Measuring Fractal Dimension ?

[David C. Ullrich]

On Mon, 22 Jun 2009 10:31:26 -0400, Charles Yeomans
<charles at declareSub.com> wrote:

>
>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.

This sounds like you didn't read his post, or totally missed the
point.

Say S is the set of (x,y) in the plane such that x^2 + y^2 = 1.
What's the "index", or "winding number", of that curve about the
origin?

(Hint: The curve c defined by c(t) = (cos(t), sin(t)) for
0 <= t <= 2pi has index 1 about the origin. The curve
d(t) = (cos(-t), sin(-t)) (0 <= t <= 2pi) has index -1.
The curve (cos(2t), sin(2t)) (same t) has index 2.)

>Charles Yeomans