Growth, geometry, and mechanics of a blooming lily

Abstract

Despite the common use of the blooming metaphor, its floral inspiration remains poorly understood. Here we study the physical process of blooming in the asiatic lily Lilium casablanca. Our observations show that the edges of the petals wrinkle as the flower opens, suggesting that differential growth drives the deployment of these laminar shell-like structures. We use a combination of surgical manipulations and quantitative measurements to confirm this hypothesis and provide a simple theory for this change in the shape of a doubly curved thin elastic shell subject to differential growth across its planform. Our experiments and theory overturn previous hypotheses that suggest that blooming is driven by differential growth of the inner layer of the petals and in the midrib by providing a qualitatively different paradigm that highlights the role of edge growth. This functional morphology suggests new biomimetic designs for deployable structures using boundary or edge actuation rather than the usual bulk or surface actuation.

Fig. 1.

Observations of and experiments on blooming in the asiatic lily Lilium casablanca. (A) A young green lily bud. The black dots separated by 1 cm allow us to measure growth strains. (B) The cross-section of a lily bud. (C) A typical opening sequence of a lily flower over a period of 4.5 days. The black line is the profile in the bud state, the transparently light blue shows the half-open state, and the white one is the fully open state.

Fig. 2.

Anatomy of the lily bud and the role of midrib. (A) The composite structure of a petal midrib: the Left panel shows a single petal; the Center panel shows the grooved structure of the midrib; the Right panel shows that when the leafy part (gray) is peeled away, the woody part straightens out, a sign that there is some relative growth between the two. (B) When the midribs are removed from a petal and a sepal, the flower can still bloom normally, with a slightly different curvature relative to the pristine petals/sepals. (C) The inner petals have rippled edges in the bud, showing clearly that their edges are growing relative to the rest of the tissue.

Fig. 3.

Experimental measurement of differential growth and numerical simulation in a single petal. (A) Longitudinal growth strain along the midrib and the edges varies in the lateral (y) direction. The edge growth strain is averaged over 6 sepals, and the midrib growth strain is averaged over 10 petals/sepals. This lateral growth gradient is sufficient to drive blooming. (B) Simulation of the blooming process in a single elliptical petal that is originally a convex spherical shell. As the edge-growth strain increases (see text for details), the curvature of the petal first reverses; i.e., it blooms. and then edge-localized ripples arise. The order of blooming and rippling can be reversed by changing the relative distribution of growth strains as can be seen in the inner and outer petals and sepals that follow opposite paths.

Fig. 4.

A simple theory for the reversal of an elliptical shell. (A) A parabolic lateral growth strain is prescribed on an elliptical concave shell with a lenticular cross-section along both the x and y axes and quadratically varying thickness (see text). (B) As the differential growth β_g increases from left to right, the shell reverses curvature; i.e., it blooms. (C) For an initially spherical shell with , the elastic strain β as function of β_g has the signature of a pitchfork bifurcation. (D) The rescaled curvatures and as function of growth strain β_g, and . (Notation: S-stable, U-unstable). (E) The critical point denoting the transition from stretch-dominated to bend-dominated [denoted by × in (D)] as a function of the bend-stretch coupling parameter α for different , and the dotted line follows the analytic solution for the spherical case with m = 1. The parameters are chosen to correspond to the dimensions of a typical lily petal: a = 3, b = 1, t₀ = 0.1, ν = 0.3.