# Symmetry

42 43 symmetries in chAos regularities in highly complex systems Invariance equates with symmetry, so on the face of it, turbulence, which is the very image of a totally disturbed system, would appear to be an unlikely candidate for symmetries of any kind. The physics of turbulent systems was for a long time one of sciences most intractable problems, it is still not completely understood, but the recognition of the role of strange attractors in the process has brought new insights and a new mathematical instrument to bear on such complex systems. The cryptic geometry of strange attractors was part of the new nonlinear maths of Chaos theories the revolution in which fractals first appeared. It involves the concept of viewing dynamical systems as occupying geometrical space, the coordinates of which are derived from the systems variables. In linear systems the geometry within this phase space is simple, a point or a regular curve in nonlinear systems it involves far more complex shapes, the strange attractors. One of the most famous of these is the Lorenz attractor 1,2, which forms the basis of chaotic models of weather prediction including IceAges. Another classic example is the dripping tap experiment 3 where beautiful regular forms are found within apparent randomness. As we have seen, Fractal geometry is intrinsic to many aspects of Chaos theoryand fractals are, predictably, firmly associated with attractors. In fact all strange attractors are fractal, as is Feigenbaum mapping, which is a sort of master attractor. The Feigenbaum number which lies at the heart of this mapping, predicts the complex, perioddoubling values across a whole range of nonlinear phenomena, including turbulence 4. The Feigenbaum value is recursive, and appears whenever there is repeated period doubling. It is, in short, a universal constant like pi or phi, and has a similar symmetrical potency. 1. The Lorenz attractor displays two symmetrical states, between which it occasionally flips 4. A dynamical system bifurcation diagram demonstrating the presence of the fractal Feigenbaum constant 2. A weaker Lorenz attractor produces a more complex zone of probabilities 3. The times between successive drips from a tap, plotted as x, y, and z, form a strange attractor in 3dimensional phase space