Deep Simplicity and Iterations

I’ve been reading Deep Simplicity: Bringing Order to Chaos and Complexity by John Gribbin. I was hoping to glean more insight into the whole “levels of organization” problem
that has long fascinated me. I also used one of Gribben’s examples of
what many call a “sensitive dependence on initial conditions” to use my
computer to plot the results. I was considering using Matplotlib / pylab – matlab style python plotting (plots, graphs, charts) but instead used Excel, which did the job fine.

What was the task in question? It was to iterate the function 2x^2-1,
where x is in between -1 and 1, to see how even a small change in an
initial x leads to diverging values as we go through the iterations.
That is, the small differences are magnified by the feedback loop of
iteration. (I know that the last two sentences are not terribly well
written alas….)

Besides getting to play with mathematically oriented computation, did I
actually glean any more insight about why there seem to be distinct
levels of organization in our cosmos? I don’t know. At first, I was
going to write that the sensitive dependence on initial conditions can
give rise to boundaries — but I can’t say such a statement makes any
sense to me. Hmmm….back to the mental drawing board.