Measuring Fractal Dimension ?

[David C. Ullrich]

On Mon, 22 Jun 2009 05:46:55 -0700 (PDT), pdpi <pdpinheiro at gmail.com>
wrote:

>On Jun 19, 8:13 pm, Charles Yeomans <char... at declareSub.com> wrote:
>> On Jun 19, 2009, at 2:43 PM, David C. Ullrich wrote:
>>
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>>
>>
>> > Evidently my posts are appearing, since I see replies.
>> > I guess the question of why I don't see the posts themselves
>> > \is ot here...
>>
>> > On Thu, 18 Jun 2009 17:01:12 -0700 (PDT), Mark Dickinson
>> > <dicki... at gmail.com> wrote:
>>
>> >> On Jun 18, 7:26 pm, David C. Ullrich <ullr... at math.okstate.edu>  
>> >> wrote:
>> >>> On Wed, 17 Jun 2009 08:18:52 -0700 (PDT), Mark Dickinson
>> >>>> Right.  Or rather, you treat it as the image of such a function,
>> >>>> if you're being careful to distinguish the curve (a subset
>> >>>> of R^2) from its parametrization (a continuous function
>> >>>> R -> R**2).  It's the parametrization that's uniformly
>> >>>> continuous, not the curve,
>>
>> >>> Again, it doesn't really matter, but since you use the phrase
>> >>> "if you're being careful": In fact what you say is exactly
>> >>> backwards - if you're being careful that subset of the plane
>> >>> is _not_ a curve (it's sometimes called the "trace" of the curve".
>>
>> >> Darn.  So I've been getting it wrong all this time.  Oh well,
>> >> at least I'm not alone:
>>
>> >> "De?nition 1. A simple closed curve J, also called a
>> >> Jordan curve, is the image of a continuous one-to-one
>> >> function from R/Z to R2. [...]"
>>
>> >> - Tom Hales, in 'Jordan's Proof of the Jordan Curve Theorem'.
>>
>> >> "We say that Gamma is a curve if it is the image in
>> >> the plane or in space of an interval [a, b] of real
>> >> numbers of a continuous function gamma."
>>
>> >> - Claude Tricot, 'Curves and Fractal Dimension' (Springer, 1995).
>>
>> >> Perhaps your definition of curve isn't as universal or
>> >> 'official' as you seem to think it is?
>>
>> > Perhaps not. I'm very surprised to see those definitions; I've
>> > been a mathematician for 25 years and I've never seen a
>> > curve defined a subset of the plane.
>>
>> I have.
>>
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>> > Hmm. You left out a bit in the first definition you cite:
>>
>> > "A simple closed curve J, also called a Jordan curve, is the image
>> > of a continuous one-to-one function from R/Z to R2. We assume that
>> > each curve
>> > comes with a fixed parametrization phi_J : R/Z ->¨ J. We call t in R/Z
>> > the time
>> > parameter. By abuse of notation, we write J(t) in R2 instead of phi_j
>> > (t), using the
>> > same notation for the function phi_J and its image J."
>>
>> > Close to sounding like he can't decide whether J is a set or a
>> > function...
>>
>> On the contrary, I find this definition to be written with some care.
>
>I find the usage of image slightly ambiguous (as it suggests the image
>set defines the curve), but that's my only qualm with it as well.
>
>Thinking pragmatically, you can't have non-simple curves unless you
>use multisets, and you also completely lose the notion of curve
>orientation and even continuity without making it a poset. At this
>point in time, parsimony says that you want to ditch your multiposet
>thingie (and God knows what else you want to tack in there to preserve
>other interesting curve properties) and really just want to define the
>curve as a freaking function and be done with it.

Precisely.