
Figure 9
The technique of the `linear' projection of Neumann illustrated using a vesuvianite crystal as an example. We choose a system of axes including the principal one, such that the origin coincides with the center. The principal axis is simultaneously the `zone axis'. Then we move all the facets parallel to it  facets that together constitute the `zone'  such that they pass through that `zone axis'. Obviously, parallel facets will coincide. The displaced planes cut the horizontal projection plane above the crystal in the form of a fanshaped star through the intersection point of the `zone axis'. The other `zones' that determine the nature of the crystal manifest themselves as groups of parallel lines. Since, in our case, the a and b axes are equivalent, given the geometry of the vesuvianite crystal, we use two a's in characterizing the proportions. In the indicated fan we therefore find back the intersection lines of the following facets: a:a:c, a':a:c, a:a:c, a:a':c, a:a:c, a':a':c, a:a':c and a':a:c. 