Isotopy classification of three-dimensional embedded nets.

The article by Power [ (2020), A, 275–301] on the isotopy classification of crystal nets is discussed.

On the origin of crystallinity: a lower bound for the regularity radius of Delone sets.

The mathematical conditions for the origin of long-range order or crystallinity in ideal crystals are one of the very fundamental problems of modern crystallography. It is widely believed that the (global) regularity of crystals is a consequence of `local order', in particular the repetition of local fragments, but the exact mathematical theory of this phenomenon is poorly known. In particular, most mathematical models for quasicrystals, for example Penrose tiling, have repetitive local fragments, but are not (globally) regular.

Polyhedra, complexes, nets and symmetry.

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).