This paper presents algorithms for determining magnetic symmetry operations of magnetic crystal structures, identifying magnetic space-group types from a given magnetic space group (MSG), searching for transformations to a Belov–Neronova–Smirnova setting, and symmetrizing the magnetic crystal structures on the basis of the determined MSGs.

An algorithm is presented for an exact calculation of patch frequencies for a family of tilings which can be obtained via dualization.

In the case of small nanocrystals (about 4 nm), with a change in electron beam energy the influence of multiple/inelastic scattering on the background signal of powder electron diffraction patterns is evident. Adopting different background removal approaches at lower (80 kV) and higher (300 kV) electron beam energies, e-PDF (electron pair distribution function) G(r) profiles are extracted. From small-box modelling of the structural parameters related to the iron oxide nanoparticles considered, the applicability of the subtraction procedures is discussed.

Several models for estimating the standard uncertainties of reflection intensities are analysed for refinement against 3D electron diffraction data. A new model is proposed which results in more accurate structure models.

This contribution outlines how to obtain perfect precise colorings of tilings, and illustrates this by obtaining perfect precise colorings of some families of k-valent semiregular planar tilings with k colors.

The face-centered cubic lattice describes the crystalline structure of various materials. Based on two weights assigned to the two types of neighbors among the nodes (representing, e.g., ions), formulae of chamfer distances are computed with the help of the simplex method.

Maximum independent sets in periodic nets are constructed as homomorphic images of Cayley colour graphs. The method applies to zeolites and aluminosilicates.

A method is proposed for transforming unit cells for a group of crystals so that they all appear as similar as possible to a selected cell.

Unit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. A review is presented of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S⁶ and the equivalent three-dimensional complex space C³.

The space C³ is explained in more detail than in the original description. Boundary transformations of the fundamental unit are described in detail. A graphical presentation of the basic coordinates is described and illustrated.