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.mechanics | curvature reversal P lant and fungal movements, although limited by the absence of dedicated motile elements such as muscles, are nevertheless varied. They span a range of length and time scales, from the rapid movements in microscopic nematophagous fungi (1), to the slow opening and closing of leaf stomata. A striking example of rapid movements is seen in the Venus flytrap, which snaps in a fraction of a second as a result of an actively controlled release of turgor across the thickness of the leaf, switching its curvature (2). This and other examples of rapid movements in plants and fungi usually involve slender geometries and may be quantified in terms of the motion of water flow in a soft porous solid (3). Morphological transitions that guide cell,tissue, leaf, branch, root, and stem shape, on the other hand, are much slower and are driven by differential growth. Here we consider a ubiquitous example of plant movement driven by differential growth, the blooming of a flower, the inspiration for much art and poetry but surprisingly little science.When a flower blossoms, its petals change curvature on a time scale of a few hours to days, consistent with the idea that these movements are growth driven. In flowers that bloom just once, differential cell proliferation is the dominant mode of growth, whereas in those that open and close repeatedly, cell elongation plays an important role (4). Frequently proposed explanations for petal movements posit a difference in growth rate between the abaxial and adaxial sides (surfaces) of a petal or an active role for the midribs (4, 5). However many flower petals have rippled edges, and this raises another qualitatively different possibility that we will explore here, namely that it is possible to change the shape of a lamina via excess growth of the margins relative to the center. This mechanism of differential planar growth has recently been shown to both explain the morphology of saddle and ripple shaped leaves and algal blades (6-10) and to e...

Curvature-Induced Instabilities of Shells

“…Instabilities and shape changes are also fundamental during the development and morphogenesis of thin tissue [3,4]. To control and evolve shape, Nature heavily relies on internal stimuli such as growth, swelling, or active stresses [5,6]. If the stimulus varies through the thickness of the shell, it generally induces a change of the spontaneous (or natural) curvature of the tissue [7].…”

Procedural techniques for simulating the growth of plant leaves and adapting venation patterns

“…A long flat lamina deforms to a saddle shape and/or develops undulations that may lead to strongly localized ripples as the growth strain is localized to the contour of the leaf. Haiyi et al [Liang and Mahadevan 2011] use a combination of surgical manipulations and quantitative measurements to confirm this hypothesis and provide a simple theory for changes in the shape of a doubly curved thin elastic shell subject to differential growth across its plan-form. This functional morphology suggests new bio-mimetic designs for deployable structures using boundary or edge actuation rather than the usual bulk or surface actuation.…”

Pine cone scale-inspired motile origami

“…The octagonal shape and the parallel alignment of fibers lead to the formation of a concave structure and to the folding motion of the pine cone scales. The curvature of the petals also has an important role in folding and blooming of flowers 36,37 . In addition, octagonal or rounded petals have lines that are similar to the fibers in pine cone scales and may influence flower blooming.…”