Tetragonal tungsten bronze (TTB) oxides are one of the most important classes of ferroelectrics. Many of these framework structures undergo ferroelastic transformations related to octahedron tilting deformations. Such tilting deformations are closely related to the rigid unit modes (RUMs). This paper discusses the whole set of RUMs in an ideal TTB lattice and possible crystal structures which can emerge owing to the condensation of some of them. Analysis of available experimental data for the TTB-like niobates lends credence to the obtained theoretical predictions.

]]>This article proposes a new theory of X-ray scattering that has particular relevance to powder diffraction. The underlying concept of this theory is that the scattering from a crystal or crystallite is distributed throughout space: this leads to the effect that enhanced scatter can be observed at the `Bragg position' even if the `Bragg condition' is not satisfied. The scatter from a single crystal or crystallite, in any fixed orientation, has the fascinating property of contributing simultaneously to many `Bragg positions'. It also explains why diffraction peaks are obtained from samples with very few crystallites, which cannot be explained with the conventional theory. The intensity ratios for an Si powder sample are predicted with greater accuracy and the temperature factors are more realistic. Another consequence is that this new theory predicts a reliability in the intensity measurements which agrees much more closely with experimental observations compared to conventional theory that is based on `Bragg-type' scatter. The role of dynamical effects (extinction *etc.*) is discussed and how they are suppressed with diffuse scattering. An alternative explanation for the Lorentz factor is presented that is more general and based on the capture volume in diffraction space. This theory, when applied to the scattering from powders, will evaluate the full scattering profile, including peak widths and the `background'. The theory should provide an increased understanding of the reliability of powder diffraction measurements, and may also have wider implications for the analysis of powder diffraction data, by increasing the accuracy of intensities predicted from structural models.

The formerly introduced theoretical *R* values [Henn & Schönleber (2013). *Acta Cryst.* A**69**, 549–558] are used to develop a relative indicator of systematic errors in model refinements, *R*^{meta}, and applied to published charge-density data. The counter of *R*^{meta} gives an absolute measure of systematic errors in percentage points. The residuals (*I*_{o}−*I*_{c})/σ(*I*_{o}) of published data are examined. It is found that most published models correspond to residual distributions that are not consistent with the assumption of a Gaussian distribution. The consistency with a Gaussian distribution, however, is important, as the model parameter estimates and their standard uncertainties from a least-squares procedure are valid only under this assumption. The effect of correlations introduced by the structure model is briefly discussed with the help of artificial data and discarded as a source of serious correlations in the examined example. Intensity and significance cutoffs applied in the refinement procedure are found to be mechanisms preventing residual distributions from becoming Gaussian. Model refinements against artificial data yield zero or close-to-zero values for *R*^{meta} when the data are not truncated and small negative values in the case of application of a moderate cutoff *I*_{o} > 0. It is well known from the literature that the application of cutoff values leads to model bias [Hirshfeld & Rabinovich (1973). *Acta Cryst.* A**29**, 510–513].

The aim of this report is to describe the Seitz notation for symmetry operations adopted by the Commission on Crystallographic Nomenclature as the standard convention for Seitz symbolism of the International Union of Crystallography. The established notation follows the existing crystallographic conventions in the descriptions of symmetry operations.

]]>Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zigzag or helical. A polyhedron or complex is *regular* if its geometric symmetry group is transitive on the flags (incident vertex–edge–face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well known to crystallographers and they are identified explicitly. There are also six infinite families of *chiral* apeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits.

Apart from solving the heavy-atom substructure in proteins and *ab initio* phasing of protein diffraction data at atomic resolution, direct methods have also been successfully combined with other protein crystallographic methods in dealing with diffraction data far below atomic resolution, leading to significantly improved results. In this respect, direct methods provide phase constraints in reciprocal space within a dual-space iterative framework rather than solve the phase problem independently. Applications of this type of direct methods to difficult SAD phasing, model completion and low-resolution phase extension will be described in detail.

Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroup *F*(*N*) of the automorphism group of NC nets (*N*, *T*). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroup *F*(*N*).

The potential of mathematical crystallography as an emerging field is examined from a sociological point of view. Mathematical crystallography is unusual as an emerging field as it is also an old field, albeit scattered, with evidence of continued substantial activity. But its situation is similar to that of an emerging field, so we analyse it as such. Comparisons with past emergent efforts suggest that a new field can grow if given an economic demand for its product and a receptive environment. Developing a field entails developing a sense of identity, developing infrastructure and recruiting practitioners.

]]>In merohedric twinning, the lattices of the individuals are perfectly overlapped and the presence of twinning is not easily detected from the diffraction pattern, especially in the case of inversion twinning (class I). In general, the investigator has to consider three possible structural models: a crystal with space-group type *H* and point group *P*, either untwinned (**H model**) or twinned through an operation *t* in vector space (** t-H model**), and an untwinned crystal with space group

The Brillouin-zone database of the *Bilbao Crystallographic Server* (http://www.cryst.ehu.es) offers **k**-vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so-called reciprocal-space groups and this classification scheme is compared with the classification of Cracknell *et al.* [*Kronecker Product Tables*, Vol. 1, *General Introduction and Tables of Irreducible Representations of Space Groups* (1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space-group representations by specifying the independent parameter ranges of general and special **k** vectors. Guides to the **k**-vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin-zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.

The form of physical property tensors of a quasi-one-dimensional material such as a nanotube or a polymer can be determined from the point group of its symmetry group, one of an *infinite* number of line groups. Such forms are calculated using a method based on the use of trigonometric summations. With this method, it is shown that materials invariant under infinite subsets of line groups have physical property tensors of the same form. For line group types of a family of line groups characterized by an index *n* and a physical property tensor of rank *m*, the form of the tensor for all line group types indexed with *n* > *m* is the same, leaving only a *finite* number of tensor forms to be determined.

X-ray free-electron lasers solve a number of difficulties in protein crystallography by providing intense but ultra-short pulses of X-rays, allowing collection of useful diffraction data from nanocrystals. Whereas the diffraction from large crystals corresponds only to samples of the Fourier amplitude of the molecular transform at the Bragg peaks, diffraction from very small crystals allows measurement of the diffraction amplitudes between the Bragg samples. Although highly attenuated, these additional samples offer the possibility of iterative phase retrieval without the use of ancillary experimental data [Spence *et al.* (2011). *Opt. Express*, **19**, 2866–2873]. This first of a series of two papers examines in detail the characteristics of diffraction patterns from collections of nanocrystals, estimation of the molecular transform and the noise characteristics of the measurements. The second paper [Chen *et al.* (2014). *Acta Cryst.* A**70**, 154–161] examines iterative phase-retrieval methods for reconstructing molecular structures in the presence of the variable noise levels in such data.

X-ray free-electron laser diffraction patterns from protein nanocrystals provide information on the diffracted amplitudes between the Bragg reflections, offering the possibility of direct phase retrieval without the use of ancillary experimental diffraction data [Spence *et al.* (2011). *Opt. Express*, **19**, 2866–2873]. The estimated continuous transform is highly noisy however [Chen *et al.* (2014). *Acta Cryst.* A**70**, 143–153]. This second of a series of two papers describes a data-selection strategy to ameliorate the effects of the high noise levels and the subsequent use of iterative phase-retrieval algorithms to reconstruct the electron density. Simulation results show that employing such a strategy increases the noise levels that can be tolerated.

The principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry. Previous work on affine extensions of the icosahedral group has revealed a new organizational principle in virus structure and assembly. This group-theoretic framework is adapted here to the physical requirements dictated by carbon chemistry, and it is shown that mathematical models for carbon onions can be derived within this affine symmetry approach. This suggests the applicability of affine symmetry in a wider context in nature, as well as offering a novel perspective on the geometric principles underpinning carbon chemistry.

]]>The fullerenes of the C_{60} series (C_{60}, C_{240}, C_{540}, C_{960}, C_{1500}, C_{2160}*etc.*) form onion-like shells with icosahedral *I*_{h} symmetry. Up to C_{2160}, their geometry has been optimized by Dunlap & Zope from computations according to the analytic density-functional theory and shown by Wardman to obey structural constraints derived from an affine-extended *I*_{h} group. In this paper, these approaches are compared with models based on crystallographic scaling transformations. To start with, it is shown that the 56 symmetry-inequivalent computed carbon positions, approximated by the corresponding ones in the models, are mutually related by crystallographic scalings. This result is consistent with Wardman's remark that the affine-extension approach simultaneously models different shells of a carbon onion. From the regularities observed in the fullerene models derived from scaling, an icosahedral infinite C_{60} onion molecule is defined, with shells consisting of all successive fullerenes of the C_{60} series. The structural relations between the C_{60} onion and graphite lead to a one-parameter model with the same Euclidean symmetry *P*6_{3}*mc* as graphite and having a *c*/*a* = τ^{2} ratio, where τ = 1.618… is the golden number. This ratio approximates (up to a 4% discrepancy) the value observed in graphite. A number of tables and figures illustrate successive steps of the present investigation.

The structure of quasicrystals is aperiodic. Their diffraction patterns, however, can be considered periodic. They are composed solely of series of peaks which exhibit a fully periodic arrangement in reciprocal space. Furthermore, the peak intensities in each series define the so-called `envelope function'. A Fourier transform of the envelope function gives an average unit cell, whose definition is based on the statistical distribution of atomic coordinates in physical space. If such a distribution is lifted to higher-dimensional space, it becomes the so-called atomic surface – the most fundamental feature of higher-dimensional analysis.

]]>The chain of algebraic geometry and topology constructions is mapped on a structural level that allows one to single out a special class of discrete helicoidal structures. A structure that belongs to this class is locally periodic, topologically stable in three-dimensional Euclidean space and corresponds to the bifurcation domain. Singular points of its bounding minimal surface are related by transformations determined by symmetries of the second coordination sphere of the eight-dimensional crystallographic lattice *E*_{8}. These points represent cluster vertices, whose helicoid joining determines the topology and structural parameters of linear biopolymers. In particular, structural parameters of the α-helix are determined by the seven-vertex face-to-face joining of tetrahedra with the *E*_{8} non-integer helical axis 40/11 having a rotation angle of 99°, and the development of its surface coincides with the cylindrical development of the α-helix. Also, packing models have been created which determine the topology of the *A*, *B* and *Z* forms of DNA.