Re: OT: White Goddess

YHL wrote:
>On Wed, 11 Apr 2001, Andreas Johansson wrote:
>
> > > >    Think of truth and fact like circles and squares -
> > > > a square is a circle, with sufficient numbers of sides
> > > > (or however that saying goes - I know I got it wrong).
> > >
> > >Wrong.  A square is a polygon.  A circle is beyond a polygon.  It has
> > >transcended infinity.  You cannot give a polygon enough sides to *be* a
> > >circle, it can only *approach* a circle.  The difference is not merely
>a
> > >matter of degree.
> >
> > Eh, seeing that I supposed to be good at math, I should probably know
>this
> > myself, but what's the difference between a circle and a regular polygon
> > with an infinite number of sides? I definitely recall being told by my
>math
> > teacher tellin' me they're the same ...
>
>Suppose you have a regular polygon with n sides.  (I think you could get
>by with a weaker condition but this will suffice.)  The "limit" of the
>polygon as n goes to infinity is a circle.
Yes, that what I've meant. I'm a bit unused to talking about math in
English.
>
>I am not certain what mathematical meaning, if any, "transcended
>infinity" has, though.
Me neither.
>
>In a sense, I suppose you're both right.  If you just sit there adding
>sides to a polygon you'll sit there forever.
Hey, this assumes I use a amount of time greater than zero to add a side!
Infinity times 0s is still 0s ...

>  ^_^  Also, it depends on
>whether you're talking geometry and ideal shapes (in which case you could
>hit the circle) or the "real world" where eventually Planck length and
>practicality would kick in.
We can safely leac Planck out of this. As I said in the reply to Ray, the
discreet size of atoms, coupled with Heisenberg and the fact that atoms
don't have "edges" in the normal sense of the word, means we can never
construct a circle that's perfect on the atom scale.

>  Even our best circles are approximations if
>you magnify them large enough...but for most practical purposes, I
>suspect "a polygon with 100 million sides" would be a quite adequate
>"circle."
Certainly. If the thing's cirumference is one metre, that'd mean each side
is 10nm. Well, that's small. The size of carbon nanotubes, roughly.

                                                    Andreas
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