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Packing theory derived from phyllotaxis and products of linear forms 0

S. E. Graiff Zurita, B. Kane, R. Oishi-Tomiyasu

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.

Packing theory derived from phyllotaxis and products of linear forms 0

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 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.

Paper Structure

This paper contains 6 sections, 7 theorems, 100 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Parastichies from a viewpoint of lattice-basis reduction and Markoff theory
  3. Lattice maps for packing of the Euclidean spaces
  4. System of PDEs and solutions provided by inviscid Burgers equation
  5. A family of solutions obtained by separation of variables
  6. General case: packing of Riemannian manifolds

Key Result

Proposition 1

For any $\varphi_1 > 0 > \varphi_2$ and $\epsilon > 0$, suppose that a lattice $L_{\varphi_1, \varphi_2, \epsilon}$ and its basis vectors ${\mathbf b}_n$ ($n \in {\mathbb Z}$) are defined as above. Let $N$ be the integer that fulfills Let $1 \le d \le a_{N-1}$ be the smallest integer that satisfies In this case, the following $u_1, u_2, u_3$ are a Selling reduced superbase of $L_{\varphi_1, \var

Figures (7)

  • Figure 1: Left: Vogel spiral $\sqrt{n} e^{2 \pi i n /(1+\gamma_1)}$ ($n > 0$: integer) and images of the lattice shortest vectors (arrows) that indicate the directions of parastichies, Right: Doyle spiral of type $(12, 24)$ dealt with in Example \ref{['exa: Doyle spiral']}.
  • Figure 2: Original Vogel spiral (left) and a packing obtained from the lattice basis $B_2$ in Theorem \ref{['thm: theorem 1']} (right). The lattice map $f$ to use is explained in Example \ref{['exa: case of (a) and (b)']}.
  • Figure 3: Left: hexagonal lattice packing in ${\mathbb R}^2$, Right: radius ratios of adjacent circles in the Doyle spiral
  • Figure 4: Packings in a disk obtained from the lattice bases (i)--(iii) of Example \ref{['exa: case of (a) and (b)']}. The parameter $s$ is set to (i), (ii) $s = 2 q_9^{(-)} + (\varphi + \overline{\varphi}) p_9^{(-)} = 47$, where $p_9^{(-)} = -21$, $q_9^{(-)} = 55$ are the ninth convergent of $-1/\overline{\varphi} = (-3+\sqrt{5})/2$, and (iii) $s = -p_{11}^{(-)} = 55$. As for (ii), the case of $s = 45 \ne 2 q_n^{(-)} + (\varphi + \overline{\varphi}) p_n^{(-)}$ ($n \in {\mathbb Z}_{>0}$) is also presented. The arrow indicates that the spirals are not smoothly connected around the $x>0$ part of the $x$-axis.
  • Figure 5: Packing of planes with logarithmic spirals. Each point is colored according to the $y$-value (birth time) of its preimage (cf. Eq.(\ref{['eq: f in case of (c) 2']})). The points with the same $y$-value form the identical shape, regardless of $0 < y \le M$. This self-similarity explains their biological shapes. The last $s = e^{2\pi}$ is also the case of an inviscid Burgers solution.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • Remark 1
  • Remark 2
  • ...and 18 more