Number-theoretic properties of two-dimensional lattices (Q1109803)

Phyllotaxis is that area of botany concerned with the genesis of the patterns of primordia (e.g., leaves, florets, scales, seedlings) observed on plants and buds, as on the sunflower and pineapple. This article is dedicated to the brothers L. and A. Bravais, botanist and mathematician, who initiated the lattice treatment of phyollotaxis in 1837. Suppose the observed points are consecutively labelled in a regular fashion, and projected onto the plane. These points can then be transformed mathematically to points of a lattice, where the eye-catching spirals become straight lines. Joining any two points x and y of the lattice determines a direction and thus a family of \(m=| x-y|\) parallel and equidistant lines partitioning the points of the lattice. A pair of such families containing m and n lines, respectively, is denoted by (m,n). The pair (m,n) is called opposed when the slopes are opposite, and is called visible when there is a lattice point at the intersection of every pair of lines. Let the lattice have divergence d, \(0<d<\), and rise r (so the first point has coordinates (d,r)), and let (x) denote the nearest integer to x. The author proves eight propositions, including: The pair (m,n) is visible iff \(| m(nd)-n(md)| =1;\) The pair (m,n) is visible and opposed iff there exist unique integers \(0<v<n\), \(0<u<m\) such that \(| mu-nv | =1\) and d is in the closed interval with endpoints u/m, v/n. He then illustrates with some botanical examples.