Cylindrical system

Système cylindrique (Fr) Sistema cilindrico (It).

Definition

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
${A_\infty \over M}C$	${\bar \infty}$	$C_{\infty h} \equiv S_{\infty} \equiv C_{\infty i}$	$\infty$	rotating finite cylinder
$A_\infty \infty A_2$	$\infty 2$	$D_{\infty }$	$\infty$	finite cylinder submitted to equal and opposite torques
$A_\infty M$	$\infty m$	$C_{\infty v}$	$\infty$	stationary cone
${A_\infty \over M} {\infty A_2 \over \infty M} C$	${\bar \infty}m \equiv {\bar \infty} {2\over m}$	$D_{\infty h} \equiv D_{\infty d}$	$\infty$	stationary finite cylinder

Note that $A_\infty M$ represents the symmetry of a force, or of an electric field and that ${A_\infty \over M}C$ represents the symmetry of a magnetic field (Curie 1894), while ${A_\infty \over M} {\infty A_2 \over \infty M} C$ 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

Curie laws
spherical system
Section 3.2.1.4 of International Tables of Crystallography, Volume A, 6^th edition
Section 1.1.4 of International Tables of Crystallography, Volume D