The filling law: A general framework for leaf folding and its consequences on leaf shape diversity

Leaves are packed in a bud in different ways, being flat, rolled, or folded, but always filling the whole bud volume. This “filling law” has many consequences, in particular on the shapes of growing folded leaves. This is shown here for different types of folding and packing. The folded volume is roughly a part of an ellipsoid, with the veins on the outside rounded face and the lamina margin on the adaxial plane. The veins on the abaxial side protect the fragile lamina inside. The first general consequence of the folds and the space limitation of the lamina growth is the presence of symmetries on the leaf shape, and the second is the quantitative relationships between the sizes of the lobes and sinuses. For particular geometries, the leaf lamina can be limited by lateral veins, creating spoon-like lobes, or tangent cuts, creating asymmetrical wavy perimeters. Changes in the packing between different cultivars correspond to changes in the mature leaf shapes. Each particular case shows how pervasive the geometrical consequences of the filling law are.

► Leaf shape seems to be the consequence of filling the bud with a certain folding.
► Fig tree and beech leaves solve two non-zero thickness “fold-and-cut" problems.
► Reiteration and confinement are two important ingredients of leaf development.