II Fractals - A Definition

The best way to define a fractal is through its attributes: a fractal is "rugged", which means that it is nowhere smooth, it is "self-similar", which means that parts look like the whole, it is "developed through iterations", which means that a transformation is repeatedly applied and it is "dependent on the starting conditions". Another characteristic is that a fractal is "complex", but nevertheless it can be described by simple algorithms - that also means that beneath most natural rugged objects there is some order.

2.1 What is a Fractal?

"Fractals are objects of any kind whose spatial form is nowhere smooth, hence termed "irregular", and whose irregularity repeats itself geometrically across many scales"[01].

In fact there are so many different types of fractals, some of which will be introduced in chapter "3 Different Fractals", that it is not possible to give one definition for all of them. Besides, when we are talking about fractals in general we should never forget that there are many which have not been found yet. Considering this circumstance, it is more useful to describe some of their characteristics.

2.2 Characteristics

... the world is chaotic, discontinuous, irregular in its superficial physical form but ... beneath this first impression lies an order which is regular, unyielding and of infinite complexity[02].

• 2.2.1 A Fractal is Rugged
• 2.2.2 A Fractal is Self-Similar
• 2.2.3 A Fractal is Infinitely Complex
• 2.2.4 A Fractal is Developed through Iterations
• 2.2.5 A Fractal Depends on Starting Conditions
• 2.2.6 A Fractal is Common in Nature

2.3 Influences

There is one important fact about the group of "general" fractals namely the natural development. This means that for the growth of natural but also for artificial objects many additional influences have to be thought of. Thus a tree or a fern can be produced by fractal geometry but these pictures nevertheless offer some differences in respect to their natural brothers. A tree standing alone on a hill for example is influenced by the wind blowing there, which forms the tree in one typical form: branches are only to be found on the side turning away from the direction the wind blows. Other influences may be soil and water conditions, kinds of plants nearby and animals.

Nevertheless "true" fractals can produce typical natural and man-made forms, but only under sterile conditions. If some random factor is added then the resulting objects come nearer to the "real" world. Likewise the development of cities underlies a couple of influences like natural barriers such as hills and rivers, but also man-made ones such as roads leading to other towns, important industrial areas and green-zones of which the growth of the city reacts. The same is true for elevations and even ground plans of buildings that react to the surrounding no matter whether it is man-made or natural.

From that follows that if we know the underlying algorithm of any object - under sterile conditions - and if some mechanisms for simulation of certain influences are added, we may determine future developments of e.g. the growth of a city.

Footnotes

[01] Batty and Longley, Fractal Cities (1994), Academic Press Inc., ISBN 0-12-4555-70-5, p.3.
[02] Fractals can be called the geometry of chaos. Batty and Longley, Fractal Cities (1994), Academic Press Inc., ISBN 0-12-4555-70-5, introduction p.v.