Measuring Fractal Dimension ?

[David C. Ullrich]

On Wed, 17 Jun 2009 07:35:35 +0200, Jaime Fernandez del Rio
<jaime.frio at gmail.com> wrote:

>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.

Nope. Not that I see the relvance here - the g_k _do_
converge uniformly.

>Jaime