Measuring Fractal Dimension ?

[Mark Dickinson]

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:

"Definition 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?

Mark