Measuring Fractal Dimension ?
Jaime Fernandez del Rio
jaime.frio at gmail.com
Wed Jun 17 07:35:35 CEST 2009
On Wed, Jun 17, 2009 at 4:50 AM, Lawrence
D'Oliveiro<ldo at geek-central.gen.new_zealand> wrote:
> In message <7x63ew3uo9.fsf at ruckus.brouhaha.com>, wrote:
>
>> Lawrence D'Oliveiro <ldo at geek-central.gen.new_zealand> writes:
>>
>>> I don't think any countable set, even a countably-infinite set, can have
>>> a fractal dimension. It's got to be uncountably infinite, and therefore
>>> uncomputable.
>>
>> I think the idea is you assume uniform continuity of the set (as
>> expressed by a parametrized curve). That should let you approximate
>> the fractal dimension.
>
> Fractals are, by definition, not uniform in that sense.
I had my doubts on this statement being true, so I've gone to my copy
of Gerald Edgar's "Measure, Topology and Fractal Geometry" and
Proposition 2.4.10 on page 69 states: "The sequence (gk), in the
dragon construction of the Koch curve converges uniformly." And
uniform continuity is a very well defined concept, so there really
shouldn't be an interpretation issue here either. Would not stick my
head out for it, but I am pretty sure that a continuous sequence of
curves that converges to a continuous curve, will do so uniformly.
Jaime
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