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Symmetry

40 41 selforGAnisinG symmetries regularities in nonlinear systems There are many natural patterns that present more subtle regularities than the highly ordered symmetries of crystals. Some of these are generated by quite simple rules, others by a complex of factors many result from some form or other of selforganisation. These Li opposite express a certain universality, their symmetries tend to be thematic and fluid rather than rigid and static. The simple ripple patterns on a sea shore, for instance, are created by a multiplicity of contributory factors, including tides, currents and windsnot to mention the more general effects of gravity and warmth from the sun. All of these are drawn into a selforganising, selflimiting order whose charm lies precisely in the fact that it is repetitive, yet infinitely variable. Rivers are also selforganising. Whether they are a gentle stream or a broad torrent they tend to follow similar meandering paths. There is an invariant quality in these loops and bends that conform to welldefined mathematical parameters. Similar constraints govern the hierarchical patterns of riverdrainage. Rivers shape the terrain that they flow through, and are in turn shaped by it, but there are many subtle factors that limit and influence their form. Scaleinvariant symmetries also appear in fracture patterns of the kind found in mud cracks and ceramic crackleglaze. Formations of this sort usually appear as a result of stresses induced by shrinkage. There are variations in the modes of cracking in different materials and in different conditions, but all are characterised by an overall consistency, and many have scaling properties. They are formed, and limited, by the release of stress, so they are progressive and selforganisingand of course they tend to be fractalline in nature.
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