined with long‑range order, non‑crystallographic rotational symmetry, and discrete scale invariance. They recast the standard cut‑and‑project (C&P) construction in a “symmetric C&P” (sC&P) language that uses a highly symmetric window or weighting function, allowing the simultaneous realization of quasi‑periodicity, exotic rotational symmetry, and self‑similarity. This formalism also introduces the notions of local versus global scale invariance and of self‑dual quasicrystals, which are new even in the Euclidean context.

Having set this groundwork, the authors move to Lorentzian geometry. They define Lorentzian lattices as integer lattices equipped with an indefinite bilinear form of signature (p,q), focusing on the physically relevant (d‑1,1) case. Self‑dual Lorentzian lattices are classified, and the two unique lattices up to dimension ten—odd self‑dual I₃,₁ and even self‑dual II₉,₁—are highlighted. The reflection groups of these lattices are infinite non‑crystallographic Coxeter groups, and their associated Salem numbers provide the discrete scaling factors.

The sC&P scheme is then generalized to spacetime. Because the non‑crystallographic symmetry group is now infinite, the “window” must respect this larger symmetry, leading to a point set that does not admit a conventional tiling. Local scale invariance is lost; instead the construction can exhibit global scale invariance or self‑duality, meaning that the whole point set rescales by a fixed algebraic factor while preserving its pattern.

Two explicit families of spacetime quasicrystals are constructed. In 1+1 dimensions the authors use the odd lattice I₃,₁ to produce a self‑dual, globally scale‑invariant point set. In 3+1 dimensions they employ the even lattice II₉,₁ to generate four distinct quasicrystals (self‑dual and three variants of global scale invariance). In each case the scaling factor is a Salem number associated with the lattice’s Coxeter eigenvalues, and the point set can be described by a weighting function that selects lattice points near an irrationally sloped hyperplane in the higher‑dimensional embedding space.

The speculative physics section proposes that our observable 4‑dimensional universe could be realized as an irrationally sloped embedding of ℝ³,¹ inside the compact 10‑dimensional torus T⁹,¹ = ℝ⁹,¹ / II₉,₁, the most symmetric toroidal compactification of the superstring found by Moore. In this picture the 3+1‑dimensional slice densely fills the torus without ever intersecting itself, so an observer perceives an infinite Minkowski spacetime even though the underlying manifold is compact. The authors argue that the discrete scaling symmetry of the quasicrystal may naturally generate the observed hierarchy between the Planck scale Mₚ, the electroweak scale M_EW, and the vacuum‑energy scale M_vac, reproducing the empirical seesaw relation Mₚ M_vac ≈ M_EW². The idea is that the torus volume sets Mₚ, the global scaling factor sets M_EW, and the choice of window size (or weighting) determines M_vac, linking the three scales through the algebraic properties of the lattice.

Finally, the paper suggests broader applications: spacetime quasicrystals could provide discrete models for scale‑invariant or conformally invariant field theories, serve as a framework for discrete holography, and offer new tools for numerical simulations of quantum gravity where an underlying discrete but self‑similar structure is advantageous. Appendices collect Vinberg’s classification of reflection symmetries for self‑dual Lorentzian lattices (including corrected Coxeter‑Dynkin diagrams) and tabulate the algebraic number fields associated with the scaling factors for all self‑dual lattices up to dimension ten. Overall, the work delivers the first rigorous construction of Lorentzian quasicrystals, highlights their distinctive mathematical features, and opens a speculative but intriguing dialogue between exotic geometry and fundamental physics.