symmetries of a cube

I had to cancel a previous reply due to failing to
understand the intended meaning of "opposite".

mitch <[EMAIL PROTECTED]> writes:
 : Relative to a cube, a 2-fold symmetry is a 180 degree rotation.  For a
 : given 2-fold rotation, the axis of symmetry is the line from the
 : midpoint of a given edge to the midpoint of the opposite edge.

That would be the DIAGONALLY opposite edge.

 :  The six 2-fold symmetries segregate into pairs according to their relation to
 : the three 4-fold symmetries.
 : 
 : To see this, consider the diagram,
 : 
 : [begin fixed width]
 : 
 :                /|
 :               / |
 :        ------/--|----------
 :       /     /   |         /
 :      /     /    |        /
 :     /     /     /       /
 :    /     |     /       /
 :   /      |    /       /
 :  --------|---/--------
 :          |  /
 :          | /
 :          |/
 : 
 : 
 : [end fixed width]

This diagram is completely irrelevant to identifying the axes
of symmetry in question, WHICH ARE DIAGONAL.
Mitch drew another diagram involving axes through
the centers of faces, but nobody needed the help
on that one.  The diagram he SHOULD have drawn
shows 3 of the 2-fold symmetries as:

 :
 :          ----------2------------
 :         /          /          /|
 :        /                     / |
 :       /           /         /  |
 :      1                     /   |
 :     / \          /        /    |
 :    /      \              /     3
 :   /          \  /       /      |
 :  -----------------------       |
 :  |             / \     |       |
 :  |                  \  |       |
 :  |            /        | \    /
 :  |                     |    \/
 :  3           /         |    /1
 :  |                     |   /
 :  |          /          |  /
 :  |                     | /
 :  |         /           |/
 :  ----------2------------

I didn't draw the line connecting the 3's because the
diagram gets unparseably busy;it is pointless to argue over whose 
ascii art manages to accurately communicate.  His point was
that there are 6 of these mid-edge diagonals and 

 : So, each 4-fold symmetry corresponds with four of the 2-fold symmetries.

This is just ridiculous.  I mean, if you are yourself observing and
DEFINING a "correspondence" then you can allege any old correspondence
you like.  But 4 is not a divisor of 6 and if there are three 4-fold
symmetries, they will not have any natural correspondence with any 4 of
the six 2-fold ones.  What WILL be natural (since there 6 of one
and of the other) is for each of the 3 lines-through-the-opposite-face-centers
to correspond to TWO of the 6 lines-through-the-diagonally-opposite-edge-midpoints.
You can then get a "natural" 4 by just taking the other 4 of those 2.
But it's all an awful lot of work for, basically, nothing, until you
can come up with an explanation that is shorter than 1537 lines.