# 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
$A_{\infty}$ $\infty$ $C_{\infty}$ $\infty$ rotating cone
$\frac{A_{\infty}}{M}C$ $\bar\infty$

$C_{\infty\,h}\equiv\,S_{\infty}\equiv\,C_{\infty\,i}$

$\infty$

rotating finite cylinder
$A_{\infty}\infty\,A_2$ $\infty2$ $D_{\infty}$ $\infty$ finite cylinder
submitted to equal and
opposite torques
$A_{\infty}M$ $\infty\,m$

$C_{\infty\,v}$

$\infty$ stationary cone

$\frac{A_{\infty}}{M}\frac{\infty\,A_2}{\infty\,M}C$

$\bar\infty\,m\equiv\bar\infty\frac{2}{m}$

$D_{\infty\,h}\equiv\,D_{\infty\,d}$

$\infty$ 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.

## History

The groups containing isotropy axes were introduced by P. Curie (1859-1906) in order to describe the symmetry of physical systems (Curie P. (1884). Sur les questions d'ordre: répétitions. Bull. Soc. Fr. Minéral., 7, 89-110; Curie P. (1894). Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et d’un champ magnétique. J. Phys. (Paris), 3, 393-415.).