Measuring Fractal Dimension ?

[Charles Yeomans]

On Jun 22, 2009, at 8:46 AM, pdpi 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:
>>
>>
>> <snick>
>>
>>
>>
>>> 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.
> -- 


But certainly the image set does define the curve, if you want to view  
it that way -- all parameterizations of a curve should satisfy the  
same equation f(x, y) = 0.

Charles Yeomans