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Fractals
Definition
Fractals are shapes which are recursively constructed. This means that the outline of the whole shape is made up of smaller and similar versions of the whole. These shapes can be observed at all scales and so are infinitely complex. The intricacy of the structure doesn't diminish with its level of magnification. The shape can also have a component of irregularity.
Characteristics
> Scaling - the same form occurs at different scales
> Self-Similarity - similarforms recur in the same structure
> Fractal Dimension - recrsive construction creates an infinite boundary
> Hidden Order - Complex behaviour results from simple rules
History
Benoit Mandelbrot pioneered the analysis of complex natural phenomena such as coastline and mountain range, through the non-linear realm and fractal geometry. Mandelbrot's research suggested that the study of certain patterns and shapes could universally explain more diverse and recurring phenomena at all scales. This new wave of thinking on universality was furthered by Feigenbaum. The mathematician Ian Stewart described how mathematics in general became more geometric. The shift was away from the rigid geometry of Euclid into visual geometry. Behaviours in nature which were thought to be only qualitative were formalised using precise tools. It began to emerge that fractal scaling was universal in morphogenesis and the understanding of the encoding and method of construction of patterns became a central issue in biology.
Significance
Fractal shapes are not smooth, flat planes; they are convoluted. This is in great contrast to the simplicity of platonic form. The self-similarity concept could be aplied to a city. In Plato's 'The Republic', the noble city has the same tripartite structure as the soul of a single noble inhabitant of the same city. The aesthetic quality of fractal forms could be described without the detailed knowledge of the mathematical method. Our innate appreciation for beauty is drawn to the harmony and balance of nature. It used to be that phenomena such as sway, current and whirlpool were neglected in the world of pragmatic dualism as something non-scientific; which cannot be calculated mathematically. We can easily observe that the shapes of natural things are the result of dynamical processes. The tendency for natural structures to optimise large surface areas indicates that the aim is to maximise exchange opportunities at the boundary.
Two of the best known fractal shapes are the Koch Snowflake and the Serpinski Gasket.