Symmetric tangled Platonic polyhedra.

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 ([Formula: see text]) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of [Formula: see text] 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, [Formula: see text] 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.

Mapping hyperbolic order in curved materials.

Nature employs an impressive range of topologically complex ordered nanostructures that occur in various forms in both natural and synthetic materials. A particular class of these exhibits negative curvature and forms periodic saddle-shaped surfaces in three dimensions. Unlike pattern formation on flat or positively curved surfaces like spherical systems, the understanding of patterning on such surfaces is highly complicated due to the structures being intrinsically intertwined in three dimensions. We present a new method for visualisation and analysis of patterns on triply periodic negatively curved surfaces by mapping to two-dimensional hyperbolic space analogous to spherical projections in cartography thus effectively creating a more accessible "hyperbolic map" of the pattern. Specifically, we exemplify the method via the simplest triply periodic minimal surfaces: the Primitive, Diamond, and Gyroid in their universal cover along with decorations from a soft materials, whose structures involve decorations of soft matter on negatively curved surfaces, not necessarily minimal.

Polyhedra and packings from hyperbolic honeycombs.

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.

Polar-Nonpolar Interfaces of Normal Bicontinuous Cubic Phases in Nonionic Surfactant/Water Systems Are Parallel to the Gyroid Surface.

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.

Hierarchical self-assembly of a striped gyroid formed by threaded chiral mesoscale networks.

Numerical simulations reveal a family of hierarchical and chiral multicontinuous network structures self-assembled from a melt blend of Y-shaped ABC and ABD three-miktoarm star terpolymers, constrained to have equal-sized A/B and C/D chains, respectively. The C and D majority domains within these patterns form a pair of chiral enantiomeric gyroid labyrinths (srs nets) over a broad range of compositions. The minority A and B components together define a hyperbolic film whose midsurface follows the gyroid minimal surface. A second level of assembly is found within the film, with the minority components also forming labyrinthine domains whose geometry and topology changes systematically as a function of composition. These smaller labyrinths are well described by a family of patterns that tile the hyperbolic plane by regular degree-three trees mapped onto the gyroid. The labyrinths within the gyroid film are densely packed and contain either graphitic hcb nets (chicken wire) or srs nets, forming convoluted intergrowths of multiple nets. Furthermore, each net is ideally a single chiral enantiomer, induced by the gyroid architecture. However, the numerical simulations result in defect-ridden achiral patterns, containing domains of either hand, due to the achiral terpolymeric starting molecules. These mesostructures are among the most topologically complex morphologies identified to date and represent an example of hierarchical ordering within a hyperbolic pattern, a unique mode of soft-matter self-assembly.

Docetaxel-Loaded Fluorescent Liquid-Crystalline Nanoparticles for Cancer Theranostics.

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.

Comment on "Discovery of a tetracontinuous, aqueous lyotropic network phase with unusual 3D-hexagonal symmetry" by M. Mahanthappa, G. Sorenson and A. Schmitt.

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.

Tiling patterns from ABC star molecules: 3-colored foams?

We present coarse-grained simulations of the self-assembly of 3-armed ABC star polyphiles. In systems of star polyphiles with two arms of equal length the simulations corroborate and expand previous findings from related miktoarm star terpolymer systems on the formation of patterns containing columnar domains whose sections are 2D planar tilings. However, the systematic variation of face topologies as the length of the third (unequal) arm is varied differs from earlier findings regarding the compositional dependence. We explore 2D 3-colored foams to establish the optimal patterns based on interfacial energy alone. A generic construction algorithm is described that accounts for all observed 2D tiling patterns and suggests other patterns likely to be found beyond the range of the simulations reported here. Patterns resulting from this algorithm are relaxed using Surface Evolver calculations to form 2D foams with minimal interfacial length as a function of composition. This allows us to estimate the interfacial enthalpic contributions to the free energy of related star molecular assemblies assuming strong segregation. We compare the resulting phase sequence with a number of theoretical results from particle-based simulations and field theory, allowing us to tease out relative enthalpic and entropic contributions as a function of the chain lengths making up the star molecules. Our results indicate that a richer polymorphism is to be expected in systems not dominated by chain entropy. Further, analysis of corresponding planar tiling patterns suggests that related two-periodic columnar structures are unlikely hypothetical phases in 4-arm star polyphile melts in the absence of sufficient arm configurational freedom for minor domains to form lens-shaped di-gons, which require higher molecular weight polymeric arms. Finally, we discuss the possibility of forming a complex tiling pattern that is a quasi-crystalline approximant for 3-arm star polyphiles with unequal arm lengths.

Metastable interwoven mesoporous metal-organic frameworks.

Three isostructural interwoven 3,4-connected mesoporous metal-organic frameworks of pto-a topology (UTSA-28-Cu, UTSA-28-Zn, and UTSA-28-Mn) were synthesized and structurally characterized. Because of their metastable nature, their gas sorption properties are highly dependent on the metal ions and activation profiles. The most stable, UTSA-28a-Cu, exhibits promising gas storage and separation capacities.

Programmable Self-Assembly of Nanoplates into Bicontinuous Nanostructures.

Self-assembly is the process by which individual components arrange themselves into an ordered structure by changing the shapes, components, and interactions. It has enabled us to construct an extensive range of geometric forms on many length scales. Nevertheless, the potential of two-dimensional polygonal nanoplates to self-assemble into extended three-dimensional structures with compartments and corridors has remained unexplored. In this paper, we show coarse-grained Monte Carlo simulations demonstrating self-assembly of hexagonal/triangular nanoplates via complementary interactions into faceted, sponge-like "bicontinuous polyhedra" (or infinite polyhedra) whose flat walls partition space into a pair of mutually interpenetrating labyrinths. Two bicontinuous polyhedra can be self-assembled: the regular (or Platonic) Petrie-Coxeter infinite polyhedron (denoted {6,4|4}) and the semi-regular Hart "gyrangle". The latter structure is chiral, with both left- and right-handed versions. We show that the Petrie-Coxeter assembly is constructed from two complementary populations of hexagonal nanoplates. Furthermore, we find that the 3D chiral Hart gyrangle can be assembled from identical achiral triangular nanoplates decorated with regioselective complementary interaction sites. The assembled Petrie-Coxeter and Hart polyhedra are faceted versions of two of the simplest triply periodic minimal surfaces, namely, Schwarz's primitive and Schoen's gyroid surfaces, respectively, offering alternative routes to those bicontinuous nanostructures, which are widespread in synthetic and biological materials.

Starfish grow extraordinary crystals.

Biomineralization in a starfish displays morphologically complex features.

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.

A novel lyotropic liquid crystal formed by triphilic star-polyphiles: hydrophilic/oleophilic/fluorophilic rods arranged in a 12.6.4. tiling.

Triphilic star-polyphiles are short-chain oligomeric molecules with a radial arrangement of hydrophilic, hydrocarbon and fluorocarbon chains linked to a common centre. They form a number of liquid crystalline structures when mixed with water. In this contribution we focus on a hexagonal liquid crystalline mesophase found in star-polyphiles as compared to the corresponding double-chain surfactant to determine whether the hydrocarbon and fluorocarbon chains are in fact demixed in these star-polyphile systems, or whether both hydrocarbon and fluorocarbon chains are miscible, leading to a single hydrophobic domain, making the star-polyphile effectively amphiphilic. We report SANS contrast variation data that are compatible only with the presence of three distinct immiscible domains within this hexagonal mesophase, confirming that these star-polyphile liquid crystals are indeed hydrophilic/oleophilic/fluorophilic 3-phase systems. Quantitative comparison with scattering simulations shows that the experimental data are in very good agreement with an underlying 2D columnar (12.6.4) tiling. As in a conventional amphiphilic hexagonal mesophase, the hexagonally packed water channels (dodecagonal prismatic domains) are embedded in a hydrophobic matrix, but that matrix is split into oleophilic hexagonal prismatic domains and fluorophilic quadrangular prismatic domains.

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.

Hyperbolic crystallography of two-periodic surfaces and associated structures.

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.

Optimal packings of three-arm star polyphiles: from tricontinuous to quasi-uniformly striped bicontinuous forms.

Star-shaped molecules with three mutually immiscible arms self-assemble to form a variety of novel structures, with conformations that attempt to minimize interfacial area between the domains composed of the different arms. The geometric frustration caused by the joining of these arms at a common centre limits the size and shape of each domain, encouraging the creation of complex and interesting solutions. Some solutions are tricontinuous, and these solutions (and others) share aspects of bicontinuous structures with amphiphilic assemblies as similar molecular segregation factors are at work. We describe both highly symmetric and balanced structures, as well as unbalanced solutions that take the form of intricately striped amphiphilic membranes. All these patterns can result in chiral assemblies with multiple networks.

Crystals: animal, vegetable or mineral?

The morphologies of biological materials, from body shapes to membranes within cells, are typically curvaceous and flexible, in contrast to the angular, facetted shapes of inorganic matter. An alternative dichotomy has it that biomolecules typically assemble into aperiodic structures in vivo, in contrast to inorganic crystals. This paper explores the evolution of our understanding of structures across the spectrum of materials, from living to inanimate, driven by those naive beliefs, with particular focus on the development of crystallography in materials science and biology. The idea that there is a clear distinction between these two classes of matter has waxed and waned in popularity through past centuries. Our current understanding, driven largely by detailed exploration of biomolecular structures at the sub-cellular level initiated by Bernal and Astbury in the 1930s, and more recent explorations of sterile soft matter, makes it clear that this is a false dichotomy. For example, liquid crystals and other soft materials are common to both living and inanimate materials. The older picture of disjoint universes of forms is better understood as a continuum of forms, with significant overlap and common features unifying biological and inorganic matter. In addition to the philosophical relevance of this perspective, there are important ramifications for science. For example, the debates surrounding extra-terrestrial life, the oldest terrestrial fossils and consequent dating of the emergence of life on the Earth rests to some degree on prejudices inferred from the supposed dichotomy between life-forms and the rest.

Periodic entanglement III: tangled degree-3 finite and layer net intergrowths from rare forests.

Entanglements of two-dimensional honeycomb nets are constructed from free tilings of the hyperbolic plane (H2) on triply periodic minimal surfaces. The 2-periodic nets that comprise the structures are guaranteed by considering regular, rare free tilings in H2. 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 in Periodic entanglement Parts I and II [Evans et al. (2013). Acta Cryst. A69, 241-261; Evans et al. (2013). Acta Cryst. A69, 262-275].

Morphology: an ambiguous indicator of biogenicity.

This paper deals with the difficulty of decoding the origins of natural structures through the study of their morphological features. We focus on the case of primitive life detection, where it is clear that the principles of comparative anatomy cannot be applied. A range of inorganic processes are described that result in morphologies emulating biological shapes, with particular emphasis on geochemically plausible processes. In particular, the formation of inorganic biomorphs in alkaline silica-rich environments are described in detail.