The intrinsic group-subgroup structures of the Diamond and Gyroid minimal surfaces in their conventional unit cells.

The intrinsic, hyperbolic crystallography of the Diamond and Gyroid minimal surfaces in their conventional unit cells is introduced and analysed. Tables are constructed of symmetry subgroups commensurate with the translational symmetries of the surfaces as well as group-subgroup lattice graphs.

Evolution of local motifs and topological proximity in self-assembled quasi-crystalline phases.

Using methods from the field of topological data analysis, we investigate the self-assembly and emergence of three-dimensional quasi-crystalline structures in a single-component colloidal system. Combining molecular dynamics and persistent homology, we analyse the time evolution of persistence diagrams and particular local structural motifs. Our analysis reveals the formation and dissipation of specific particle constellations in these trajectories, and shows that the persistence diagrams are sensitive to nucleation and convergence to a final structure.

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).

Trading spaces: building three-dimensional nets from two-dimensional tilings.

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S (2)), Euclidean (E (2)) and hyperbolic (H (2)) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.

Finding auxetic frameworks in periodic tessellations.

It appears that most models for micro-structured materials with auxetic deformations were found by clever intuition, possibly combined with optimization tools, rather than by systematic searches of existing structure archives. Here we review our recent approach of finding micro-structured materials with auxetic mechanisms within the vast repositories of planar tessellations. This approach has produced two previously unknown auxetic mechanisms, which have Poisson's ratio νss=-1 when realized as a skeletal structure of stiff incompressible struts pivoting freely at common vertices.

Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images.

We present an algorithm for determining the Morse complex of a two or three-dimensional grayscale digital image. Each cell in the Morse complex corresponds to a topological change in the level sets (i.e.

Betti number signatures of homogeneous Poisson point processes.

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: beta{0} is the number of connected components and beta{k} effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centered spheres with radius alpha.