Measuring Fractal Dimension ?

[Charles Yeomans]

On Jun 19, 2009, at 2:43 PM, David C. Ullrich wrote:

> 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
> <dickinsm 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.

>
>
> 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.

>  Then later in the same paper
>
> "Definition 2. A polygon is a Jordan curve that is a subset of a
> finite union of
> lines. A polygonal path is a continuous function P : [0, 1] ->¨ R2
> that is a subset of
> a finite union of lines. It is a polygonal arc, if it is 1 . 1."
>

These are a bit too casual for me as well...
>
> By that definition a polygonal path is not a curve.
>
> Worse: A polygonal path is a function from [0,1] to R^2
> _that is a subset of a finite union of lines_. There's no
> such thing - the _image_ of such a function can be a
> subset of a finite union of lines.
>
> Not that it matters, but his defintion of "polygonal path"
> is, _if_ we're being very careful, self-contradictory.
> So I don't think we can count that paper as a suitable
> reference for what the _standard_ definitions are;
> the standard definitions are not self-contradictory this way.


Charles Yeomans