Cover illustration: Viruses and fullerenes obey similar mathematical constraints. Point arrays derived from affine symmetries provide blueprints both for the organization of material in a virus (here Pariacoto virus, middle) and for the atomic positions in nested fullerene cages called carbon onions (here the carbon onion C60-C240-C540, C60 not shown), showing that affine symmetry is a common thread in biology and chemistry [Dechant et al. (2014). Acta Cryst. A70, 162-167].
Considered as an emerging field, how mathematical crystallography grows during the 21st century may depend on how it addresses demand and attracts recruits within the chemical, physical and mathematical communities.
The effect of merohedric twinning on the reflection conditions and the symmetry of the diffraction pattern is analysed systematically and criteria to confirm or exclude the presence of twinning are presented.
The Brillouin-zone database of the Bilbao Crystallographic Server (https://www.cryst.ehu.es) is presented. Recent improvements and modifications of the database are discussed and illustrated by several examples.
A selective sampling scheme is described to improve the noise tolerance of direct phasing based on shape transform diffraction between Bragg reflections in nanocrystallography using X-ray free-electron lasers.
The carbon atomic positions obtained by a quantum-chemical computation are compared with model ones having six integral indices. In one approach these indices follow from affine extensions of the icosahedral group and in another one from crystallographic scaling transformations.
The chain of algebraic geometry and topology constructions permits one to single out a special class of discrete helicoidal structures. The symmetry of these structures determines the structural parameters of the α-helix and topology of the A, B and Z forms of DNA.