Exceptional algebraic properties of the three quadratic irrationalities observed in quasicrystals

There are only three irrationalities directly related to experimentally obs erved quasicrystals, namely, those which appear in extensions of rational n umbers by root5, root2, root3. In this article, we demonstrate that the alg ebraically defined aperiodic point sets with precisely these three irration al numbers play an exceptional role. The exceptional role stems from the po ssibility of equivalent characterization of these point sets using one bina ry operation.