The goal of this chapter is to characterise the shortest vectors in a cylindrical lattice. This will allow us, in the next Chapter, to classify lattices by their shortest vectors in a way that has biological relevance. Cylindrical lattices have a natural representation as ‘unrolled’ plane lattices in which the cylinder circumference corresponds to a vector of the plane lattice encoding the periodicity; the height of a point in the cylindrical lattice is the component perpendicular to the periodicity vector. Any vector in the lattice has a parastichy number corresponding to the size of this component in units of node height. The principal parastichyPairs of parastichy vectorsprincipal pair vectors are the two shortest vectors in the plane lattice. We show how the the cylindrical spirals formed by these vectors—and the two corresponding parastichyParastichy number numbers—provide a natural model for biological observations of spiral counts. In order to compute these parastichy numbers in a given lattice we will consider generating pairs of vectors as the building blocks of the lattice, and carry out the main analysis of this chapter: how to deduce from the generating pairs what the lattice parameters are and, especially, what the shortest vectors in the lattice are. The central and suggestive result of this Chapter is Turing’s theorem: that the third-shortest vector in a lattice has a parastichy number which is the sum or difference of the two principal parastichy numbers.