If men define situations as real, they are real in their consequences. – Thomas theorem

Fractality

Posted in Today by bendyourcircuit on 23/06/2009

Fractal is a word Mandelbrot coined to describe the geometry of the rough and broken—from the Latin fractus, the origin of fractured. Frac­tality is the repetition of geometric patterns at different scales, revealing smaller and smaller versions of themselves. Small parts resemble, to some degree, the whole. […]
[…] The veins in leaves look like branches; branches look like trees; rocks look like small mountains. There is no qualitative change when an object changes size. […] Mandelbrot designed the mathematical object now known as the Mandelbrot set, the most famous object in the history of mathematics. It became popular with followers of chaos theory because it generates pic­tures of ever increasing complexity by using a deceptively minuscule recur­sive rule; recursive means that something can be reapplied to itself infinitely. You can look at the set at smaller and smaller resolutions with­out ever reaching the limit; you will continue to see recognizable shapes. The shapes are never the same, yet they bear an affinity to one another, a strong family resemblance.
These objects play a role in aesthetics. Consider the following applica­tions:
Visual arts: Most computer-generated objects are now based on some version of the Mandelbrotian fractal. We can also see fractals in architec­ture, paintings, and many works of visual art—of course, not consciously incorporated by the work’s creator.
Music: Slowly hum the four-note opening of Beethoven’s Fifth Sym­phony: ta-ta-ta-ta. Then replace each individual note with the same fournote opening, so that you end up with a measure of sixteen notes. You will see (or, rather, hear) that each smaller wave resembles the original larger one. Bach and Mahler, for instance, wrote submovements that resemble the larger movements of which they are a part.
Poetry: Emily Dickinson’s poetry, for instance, is fractal: the large re­sembles the small. It has, according to a commentator, „a consciously made assemblage of dictions, metres, rhetorics, gestures, and tones.“

Nassim Taleb, The Black Swan: The Impact of the Highly Improbable

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