[IUCr Home Page][Commission Home Page]

next up previous
Next: 6. Calculations Based on the Stereogram Up: The Stereographic Projection Previous: 4. The Use of Stereographic Nets

5. Representation of Symmetry on the Stereographic Projection

A symmetry plane of a crystal necessarily cuts the spherical projection of the crystal along a great circle. It is therefore represented on the stereogram by a line which will be either the primitive, a diameter of the primitive, or one of the (nearly) vertical arcs of the Wulff net (after rotation into an appropriate orientation). It is represented by a thick line. The operation of reflection of a point p in such a reflection plane to $p^{\prime}$ can be carried out on the Wulff net. Rotate the stereogram so that the line representing the mirror plane coincides with a great circle on the net: then count divisions from p along a small circle until the representation of the mirror is reached. The position of $p^{\prime}$ is then a further equal number of divisions along the small circle beyond the mirror. If the primitive is reached before $p^{\prime}$ then continue counting divisions inwards until the total is correct, but mark the position $\circ$ instead of $\times$.

An axis of crystallographic symmetry (2-, 3-, 4- or 6-fold) necessarily intersects the sphere in a point which can be represented by its projection on the stereogram; and marked with the appropriate symbol (sym1.gif respectively). The same is also true for the direction of the component rotation axis of an axis of rotation-inversion $\overline{3}$, $\overline{4}$ or $\overline{6}$ (marked as sym2.gif respectively). The operation of an n-fold symmetry axis on a point p is to rotate p through graduations equivalent to 360$^{\circ}$/n along a small circle round the pole of the axis. If the latter is at the centre of the stereogram then this is a straight-forward rotation of 360$^{\circ}$/n. If the pole of the symmetry axis lies on the primitive, then place the stereogram so that this point lies at the bottom of the Wulff net. Increments of 360$^{\circ}$/n may then be counted in terms of the graduations along the small circle through p. When the primitive is reached counting continues as the path is retraced but the next point is plotted as $\circ$ instead of $\times$ (or vice-versa).

If the symmetry axis is in a more general position it would seem that one would require a stereographic net of the kind shown in Fig. 13 with the intersection of the meridians at the position of the symmetry axis. However, it is perfectly possible to use the Wulff net in the following way, repositioning the stereogram on the net as necessary.

(i) Measure the angular distance from the axis of symmetry (s) of the point (p) that is to be repeated by the symmetry.

(ii) Construct the small circle (l) of this radius about s.

(iii) Construct the circle (e) of radius 90$^{\circ}$ about s. To construct this place s on a diameter of the net, find the point 90$^{\circ}$ away along that diameter and the two ends of the perpendicular diameter, and draw a circle through these three points. Although constructed as a small circle, this is in fact a great circle, so that angular distances can be measured along it.

(iv) Mark the intersection of e with the great circle through s and p. Count increments of 360$^{\circ}$/n from this point along e. From each of these incremental points find the great circle that goes through s; the points in which these great circles cut l are the points in which p is repeated by the axis of symmetry.

If the poles of the faces of a crystal of unknown symmetry are plotted in an arbitrary orientation on a stereogram it will often be possible to recognise the orientation of the symmetry elements. The stereogram can then be re-plotted in one of the standard orientations, shown in text-books of morphological crystallography to demonstrate the symmetry of the seven crystal systems and the thirty two crystal classes.


next up previous
Next: 6. Calculations Based on the Stereogram Up: The Stereographic Projection Previous: 4. The Use of Stereographic Nets

Copyright © 1984, 1997 International Union of Crystallography

IUCr Webmaster