🚀 go-pugleaf

In article <wpenrose.354.00550F9D@interaccess.com>,
William R. Penrose <wpenrose@interaccess.com> wrote:
>
>But this result doesn't depend on the dimensions of atoms or anything,  It is 
>one of those weird things that calculus does, when you calculate surfaces and 
>volumes of revolution.  Even your infinitely thin paint wouldn't cover it.
>
>The lesson is:  If you are a painter, be careful of what you bid on.

Hmmm, my real analysis is years old, and I didn't do very well on it
(certainly not at the sci.math level this is crossposted to), but doesn't 
it matter what you mean by "paint"?  

I can take a sphere, i.e., finite surface area, and do a mapping of it
onto a plane.  There's a point missing on the sphere, which would make
the surface unbound; I think you have to be careful with that.  Similarly,
I'd speculate you can map the volume of revolution onto the surface;
is that "painting"?

More interesting artifacts are, say, Peano curves (?): I can take a bound
interval on the number line, say [0,1], and map it to a bound area, say 
a square, such that my curve passes through every point of the square.
This curve, BTW, is everywhere continuous but nowhere differentiable.