Discrete Corkscrew Columns -- Five Types

There are three column types having ${\bf N}_\pi$ symmetry. These columns remain the same when translated in the axial direction a unit distance and simultaneously rotated through the angle $180^\circ$ ($\pi/k$ ,in general). Among these columns are those that are invariant under reflection through the centerplane of the column ( ${\bf Z}_2(\kappa)$ ), those that are invariant under reflection through a plane containing the cylinder axis(${\bf Z}_2(\tau)$ )and those that are invariant under both reflections. SeeFigures  18, 19 and  20.

   

Figure:Corkscrew column with $\pi /2$ rotation and left-right reflection.

   

Figure:Corkscrew column with $\pi /2$ rotation and up-down reflection.

   

Figure:Corkscrew column with $\pi /2$ rotation and left-right and up-down reflections.

There are two continuous families depending on $\omega$ with discrete corkscrew motions (those with ${\bf N}_\omega$ symmetry). See Figures 21 and  22.

   

Figure:Corkscrew column with $\omega$ rotations where $0^\circ\le\omega\le\frac{180^\circ}{k}$ and no additional symmetry.

  

Figure:Corkscrew column with $\omega$ rotations where $0^\circ\le\omega\le\frac{180^\circ}{k}$ and up-down rotation.