A complete set of unstable 3-periodic nets of genus 4 is derived and fully described.

Chains of TO₄ⁿ⁻ tetrahedra are represented as chain graphs in which tetrahedra are represented as vertices and the linkage between tetrahedra is represented as edges. Topologically distinct chain graphs are generated for all possible chain stoichiometries (up to a boundary number of tetrahedra) using the formalisms of graph theory, making possible the comparison of observed chain arrangements with all topologically possible chain arrangements.

The one-to-one correspondence between a set of cosets of the Mathieu group M₁₁, block sets of the Steiner system S(4, 5, 11) and 11-vertex equi-edged triangulated clusters determines symmetry-allowable transformations between different 11-vertex triangulated clusters as automorphisms of the Steiner system S(4, 5, 11). The transformations can describe local reconstructions corresponding to polymorphic transformations in metals.

A growth form and an asymptotic formula for the coordination sequence of the vertex graph of the Ammann–Beenker tiling are obtained.

Labelled quotient graphs can be used to analyse vertex-connectivity in periodic graphs and assign a topological class to crystal structures.

As part of the study of aperiodic tilings and tiling spaces, the frequency module of the hull of a primitive substitution tiling is computed.

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.

A number of theorems and hypotheses regarding local groups in arbitrary Delone sets on the plane and in 3D space are considered. Some of the statements generalize a classical theorem on the impossibility of fivefold symmetry in a lattice.

The synthesis of molecular Borromean rings and other links is an active area of chemical research. Hypothetical 1-, 2- and 3-periodic molecular structures are described with the Borromean property that, although no two rings or chains are linked, the structures are interlocked and do not fall apart.

A computer program that implements a new approach to quantifying distortions in coordination polyhedra, molecules and cages by optimizing dynamic model shapes is presented.