Prospects for mathematical crystallography

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.

Effects of merohedric twinning on the diffraction pattern

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.

Brillouin-zone database on the Bilbao Crystallographic Server

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.

On physical property tensors invariant under line groups

Materials invariant under infinite subsets of line groups are shown to have physical property tensors of the same form.

Direct phasing in femtosecond nanocrystallography. I. Diffraction characteristics

The diffraction of X-ray free-electron laser pulses by protein nanocrystals and the implications for direct phasing are investigated.

Direct phasing in femtosecond nanocrystallography. II. Phase retrieval

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.

Viruses and fullerenes – symmetry as a common thread?

Carbon onions from affine extensions are considered in analogy with virus work.

Alternative approaches to onion-like icosahedral fullerenes

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.

What periodicities can be found in diffraction patterns of quasicrystals?

The structure of quasicrystals is aperiodic; however, their diffraction patterns comprise periodic series of peaks, which can be used to retrieve essential features of the quasicrystalline structure.

Symmetry of helicoidal biopolymers in the frameworks of algebraic geometry: α-helix and DNA structures

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.

Essentials of Crystallography