6.3: Cylindrical system
The cylindrical system contains non-crystallographic point groups with one axis of revolution (or isotropy axis). There are five groups in the spherical system:
| Hermann-Mauguin symbol | Short Hermann-Mauguin symbol | Schönfliess symbol | order of the group | general form |
|---|---|---|---|---|
| \[A_{\infty} \nonumber \] | \[\infty \nonumber \] | \[C_{\infty} \nonumber \] | \[\infty \nonumber \] | rotating cone |
| \[\frac{A_{\infty}}{M}C \nonumber \] | \[\bar\infty \nonumber \] |
\[C_{\infty\,h}\equiv\,S_{\infty}\equiv\,C_{\infty\,i} \nonumber \] |
\[\infty \nonumber \] |
rotating finite cylinder |
| \[A_{\infty}\infty\,A_2 \nonumber \] | \[\infty2 \nonumber \] | \[D_{\infty} \nonumber \] | \[\infty \nonumber \] |
finite cylinder
submitted to equal and opposite torques |
| \[A_{\infty}M \nonumber \] | \[\infty\,m \nonumber \] |
\[C_{\infty\,v} \nonumber \] |
\[\infty \nonumber \] | stationary cone |
|
\[\frac{A_{\infty}}{M}\frac{\infty\,A_2}{\infty\,M}C \nonumber \] |
\[\bar\infty\,m\equiv\bar\infty\frac{2}{m} \nonumber \] |
\[D_{\infty\,h}\equiv\,D_{\infty\,d} \nonumber \] |
\[\infty \nonumber \] | stationary finite cylinder |
Note that \(A_{\infty}M\) represents the symmetry of a force, or of an electric field and that \(\frac{A_{\infty}}{M}C\) represents the symmetry of a magnetic field (Curie 1894), while \(\frac{A_{\infty}}{M}\frac{\infty\,A_2}{\infty\,M}C\) represents the symmetry of a uniaxial compression.