The filling law: a general framework for leaf shape diversity and its consequences on folded leaves

TL;DR

This paper introduces the filling law, a framework explaining how leaf shapes and folding patterns are constrained by volume filling in buds, revealing geometric relationships and symmetries across diverse leaf types.

Contribution

It provides a general geometric framework linking leaf folding, packing, and shape diversity, with implications for understanding leaf morphology.

Findings

Folded leaf volume fits within an ellipsoid shape.

Symmetries and quantitative relationships in leaf shape are derived.

Variations in packing influence leaf shape across cultivars.

Abstract

Leaves are packed in a bud in different ways, being flat, enrolled, or folded, but always filling the whole bud volume. This {\guillemotleft} filling law {\guillemotright} has many consequences, in particular on the shape of growing folded leaves. This is shown here for different types of folding and packing. The folded volume is always part of an ellipsoid, with the veins on the outside rounded face, and the lamina margin on an adaxial plane or axis. The veins on the abaxial side protect the more fragile lamina. The first general consequence of the folds is the presence of symmetries on the leaf shape, but also quantitative relationships between lobes and sinus sizes. For particular geometries, the leaf lamina can be limited by lateral veins, creating spoon-like lobes, or transverse cuts, creating asymmetrical wavy perimeters. A change in the packing between cultivars induces the corresponding change in the leaf shape. Each particular case shows how pervasive are the geometrical consequences of the filling law.