Fractal is a term coin by their originator, Benoit Mandelbrot, in 1975. They are objects whose spatial form is not even nor fluid, hence irregular in nature, but in essence its irregularity as a non-smooth form repeats itself geometrically across in many different scales. The distinctive feature of fractals is that they are ‘self-similar’. It is the geometry of such object which is fractal, and any system which can be visualised or analysed geometrically, can be fractal if it inherit these characteristics.
Fractal geometry is a portrayal as geometry of nature. A geometry of nature evolved from the hypothesis which objects with geometry whose structure is irregular in terms of Euclidean geometry, but within this irregularity lays a pattern which is as order as those in simpler objects composed of straight lines.
A coastline or mountain ranges are examples of natural fractals. These fractals displays a sense of irregularity which characterises these objects, but is not entirely without order. Any part of a fratal, if enlarged or reduced in size, presents more or less the same appearance as a whole, the architectural forms or urbanistic models that are derived from them are characterised by a scalar ambiguity, by what we might call an “self similarity”[1].
Geometry is no longer composed and conceived by straight lines, the geometry of Euclid, but can now admit irregularity without abandoning continuity. Objects composed of a multitude of lines which are nowhere smooth may well manifest order in more accumulative terms than the sorts of simple objects which are dealt with in mathematics. This order can be discovered in fractals in terms of the following three principals.
Firstly, fractals are always self similar, at least in some general sense as this is the bases of its definition. On an enlarged or reduced scale, and within a given range you examine a fractal, it will always appear to be similar to the ‘whole’ with some degree of irregularity. The ‘whole’ will always be manifest in the ‘parts’. A great example is looking at a piece of rock that broke off from a mountain. You will see the mountains in the rock to some degree of familiarity[2].
Secondly, fractals can sometimes be portrayed in terms of a hierarchy of self similar components. Fractals are ordered hierarchically across many scales and the tree is a classic example. In fact, the tree is a literal interpretation of the term hierarchy and as such, it presents the most fundamental of fractals. Looking at the twigs on the branches of a tree and you can see the whole tree in these twigs, even thought at a much reduced scale. The organisation and spacing of cities as central places is such an order while the configuration of districts and neighbourhoods, and spatial distribution of roads and other communications are hierarchically structured[3].
The third principles relates to the irregularity of form. In terms of irregularity, we mean forms which are continuous but nowhere smooth, hence non-differentiable in terms of calculus. Take coastline as an example, “if you measure its length from a map, the map will have been constructed at a scale which omits lower level detail. If you actually measure the length by walking along the beach, you will face a problem of knowing what scale or yardstick to se and deciding whether to measure around every rock and pebble.” In the end, you will obtain a length that is relative and very dependent on the scale of measurement, and as the scales gets finer and finer down to the microscopic level, the length of the coastline will continue to grow. Therefore we are forced to conclude with the idea that the coastline’s length is ‘infinitely’ long or rather that its absolute length has no meaning and the length given is always relative to the scale of measurement[4].
[1] Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994) p.59
[2] Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994) p.60
[3] Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994) p.60
[4] Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994) p.60
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