In this connection the point where the normal to a face cuts the sphere is called the pole of the face. This symmetry can then also be recognised in the stereographic projection of these points.įigure 2: The spherical projection of a crystal. ![]() The points of intersection with the sphere represent the faces of the crystal in terms of direction, entirely uninfluenced by their relative sizes, and the symmetry of the arrangement of these points on the surface of the sphere reveals the true symmetry of the crystal, whether or not it be well-formed. If the crystal is imagined to lie within a sphere that is centred on some arbitrary point inside the crystal, then normals to the faces can be constructed from this point and extended to cut the sphere (Fig. However these accidents do not affect the angles between the faces-only their relative sizes. However, real crystals are very rarely well-formed in this way accidents of crystal growth such as unequal access of the crystallising liquid, or interference by adjacent crystals or other objects, may have impeded the growth of the crystal in certain directions in such a way as to mask the true symmetry. On a `well-formed` crystal all the faces that are related to one another by the symmetry of the crystal structure are developed to an equal extent, and the shape of the crystal therefore reveals its true symmetry. The most common application is that of representing the angles between the faces of a crystal, and the symmetry relations between them. ![]() Figure 1: The point p is the stereographic projection of the point P on the sphere.
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