Packing theory derived from phyllotaxis and products of linear forms 0

TL;DR

This paper develops a generalized mathematical framework for packing points on Riemannian manifolds, inspired by natural spiral patterns, and demonstrates its effectiveness in various dimensions with specific density bounds.

Contribution

It introduces a novel approach combining Markoff theory and linear forms to derive packing patterns on general Riemannian manifolds, including new spiral packings in 2D and 3D.

Findings

New spiral packings in 2D and 3D spaces.

Almost uniform point distribution on real analytic Riemannian surfaces.

Lower bounds on packing density: ~0.7 for surfaces and 0.38 for 3-manifolds.

Abstract

\textit{Parastichies} are spiral patterns observed in plants and numerical patterns generated using golden angle method. We generalize this method by using Markoff theory and the theory of product of linear forms, to obtain a theory for packing of Riemannian manifolds of general dimensions $n$ with a locally diagonalizable metric, including the Euclidean spaces. For example, packings in a plane with logarithmic spirals and in a 3D ball (3D analogue of the Vogel spiral) are newly obtained. Using this method, we prove that it is possible to generate almost uniformly distributed point sets on any real analytic Riemannian surfaces in a local sense. We also discuss how to extend the packing to the whole manifold in some special cases including 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.