Olaf Delgado-Friedrichs

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.

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 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.

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.

Isohedral simple tilings: binodal and by tiles with ≤16 faces

All isohedral simple tilings of 3D Euclidian space by tiles with </=16 faces have been determined. There are no such tilings by polyhedra with less than 14 faces. For simple polyhedra with 14, 15 and 16 faces, there are respectively 10, 65 and 434 that fill space and a total of 23, 136 and 710 distinct tilings. The tilings by 14-face polyhedra are described in detail as are binodal (vertex 2-transitive) isohedral simple tilings.

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.

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.

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.

Percolating length scales from topological persistence analysis of micro-CT images of porous materials

This work was funded in part by ARCgrant DP110102888 (to A.P.S. and V.R.),and ARC Future FellowshipsFT100100470 (A.P.S.) and FT140100604(V.R.). Financial support was alsoprovided by the member companiesof the ANU/UNSW DigicoreConsortium, who have alsocontributed samples for this study.