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In this paper we present a general setting for aperiodic Jordan algebrasarising from icosahedral quasicrystals that are obtainable as model sets of acut-and-project scheme with a convex acceptance window. In these hypothesis, weshow the existence of an aperiodic Jordan algebra structure whose generatorsare in one-to-one correspondence with elements of the quasicrystal. Moreover,if the acceptance window enjoys a non-crystallographic symmetry arising fromH2, H3 or H4 then the resulting Jordan algebra enjoys the same H2, H3 or H4symmetry. Finally, we present as special cases some examples of Jordan algebrasover a Fibonacci-chain quasicrystal, a Penrose tiling, and the Elser-Sloanequasicrystal.

Jordan algebras over icosahedral cut-and-project quasicrystals