In crystal geometry, the most important aspect of the lattice is the angular relationship between various planes and other symmetry elements, not the relative translational position of planes. The importance of this is demonstrated by the expression of the crystal structure through the outside planes of a macrocrystal, for example as in the fluorite crystal shown here:
(It should be noted that this is not common. To grow crystals so that their external structure represents their internal structure requires a very controlled set of conditions. This makes it only the more important to understand angular relationships, as they cannot be seen simply by looking at a crystal.)
We need to be able to describe these angular relationships in an easily understandable manner, and so we use the stereographic projection, which presents a 3D structure on the surface of a sphere. This can be extrapolated to a 2D structure, which allows direct measurement of angles between various rotational axes or normals to planes, and this is very useful. This has a long history, being first used by Neumann in 1823.
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