Abstract
An algebraic binary operation is introduced into quasicrystals admitting five-fold symmetry. In terms of this many quasicrystals displaying full pentagonal or icosahedral symmetry are seen to be finitely generated. Examples are given in dimensions 1, 2, 3 and 4. The operation of left quasicrystal addition is affine-linear. The monoid generated by these operators is discussed and a presentation for it given in the generic case.