Systematic variations in divergence angle

Takuya Okabe

doi:10.1016/j.jtbi.2012.08.007

Abstract

Practical methods for quantitative analysis of radial and angular coordinates of leafy organs of vascular plants are presented and applied to published phyllotactic patterns of various real systems from young leaves on a shoot tip to florets on a flower head. The constancy of divergence angle is borne out with accuracy of less than a degree. It is shown that apparent fluctuations in divergence angle are in large part systematic variations caused by the invalid assumption of a fixed center and/or by secondary deformations, while random fluctuations are of minor importance.

Highlights

It has long since been recognized that divergence angles between successive leafy organs of vascular plants are accurately regulated at one of special constant values
The center of parastichy is not fixed in space, but it may float around in the pattern as primordia arise one after another
The floating center can be the main source of apparent variations of divergence angle

Summary

Introduction

It has long since been recognized that divergence angles between successive leafy organs of vascular plants are accurately regulated at one of special constant values. Deviations from the the constant angle are normally so small that this botanical phenomenon, phyllotaxis, should rather be regarded as a genuine subject of exact science. In a polar coordinate system, position of the n-th leaf is specified by the radial and angular coordinates (rn, θn). The angular regularity is expressed by the equation θn = nd, (1). A spiral pattern is made when the radial component rn is a monotonic function of n, i.e., rn preserves the order of the leaf index n. Important is a logarithmic spiral given by rn = an,

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