Measuring Fractal Dimension ?

David C. Ullrich ullrich at math.okstate.edu
Fri Jun 19 20:43:11 CEST 2009


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.

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... 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."

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.

Then the second definition you cite: Amazon says the
prerequisites are two years of calculus. The stanard
meaning of log is log base e, even though it means
log base 10 in calculus.

>Mark




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