Exceptional algebraic properties of the three quadratic irrationalities observed in quasicrystals

There are only three irrationalities directly related to experimentally observed quasicrystals, namely, those which appear in extensions of rational numbers by ?5, ?2, ?3. In this article, we demonstrate that the algebraically defined aperiodic point sets with precisely these three irrational numbers play an exceptional role. The exceptional role stems from the possibility of equivalent characterization of these point sets using one binary operation.