Lorentzian Quasicrystals and the Irrationality of Spacetime

Ordered structures that tile the plane in an aperiodic fashion - thus lacking translational symmetry - have long been considered in the mathematical literature. A general method for the construction of quasicrystals is known as cut-and-project (CNP for short), where an irrational slice 'cuts' a higher-dimensional space endowed with a lattice and suitably chosen lattice points are further 'projected' down onto the subspace to form the vertices of the quasicrystal. However, most of the known examples of CNP quasi-tilings are Euclidean. In this talk, after presenting the main ingredients of the Euclidean prescription, we will extend it to Lorentzian spacetimes and develop Lorentzian CNP. This will allow us to discuss the first ever examples of Lorentzian quasicrystals, one in (1+1)- and another in (1+3)-dimensional spacetimes. Finally, we will argue why the latter construction might be relevant for our beloved Minkowski spcae. In particular, we shall appreciate how the picture of a quasi-crystalline spacetime could provide a potentially new string-compactification scheme that can naturally accommodate for the hierarchy problem and the smallness of our cosmological constant. Lastly, we will comment on its relevance to quantum geometry and quantum gravity; first, as a conformal Lorentzian structure of no intrinsic scale, and second through the connection of quasicrystals to quantum error-correcting codes.