Packing Theory Derived from Phyllotaxis and Products of Linear Forms

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 diagonal...

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Bibliographic Details
Published in:Constructive approximation 2024-12, Vol.60 (3), p.515-545
Main Authors: Graiff Zurita, S. E., Oishi-Tomiyasu, R.
Format: Article
Language:English
Subjects:
Analysis
Euclidean geometry
Mathematics
Mathematics and Statistics
Numerical Analysis
Packing density
Riemann manifold
Riemann surfaces
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Summary: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.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-024-09691-3