Figure 9

Collapsing the ADD of 500 sentinel patterns for a scatterer, whose diffraction volumes is centro-symmetric and has octahedral symmetry, into the fundamental domain: (A–D). Starting clockwise from (A), which shows a projection of the ADD onto two components of each quaternion (\(Q = (Q_0, Q_1, Q_2, Q_3)\)), we collapsed the points related by centro-symmetry (since 2D patterns have sufficiently low resolution) to obtain a sharper distribution in (B). The red disk throughout the panels represent the average quaternion \({\overline{Q}}_{AB}\) of the ADD. In (C), we rotate the ADD such that \({\overline{Q}}_{AB} = (1,0,0,0)\) for clarity. The histogram of the ADD vs \(Q_0\) is shown above panel (C), can sometimes reveal the flavor of symmetry in W. Finally, using the particle’s known symmetry group operations we can fold the ADD into the fundamental domain in (D).