What is a fractal?

This is the Mandelbrot set. I've generated it with Fractal Explorer and used the Zoom tool to analyze the autosimilarity. Every  part of it has the same organization in hierarchical levels. Personal archives

I've generated this fractal as a variation of a flipped Mandelbrot. This fractal has only local organizations. Personal archives.

This fractal is a variation of the previous fractal, same origin, a flipped Mandelbrot. Personal archives.

Fractals represent the dynamic systems, the geometry of Nature, the infinite retro-feedings, in synthesis, what cannot be measured in Euclidian terms. A fractal is a way of seeing the infinite.

The term that in Latin means fragmentary or cut, was presented by the Polish mathematician Benoît Mandelbrot (1924 -) for the first time in his book "Les Objets Fractals: Forme, Hasard et Dimension" (1975).

Fractals have not a definition yet, but their characteristic properties are enumerated as follows:

A fractal has a fine structure; this is detail in arbitrarily small scales.

A fractal is too irregular to be described with the traditional Euclidian geometry, so much local as globally.

Frequently, a fractal has a certain form of autosimilarity, maybe approximated or statistical.

In general, their "fractal dimension" is bigger than their topological dimension. Geometric fractals can be generated in a very simple recursive way. (Spinadel V., Perera J. G, Perera J.H. p. 2, 2000).

Many objects in Nature are better geometrically described as fractals, with autosimilarity in all scales. For instance, the universe consist on clusters of galaxies, organized in clusters of clusters of galaxies, and so on.

Although an effort has been made to characterize the fractals geometrically, there has not been a lot of progress in understanding its dynamic origin. We have a tendency to think that the universe is formed starting from static structures because the dynamics that forms these structures has bigger scale than the period of observation that can be the life of a human being. The earthquakes that we observe last some few seconds, and the formation of the crack seems static, but it is built in millions of years.

Therefore, the origin of the fractals is rather a dynamic problem, not a geometric one. The laws of their physics are local, fractals usually are not organized in the biggest distances.

Systems are fractal when at some critical point, a phase of continuous transition takes place. Fractals and chaotic systems possess the property of autosimilarity.

In geographical terms it is first spoken of SPACING that refers to the population or the activities of the centers, but not at the physical distance. As we move away from the center, the level of complexity increases and it is governed under a HIERARCHICAL organization, typical of fractality. And the MULTI-SCALE organization, of evolutionary processes from the center to the periphery also contains fractal characters. Chaotic groups have tendency to present fractional dimensions.

Summarizing, both concepts chaos and fractality, are supplemented. If we relate the urban morphology with the corresponding society, we would have the following theoretical structure:

MORFOLOGY /Fractal Geometry/ Simple rules(formula) /Complex behaviour

SOCIETY /Auto-organization/ Complex grouping /Simple emergent behaviour

Watch a video with a brief explanation and more examples: