It's fall. The curious look at the dead leaves, dissect them and discover the complex networks formed by their veins. These veins allow the leaves to resist the wind and the weight of visiting insects.
In this image, a leaf of the pagoda fig, a species of tree in the genus Ficus . We see a thick and solid vein, which constitutes the central axis of the leaf, from which branch out secondary veins parallel to each other. Together, they serve as the main mechanical reinforcement.
Between the secondary veins, finer veins have a large number of connections, and loop at the edges of the leaf. Because in addition to their role of mechanical support, the ribs allow the transport of water and the products of photosynthesis, sugars in particular, such as glucose. They are interconnected: their network can serve as a diversion which makes it possible to maintain the irrigation of the leaf as a whole, if the leaf is damaged locally, for example by insect attacks.
The hierarchy of veins of different sizes corresponding to successive formations during the growth of the leaf. By observing a region of the leaf more precisely, regularities appear in the morphology. The density of the ribs thus appears to be spatially uniform. This uniformity exists at all stages of leaf development.
Lily of the valley, very different from Ficus
There are two main classes of flowering plants (called angiosperms): monocots and dicotyledons. They are particularly distinguished at the level of the seed, which has two “cotyledons” (leaf embryo) for dicotyledons (apple trees, sunflowers, legumes, rosaceae, etc.) and only one for monocotyledons (orchids, palm trees, grasses, etc.).
They are also distinguished by the veins of their leaves. Dicotyledons show a hierarchical architecture, like that visible in the photo.
Monocots – like lily of the valley – generally have one main vein which forms the axis of the leaf, while the other veins line up parallel to the first. As you approach the tip, the space for the ribs becomes narrower, forcing some of the ribs to stop. In this case, when a rib stops growing, it connects to its closest neighbors by forming a small curve.
Open research questions about ribs and cracks
There is a striking visual similarity between leaf vein morphologies and the figures formed by cracks in thin layers, typical of ceramics or glazes. This geometric similarity made it possible to suggest similar formation mechanisms between ribs and cracks .
Indeed, cracks form according to well-defined rules: their propagation is guided by mechanical constraints, for example when paints age . We study their formation with model materials, which contain well-calibrated pigments, which we dry in a controlled manner to study the solidification phase during which cracks form. The formation of cracks in a drying layer is a phenomenon easily observed on a time and space scale that can be adapted depending on the environment and drying. This is why the formation of cracks constitutes a means of study allowing us to better understand certain spatial networks.
The analysis of spatial networks, whether they are formed of interconnected lines or not, and the understanding of their formation dynamics remain open research questions, initiated by Yves Couder .
Indeed, crack structures are not exclusive to solid materials such as paints, but are also found in natural environments such as in dry clay soils. The formation of cracks opens up new surfaces in the environment, particularly at depth. The presence of these surfaces thus promotes the penetration of rainwater into the pores of the soil .
These patterns are also observed on a larger scale in urban areas . Thus, street networks appeared through successive divisions of agricultural areas into polygonal areas , and the borders of these areas became routes (paths, streets, etc.) which interconnected in order to best ensure transport.
The formation of a crack network can be demonstrated simply during the drying of a layer of cornstarch or clay . Cracks propagate in the layer relieving mechanical stresses and create complex interconnected hierarchical networks. 
Ludovic Pauchard , CNRS Researcher at the FAST laboratory (Fluids, Automation and Thermal Systems), Paris-Saclay University
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