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Cylindrical system

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  • 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
    \[\frac{A_{\infty}}{M}C\] \[\bar\infty\]



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


    \[\infty\] stationary cone




    \[\infty\] 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.


    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