Spiral Columns -- Two Types

There are two types of spirals -- both of which have twisted translation symmetry. There are the spirals that are symmetric when the column is rotated by $180^\circ$ in a plane containing the axis of the cylinder ( ${\bf Z}_2(\tau\kappa)\dot{+}{\bf L}$) and those that do not have this symmetry (${\bf L}$). See Figures 8 and 9.

  

Figure 8: Spiral column with no up-down rotation.

  

Figure 9: Spiral column with up-down rotation.