Three-periodic nets, tilings and surfaces. A short review and new results.

A brief introductory review is provided of the theory of tilings of 3-periodic nets and related periodic surfaces. Tilings have a transitivity [p q r s] indicating the vertex, edge, face and tile transitivity. Proper, natural and minimal-transitivity tilings of nets are described. Essential rings are used for finding the minimal-transitivity tiling for a given net. Tiling theory is used to find all edge- and face-transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity [1 1 1 1], [1 1 1 2], [2 1 1 1] and [2 1 1 2], respectively. These are all minimal-transitivity tilings. This work identifies the 3-periodic surfaces defined by the nets of the tiling and its dual and indicates how 3-periodic nets arise from tilings of those surfaces.

Isogonal non-crystallographic periodic graphs based on knotted sodalite cages.

This work considers non-crystallographic periodic nets obtained from multiple identical copies of an underlying crystallographic net by adding or flipping edges so that the result is connected. Such a structure is called a `ladder' net here because the 1-periodic net shaped like an ordinary (infinite) ladder is a particularly simple example. It is shown how ladder nets with no added edges between layers can be generated from tangled polyhedra. These are simply related to the zeolite nets SOD, LTA and FAU. They are analyzed using new extensions of algorithms in the program Systre that allow unambiguous identification of locally stable ladder nets.

Surface embeddings of the Klein and the Möbius-Kantor graphs.

This paper describes an invariant representation for finite graphs embedded on orientable tori of arbitrary genus, with working examples of embeddings of the Möbius-Kantor graph on the torus, the genus-2 bitorus and the genus-3 tritorus, as well as the two-dimensional, 7-valent Klein graph on the tritorus (and its dual: the 3-valent Klein graph). The genus-2 and -3 embeddings describe quotient graphs of 2- and 3-periodic reticulations of hyperbolic surfaces. This invariant is used to identify infinite nets related to the Möbius-Kantor and 7-valent Klein graphs.

Edge-2-transitive trinodal polyhedra and 2-periodic tilings.

All trinodal, edge-2-transitive polyhedra and 2-periodic tilings are enumerated and described. These are of special interest for the design and synthesis of materials such as metal-organic polyhedra and frameworks.

2-Periodic self-dual tilings.

2-Periodic self-dual tilings are important in fields ranging from crystal chemistry to mathematical physics. They have been systematically enumerated using combinatorial tiling theory, and 1, 5 and 62 uninodal, binodal and trinodal self-dual tilings have been found. This paper illustrates all uninodal and binodal self-dual tilings and selected trinodal self-dual tilings. Most of these structures are described for the first time.

Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory.

We show how discrete Morse theory provides a rigorous and unifying foundation for defining skeletons and partitions of grayscale digital images. We model a grayscale image as a cubical complex with a real-valued function defined on its vertices (the voxel values). This function is extended to a discrete gradient vector field using the algorithm presented in Robins, Wood, Sheppard TPAMI 33:1646 (2011). In the current paper we define basins (the building blocks of a partition) and segments of the skeleton using the stable and unstable sets associated with critical cells. The natural connection between Morse theory and homology allows us to prove the topological validity of these constructions; for example, that the skeleton is homotopic to the initial object. We simplify the basins and skeletons via Morse-theoretic cancellation of critical cells in the discrete gradient vector field using a strategy informed by persistent homology. Simple working Python code for our algorithms for efficient vector field traversal is included. Example data are taken from micro-CT images of porous materials, an application area where accurate topological models of pore connectivity are vital for fluid-flow modelling.

Nets with collisions (unstable nets) and crystal chemistry.

Nets in which different vertices have identical barycentric coordinates (i.e. have collisions) are called unstable. Some such nets have automorphisms that do not correspond to crystallographic symmetries and are called non-crystallographic. Examples are given of nets taken from real crystal structures which have embeddings with crystallographic symmetry in which colliding nodes either are, or are not, topological neighbors (linked) and in which some links coincide. An example is also given of a crystallographic net of exceptional girth (16), which has collisions in barycentric coordinates but which also has embeddings without collisions with the same symmetry. In this last case the collisions are termed unforced.

Simple tilings by polyhedra with five- and six-sided faces.

Thirteen tilings of space by simple polyhedra with five- and six-sided faces ('fullerenes') are reported in which there are up to 11 kinds of vertex (vertex 11-transitive). All tilings contain dodecahedra and one or more of nine other kinds of tile. The duals are tilings by tetrahedra and include the four simplest of the known Frank-Kasper intermetallic structure phases. A fifth structure involving just the Frank-Kasper coordination polyhedra has a higher average coordination number than any known or postulated Frank-Kasper phase.

Space fullerenes: a computer search for new Frank-Kasper structures.

A Frank-Kasper structure is a 3-periodic tiling of the Euclidean space E3 by tetrahedra such that the vertex figure of any vertex belongs to four specified patterns with, respectively, 20, 24, 26 and 28 faces. Frank-Kasper structures occur in the crystallography of metallic alloys and clathrates. A new computer enumeration method has been devised for obtaining Frank-Kasper structures of up to 20 cells in a reduced fundamental domain. Here, the 84 obtained structures have been compared with the known 27 physical structures and the known special constructions by Frank-Kasper-Sullivan, Shoemaker-Shoemaker, Sadoc-Mosseri and Deza-Shtogrin.

Edge-transitive lattice nets.

Lattice nets have one vertex in the topological unit cell. Some two- and three-periodic lattice nets with one kind of edge (edge-transitive) are described. Simple expressions for the topological density of the two-periodic nets are found empirically. Thirteen infinite families of three-periodic cubic lattice nets and hexagonal, trigonal and tetragonal families are identified.

Three-periodic nets and tilings: natural tilings for nets.

Rules for determining a unique natural tiling that carries a given three-periodic net as its 1-skeleton are presented and justified. A computer implementation of the rules and their application to tilings for zeolite nets and for the nets of the RCSR database are described.

Three-periodic tilings and nets: face-transitive tilings and edge-transitive nets revisited.

Systematic generation of face-transitive tilings by size of Delaney-Dress symbol has recovered by dualization all the edge-transitive nets previously described and has led to the discovery of six new binodal edge-transitive nets which are described and illustrated.

Taxonomy of periodic nets and the design of materials.

The concept of a natural tiling for a periodic net is introduced and used to derive a transitivity associated with the structure. It is accordingly shown that the transitivity provides a useful method of classifying polyhedra and nets. For design of materials to serve as targets for synthesis, structures with one kind of edge (edge transitive) are particularly important. Edge-transitive polyhedra, layers and 3-periodic nets are then described. Some other nets of special importance in crystal chemistry are also identified.

Three-periodic nets and tilings: edge-transitive binodal structures.

28 three-periodic nets with two kinds of vertex and one kind of edge are identified. Some of their crystallographic properties and their natural tilings are described. Restrictions on site symmetry and coordination number of such nets are discussed and examples of their occurrence in crystal structures are given.

On a simple tiling of Deza and Shtogrin.

Some properties of an exceptional simple tiling by fullerenes, first described by Deza & Shtogrin [Southeast Asian Math. Soc. Bull. (1999), 23, 1-11], are presented.

Hypothetical binodal zeolitic frameworks.

Hypothetical binodal zeolitic structures (structures containing two kinds of tetrahedral sites) were systematically enumerated using tiling theory and characterized by computational chemistry methods. Each of the 109 refineable topologies based on "simple tilings" was converted into a silica polymorph and its energy minimized using the GULP program with the Sanders-Catlow silica potential. Optimized structural parameters, framework energies relative to alpha-quartz and volumes accessible to sorption have been calculated. Eleven of the 30 known binodal topologies listed in the Atlas of Zeolite Framework Types were found, leaving 98 topologies that were unknown previously. The chemical feasibility of each structure as a zeolite was evaluated by means of a feasibility factor derived from the correlation between lattice energy and framework density. Structures are divided into 15 families, based on common structural features. Many "feasible" structures contain only small pores. Several very open structures were also enumerated, although they contain three-membered rings which are thermodynamically disfavoured and not found in conventional zeolites. We believe that such topologies may be realizable as framework materials, but with different elemental compositions to those normally associated with zeolites.

Isohedral simple tilings: binodal and by tiles with

All isohedral simple tilings of 3D Euclidian space by tiles with

Reticular chemistry: occurrence and taxonomy of nets and grammar for the design of frameworks.

The structures of all 1127 three-periodic extended metal-organic frameworks (MOFs) reported in the Cambridge Structure Database have been analyzed, and their underlying topology has been determined. It is remarkable that among the almost infinite number of net topologies that are available for MOFs to adopt, only a handful of nets are actually observed. The discovery of this inversion between expected and observed nets led us to deduce a system of classification "taxonomy" for interpreting and rationalizing known MOF structures, as well as those that will be made in future. The origin of this inversion is attributed to the different modes with which MOF synthesis has been approached. Specifically, three levels of complexity are defined that embody rules "grammar" for the design of MOFs and other extended structures. This system accounts for the present proliferation of MOF structures of high symmetry nets, but more importantly, it provides the basis for designing a building block that "codes" for a specific structure and, indeed, only that structure.

Three-periodic nets and tilings: minimal nets.

The 15 3-periodic minimal nets of Beukemann & Klee [Z. Kristallogr. (1992), 201, 37-51] have been examined. Seven have collisions in barycentric coordinates and are self-entangled. The other eight have natural tilings. Five of these tilings are self-dual and the nets are the labyrinth nets of the P, G, D, H and CLP minimal surfaces of genus 3. Twelve ways have been found for subdividing a cube into smaller tiles without introducing new vertices. Duals of such tilings with one vertex in the primitive cell have nets that are one of the minimal nets. Minimal nets without collisions are uniform.

Chemical evaluation of hypothetical uninodal zeolites.

Optimized structural parameters, framework energies relative to alpha-quartz, and volumes accessible to sorption have been calculated for the systematically enumerated hypothetical uninodal zeolitic structures (structures in which all tetrahedral sites are equivalent). The structures were treated as silica polymorphs, and their energies were minimized using the GULP program with the Sanders-Catlow silica potential. Results are given for 164 structures, which include all 21 known uninodal zeolites, two known minerals (tridymite and cristobalite), and 78 unknown zeolite topologies. Twenty-three hypothetical structures were identified as chemically feasible. Complete structural information is provided, and several structures are discussed in detail. The results will assist in the design of new synthetic routes and in the identification of newly synthesized materials.