>
> Iterative systems like nonlinear systems produce remarkable patterns. In
>mathematics the Mandelbrot set, Julia sets, Sierpinski's gasket and the
>Duffing equation among others produce amazing complexity (I know Brian Harper
>you are going to say "ordered" But that is another argument)
>
Well, I think everyone is probably aware of my objections to using things
like Mandelbrot or Julia sets as illustrations of complexity. I've been
thinking a lot about what would be a good yet simple illustration of
the evolution of complexity. I had thought for some time that a certain
class of one-dimensional cellular automata (CA) studied extensively by
Stephen Wolfram (of Mathematica fame) would do the trick. In this type
of CA (referred to as class IV) one is supposed to observe
a continual emergence of new patterns for ever, without the more
common collapse (after some time) into a static or regularly repeating
pattern. I read a recent paper where the author mentioned another
work which had proven that no class IV 1-D CA exist. Bummer ;-).
Another possibility I've been looking into over the past few weeks
is Per Bak's infamous sand pile "paradigm" for self-organized
criticality. I had looked into this some time ago and was not
overly impressed. My interest has been renewed after reading
Bak's new book <How Nature Works> [yes, I know, its an
outrageous and presumptuous title :) ]. I'm starting to suspect
now that some of my former objections were due to a lack of
appreciation and understanding of the model. It seems to me
now that this may be a really simple example of how an iterative
process can organize itself into an irreducibly complex critical
state. The model is really simple so I plan to write a little computer
program so I can play around with it some.
I don't want to go into any elaborate details at this time into
what happens in this model, however, in case anyone's interested,
the model can be described humorously in terms of an equivalent
model of governmental bureaucracy. Suppose there are 100 bureaucrats
sitting at their desks in 10 rows of ten. Each has four neighbors
(front, back, left, right) except those on the edges of the room.
These b's have three neighbors and one adjacent window (the b's
on the edge obviously have seniority). Now, pieces of paper begin to
mysteriously appear and land on one of the b's
desks (chosen at random or deterministically, it doesn't matter).
The simple rule is that whenever a b has four pieces of paper on
her desk, she immediately gives one to each of her four neighbors.
This may in turn produce 4 pieces on one of the neighbors desks,
so this neighbor immediately redistributes his papers and so on
until no one has four pieces of paper. Only after all these shufflings
are finished does another piece of paper appear from the beyond.
The shuffling will always terminate at some point since the fellows
on the edge will pass a piece to each of their three neighbors, the
other is tossed out the window ;-).
A really simple model yet lots of really interesting stuff happens,
none of which I'm going to describe ;-). I'm looking forward to
writing a program so I can watch it all happen.
Brian Harper | "If you don't understand
Associate Professor | something and want to
Applied Mechanics | sound profound, use the
The Ohio State University | word 'entropy'"
| -- Morrowitz
Bastion for the naturalistic |
rulers of science |