A strategy is described for regularizing ill posed structure and nanostructure scattering inverse problems (*i.e.* structure solution) from complex material structures. This paper describes both the philosophy and strategy of the approach, and a software implementation, DiffPy Complex Modeling Infrastructure (*DiffPy-CMI*).

The architecture of infinite structures with non-crystallographic symmetries can be modelled *via* aperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. This paper presents a group theoretical method for the construction of finite nested point sets with non-crystallographic symmetry. Akin to the construction of quasicrystals, a non-crystallographic group *G* is embedded into the point group of a higher-dimensional lattice and the chains of all *G*-containing subgroups are constructed. The orbits of lattice points under such subgroups are determined, and it is shown that their projection into a lower-dimensional *G*-invariant subspace consists of nested point sets with *G*-symmetry at each radial level. The number of different radial levels is bounded by the index of *G* in the subgroup of . In the case of icosahedral symmetry, all subgroup chains are determined explicitly and it is illustrated that these point sets in projection provide blueprints that approximate the organization of simple viral capsids, encoding information on the structural organization of capsid proteins and the genomic material collectively, based on two case studies. Contrary to the affine extensions previously introduced, these orbits endow virus architecture with an underlying finite group structure, which lends itself better to the modelling of dynamic properties than its infinite-dimensional counterpart.

Tilings with a singular point are obtained by applying conformal maps on regular tilings of the Euclidean plane and their symmetries are determined. The resulting tilings are then symmetrically colored by applying the same conformal maps on colorings of regular tilings arising from sublattice colorings of the centers of the tiles. In addition, conditions are determined in order that the coloring of a tiling with singularity that is obtained in this manner is perfect.

]]>Uniqueness of the phase problem in macromolecular crystallography, and its relationship to the case of single particle imaging, is considered. The crystallographic problem is characterized by a constraint ratio that depends only on the size and symmetry of the molecule and the unit cell. The results are used to evaluate the effect of various real-space constraints. The case of an unknown molecular envelope is considered in detail. The results indicate the quite wide circumstances under which *ab initio* phasing should be possible.

Entanglements of two-dimensional honeycomb nets are constructed from free tilings of the hyperbolic plane () on triply periodic minimal surfaces. The 2-periodic nets that comprise the structures are guaranteed by considering regular, rare free tilings in . This paper catalogues an array of entanglements that are both beautiful and challenging for current classification techniques, including examples that are realized in metal–organic materials. The compactification of these structures to the genus-3 torus is considered as a preliminary method for generating entanglements of finite θ-graphs, potentially useful for gaining insight into the entanglement of the periodic structure. This work builds on previous structural enumerations given in *Periodic entanglement* Parts I and II [Evans *et al.* (2013). *Acta Cryst.* A**69**, 241–261; Evans *et al.* (2013). *Acta Cryst.* A**69**, 262–275].

Based on the rigorous Green function formalism to describe the grazing-incidence small-angle X-ray scattering (GISAXS) problem, a system of two linked integral equations is derived with respect to amplitudes of the reflected and transmitted plane **q**-eigenwaves (*eigenstate* functions) propagating through two homogeneous media separated from each other by a rough surface interface. To build up the coupled solutions of these basic equations beyond the perturbation theory constraint 2*k*σθ_{0} < 1, a simple iteration procedure is proposed as opposed to the self-consistent wave approach [Chukhovskii (2011). *Acta Cryst.* A**67**, 200–209; Chukhovski (2012). *Acta Cryst.* A**68**, 505–512]. Using the first-order iteration, analytical expressions for the averaged specular and non-specular scattering intensity distributions have been obtained. These expressions are further analysed in terms of the GISAXS parameters {*k*, θ, θ_{0}} and surface finish ones , where θ and θ_{0} are the scattering and incidence angles of the X-rays, respectively, σ is the root-mean-square roughness, is the correlation length, *h* is the fractal surface model index, *k* = 2π/λ, and λ is the X-ray wavelength. A direct way to determine the surface finish parameters from the experimental specular and diffuse scattering indicatrix scan data is discussed for an example of GISAXS measurements from rough surfaces of α-quartz and CdTe samples.

A correction is made to the article by Oishi-Tomiyasu [Acta Cryst. (2013), A69, 603610].

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