Tables for
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 2.1, p. 169


Th. Hahna and A. Looijenga-Vosb

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Projections of crystallographic symmetry elements

Symmetry element in three dimensionsSymmetry element in projection
Arbitrary orientation
[\!\!\!\!\!\!\!\matrix{{\rm Symmetry\ centre}\!\!\!\!\hfill &\bar{1}\hfill\cr {\rm Rotoinversion\ axis} &\bar{3} \equiv 3 \times \bar{1}\hfill}\bigg\}] Rotation point 2 (at projection of centre)
Parallel to projection direction
Rotation axis 2; 3; 4; 6 Rotation point 2; 3; 4; 6
[\!\matrix{\hbox{Screw axis}\!\!\!\!\hfill& 2_{1}\hfill\cr & 3_{1},3_{2}\hfill\cr & 4_{1},4_{2},4_{3}\hfill\cr & 6_{1},6_{2},6_{3},6_{4},6_{5}\hfill\cr}] [\!\matrix{\hbox{Rotation point}\hfill &\!\!\!2\hfill\cr & \!\!\!3\hfill\cr & \!\!\!4\hfill\cr & \!\!\!6\hfill\cr}]
[\!\matrix{\hbox{Rotoinversion axis}\hfill &\!\!\!\bar{4}\hfill\cr & \!\!\!\bar{6} \equiv 3/m\hfill\cr\cr & \!\!\!\bar{3} \equiv 3 \times \bar{1}\hfill\cr}] [\!\matrix{\hbox{Rotation point}\hfill &\!\!\!4\hfill\cr &\!\!\!3, \hbox{with overlap}\hfill\cr & \quad\!\!\! \hbox{of atoms}\hfill\cr &\!\!\!6\hfill\cr}]
Reflection plane m Reflection line m
Glide plane with [\perp] component Glide line g
Glide plane without [\perp] component Reflection line m
Normal to projection direction
[\!\matrix{\hbox{Rotation axis}\hfill &\!\!\!2\semi\ \!\!\!4\semi \ \!\!\!6\hfill\cr & \!\!\!3\hfill\cr}] [\!\matrix{\hbox{Reflection line } m\hfill\cr \hbox{None}\hfill\cr}]
[\!\matrix{\hbox{Screw axis}\hfill & \!\!\!4_{2}\semi\ 6_{2},6_{4}\hfill\cr & \!\!\!2_{1}\semi\ 4_{1},4_{3}\semi\ 6_{1},6_{3},6_{5}\hfill\cr & \!\!\!3_{1},3_{2}\hfill\cr}] [\!\matrix{\hbox{Reflection line } m\hfill\cr \hbox{Glide line }g\hfill\cr \hbox{None}\hfill\cr}]
[\!\matrix{\hbox{Rotoinversion axis}\hfill &\!\!\!\bar{4}\hfill\cr & \!\!\!\bar{6} \equiv 3/m\hfill\cr\cr\cr &\!\!\!\bar{3} \equiv 3 \times \bar{1}\hfill}] [\!\matrix{\hbox{Reflection line }m \hbox{ parallel to axis}\hfill\cr \hbox{Reflection line }m \hbox{ perpendicular}\hfill\cr\quad\hbox{to axis (through projection of}\hfill\cr\quad\hbox{inversion point)}\hfill\cr \hbox{Rotation point 2 (at projection}\hfill\cr\quad\hbox{of centre)}\hfill\cr}]
Reflection plane m None, but overlap of atoms
Glide plane with glide vector t Translation with translation vector t
The term `with [\perp] component' refers to the component of the glide vector normal to the projection direction.