I wrote:
> That is, given a series whose plot is obviously self-similar,
> but whose generating function is not immediately clear, how to discover
> that generating function.
Raul responded:
> One presumably obvious tactic would be to describe the
> self-similarity. (In english, or mathematically, or [ideally] both.)
That is the goal. The question is how to reach it. Or, rather, what tools
can I use to reach it? I am seeking specific advice, tips, and techniques.
I am comfortable enough with J that I do not foresee problems translating a
structured, mechanical description into code. The problem is writing the
structured, mechanical description.
The best I have right now, is "it looks like a hybrid between a castle and
a mountain range. The castles grow, and their turrets are composed of
smaller castles. The growth is not monotonic; small castles appear
interspersed among large castles. But the interspersion is clearly
repetitive, itself in a self-similar way.".
The problem for me, here, is figuring out exactly what is repeating. Every
time I think I find the fundamental unit, I try describing the entire plot
in terms of that unit, and it doesn't quite fit. For example: at the
first glance, it looks like the the castle is repeating (i.e. the plot is
a pattern of variously scaled castles). But if you try to be strict in
this description, you quickly find you can't apply it to the connections
between castles. So, you look a little deeper, and discover the castle
itself appears to be composed of repetitions of a single line segment, as
are the connections between castles. You try that route, only to discover
the various dilations and rotations of the segment become problematic.
But encouragement can be drawn from my earlier experience: before my
revelation about the first fractal, I had thought that the fractal was
composed of variations of a single line segment. But the eureka moment
that gave me joy was realizing that I was wrong; I could reduce my
generating function's input to a single number: 0 (whereas before the
input had been the sequence underlying the line segment I had considered
fundamental; it turned out that sequence was just a few more applications
of the iterating function to the true initial condition, 0). So I'm
hoping for a similar insight here. I just don't know how to work towards
it.
To get a feel for the obstacles I'm encountering, and what might make it
hard to give voice to a fractal, try writing the Dragon Curve in J:
http://rosettacode.org/wiki/Dragon_curve
However, do so only by looking at the plot:
http://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Fractal_dragon_curve.jpg/263px-Fractal_dragon_curve.jpg
That is: do not read the algorithmic description of the curve in Wikipedia,
and don't make reference to the programs listed on the Dragon Curve
rosettacode task.
If you can describe the Dragon Curve mechanicistically in English, the
method of description and the tools you employ might help me do the same
for my own fractal.
-Dan
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