Introduction

General aspects

A spatially extended dynamical system exhibits geometrical patterns when its state variables show spatial coherence, i.e., their spatial distribution is neither uniform nor random but exhibits organized spatial structures characterized by a certain regularity. The occurrence of patterns is an important feature of spatiotemporal systems, as it is a signature of the existence of order embedded in the underlying dynamics. This fact, along with the beauty of some of their geometrical features, explains why patterns have always drawn the attention of the science community.

Patterns are in fact ubiquitous in nature. In particular, a number of environmental processes are known for their ability to develop highly organized spatial features (e.g., Greig-Smith, 1979). For example, remarkable degrees of organization can be found in the spatial distribution of convective clouds (e.g., Lovejoy, 1982), dryland and riparian vegetation (e.g., Macfadyen, 1950), sand ripples and dunes (e.g., Lancaster, 1995), river channels (e.g., Allen, 1984; Ikeda and Parker, 1989; Rodriguez-Iturbe and Rinaldo, 2001), soil cracks (Lachenbruch, 1961), coastlines (Bird, 2000; Ashton et al., 2001), peatlands (Eppinga et al., 2008, e.g.), and arctic hummocks and patterned ground (Gleason et al., 1986).

These features are often undetectable on the ground, but have become visible with the advent of aerial photography (e.g., Macfadyen, 1950). Figures 5.1, 5.2, 5.3, 5.4, and 5.5 show some examples of spectacular spatially periodic natural patterns in a number of landscapes around the world.