Re: Zooming the fractal

From: scerir (
Date: Sun Apr 15 2001 - 11:40:54 MDT

> Jim Fehlinger wrote:
> The analogy here, of course, is a view of the nested processes
> of selection, with evolution as the outermost level and the
> fleeting activation patterns of consciousness as the innermost,
> as all parts of a unified fractal pattern spread out in time,
> spanning timescales ranging from milliseconds to billions of
> years, and broadly self-similar across all those scales.

Beyond the (qualitative) unified fractal pattern there are also fractal
functions. One of the properties of a fractal function is that it does not
possess a characteristic scale length, and consequently its derivatives

Fortunately, the dynamics of *complex* phenomena - described by fractal
functions - can be expressed, often, in terms of fractional differential
equations of motion.

Fractional diffusion equations, in example, have been used to model the
evolution of stochastic phenomena with *long-time* memory, that is,
phenomena with correlations that decay as inverse power laws (rather than
exponentially in time).

Actually, fractal functions have, often, fractional derivatives. Therefore
*complex* phenomena - having a fractal dimension - are now reasonably
modeled using fractional equations of motion.

- P. Meakin
"Fractals, Scaling and Growth far from Equilibrium",
Cambridge Nonlinear Science, Series 5,
Cambridge University Press, Cambridge (1998).

- M. Schroeder
"Fractals, Chaos, Power Laws",
W.H. Freeman and Comp., New York (1991).

- B.B. Mandelbrot
"The Fractal Geometry of Nature",
W.H. Freeman and Co., San Francisco (1977).

- B.J. West
"Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails",
Studies of Nonlinear Phenomena in the Life Sciences, Vol. 9,
World Scientific, Singapore (1998).


Entropy can also be regarded as the fractal dimension of an appropriate
compact set.
- Chris Hillman

This archive was generated by hypermail 2b30 : Mon May 28 2001 - 09:59:46 MDT