Re: Measuring Fractal Dimension ?

On Fri, 19 Jun 2009 12:40:36 -0700 (PDT), Mark Dickinson
<dickinsm@xxxxxxxxx> wrote:

On Jun 19, 7:43 pm, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:
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...

Judging by this thread, I'm not sure that much is off-topic
here. :-)

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.

That in turn surprises me. I've taught multivariable
calculus courses from at least three different texts
in three different US universities, and as far as I
recall a 'curve' was always thought of as a subset of
R^2 or R^3 in those courses (though not always with
explicit definitions, since that would be too much
to hope for with that sort of text). Here's Stewart's
'Calculus', p658:

"Examples 2 and 3 show that different sets of parametric
equations can represent the same curve. Thus we
distinguish between a *curve*, which is a set of points,
and a *parametric curve*, in which the points are
traced in a particular way."

Again as far as I remember, the rest of the language
in those courses (e.g., 'level curve', 'curve as the
intersection of two surfaces') involves thinking
of curves as subsets of R^2 or R^3. Certainly
I'll agree that it's then necessary to parameterize
the curve before being able to do anything useful
with it.

[Standard question when teaching multivariable
calculus: "Okay, so we've got a curve. What's
the first thing we do with it?" Answer, shouted
out from all the (awake) students: "PARAMETERIZE IT!"
(And then calculate its length/integrate the
given vector field along it/etc.)
Those were the days...]

Surely you don't say a curve is a subset of the plane and
also talk about the integrals of verctor fields over _curves_?

This is getting close to the point someone else made,
before I had a chance to: We need a parametriztion of
that subset of the plane before we can do most interesting
things with it. The parametrization determines the set,
but the set does not determine the parametrization
(not even "up to" some sort of isomorphism; the
set does not determine multiplicity, orientation, etc.)

So if the definition of "curve" is not as I claim then
in some sense it _should_ be.

Hales defines a curve to be a set C and then says he assumes
that there is a parametrization phi_C. Does he ever talk
about things like the orientation of a curve a about a point?
Seems likely. If so then his use of the word "curve" is
simply not consistent with his definition.

A Google Books search even turned up some complex
analysis texts where the word 'curve' is used to
mean a subset of the plane; check out
the book by Ian Stewart and David Orme Tall,
'Complex Analysis: a Hitchhiker's Guide to the
Plane': they distinguish 'curves' (subset of the
complex plane) from 'paths' (functions from a
closed bounded interval to the complex plane).

Hmm. I of all people am in no position to judge a book
on complex analysis by the silliness if its title...

"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.

Right. I'm much more willing to accept 'path' as standard
terminology for a function [a, b] -> (insert_favourite_space_here).

Not that it matters, but his defintion of "polygonal path"
is, _if_ we're being very careful, self-contradictory.

Agreed. Surprising, coming from Hales; he must surely rank
amongst the more careful mathematicians out there.

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.

I just don't believe there's any such thing as
'the standard definition' of a curve. I'm happy
to believe that in complex analysis and differential
geometry it's common to define a curve to be a
function. But in general I'd suggest that it's one
of those terms that largely depends on context, and
should be defined clearly when it's not totally
obvious from the context which definition is
intended. For example, for me, more often than not,
a curve is a 1-dimensional scheme over a field k
(usually *not* algebraically closed), that's at
least some of {geometrically reduced, geometrically
irreducible, proper, smooth}. That's a far cry either
from a subset of an affine space or from a
parametrization by an interval.

Ok.

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 means
log base 10 in calculus.

Sorry, I've lost context for this comment. Why
are logs relevant? (I'm very well aware of the
debates over the meaning of log, having frequently
had to help students 'unlearn' their "log=log10"
mindset when starting a first post-calculus course).

The point is that a calculus class is not mathematics.
In my universe the standard definition of "log" is different
froim what log means in a calculus class, and my point
was that a definition of "curve" in a book that specifies
it's supposed to be accessible to calculus students
doesn't seem to me like much evidence regarding
the standard definition in mathematics.

Mark

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