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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.

The absolute permeability of porous media is an important parameter for various technological applications ranging from ground water hydrology to hydrocarbon recovery to microfluidics. We investigate the max-flow min-cut theorem which states that the maximum flow through any network is exactly the sum of the edge weights that define the minimum cut. We hypothesize that the min-cut can be related to the absolute permeability of a network. We find correspondence between the absolute permeability measured from pore network modeling (${K}_{p\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}m}$) to that measured from the min-cut (${K}_{e\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}m}$) and provide a means to identify structural regions that result in significant energy dissipation.

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. Identification of local motifs allows quantification of the similarities between the final structures in a topological sense. This analysis reveals a continuous variation with density between crystalline clathrate, quasi-crystalline, and disordered phases quantified by ‘topological proximity’, a visualization of the Wasserstein distances between persistence diagrams. From a topological perspective, there is a subtle, but direct connection between quasi-crystalline, crystalline and disordered states. Our results demonstrate that topological data analysis provides detailed insights into molecular self-assembly.

We gratefully acknowledge funding from the member companies of the ANU/UNSW Digicore Research Consortium, as well as the Australian Research Council. Adrian Sheppard is supported by Discovery Project DP160104995, Vanessa Robins is supported by ARC Future Fellowship FT140100604, and Anna Herring is supported by ARC Discovery Early Career Fellowship DE180100082.

Suppose an orientation-preserving action of a finite group G on the closed surface \(\Sigma _g\) of genus \(g>1\) extends over the 3-torus \(T^3\) for some embedding \(\Sigma _g\subset T^3\). Then \(|G|\le 12(g-1)\), and this upper bound \(12(g-1)\) can be achieved for \(g=n^2+1, 3n^2+1, 2n^3+1, 4n^3+1, 8n^3+1, n\in {\mathbb {Z}}_+\). The surfaces in \(T^3\) realizing a maximal symmetry can be either unknotted or knotted. Similar problems in the non-orientable category are also discussed. The connection with minimal surfaces in \(T^3\) is addressed and the situation when the maximally symmetric surfaces above can be realized by minimal surfaces is identified.

The mechanical properties of closed-cell aluminium foams are governed by their geometrical and topological evolution during impact. Here we non-destructively investigate the deformation mechanisms of a closed-cell aluminium foam sample at the cell scale. The sample has been compressed with 21 interrupted drop-weight impacts at a nominal strain rate of 40 s−1 and the post-impacted sample has been imaged at four pertinent strain states with high-resolution X-ray micro-computed tomography (XCT). Moreover, a number of qualitative and quantitative structural analyses are carried out using advanced 3D image analyses to understand the effect of foam geometry/topology on its mechanical response. Our results show that the deforming microstructure creates strong correlations across a range of geometrical, topological and shape characteristics. Quantitative image analyses of the sample at four different strain states reveal that the regions with large void fractions predominantly undergo collapse and subsequently reduces the structural heterogeneity. Further, we show that the deformation mechanism leaves topological signatures in the evolving microstructure, which can be used to better understand the mechanical response of the sample at various stages of impact-deformation. We demonstrate here, for the first time, how the topological quantities can be used to explore the foam deformation mechanisms and to correlate the deformation with mechanical response during impact.

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.

The mechanism of crystallisation in highly dissipative materials such as foams or granular materials is still widely unknown. In macroscopic granular materials high levels of energy need to be injected to overcome the natural propensity of these dissipative materials to form amorphous structures [1, 2]. The transition from disordered to ordered packings in such systems triggers a wide range of geometrical, topological and mechanical changes at multi length scales [3]. Formation of cavities and patterns by aggregates of grains and their evolution during this transition requires a complete topological description of the system. Here, crystallisation of three-dimensional packings of frictional spheres is studied at the grain scale with x-ray tomography. Using a novel and powerful topological tool, Persistent Homology, we describe the complete formation process of perfect tetrahedral and octahedral patterns: the two building blocks of FCC and HCP crystalline arrangements. Additionally we present possible and allowable deformations of these components that accurately reproduce the main topological features of the system. These results give new insights into the crystallisation of these highly dissipative materials.

The concept of an orbifold is particularly suited to classification and enumeration of crystalline groups in the euclidean (flat) plane and its elliptic and hyperbolic counterparts. Using Conway's orbifold naming scheme, this article explicates conventional point, frieze and plane groups, and describes the advantages of the orbifold approach, which relies on simple rules for calculating the orbifold topology. The article proposes a simple taxonomy of orbifolds into seven classes, distinguished by their underlying topological connectedness, boundedness and orientability. Simpler `crystallographic hyperbolic groups' are listed, namely groups that result from hyperbolic sponge-like sections through three-dimensional euclidean space related to all known genus-three triply periodic minimal surfaces (i.e.theP,D,Gyroid,CLPandHsurfaces) as well as the genus-fourI-WPsurface.

High-symmetry free tilings of the two-dimensional hyperbolic plane ({\bb H}^2) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.

Ordered arrays of cylinders, known as rod packings, are now widely used in descriptions of crystalline structures. These are generalized to include crystallographic packed arrays of filaments with circular cross sections, including curvilinear cylinders whose central axes are generic helices. A suite of the simplest such general rod packings is constructed by projecting line patterns in the hyperbolic plane ({\bb H}^2) onto cubic genus-3 triply periodic minimal surfaces in Euclidean space ({\bb E}^3): the primitive, diamond and gyroid surfaces. The simplest designs correspond to `classical' rod packings containing conventional cylindrical filaments. More complex packings contain three-dimensional arrays of mutually entangled filaments that can be infinitely extended or finite loops forming three-dimensional weavings. The concept of a canonical `ideal' embedding of these weavings is introduced, generalized from that of knot embeddings and found algorithmically by tightening the weaving to minimize the filament length to volume ratio. The tightening algorithm builds on the SONO algorithm for finding ideal conformations of knots. Three distinct classes of weavings are described.

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.

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. One of these, baptized Triangle-Square Wheels, has been produced as a linear-elastic cellular structure from Ti-6Al-4V alloy by selective electron beam melting. Its linear-elastic properties were measured by tensile experiments and yield an effective Poisson's ratio νLE≈-0.75, also in agreement with finite element modeling. The similarity between the Poisson's ratios νSS of the skeletal structure and νLE of the linear-elastic cellular structure emphasizes the fundamental role of geometry for deformation behavior, regardless of the mechanical details of the system. The approach of exploiting structure archives as candidate geometries for auxetic materials also applies to spatial networks and tessellations and can aid the quest for inherently three-dimensional auxetic mechanisms.

Crystalline frameworks in 3D Euclidean space can be constructed by projecting tilings of 2D hyperbolic space onto three-periodic minimal surfaces, giving surface reticulations. The technique involves Delaney–Dress tiling theory, group theory, differential and non-Euclidean geometry. Preliminary results of this approach, found at http://epinet.anu.edu.au , are discussed and compared with other approaches.

Our study of the gyroid minimal surface has revealed that there are two distinct covering maps from the hyperbolic plane onto the surface that respect its intrinsic symmetries. We show that if a decoration of H 2 is chiral, the projection of this pattern via the two covering maps gives rise to distinct structures in E 3 .

We describe a systematic approach to generate nets that arise from decorations of periodic minimal surfaces. Such surfaces are covered by the hyperbolic plane, in the same way that the euclidean plane covers a cylinder. Thus, a symmetric hyperbolic network can be wrapped onto an appropriate minimal surface to obtain a 3D periodic net. This requires symmetries of the hyperbolic net to match the symmetries of the minimal surface. Here, we tabulate all such symmetry groups that are compatible with the H minimal surface. c

Many crystalline networks can be viewed as decorations of triply periodic minimal surfaces. Such surfaces are covered by the hyperbolic plane in the same way that the Euclidean plane covers a cylinder. Thus, a symmetric hyperbolic network can be wrapped onto an appropriate minimal surface to obtain a 3d periodic net. This requires symmetries of the hyperbolic net to match the symmetries of the minimal surface. We describe a systematic algorithm to find all the hyperbolic symmetries that are commensurate with a given minimal surface, and the generation of simple 3d nets from these symmetry groups.

We consider the role of topology on drainage relative permeabilities derived from network models. We describe the topological properties of rock networks derived from a suite of tomographic images of Fontainbleau sandstone (Lindquist et al., 2000, J. Geophys. Res.105B, 21508). All rock networks display a broad distribution of coordination number and the presence of long-range topological bonds. We show the importance of accurately reproducing sample topology when deriving relative permeability curves from the model networks. Comparisons between the relative permeability curves for the rock networks and those computed on a regular cubic lattice with identical geometric characteristics (pore and throat size distributions) show poor agreement. Relative permeabilities computed on regular lattices and on diluted lattices with a similar average coordination number to the rock networks also display poor agreement. We find that relative permeability curves computed on stochastic networks which honour the full coordination number distribution of the rock networks produce reasonable agreement with the rock networks. We show that random and regular lattices with the same coordination number distribution produce similar relative permeabilities and that the introduction of longer-range topological bonds has only a small effect. We show that relative permeabilities for networks exhibiting pore–throat size correlations and sizes up to the core-scale still exhibit a significant dependence on network topology. The results show the importance of incorporating realistic 3D topologies in network models for predicting multiphase flow properties.

Here we present an experimental and numerical investigation on the grain-scale geometrical and mechanical properties of partially crystallized structures made of macroscopic frictional grains. Crystallization is inevitable in arrangements of monosized hard spheres with packing densities exceeding Bernal's limiting density ϕ(Bernal)≈0.64. We study packings of monosized hard spheres whose density spans over a wide range (0.59

Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the (persistent homology) rank functions. For a point pattern $X$ we construct a filtration of spaces by taking the union of balls of radius $a$ centered on points in $X$, $X_a = \cup_{x\in X}B(x,a)$. The rank function ${\beta}_k(X):{\{(a,b)\in \mathbb{R}^2: a\leq b\}} \to \mathbb{R}$ is then defined by ${\beta}_k(X)(a,b) = rank ( \iota_*:H_k(X_a) \to H_k(X_b))$ where $\iota_*$ is the induced map on homology from the inclusion map on spaces. We consider the rank functions as lying in a Hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace. This enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and of sphere packings with different volume fractions. We also investigate the potential of rank functions in providing a test of complete spatial randomness of 2D point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square., Comment: 25 pages, 15 figures

We develop topologically accurate and compatible definitions for the skeleton and watershed segmentation of a 3D digital object that are computed by a single algorithm. These definitions are based on a discrete gradient vector field derived from a signed distance transform. This gradient vector field is amenable to topological analysis and simplification via For-man's discrete Morse theory and provides a filtration that can be used as input to persistent homology algorithms. Efficient implementations allow us to process large-scale x-ray micro-CT data of rock cores and other materials.

We consider finite point-set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set. We use the minimal spanning tree of the finite set of points to compute the number and size of its -connected components. By extrapolating the limiting behavior of these quantities as ! 0 we can say whether the underlying set appears to be connected, totally disconnected, or perfect. We demonstrate the effectiveness of our techniques for a number of examples, including a family of fractals related to the Sierpinski triangle, Cantor subsets of the plane, the Henon attractor, and cantori from four-dimensional symplectic sawtooth maps. For zero-measure Cantor sets, we conjecture that the growth rate of the number of -components as ! 0 is equivalent to the box-counting dimension. © 2000 Elsevier Science B.V. All rights reserved.

We reformulate the notion of connectedness for compact metric spaces in a manner that may be implemented computationally. In particular, our techniques can distinguish between sets that are connected, have a finite number of connected components, have infinitely many connected components, or are totally disconnected. We hope that this approach will prove useful for studying structures in the phase space of dynamical systems.

Recent advances in the cataloguing of three-dimensional nets mean a systematic search for framework structures with specific properties is now feasible. Theoretical arguments about the elastic deformation of frameworks suggest characteristics of mechanically isotropic networks. We explore these concepts on both isotropic and anisotropic networks by manufacturing porous elastomers with three different periodic net geometries. The blocks of patterned elastomers are subjected to a range of mechanical tests to determine the dependence of elastic moduli on geometric and topological parameters. We report results from axial compression experiments, three-dimensional X-ray computed tomography imaging and image-based finite-element simulations of elastic properties of framework-patterned elastomers.

We present a method for geometric construction of periodic three-dimensional Euclidean nets by projecting two-dimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & Delgado-Friedrichs to enumerate simple reticulations of these TPMSs. We include a taxonomy of all networks arising from kaleidoscopic hyperbolic tilings with up to two distinct tile types (and their duals, with two distinct vertices), mapped to three related TPMSs, namely Schwarz's primitive (P) and diamond (D) surfaces, and Schoen's gyroid (G).

Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations in vivo . This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the Shrink-On-No-Overlaps (SONO) (Pieranski 1998 In Ideal knots (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20–41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including θ -graphs and entangled tetrahedron- and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_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-centred spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha-shapes, a construction from computational geometry that deserves to be more widely known in the physics community., Submitted to PRE. 11 pages, 10 figures

Traditional noise-filtering techniques are known to significantly alter features of chaotic data. In this paper, we present a noncausal topology-based filtering method for continuous-time dynamical systems that is effective in removing additive, uncorrelated noise from time-series data. Signal-to-noise ratios and Lyapunov exponent estimates are dramatically improved following the removal of the identified noisy points.

The problem considered belowis that of determining information about the topology of a subset X ⊂ ℝn given only a finite point approximation to X. The basic approach is to compute topological properties — such as the number of components and number of holes — at a sequence of resolutions, and then to extrapolate. Theoretical foundations for taking this limit come from the inverse limit systems of shape theory and Cech homology. Computer implementations involve constructions from discrete geometry such as alpha shapes and the minimal spanning tree.

We systematically analyse the mechanical deformation behaviour, in particular Poisson's ratio, of floppy bar-and-joint frameworks based on periodic tessellations of the plane. For frameworks with more than one deformation mode, crystallographic symmetry constraints or minimization of an angular vertex energy functional are used to lift this ambiguity. Our analysis allows for systematic searches for auxetic mechanisms in archives of tessellations; applied to the class of one- or two-uniform tessellations by regular or star polygons, we find two auxetic structures of hexagonal symmetry and demonstrate that several other tessellations become auxetic when retaining symmetries during the deformation, in some cases with large negative Poisson ratios ν