Rudimentary quasicrystallography: The icosahedral and decagonal reciprocal lattices

We prove that there are precisely three distinct icosahedrally symmetric lattices in three dimensions that are integral linear combinations of six vectors. By lattice, we mean a set of vectors which is closed under addition and subtraction. These three lattices can be represented as projections of the six-dimensional simple, face-centered, and body-centered hypercubic lattices into three dimensions. We also show that there is only one distinct three-dimensional decagonal lattice that is integrally spanned by five vectors.