Characterization of cut-and-project sets using a binary operation

Cut-and-project sets with convex acceptance windows, based on irrationaliti es tau =1/2(1+root5), beta =1+root2, mu =2+root3 are models for experimenta lly observed quasicrystals - materials with diffraction patterns consisting of sharp Bragg peaks in crystallographically disallowed patterns. We show that for each of these three irrationalities there exists a unique binary o peration of the type x proves (s)y:=sx+(1-s)y, such that one-dimensional cu t-and-project sets are precisely Delone sets closed under this operation.