Fractals and the notion of Infinity

A fractal is, according to Mandelbrot (apud Wikipedia): "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole".
Fractal is a word that derives from the latin Fractalus (fractured) and therefore we can deduct that a fractal image has many parts. The interesting thing about them is that each part (or sub-structure) repeats the same pattern as the entire image itself. We could say then, that a fractal is a macrocosmic structure with infinite micro systems that mimic the larger one. Take these fractal for example:




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Some fractals are not as clearly subdivided as Sol de Cobre: look at "En clave de sol" (In trebble), by Oscar Portela:


Both these fractals have been created by mahematicians, but take now The great wave at Kanagawa, a japanese ukiyo-e by Katsushika Hokusai (1831):
In this case, a plastic masterpiece uses the notion of fractal without even knowing it. Obviously Hokusai had no idea what fractals are, and yet this beautiful painting is created with the same concept in mind. Hokusai's work ws based on an old tradition of ukiyo-e making, and yet he was an innovator. Maybe that is why The Great Wave at Kanagawa has influenced so many artists in the west, and astonished as many viewers.

Learning about fractls has made me think about the metaphorical connotations of the word. How many fractal realities do we know? In many ways reality unfolds in different dimenssions just like a fractal. Even love, our own bodies or imagination (in the abstract sense) can be understood as a macro-cosmos... Fractals are the visual representation of a notion that was already expressed by Zenon of Elea (Eleatic school in Philosophy): the infinite subdivision of space and time. Just as he proves theoretically in his paradox of Achilles vs. the Turtoise:

(Wikipedia, sic)
In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is such a fast runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, in which said period the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.

In this case, his best known paradox of motion show how, at least in theory, the logical contradiction between real motion and the concept of motion itself. For Zenon, as for Parmenides (and according to fractal theory) reality develops ad infinitum and has nothing to do with what is obvious to the senses.
How interesting that Zenon's paradoxes had to wait almost 2500 years to find a suitable debate with Modern Physics. Until very recently, his theorical postulate had not been refuted succesfully by any other scientific or philosophical notion..