Branching in Nature

Branching is probably the most common mode of growth in Nature. From plants to river networks, from lung and kidney to snow-flakes or lightning sparks, branches grow and blossom everywhere, in every realm of Nature. When Galileo Galilei stated that the geometry ofNature was written in terms ofplanes, cones and spheres, he missed one essential pattern of Nature: the tree. However, the tree has been recognized as a "scientific" objet very early, ever since the classical times. Pliny, Strabo or Theophrastus were weIl aware ofthe existence of"dendrites", i. e. , stones in the shape of plants or corals, although the existence of such stones was a puzzle to them. In his book Prodromus Cristallographiae, in which the very word cristallography appears in print for the first time (1711), Mauricius CapeIler, a Swiss naturalist, shows in between facetted crystals, several examples of dendritic crystals. He seemed already to believe in the existence of a general class of branching structures (arbusculatum in modum) in Nature. In the same spirit, his friend Jean-Jacques Scheuchzer had demonstrated experimentally that viscous fingering could generate geological dendrites (1699). At the time of Renaissance, Leornardo da Vinci was quite interested in the resemblance between trees and vessels (1508), and later, Nicola Steno did not hesitate in considering branched deposits of silver found in mines as a close relative of snow.

Includes supplementary material: sn.pub/extras

Résumé
Les Houches School, October 11-15, 1999

Contenu
Patterns with Open Branches or Closed Networks: Growth in Scalar or Tensorial Fields.- Plants.- I Branching in Plants.- II Inside the Buds: The Meristems.- Contributed Paper 1 Two Symmetries Linking Biological and Physical Branching Morphogenesis.- III Establishing a Growth Axis in Fucoid Algae.- Contributed Paper 2 Shape Stability during Osmotic Growth.- Contributed Paper 3 On Transcellular Ionic Currents.- Rivers.- IV Branched Patterns in Geology: Rivers and Other Systems.- Neurons.- V Neuronal Arborization.- Contributed Paper 4 Chemical Waves and Dendrites Navigation during Self-Wiring of Neural Nets.- Branching Organs.- VI The Mouse Embryonic Lung: A Biological Example of Branching Morphogenesis.- VII Branched Structures, Acinus Morphology and Optimal Design of Mammalian Lungs.- Contributed Paper 5 Quantitative Studies of Branching Morphogenesis in the Developing Kidney.- Contributed Paper 6 Morphogenic Responses of Mammary Epithelial Cells Grown in Biological Semi-Solid Substrates.- Vessels.- VIII Vascular Development: Design Principles and Morphometric Analysis of a Branched Vascular Tree.- IX Mechanics of the Large Artery Vascular Wall.- Contributed Paper 7 A Link Between Dendritic Growth and Remodeling of Blood Vessels.- Bacteria.- X Adaptive Branching During Colonial Development of Lubricating Bacteria.- Contributed Paper 8 Pattern Formation Modeling of Bacterial Colonies.- Crystals.- XI Dendritic Growth.- Contributed Paper 9 Sidebranching in Solutal Dendritic Growth.- Contributed Paper 10 Experimental Study of Sidebranching in Directional Solidification.- Fluids.- XII Branching during Dewetting and Wetting.- Contributed Paper 11 Viscous Fingering in a Gel.- Contributed Paper 12 Branching Transition in Viscous Fingering with a Liquid Crystal.- Mathematics.- Contributed Paper 13 Mathematical Meristems: The Singularities of Laplacian Growth.- List of Participants.- Color Plates.