Measuring Fractal Dimension ?

[pdpi]

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...
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> > 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.