Re: terminology for symmetry of cylinders

[John Denker <jsd@MONMOUTH.COM>]

At 02:07 PM 1/22/01 -0500, Carl E. Mungan wrote:
> *What, if anything, is the distinction between cylindrical, axial,
> and azimuthal symmetry?

I don't use those terms, and I don't recommend them.  For constructive
counterproposals, see below.

> In your reply, consider the following three
> shapes for instance:
>
> (a) an infinitely long, straight, constant-diameter piece of spaghetti
> (b) a finite-length, straight, constant-diameter piece of spaghetti
> (c) an infinitely long, straight piece of spaghetti whose diameter
> varies sinusoidally with distance along its axis

An excellent question.  Let me take them out of order:

b) This is the symmetry of an N2 molecule.
   This can be constructed as the product of two symmetries:
   b1) rotational symmetry around the z-axis
   b2) reflection symmetry in the plane perpendicular to the
       cylinder at its midpoint.

If you want to get fancy, you can use the standard notation for point
groups.  Crystallographers love this stuff.  In particular, we can use the
notation D_infinity_h to express the combination of (b1) and (b2)

Note that the foregoing imply other symmetries, including inversion
symmetry and infinitely many reflection symmetries in planes containing the
axis.

a) 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

The foregoing imply uncountably many copies of D_infinity_h -- one for each
point along the axis.  And again there are the implied inversion and
reflection symmetries.

c) Again multiple symmetries.  This can be constructed from
  c1) Rotational symmetry around the z axis.
  c2) Reflection symmetry in a plane through an extremum in the sine wave.
  c3) One-dimensional crystal symmetry along the z axis.
  c4) There is fancier symmetry (which has a name and a symbol that escape
me at the moment) which involves shifting half a lattice vector and
flipping the sign.

The foregoing imply countably many copies of D_infinity_h, and other
symmetries.

  Somewhat more detail can be found at:
    http://www.wfu.edu/~haefnesc/symmetry/pointgroup/tutorial.html
    http://w3.rz-berlin.mpg.de/~horn/SemiconductorSurfaces/Symmetry-HTML/Sy
http://w3.rz-berlin.mpg.de/~horn/SemiconductorSurfaces/Symmetry-HTML/Symmetr
ie98.html


Just to clarify the notation by some examples:
 -- There exist lesser rotational symmetries.
    For instance, benzene is D_6_h.
 -- N2 is D_infinity_h.
    CO is the same general shape as N2, but the two ends are different,
    so it is just C_infinity_v.

The "h" means "horizontal" reflection, perpendicular to the z axis.
The "v" means "vertical" reflection, in a plane containing the z axis.

=========================================

Completely unrelated pet peeve:
  *) The plural of "virus" is "viruses".
     It's an English word with an English plural.
  *) In particular, "virii" is not, and never has been, a word
     in English or Latin, plural or otherwise.
  *) The Latin word [virus] means "venom" and has no plural.
  *) The Latin word [viri] means "men" -- the plural of [vir].