Columns of Revolution -- Four Types

There are four types of column with rotational ${\bf SO(2)}$ symmetry. Two types are periodic in the axial direction and two are not. The nonperiodiccolumns may have a reflection symmetry in the horizontal plane (${\bf D}_2\dot{+}{\bf SO(2)}$ -- for example a column which is bowed out at the center) or not ( ${\bf Z}_2(\tau)\dot+{\bf SO(2)}$ -- a column which widens at the base). See Figures  2 and 3.

   

Figure 2:Nonperiodic column of revolution with no up-down reflection.

   

Figure 3:Nonperiodic column of revolution with up-down reflection.

The periodic columns of revolution may have an up-down symmetry (${\bf D}_2\dot{+}({\bf SO(2)}\oplus{\bf Z})$ )or not ( ${\bf Z}_2(\tau)\dot{+}({\bf SO(2)}\oplus{\bf Z})$ ).See Figures 4 and 5.

   

Figure 4:Periodic column of revolution with no up-down reflection.

   

Figure 5:Periodic column of revolution with up-down reflection.