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genetheil@[EMAIL PROTECTED]
all,
Been lurking here for a few years, but since I am
neither a programmer nor an educator, only a layman
with a slightly philosophical bent who has been
interested in Logo since I first encountered it about
25 years ago, I have resisted adding any cents (pun
intended) to these discussions. Now, I am stimulated
by the discussion on recursion to relent from that
policy.
This is a question for Gene Sullivan: In your post on
Jan 31, you wrote:
"BTW. If one thinks about it, circles are typically
drawn in Logo as `fractals'.
If one regards the problem of measuring coastlines
and perimeters as fundamental to fractals, then
sides of any regular polygon used to represent a
portion of the circumference of a circle ARE being
used as in/with fractals, are they not?"
Every regular polygon approximates a circle
`fractally',
it seems.
Though I may have violated a few notions of what
mathematicians `know' about fractals.
I really don't know.
Its fun for me to think about polygons this way
though."
I have always thought of fractals as having the
fundamental property of self-similarity. I see how
drawing a series of short straight lines followed by
small turns exhibits this property. However, it seems
to me what you get is a closer and closer
approximation to a curve which in the limit as line
length goes to zero, can be thought of as a curved
line. This fundamental insight was the basis for the
invention of calculus by Newton (and Leibniz). Nowhere
in the discussions by Mandelbrot and others that I
have been able to find does the mere iteration of a
simple procedure necessarily result in a fractal.
As I understand it, in order for a curve to be a
fractal, it must necessarily have a dimensional
measure
between one and two. A circle or other simple plane
curve has a dimensional measure of one. So, I'm
confused about how you are using the word "fractal".
If you intend an analogical use, then perhaps you
could explain further. But, perhaps this question
oversteps the boundaries of this forum. If so, I
apologize. In any case, I'm glad a topic arose which
stimulated an emission from me. Whether the emission
was coherent enough to amplify the light shed on the
topic, I must leave for others to judge.
Sincerely
Gene Theil
Genetheil@[EMAIL PROTECTED]
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