Jordan algebras over icosahedral cut-and-project quasicrystals

Abstract

In this paper we present a general setting for aperiodic Jordan algebras arising from icosahedral quasicrystals that are obtainable as model sets of a cut-and-project scheme with a convex acceptance window. In these hypothesis, we show the existence of an aperiodic Jordan algebra structure whose generators are in one-to-one correspondence with elements of the quasicrystal. Moreover, if the acceptance window enjoys a non-crystallographic symmetry arising from H2, H3 or H4 then the resulting Jordan algebra enjoys the same H2, H3 or H4 symmetry. Finally, we present as special cases some examples of Jordan algebras over a Fibonacci-chain quasicrystal, a Penrose tiling, and the Elser-Sloane quasicrystal.