High-symmetry free tilings of the two-dimensional hyperbolic plane (H 2 ) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The threedimensional 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.

Inverse bicontinuous cubic phases with two aqueous network domains separated by a smooth bilayer are firmly established as equilibrium phases in lipid/water systems. The purpose of this article is to highlight the generalisations of these bicontinuous geometries to polycontinuous geometries, which could be realised as lipid mesophases with three or more network-like aqueous domains separated by a branched bilayer. An analysis of structural homogeneity in terms of bilayer width variations reveals that ordered polycontinuous geometries are likely candidates for lipid mesophase structures, with similar chain packing characteristics to the inverse micellar phases (that once were believed not to exist due to high packing frustration). The average molecular shape required by global geometry to form these multi-network phases is quantified by the surfactant shape parameter, v/(al); we find that it adopts values close to those of the known lipid phases. We specifically analyse the 3etc(187 193) structure of hexagonal symmetry P6 3 /mcm with three aqueous domains, the 3dia(24 220) structure of cubic symmetry I 43d composed of three distorted diamond networks, the cubic chiral 4srs(24 208) with cubic symmetry P4 2 32 and the achiral 4srs(5 133) structure of symmetry P4 2 /nbc, each consisting of four intergrown undistorted copies of the srs net (the same net as in the Q G II gyroid phase). Structural homogeneity is analysed by a medial surface approach assuming that the headgroup interfaces are constant mean curvature surfaces. To facilitate future experimental identification, we provide simulated SAXS scattering patterns that, for the 4srs(24 208) and 3dia(24 220) structures, bear remarkable similarity to those of bicontinuous Q G II -gyroid and Q D II -diamond phases, with comparable lattice parameters and only a single peak that cannot be indexed to the wellestablished structures. While polycontinuous lipid phases have, to date, not been reported, the likelihood of their formation is further indicated by the reported observation of a solid tricontinuous mesoporous silicate structure, termed IBN-9, which formed in the presence of surfactants [Han et al

We describe the development of a new software tool, called "Pomelo", for the calculation of Set Voronoi diagrams. Voronoi diagrams are a spatial partition of the space around the particles into separate Voronoi cells, e.g. applicable to granular materials. A generalization of the conventional Voronoi diagram for points or monodisperse spheres is the Set Voronoi diagram, also known as navigational map or tessellation by zone of influence. In this construction, a Set Voronoi cell contains the volume that is closer to the surface of one particle than to the surface of any other particle. This is required for aspherical or polydisperse systems. Pomelo is designed to be easy to use and as generic as possible. It directly supports common particle shapes and offers a generic mode, which allows to deal with any type of particles that can be described mathematically. Pomelo can create output in different standard formats, which allows direct visualization and further processing. Finally, we describe three applications of the Set Voronoi code in granular and soft matter physics, namely the problem of packings of ellipsoidal particles with varying degrees of particle-particle friction, mechanical stable packings of tetrahedra and a model for liquid crystal systems of particles with shapes reminiscent of pears. The analysis of geometries and structures on a mi-cro scale level is an important aspect of granular and soft matter physics to attain knowledge about many interesting properties of particle packings, including contact numbers , anisotropy, local volume fraction, etc. [1-3]. A well-established concept is the so called Voronoi Diagram. Here, the system is investigated by dividing the space into separate cells in respect to the positions of the center of the particles. A cell assigned to a certain particle is defined as the space (or region of space) that contains all the volume closer to the center of this specific particle than to any other one (see figure 1 left). This partition of space, however , only yields precise results for monodisperse spheres as the construction fails otherwise due to morphological properties of the objects. For nonspherical or polydisperse particles the classical Voronoi diagram is of limited use-fullnes, as shown in figure 1 (center) for a system of bidis-perse spheres. A generalized version of the Voronoi Diagram , the Set Voronoi Diagram [4], also known as nav-igational map [5] or tessellation by zone of influence [6], has to be applied. In this case the cells contain all space around the particle which is closer to the particle's surface than to the surface of any other particle. Figure 1 (right) shows the Set Voronoi Diagram of a mixture of differently shaped particles. SW and PS have contributed equally to the work described in this project Here we introduce a software tool called Pomelo, which calculates Set Voronoi Diagrams based on the algorithm described in [4]. Pomelo is particularly versatile due to its ability to generically handle any arbitrary shape which can be described mat...

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 (H 2 ) onto cubic genus-3 triply periodic minimal surfaces in Euclidean space (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.

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. chirality | liquid crystals | entanglement | hyperbolic tilings | miktoarm copolymers L iquid crystals formed by molecular self-assembly provide fascinating examples of complicated space partitions in softmaterial science. Relatively complex examples are the bicontinuous mesostructures found ubiquitously in both natural and synthetic soft matter, including lipid-water systems and block copolymer melts, namely the double diamond (symmetry Pn3m), the primitive ðIm3mÞ, and, particularly, the gyroid ðIa3dÞ mesophases. The structure of these mesophases can be described by a molecular membrane folded onto one of the three simplest triply periodic minimal surfaces (TPMS), namely the D, P, and G(yroid) surfaces, named by Schoen in the 1960s (1). From a 3D perspective, these structures are characterized by the nets describing the pair of mutually threaded labyrinths carved out of space by the convoluted hyperbolic architecture of the TPMS. For the gyroid, this is a racemic mixture of two chiral srs nets, one left-and the other right-handed [the three-letter nomenclature follows the Reticular Chemistry Structure Resource naming convention for 3D nets (2)]. This leads to an overall achiral structure when the two nets are chemically identical, which is the case in most experimentally identified gyroid liquidcrystal structures. One such structure recently reported is a gyroid assembly found in an ABC three-miktoarm star terpolymer melt (3). In this structure, the majority C component constitutes the two labyrinth nets while the...

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We investigate the possibility of forming achiral knottings of polyhedral (3-connected) graphs whose minimal embeddings lie in the genus-one torus. Various analyses to show that all examples are chiral. This result suggests a simple route to forming chiral molecules via templating on a toroidal substrate.

Surface curvature both emerges from, and influences the behavior of, living objects at length scales ranging from cell membranes to single cells to tissues and organs. The relevance of surface curvature in biology has been supported by numerous recent experimental and theoretical investigations in recent years. In this review, we first give a brief introduction to the key ideas of surface curvature in the context of biological systems and discuss the challenges that arise when measuring surface curvature. Giving an overview of the emergence of curvature in biological systems, its significance at different length scales becomes apparent. On the other hand, summarizing current findings also shows that both single cells and entire cell sheets, tissues or organisms respond to curvature by modulating their shape and their migration behavior. Finally, we address the interplay between the distribution of morphogens or micro-organisms and the emergence of curvature across length scales with examples demonstrating these key mechanistic principles of morphogenesis. Overall, this review highlights that curved interfaces are not merely a passive by-product of the chemical, biological and mechanical processes but that curvature acts also as a signal that co-determines these processes.