**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 fan-shaped 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*. |