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Re: terminology for symmetry of cylinders



At 03:05 PM 1/22/01 -0600, Joel Rauber asked about the symmetry of:
> >
> > an infinitely long, straight, constant-diameter piece of
> spaghetti

And about my statement that:
> This can be constructed as the product of three symmetries:
> a1) rotational symmetry around the z-axis
> a2) reflection symmetry in any plane perpendicular to the z axis
> a3) translational invariance along the z-axis
>

Doesn't a2 (particularly with the word *any* in it) imply a3?

Well, depending on how you parse things, (a2) might imply the existence of
translational symmetry. More specifically:
-- If I had stated (a2) as "reflection symmetry in the z=0 plane" then
it would not have implied anything about translational symmetry and
an explicit assertion of (a3) would have been 100% required.
-- If I had said "every" instead of "any" then clearly this would have
implied the existence (!) of translational symmetry.
-- Here's what I was thinking yesterday: note that the main verb
in my statement was "constructed". If you are trying to construct
a house, it is one thing to prove that a million bricks exist, and
it is quite another to actually possess a million bricks. There is
a whole sub-field of mathematics that revolves around the distinction
between "proof" and "proof by construction". In this
case, to actually _construct_ all the members of the group, you would
need to lay hands on more than one reflection operator, not just
"any" old single reflection operator.
-- It doesn't really matter. We all know what the final symmetry is;
we are just quibbling about the most elegant basis for constructing
it. The choice of basis is not unique.
-- In the real world of crystallographic point groups, it really, really
doesn't matter, because molecules aren't infinitely long, so they
never have more than one horizontal plane of symmetry.

And, regarding "cylindrical, axial, and azimuthal" ...
I'm not a strict adherent to John's statement as the words above are nice
and short and descriptive and don't involve arcane crystallographic
nomenclature.

If you don't like the point-group nomenclature, don't use it. In my note I
gave two complete systems. The non-arcane system uses terms like
*) rotational symmetry around a given axis
*) reflection symmetry in a given plane
*) et cetera.

These are strongly preferable to terms like
-- cylindrical symmetry, which is ambiguous because mathematicians define
a cylinder to be infinite in length with an arbitrary figure as its base,
while everybody else assumes it to be finite and round.
-- axial symmetry, which is ambiguous because it could mean invariant
with respect to translation along the axis, or rotation around the axis, or
perhaps both, or ????
-- radial symmetry, which is defined by the biologists. For instance the
"radial" symmetry of a starfish is really an approximate 5-fold rotation
symmetry, C_5 or perhaps C_5_v.