Hi

> The following question was posted on the Topology Atlas's" Ask a
> topologist ".
> Question:  If mirrors reverse left and right, why don't they
> reverse up and down?

This is a nice problem that has had me puzzled for a long time as
well.  I had never actually resolved this to my satisfaction so just
now I spent some time thinking about it and I think I've come up with
an explanation.  (Unfortunately this is a bit tricky to explain in
writing so my explanation is somewhat long winded. I hope the ascii
graphics are readable.  They were when I drew them.)

The premise of the statement is false.  Mirrors don't reverse left
and right but instead they reverse left handedness and right
handedness.  Left and right handedness are intrinsically 3
dimensional notions whereas up and down can be made sense of in 2
dimensions ( in fact I think up and down are 1-dimensional notions
which only make sense in the presence of a gravitational field, i.e.
down means "in the direction of the gravitational (vector) field"
whereas up specifies the opposite direction).  Mirrors leave all
relations which lie in a 2-d plane parallel to the mirror intact and
only mess with things that involve the 3rd dimension.  That's why up
and down, which are parallel to the mirror, are not affected (and
neither is left and right) but left handedness and right handedness
are.  I will explain what I mean in more detail below.

I will take a mirror to be a vertical plane that reflects images.
You are standing in front of the mirror facing it.   Then left,
right, up and down specify directions in a plane through
your body parallel to the mirror, i.e. right and up form an
orthogonal basis for all vectors in that plane.  It is then not true
that the mirror reverses left and right.  If you don't believe me then
take your arm or a stick pointing in any direction in this plane and
the person in the mirror will point in the same direction in the
plane.  As mentioned above it is only the handedness of any 3
dimensional ordered basis that is reversed as I will show below.
There is a natural 3d ordered basis (right, forward, up) which in
this order is in fact right handed.  I will try to show how the
orientation, i.e. the handedness, of this basis is reversed in the
diagram below (supressing the up direction)

<p><pre>

                          mirror
                            |
                            |
           forward          |       forward "
             ---->          |        <----
           |                |             |
    right  |                |             | right"
           \/               |            \/
                            |
                            |

</pre></p>

Thus the right direction is unchanged and similarly the up direction
is also unchanged.  Only the direction orthogonal the the plane of
the mirror, i.e. the forward direction, is reversed.  Now if you
reverse one out of 3 basis vectors of an ordered basis then the
handedness of the basis is reversed which is why right handed becomes
left handed and vice versa.

This explains why your right arm becomes your mirror images' left arm
and vice versa since the nose of the mirror man (which defines the
forward direction) points in the opposite direction to your nose.
That's nothing new.  A perhaps suprising fact is that anything which
doesn't involve a direction orthogonal to the mirror remains
unchanged.  "Rubbish" you say, "if I hold a book up to a mirror the
writing on the cover is reversed and I can't read it."  Well that's
because you're holding the book the wrong way around, i.e. the mirror
is showing you exactly what you would see if you could look through
the book, i.e. if you could read the front cover by looking at the
back of the book.  If you had letters which you didn't have to write
on a piece of paper ( like the letters in children's alphabet cereal
or alphabet soup ) and you wrote something with these letters so that
you could read it and then held it up in front of a mirror then you
would be able to read the mirror image as well.

Conclusion:
Mirrors reverse the 'forward' and 'backward' directions and leave
left, right, up and down unchanged.  As a result left handedness and
right handedness are interchanged.

Tobias