Rudimentary quasicrystallography: The icosahedral and decagonal reciprocal lattices

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Abstract
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.

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