Measuring Fractal Dimension ?

[Charles Yeomans]

On Jun 22, 2009, at 2:16 PM, David C. Ullrich wrote:

> 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.)


That is to say, the "winding number" is a property of both the curve  
and a parameterization of it.  Or, in other words, the winding number  
is a property of a function from S1 to C.

Charles Yeomans