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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.).

See also

Section 10.1.4 of International Tables of Crystallography, Volume A
Section 1.1.4 of International Tables of Crystallography, Volume D