Measuring Fractal Dimension ?
pdpinheiro at gmail.com
Wed Jun 17 09:18:17 EDT 2009
On Jun 17, 1:26 pm, Jaime Fernandez del Rio <jaime.f... at gmail.com>
> P.S. The snowflake curve, on the other hand, is uniformly continuous, right?
The definition of uniform continuity is that, for any epsilon > 0,
there is a delta > 0 such that, for any x and y, if x-y < delta, f(x)-f
(y) < epsilon. Given that Koch's curve is shaped as recursion over the
transformation from ___ to _/\_, it's immediately obvious that, for a
delta of at most the length of ____, epsilon will be at most the
height of /. It follows that, inversely, for any arbitrary epsilon,
you find the smallest / that's still taller than epsilon, and delta is
bound by the respective ____. (hooray for ascii demonstrations)
Curiously enough, it's the recursive/self-similar nature of the Koch
curve so easy to prove as uniformly continuous.
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