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We present a new method for visualisation and analysis of patterns on triply periodic negatively curved surfaces by mapping to two-dimensional hyperbolic space.

Significance Tangled tetrahedra, octahedra, cubes, icosahedra, and dodecahedra are generalizations of classical—untangled—Platonic polyhedra. Like the Platonic polyhedra, all vertices, edges, and faces are symmetrically equivalent. However, the edges of tangled polyhedra are curvilinear, or kinked, to allow entanglement, much like warps and wefts in woven fabrics. We construct the most symmetric entanglements of these polyhedra via assemblies of multistrand helices wound around edges of the conventional polyhedra; they are all necessarily chiral. The construction gives self-entangled chiral polyhedra and compound polyhedra containing catenated multiple tetrahedra or “generalized θ-polyhedra.” An unlimited variety of tangling is possible for any given topology. Related structures have been observed in synthetic materials and clathrin assemblies within cells., Conventional embeddings of the edge-graphs of Platonic polyhedra, {f, z}, where f, z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway’s two-dimensional (2D) orbifold notation (equivalent to Schönflies symbols Ih, Oh, and Td). Tangled Platonic {f, z} polyhedra—which cannot lie on the sphere without edge-crossings—are constructed as windings of helices with three, five, seven,… strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the “θz” polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.

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.

Biomineralization in a starfish displays morphologically complex features

We develop tools to explore and catalogue the topologies of knotted or pseudoknotted circular folds due to secondary and tertiary interactions within a closed loop of RNA which generate multiple double-helices due (for example) to strand complementarity. The fold topology is captured by a ‘contracted fold’ which merges helices separated by bulges and removes hairpin loops. Contracted folds are either trivial or pseudoknotted. Strand folding is characterised by a rigid-vertex ‘polarised strand graph’, whose vertices correspond to double-helices and edges correspond to strands joining those helices. Each vertex has a plumbline whose polarisation direction defines the helical axis. That polarised graph has a corresponding circular ribbon diagram and canonical alphanumeric fold label. Key features of the ‘fully-flagged’ fold are the arrangement of complementary domains along the strand, described by a numerical bare fold label, and a pair of binary ‘flags’: a parity flag that specifies the twist in each helix (even or odd half-twists), and an orientation flag that characterises each double-helix as parallel or antiparallel. A simple algorithm is presented to translate an arbitrary fold label into a polarised strand graph. Any embedding of the graph in 3-space is an admissible fold geometry; the simplest embeddings minimise the number of edge-crossings in a planar graph drawing. If that number is zero, the fold lies in one of two classes: (a)-type ‘relaxed’ folds, which contain conventional junctions and (b)-type folds whose junctions are described as meso-junctions in H. Wang and N.C. Seeman,Biochem, vol. 34, pp920-929. (c)-type folds induce polarised strand graphs with edge-crossings, regardless of the planar graph drawing. Canonical fold labelling allows us to sort and enumerate all ‘semi-flagged’ folds with up to six contracted double-helices as windings around the edges of a graph-like fold skeleton, whose cyclomatic number - the ‘fold genus’ - ranges from 0 – 3, resulting in a pair of duplexed strands along each skeletal edge. Those semi-flagged folds admit both even and odd double-helical twists. Appending specific parity flags to those semi-flagged folds gives fully-flagged (a)-type folds, which are also enumerated up to genus-3 cases. We focus on all-antiparallel folds, characteristic of conventional ssRNA and enumerate all distinct (a), (b) and (c)-type folds with up to five double-helices. Those circular folds lead to pseudoknotted folds for linear ssRNA strands. We describe all linear folds derived from (a) or (b)-type circular folds with up to four contracted double-helices, whose simplest cases correspond to so-calledH,KandLpseudoknotted folds, detected in ssRNA. Fold knotting is explored in detail, via constructions of so-called antifolds and isomorphic folds. We also tabulate fold knottings for (a) and (b)-type folds whose embeddings minimise the number of edge-crossings and outline the procedure for (c)-type folds. The inverse construction - from a specific knot to a suitable nucleotide sequence - results in a hierarchy of knots. A number of specific alternating knots with up to 10 crossings emerge as favoured fold designs for ssRNA, since they are readily constructed as (a)-type all-antiparallel folds.

We investigated the structures of normal (type I) bicontinuous cubic phases in hexa-, hepta-, and octaethylene glycol dodecyl ether/water mixtures by small-angle X-ray crystallography of single-crystal domains. Reconstructed electron densities showed that the hydrophilic chains with high electron density are confined to a film centered on the surface of the Gyroid (a triply periodic minimal surface), while hydrophobic chains with low electron density are distributed within the pair of interwoven labyrinths carved out by the Gyroid. Further, the local minimum within the high electron density region, due to bulk water, coincides precisely with the Gyroid. This minimum is less pronounced in mixtures with longer ethylene glycol chains, consistent with their decreased water content. Our analysis clearly shows that the polar-nonpolar interfaces are parallel to the Gyroid surface in all mixtures. The repulsive hydration or overlapping force between the pair of facing monolayers of ethylene glycol chains on either side of the Gyroid surface is the likely origin of the parallel interfaces.

We investigated two distinct lyotropic liquid crystal inverse bicontinuous cubic phases of phytantriol/water mixtures by small-angle X-ray crystallography of the single-crystal regions. Reconstructed electron density maps revealed hydrophilic head and hydrophobic tail regions of the phytantriol bilayer membranes and water regions. The bilayer membranes are shown to be located on the D and gyroid triply periodic minimal surfaces. To investigate the structures of the polar-nonpolar interfaces, we optimized two models: a parallel surface model and a constant mean curvature surface model. The parallel surface model agreed well with the X-ray data, and the R factors, which show the degree of agreement between those structural models and the data, were less than 0.04. In stark contrast, the constant mean curvature surface model deviated significantly from the data, and the R factors were around 0.15. We therefore conclude that the polar-nonpolar interface of the inverse bicontinuous cubic phase of the phytantriol/water system is close to a parallel surface to a triply periodic minimal surface.

We derive more than 80 embeddings of 2D hyperbolic honeycombs in Euclidean 3 space, forming 3-periodic infinite polyhedra with cubic symmetry. All embeddings are "minimally frustrated," formed by removing just enough isometries of the (regular, but unphysical) 2D hyperbolic honeycombs [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] to allow embeddings in Euclidean 3 space. Nearly all of these triangulated "simplicial polyhedra" have symmetrically identical vertices, and most are chiral. The most symmetric examples include 10 infinite "deltahedra," with equilateral triangular faces, 6 of which were previously unknown and some of which can be described as packings of Platonic deltahedra. We describe also related cubic crystalline packings of equal hyperbolic discs in 3 space that are frustrated analogues of optimally dense hyperbolic disc packings. The 10-coordinated packings are the least "loosened" Euclidean embeddings, although frustration swells all of the hyperbolic disc packings to give less dense arrays than the flat penny-packing even though their unfrustrated analogues in [Formula: see text] are denser.

This communication briefly reviews why network topology is an important tool (for understanding, comparing, communicating, designing, and solving crystal structures from powder diffraction data) and then discusses the terms of an IUPAC project dealing with various aspects of network topology. One is the ambiguity in node assignment, and this question is addressed in more detail. First, we define the most important approaches: the “all node” deconstruction considering all branch points of the linkers, the “single node” deconstruction considering only components mixed, and the ToposPro “standard representation” also considering linkers as one node but, if present, takes each metal atom as a separate node. These methods are applied to a number of metal–organic framework structures (MOFs, although this is just one example of materials this method is applicable on), and it is concluded that the “all node” method potentially yields more information on the structure in question but cannot be recommended as the o...

This paper describes the families of the simplest, two-periodic constant mean curvature surfaces, the genus-two HCB and SQL surfaces, and their isometries. All the discrete groups that contain the translations of the genus-two surfaces embedded in Euclidean three-space modulo the translation lattice are derived and enumerated. Using this information, the subgroup lattice graphs are constructed, which contain all of the group–subgroup relations of the aforementioned quotient groups. The resulting groups represent the two-dimensional representations of subperiodic layer groups with square and hexagonal supergroups, allowing exhaustive enumeration of tilings and associated patterns on these surfaces. Two examples are given: a two-periodic [3,7]-tiling with hyperbolic orbifold symbol {\sf {2223}} and a {\sf {22222}} surface decoration.

We enumerate trivalent reticulations of two- and three-periodic hyperbolic surfaces by equal-sided n -gonal faces, ( n , 3), where n = 7, 8, 9, 10, 12. These are the simplest hyperbolic generalizations of the planar graphene net, (6, 3) and dodecahedrane, (5, 3). The enumeration proceeds by deleting isometries of regular reticulations of two-dimensional hyperbolic space until the ( n , 3) nets can be embedded in euclidean three-space via periodic hyperbolic surfaces. Those nets are then symmetrized in euclidean space retaining their net topology, leading to explicit crystallographic net embeddings whose edges are as equal as possible, affording candidtae patterns for graphitic schwarzites. The resulting schwarzites are the most symmetric examples. More than one hundred topologically distinct nets are described, most of which are novel.

We call attention to methods of enumerating periodic structures and to the databases that contain them. These provide information essential to the systematic design of crystalline materials. The underlying topology is uniquely specified and identifiable from the Systre key which is thus the topological genome.

Examples are given of vertex and/or face-transitive p 3 (p = 7, 8, 9) tilings of periodic minimal surfaces. Infinite families of tilings, including aperiodic structures, are readily derived from some of these. They are of particular interest for theoretical studies of potential carbon allotropes.

Metal–organic frameworks (MOFs) are often comprised of multiple disjoint molecular networks. The catenated loops and tangles of these patterns typically defy simple enumeration and explanation. Here we contrast some planar layer MOFs (built of (4,4) layers) with the three-dimensional crystals MOF-14, MOF-388 and UTSA-28. The analysis allows planar weavings to be warped, giving analogous three-dimensional frameworks. The technique allows us to warp the most common (4,4) layer MOFs. The resulting three-dimensional weavings are those of MOF-14, MOF-388 and UTSA-28. So two- and three-dimensional MOFs are weavings of layered nets. In the former case, the layers are conventional flat sheets, in the latter they are hyperbolically curved layers. This recognition suggests a simple enumeration scheme for generating and cataloguing MOF weavings via two-dimensional surfaces, regardless of their apparent dimensionality.

Here, we describe a novel monoolein-based cubosome formulation engineered for possible theranostic applications in oncology. The Docetaxel-loaded nanoparticles were stabilized in water by a mixture of commercial Pluronic (poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) triblock copolymer) F108 (PF108) and rhodamine- and folate-conjugated PF108 so that the nanoparticles possess targeting, therapeutic, and imaging properties. Nanoparticles were investigated by DLS, cryo-TEM, and SAXS to confirm their structural features. The fluorescent emission characterization of the proposed formulation indicated that the rhodamine conjugated to the PF108 experiences an environment less polar than water (similar to chloroform), suggesting that the fluorescent fragment is buried within the poly(ethylene oxide) corona surrounding the nanoparticle. Furthermore, these nanoparticles were successfully used to image living HeLa cells and demonstrated a significant short-term (4 h incubation) cytotoxicity effect against these cancer cells. Furthermore, given their analogy as nanocarriers for molecules of pharmaceutical interest and to better stress the singularities of these bicontinuous cubic nanoparticles, we also quantitatively evaluated the differences between cubosomes and multilamellar liposomes in terms of surface area and hydrophobic volume.

Reactions of the flexible N,N′-di(3-pyridyl)suberoamide (L) with Cu(II) salts in the presence of the isomeric phenylenediacetic acids under hydrothermal conditions afforded three new coordination networks, {[Cu(L)(1,2-pda)]·H2O}n (1,2-H2pda = 1,2-phenylenediacetic acid), 1, {[Cu(L)(1,3-pda)]·2H2O}n (1,3- H2pda = 1,3-phenylenediacetic acid), 2, and {[Cu(L)(1,4-pda)]·2H2O}n (1,4-H2pda = 1,4-phenylenediacetic acid), 3, which have been structurally characterized by X-ray crystallography. Complex 1 forms a single 3,5-coordinated 3D net with the (42·65·83)(42·6)-3,5T1 topology, which can be further simplified as a 6-coordinated (412·63)-pcu topology. Complex 2 is a 5-fold interpenetrated 3D structure with the (65·8)-cds topology, which exhibits the maximum number of interpenetration presently known for cds and complex 3 is the first 1D self-catenated coordination network. The ligand isomerism of the phenylenediacetate ligands is important in determining the structural types of the Cu(II) coordination networks based on the flexible L ligands.

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.

Self-assembly remains the most efficient route to the formation of ordered nanostructures, including the double gyroid network phase in diblock copolymers based on two intergrown network domains. Here we use self-consistent field theory to show that a tricontinuous structure with monoclinic symmetry, called 3ths(5), based on the intergrowth of three distorted ths nets, is an equilibrium phase of triblock star-copolymer melts when an extended molecular core is introduced. The introduction of the core enhances the role of chain stretching by enforcing larger structural length scales, thus destabilizing the hexagonal columnar phase in favor of morphologies with less packing frustration. This study further demonstrates that the introduction of molecular cores is a general concept for tuning the relative importance of entropic and enthalpic free energy contributions, hence providing a tool to stabilize an extended repertoire of self-assembled nanostructured 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.

The 3-periodic nets of genus 3 (`minimal nets') are reviewed and their symmetries re-examined. Although they are all crystallographic, seven of the 15 only have maximum-symmetry embeddings if some links are allowed to have zero length. The connection between the minimal nets and the genus-3 zero-mean-curvature surfaces (`minimal minimal' surfaces) is explored by determining the surface associated with a net that has a self-dual tiling. The fact that there are only five such surfaces but 15 minimal nets is rationalized by showing that all the minimal nets can serve as the labyrinth graph of one of the known minimal minimal surfaces.

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.

D’Arcy Thompson’s views on the forms of biomaterials are assessed in the light of current thinking on biomorphology in selected areas of biology. It is clear that his guiding concepts — that biological materials are structured in response to physical forces, and that the biological and abiotic realms share many common features — remain valid. Advances in the physical and biological sciences are discussed, from quantum mechanics and molecular biology to liquid crystalline materials and macroscopic forms. These reveal Thompson’s clear-sighted view of the role of physical and mathematical sciences in biology, as well as his blind-spots.

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.

The structure of the porous three-dimensional reticulated pattern in the wing scales of the butterfly Callophrys rubi (the Green Hairstreak) is explored in detail, via scanning and transmission electron microscopy. A full 3D tomographic reconstruction of a section of this material reveals that the predominantly chitin material is assembled in the wing scale to form a structure whose geometry bears a remarkable correspondence to the srs net, well-known in solid state chemistry and soft materials science. The porous solid is bounded to an excellent approximation by a parallel surface to the Gyroid, a three-periodic minimal surface with cubic crystallographic symmetry I4₁32, as foreshadowed by Stavenga and Michielson. The scale of the structure is commensurate with the wavelength of visible light, with an edge of the conventional cubic unit cell of the parallel-Gyroid of approximately 310 nm. The genesis of this structure is discussed, and we suggest it affords a remarkable example of templating of a chiral material via soft matter, analogous to the formation of mesoporous silica via surfactant assemblies in solution. In the butterfly, the templating is achieved by the lipid-protein membranes within the smooth endoplasmic reticulum (while it remains in the chrysalis), that likely form cubic membranes, folded according to the form of the Gyroid. The subsequent formation of the chiral hard chitin framework is suggested to be driven by the gradual polymerisation of the chitin precursors, whose inherent chiral assembly in solution (during growth) promotes the formation of a single enantiomer.

We suggest constructive definitions for the determination of untangled finite graphs and three-periodic nets, using barycentric embeddings in two and three dimensions. The possibility of deliberately constructing tangled graphs and nets is canvassed, and we conclude that tangled patterns offer a novel class of nano- and meso-structured materials with useful features, including high internal surface area and volume and chirality.

A novel technique to generate three-dimensional Euclidean weavings, composed of close-packed, periodic arrays of one-dimensional fibres, is described. Some of these weavings are shown to dilate by simple shape changes of the constituent fibres (such as fibre straightening). The free volume within a chiral cubic example of a dilatant weaving, the ideal conformation of the G 129 weaving related to the Σ + rod packing, expands more than fivefold on filament straightening. This remarkable three-dimensional weaving, therefore, allows an unprecedented variation of packing density without loss of structural rigidity and is an attractive design target for materials. We propose that the G 129 weaving (ideal Σ + weaving) is formed by keratin fibres in the outermost layer of mammalian skin, probably templated by a folded membrane.

I explore the terrain that lies between architecture and geometry, from the perspective of a structural scientist with no professional architectural expertise. The divide between these disciplines perhaps stems from an ancient dichotomy between the art versus engineering schools of architecture, fertilised by the current dogma that art and science can never meet. Architects stand to gain much from study of the spectacular advances in geometry in recent decades, such as the growing understanding of cellular patterns in space, tiles, nets and curved surfaces. Some examples of those advances are discussed in detail. I conclude that both architecture and geometry would benefit from a renewed mutual interest.

Crystallisation of barium carbonate from alkaline silica solutions results in the formation of extraordinary micron-scale architectures exhibiting non-crystallographic curved shapes, such as helical filaments and worm-like braids. These so-called "silica biomorphs" consist of a textured assembly of uniform elongated witherite nanocrystallites, which is occasionally sheathed by a skin of amorphous silica. Although great efforts have been devoted to clarifying the physical origin of these fascinating materials, to date little is known about the processes underlying the observed self-organisation. Herein, we describe the effect of two selected additives, a cationic surfactant and a cationic polymer, on the morphology of the forming crystal aggregates, and relate changes to experiments conducted in the absence of additives. Minor amounts of both substances are shown to exert a significant influence on the growth process, leading to the formation of predominantly flower-like spherulitic aggregates. The observed effects are discussed in terms of feasible morphogenesis pathways. Based on the assumption of a template membrane steering biomorph formation, it is proposed that the two additives are capable of performing specific bridging functions promoting the aggregation of colloidal silica which constitutes the membrane. Morphological changes are tentatively ascribed to varying colloid coordination effecting distinct membrane curvatures. (C) 2009 Elsevier B.V. All rights reserved. (Less)

Herein we describe some properties and the occurrences of a beautiful geometric figure that is ubiquitous in chemistry and materials science, however, it is not as well-known as it should be. We call attention to the need for mathematicians to pay more attention to the richly structured natural world, and for materials scientists to learn a little more about mathematics. Our account is informal and eschews any pretence of mathematical rigor, but does start with some necessary mathematics. Regular figures such as the five regular Platonic polyhedra are an enduring part of human culture and have been known and celebrated for thousands of years. Herein we consider them as the five regular tilings on the surface of a sphere (a two-dimensional surface of positive curvature). A flag of a tiling of a two-dimensional surface consists of a combination of a coincident tile, edge, and vertex. A generally accepted definition of regularity is flag transitivity, which means that all flags are related by symmetries of the tiling (i.e. there is just one kind of flag). In addition to the five Platonic solids, there are three regular tilings of the plane (a surface of zero curvature), and these are the familiar coverings of the plane by triangles, squares, or hexagons tiled edge-to-edge. The corresponding regular tilings of three-dimensional space are also wellknown. Flags are now a polyhedron (tile) with a coincident face, edge, and vertex, and the regular tilings of the three-sphere are the six nonstellated regular polytopes of four dimensional space. We remark that four dimensions is the richest space in this regard; higher dimensions have only three regular polytopes (and of course three dimensions has five). However, in flat threedimensional (Euclidean) space, the space of our day-to-day experience, there is disappointingly only one regular tiling—the familiar space filling by cubes sharing faces (face-to-face). The classic reference to these figures is Coxeter!s Regular Polytopes, in which he remarks on the tilings of threedimensional Euclidean space: “For the development of a general theory, it is an unhappy accident that only one honeycomb [tiling] is regular...”. Unhappy indeed, because, perhaps as a consequence, the rich world of periodic graphs, which are the underlying topology of crystal structures, has been largely neglected by mathematicians. The graph associated with (carried by) the regular tiling by cubes is the set of edges and vertices. It is notably the structure of a form of elemental polonium, and chemists often refer to it as the a-Po net. Recently a system of symbols for nets has been developed and this net has the symbol pcu. Our review is concerned with another such periodic graph, and an associated surface.

Cubic membranes are soft three-dimensional crystals found within cell organelles in a variety of living systems, despite the aphorism of Fedorov: ‘crystallization is death’. They consist of multi-bilayer lipid–protein stacks, folded onto anticlastic surfaces that resemble triply periodic minimal surfaces, forming highly swollen crystalline sponges. Although cubic membranes have been observed in numerous cell types and under different pathophysiological conditions, knowledge about the formation and potential function(s) of non-lamellar, cubic structures in biological systems is scarce. We report that mitochondria with this cubic membrane organization isolated from starved amoeba Chaos carolinense interact sufficiently with short segments of phosphorothioate oligonucleotides (PS-ODNs) to give significant ODNs uptake. ODNs condensed within the convoluted channels of cubic membrane by an unknown passive targeting mechanism. Moreover, the interaction between ODNs and cubic membrane is sufficient to retard electrophoretic mobility of the ODN component in the gel matrix. These ODN–cubic membrane complexes are readily internalized within the cytoplasm of cultured mammalian cells. Transmission electron microscopic analysis confirms ODNs uptake by cubic membranes and internalization of ODN–cubic membrane complexes into the culture cells. Cubic membranes thus may offer a new, potentially benign medium for gene transfection.

This article explores global geometric features of bicontinuous space-partitions and their relevance to self-assembly of block-copolymers. Using a robust definition of `local channel radius', based on the concept of a medial surface [Schroder et al., Eur. Phys. J. B 35, 551 (2003)], we relate radius variations of the space-partition to polymolecular chain stretching in bicontinuous diblock- and terblock copolymer assemblies. We associate local surface patches with corresponding cellular volume elements, to define local volume-to-surface ratios. The distribution of these v/a ratios and of the channel radii are used to quantify the degree of packing frustration of molecular chains as a function of the specific bicontinuous geometry, modelled by triply-periodic minimal surfaces and related parallel interfaces. The Gyroid geometry emerges as the most nearly homogeneous bicontinuous form, with the smallest heterogeneity of channel radii, compared to the cubic Primitive and Diamond surfaces. We clarify a geometric feature of the Gyroid geometry: the three-coordinated nodes of the graph are not the widest points of the labyrinths; the widest points are at the midpoints of the edges. We also explore a more complex cubic triply-periodic surface, the I-WP surface, containing two geometrically distinct channel subdomains. One of the two channel systems is nearly as homogeneous in local channel diameters as the Gyroid, the other is more heterogeneous than the Primitive surface. Its hybrid nature suggests the possibility of an “alternating I-WP” phase in polydisperse linear ABC-terpolymer blends, with monodisperse molecular weight distributions (MWD) in the A and B blocks and a more polydisperse C block.

We explore the use of tetraethoxysilane (TEOS) as a silica source for the formation of carbonate-silica composite materials known as ‘biomorphs’. The basic hydrolysis of TEOS furnishes silica in a controllable fashion, allowing a significantly higher reproducibility of the obtained silica–barium and silica–strontium carbonate co-precipitates compared to commercial water glass silica used so far. We further discuss the influence of ethanol used as a co-solvent on the morphologies of biomorphs, which are examined by optical microscopy, field emission scanning electron microscopy (FESEM) and energy dispersive X-ray analysis (EDX).

The 'simplest' entanglements of the graph of edges of the cube are enumerated, forming two-cell {6, 3} (hexagonal mesh) complexes on the genus-one two-dimensional torus. Five chiral pairs of knotted graphs are found. The examples contain non-trivial knotted and/or linked subgraphs [(2, 2), (2, 4) torus links and (3, 2), (4, 3) torus knots].

Entanglements of two-dimensional honeycomb nets are constructed from free tilings of the hyperbolic plane ({\bb H}^2) on triply periodic minimal surfaces. The 2-periodic nets that comprise the structures are guaranteed by considering regular, rare free tilings in {\bb H}^2. This paper catalogues an array of entanglements that are both beautiful and challenging for current classification techniques, including examples that are realized in metal–organic materials. The compactification of these structures to the genus-3 torus is considered as a preliminary method for generating entanglements of finite θ-graphs, potentially useful for gaining insight into the entanglement of the periodic structure. This work builds on previous structural enumerations given inPeriodic entanglementParts I and II [Evanset al.(2013).Acta Cryst.A69, 241–261; Evanset al.(2013).Acta Cryst.A69, 262–275].

The article by Sorenson et al. (Soft Matter10, 8229, 2014) reports a novel phase formed by gemini surfactants in water, of symmetry P63/mcm and based on a triple intergrowth of three identical degree-three networks, known as 3etc(193). This phase is the first lyotropic liquid crystalline phase based on the intergrowth of a triplet of network- or labyrinth-like hydrophobic domains. We provide here results from self-consistent field theory that demonstrate that the same morphology is almost stable in standard AB diblock copolymer melts; at the phase transition between the double gyroid phase and the hexagonal columnar phase, the 3etc(193) morphology only incurs a marginal free energy penalty compared to the equilibrium phases. Interestingly, the ratio of lattice parameters c/a = 0.955 of the 3etc(193) as a diblock morphology is very close to that of the gemini surfactant phase and of the related IBN-9 mesoporous silicate phase (Han et al., Nat. Chem.1, 123, 2009). Based on the combination of these results, we hypothesise that the 3etc(193) morphology is likely a generic phase in soft materials, rather than an oddity.

Balanced infinite periodic minimal surface families that contain the cubic Gyroid (G), Diamond (D) and Primitive (P) surfaces are studied in terms of their global packing and local curvature properties. These properties are central to understanding the formation of mesophases in amphiphile and copolymer molecular systems. The surfaces investigated are the tetragonal, rhombohedral and hexagonal tD, tP, tG, rG, rPD and H surfaces. These non-cubic minimal surfaces furnish topology-preserving transformation pathways between the three cubic surfaces. We introduce `packing (or global) homogeneity', defined as the standard deviation Δd of the distribution of the channel diameter throughout the labyrinth, where the channel diameter d is determined from the medial surface skeleton centered within the labyrinthine domains. Curvature homogeneity is defined similarly as the standard deviation ΔK of the distribution of Gaussian curvature. All data are presented for distinct length normalisations: constant surface-to-volume ratio, constant average Gaussian curvature and constant average channel diameter. We provide first and second moments of the distribution of channel diameter for all members of these surfaces complementing curvature data from [A. Fogden, S. Hyde, Eur. Phys. J. B 7, 91 (1999)]. The cubic G and D surfaces are deep local minima of Δd along the surface families (with G more homogeneous than D), whereas the cubic P surface is an inflection point of Δd with adjacent, more homogeneous surface members. Both curvature and packing homogeneity favour the tetragonal route between G and D (via tG and tD surfaces) in preference to the rhombohedral route (via rG and rPD).

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.

Carbon nanoclusters produced by high-repetition-rate laser ablation of glassy carbon in Ar exhibits para- and ferromagnetic behaviour up to 90 K, showing a narrow hysteresis curve with a coercive force H c = 420 Oe, remnant magnetisation of 5 × 10 −3 emu/g, susceptibility of the order of 10 −5 emu/g Oe and saturation magnetization 0.3–0.8 emu/g at 1.8 K. We consider briefly the mechanism of formation of clusters in the laser-ablated vapour and structural characteristics of the nanoclusters, which show the presence of graphite-like sheets with hyperbolic curvature, as proposed for “schwarzite”. We postulate that localized unpaired spins occur because of topological and bonding defects associated with the sheet curvature, and that these spins are stabilized for >1 year due to the steric protection offered by the convoluted sheets.

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 propose a general method to study the hierarchical organization of financial data. The statistical, geometrical and topological properties of such data are analyzed by embedding the structure of their correlations in metric graphs in multi-dimensional spaces. We show an application to two dierent sets of interest rates data. In this case we construct triangular embeddings on the sphere. The resulting graph contains the minimum spanning tree as sub-graph and it preserves its hierarchical structure. This results in a clear cluster dierentiation

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

A medial surface (MS) analysis of the rhombohedral infinite periodic minimal surface family rPD is presented. The rPD family of bicontinuous surfaces has been suggested as a pathway for transitions between its two cubic members, the P and the D surface, in mesophases in liquid-crystalline self-assembly. The MS is a representation of a labyrinth as a centered 2D skeleton. By providing a definition of a pointwise channel diameter, the MS allows for an analysis of homogeneity of such surfaces including chain stretching frustration. For the rPD surface, variations of this channel diameter are locally minimal for the D surface, and a horizontal inflection point for the P surface. This may have implications for the phase stability of the corresponding liquid-crystalline mesophases. The MS can be further reduced to a 1D line graph. For the rPD surface, this graph contains curved edges and cannot be deduced from symmetry considerations alone.