COLORINGS OF QUASI-CRYSTALS

We introduce a notion of colouring the points of a quasicrystal analog ous to the idea of colouring or grading of the points of a lattice. Ou r results apply to quasicrystals that can be coordinatized by the ring R of integers of the quadratic number field Q(root 5) and provide a u seful and wide ranging tool for determining of sub-quasicrystals of qu asicrystals. Using the arithmetic properties of R we determine all pos sible finite colourings. As examples we discuss the 4-colours of verti ces of a Penrose tiling arising as a subset of 5-colouring of an R lat tice, and the 4-colouring of quasicrystals arising from the D-6 weight lattice.