the stereographic projection

How do you draw the three-dimensional symmetry of a crystal on a flat sheet of paper without distortion tricks? The stereographic projection is the standard answer. Imagine the crystal shrunk to the centre of a transparent globe; each face or symmetry axis is marked as a point where its direction pierces the sphere. Then project every point in the top half down to the south pole, and read off where the line crosses the equatorial disc. The result is a flat map, a stereogram, of directions in the crystal.

The projection has a useful property: it is conformal, meaning it preserves angles. Two directions 30 degrees apart on the sphere stay 30 degrees apart on the paper, so the true angular relationships between crystal faces are read directly. By convention, points from the upper hemisphere are drawn as dots and points from the lower hemisphere as open circles, both landing inside one circle (the primitive). Symmetry elements get standard marks: a filled hexagon for a 6-fold axis, a solid line for a mirror, an open circle at the centre for an inversion.

Stereograms are how the 32 point groups are traditionally displayed in the International Tables, one glance shows every axis and mirror and how the general points multiply. Beyond textbooks they are working tools: metallurgists plot crystal orientations and slip systems on them, and pole figures (used to describe texture in rolled metals) are stereographic projections of where particular plane-normals point. It is the crystallographer's map projection, the same idea as flattening the globe onto a chart.