Fluted Columns -- Two Types

All remaining symmetry groups have at least ${\bf Z}_k$ symmetry for some k, that is, rotation symmetry through an angle $2\pi/k$. In our description of this classification we now set k=1 with the understanding that there is a version of each of the remaining columns for each natural number k. Indeed, the pictures we show all have k=2.

There are two types of columns with axial translation symmetry: those which have a plane of reflection across a plane containing the axis of the cylinder ( ${\bf D}_2\dot{+}{\bf R}$) and those that do not ( ${\bf Z}_2\dot{+}{\bf R}$). See Figures 6 and 7.

  

Figure 6: Fluted column with no left-right reflection.

  

Figure 7: Fluted column with left-right reflection.