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This page last updated 23 February 2003

 


More about the Mandelbrot Set

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The intricate non-repeating moss-like pattern used in the Society's house style is a representation of part of the Mandelbrot Set, which has been described as 'the most complex object in mathematics', named after the mathematician Benoit Mandelbrot. The Mandelbrot Set encapsulates aspects of recent developments in the study of Fractals and Chaos that have revolutionised many areas of study, including biology, economics and meteorology. It also lends itself to the generation of beautiful coloured pictures of which this is one example.

The significance of the set is not immediately apparent from the mathematics behind its generation. It is introduced as a mapping onto the complex plane, where each point corresponds to a complex number (x + yi) with x and y the respective co-ordinates and i the square root of -1. When the complex number representing a point is subjected to a certain repeated transformation, the resulting value may diverge to infinity or may remain finite. The condition for a point's membership of the Mandelbrot Set is that the value remains finite, and a monochrome version of the representation of the set is formed by making the corresponding pixel black if the point belongs to the set, or white otherwise.

The graphic representation can be generated by a computer program that is simple, but requires a substantial amount of time to run since it must operate sequentially on each pixel. It also has to repeat a simple operation on each complex number until the behaviour at infinity can be inferred. An upper limit to the number of iterations is assumed. Coloured versions of the representation are formed by assigning colours other than white to points that do not belong to the set, the choice of colour being a function (usually following a spectral sequence) of the number of iterations performed before divergence becomes apparent. Ways of reducing the computational complexity have been devised, exploiting symmetries of the mapping.

The standard way of deciding whether or not a point belongs to the set is simple in principle. Where c is the complex number representing the point, a complex value z is initially set equal to c and is then repeatedly transformed according to zi+1 = zi² + c, where the operation of squaring is according to the rules of complex arithmetic. The membership of the point corresponding to c depends on whether the value of z remains finite or diverges.

The Mandelbrot Set has an interesting relationship to an earlier type of set also allowing graphic representation, termed 'Julia Sets' after the mathematician Gaston Julia. A Julia Set depends on an iteration exactly like that for the Mandelbrot Set except that the initial value of z is the complex number representing the point whose membership is to be tested, and c is a parameter of the set. The Mandelbrot Set can be seen as a kind of catalogue or 'road map' of Julia Sets, since a Julia Set is a connected or fragmented pattern according to whether or not its parameter c belongs to the Mandelbrot Set.

The relevance of the Mandelbrot Set to cybernetics falls under two related headings. One is that the image of the set is fractal, in that it has a similar appearance no matter how much it is enlarged. In any graphic representation, finite resolution requires that finer details are rounded off, and there is no scale at which there will be no rounding-off. A map of an island is arguably fractal since it shows a fairly smooth coastline, with headlands and bays, although an enlarged picture would show that these contained their own smaller headlands and bays, eventually down to the boundaries of individual pebbles or grains of sand and then the rugosities of these. The fractal nature of the Mandelbrot Set can be verified by computer generation of the figure with successively greater magnification, and this has been taken to extremes where extra precision of numerical representation is necessary and the enlargement is comparable to that which would reveal individual molecules of a solid object.

Fractals are important as models of natural processes, including those of geology as illustrated by the reference to a coastline, and particularly those of biology. Many living organisms have fractal character, illustrated by the readiness with which fractal processes produce compelling representations of them, especially ferns and other plants.

Even more significant is the relevance of these fractal patterns to modern Chaos Theory. This is the study of non-linear systems whose behaviour is in principle unpredictable even though that of all their parts may be well understood and deterministic. The study is sometimes denoted as Complexity Theory, and usually this is appropriate, though it can also be misleading since chaotic behaviour is shown by some quite simple systems. Certain 'executive toys' are examples.

Chaotic processes are characterised by a critical dependence on initial conditions and other external influences. This is sometimes illustrated by the 'butterfly effect' in meteorology, where it is argued that the disturbance caused by a flap of a butterfly's wings may make the difference between calm weather and a typhoon. Attempts to model meteorological and other phenomena by computer showed that the difficulties in obtaining accurate results were not simply due to insufficient precision of modelling. With increased precision the results remained equally unpredictable.

The state of a system can be represented by a point in a multidimensional 'phase space', and within this there are 'basins of attraction' towards different types of behaviour. A characteristic of chaotic systems is that the basins of attraction have a complicated interface like that between the dark (member) and light (non-member) areas of the Mandelbrot Set. As with the Mandelbrot Set, the interface has similar form when viewed at any level of resolution, so the basins of attraction are intimately mixed no matter how precisely the co-ordinates are specified. This means there is no limit to the smallness of the disturbance, or deviation of an initial setting, that can drastically change the system behaviour. The behaviour is unpredictable in principle, irrespective of the precision of the methods of measurement and manipulation.

The term 'strange attractor' has been used to indicate a basin of attraction having this property of indefinitely fine interdigitation with its alternatives. The study of Chaos, or Complexity, Theory has affected many disciplines, not least biology, with the encouraging feature that principles, some of them quantitative, have been discovered that apply equally well to systems of apparently quite different kinds. Some relatively straightforward applications are to turbulent fluid flow and convection, and to the chemical reactions showing periodicity (in either space or time) studied by Prigogine and his school. Prediction of the weather is impossible with precision since it depends on a range of chaotic effects including turbulence and convection.

The example shown is just one of many attractive illustrations of the Mandelbrot Set and other fractals, and at least one illustrated calendar has featured them. Further examples, and full treatment of the topic with numerous references, can be found in the book by James Gleick (Chaos: Making a New Science, Viking Penguin, New York, 1987) or that by Heinz-Otto Peitgen, Hartmut Jørgens and Dietmar Saupe (Chaos and Fractals: New Frontiers of Science, Springer-Verlag, New York, 1992) or in the article by A.K. Dewdney ("Computer Recreations: A computer microscope zooms in for a look at the most complex object in mathematics", Scientific American, August 1985, pp. 16-25).

Alex Andrew

 


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