Re: mirrors in D-space

["John S. Denker" <jsd@MONMOUTH.COM>]

At 04:02 PM 8/4/00 -0500, Jack Uretsky wrote:
> There can only be one inversion in 3-space

...

> because two co-ordinates determine a plane and the "handednedness"
> of the co-ordinate system is determined by the orientation of
> the 3rd co-ordinate.

Depending on how one parses the sentence, that's narrowly true, but it can
easily be misinterpreted.  Two-dimensional creatures in Flatland can
exhibit handedness, so "the 3rd co-ordinate" can't be the general
definition of what determines handedness.

Here's another way of looking at it:

a) Under "normal" conditions, for an object of dimensionality d in a space
of dimensionality D, the object can't be chiral if d is less than D -- you
can always pick it up and flip it over.

b) This implies that for any object it is the last dimension (the step from
d=D-1 to d=D) that breaks the symmetry.

But this begs the question of how you prove statement (a).  Under what
conditions does statement (a) hold?  Before you assert that statement (a)
is obvious, be warned that it is not always true!  It's bad luck to prove
things that aren't true.