Packing Theory Derived from Phyllotaxis and Products of Linear Forms

Abstract

Parastichies are spiral patterns observed in plants and numerical patterns generated using golden angle method. We generalize this method using Markoff theory and the theory of product of linear forms, to obtain a packing theory on Riemannian manifolds of general dimensions n with a locally diagonalizable metric, including the Euclidean spaces. From the theory, point packings on a plane with logarithmic spirals and on a 3D ball (3D analogue of the Vogel spiral) are newly obtained. We prove that the method is applicable to generate almost uniformly distributed point sets on any smooth Riemannian surfaces in a local sense. We also discuss how to extend it to a global packing in some special cases including the case of packing on a disc such as the Vogel spiral. The packing density is bounded below by approximately 0.7 for surfaces and 0.38 for 3-manifolds under the most general assumption.