Thursday, August 23, 2012

The Beauty of Fractals.

As everyone knows by now, Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far"-- Fractals may be exactly the same at every scale, or as illustrated in Figure 1, they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.
As mathematical equations, fractals are usually nowhere differentiable, which means that they cannot be measured in traditional ways. An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.  A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. (wiki)

Jack (John) OShea was my art teacher the year Fractals became envogue. we were very hip.

 Perspective and mysterious things HERE

Figure 1a. The Mandelbrot set illustrates self-similarity. As you zoom in on the image at finer and finer scales, the same pattern re-appears so that it is virtually impossible to know at which level you are looking.
Figure 1b. Mandelbrot zoomed 6x
Figure 1c. Mandelbrot zoomed 100x
Figure 1d. Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.

Oh, The Mathematics.  Sigh

On Pondering the twenty fields of study i loved but did not pursue. 

Math Beauty
The Helix moves:

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