Leaves, Flowers and Garbage Bags: Making Waves Rippled fractal patterns on thin plastic sheets and biological membranes offer elegant examples of the spontaneous breaking of symmetry

the world’s most durable mysteries. Some patterns—clouds, snowflakes— form in space. Others—the ebb and flow of tides, seasonal wet and dry spells—are patterns that form in time. Natural patterns are mysterious because they are complex, organized and interconnected, even though the laws of physics on which they rest—Newton’s classical laws of motion—are simple. The living world presents the ultimate examples of pattern formation. The patterns in biological systems are the most stunningly complex of any we encounter. Consider: In order to form a complex organism from an initial featureless collection of identical cells, a system must undergo myriad transitions that break its spatial symmetries and trigger the differentiation of cells at selected sites. How are these sites selected? How complex and controlled must a growth process be to direct that particular things happen in sequence and at the right sites? It is difficult to imagine how the impersonal interactions of atoms can lead to the growth of a plant or an animal from inanimate matter. Yet in fact this is what happens with the birth and development of every living thing. Some of the simplest features of biological shapes can be explained by basic physical laws. We will describe here an elegant example: the edges of flowers and leaves, where complex rippled shapes give the impressions of ruffles and frills. We suspected that very simple growth processes might provide the mechanism that shapes thin membranes and sheets into complex shapes in space, and indeed we have found that they do. By themselves, these processes do not break