Jordan algebras over icosahedral cut-and-project quasicrystals

This is the authors' abstract. We don't add key points for this paper.

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 hypotheses, 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 H 2 , H 3 or H 4 then the resulting Jordan algebra enjoys the same H 2 , H 3 or H 4 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.