Stephen T. Hyde

We investigate the self-assembly of compositionally asymmetric AB diblock co-polymers confined to thin hyperbolic films on the three cubic minimal surfaces: the primitive, the diamond and the gyroid. To analyse and quantify the assemblies, we use a computational pipeline where decorations of the curved surfaces from (flat) three-dimensional space are visualized in curved two-dimensional space using the Poincaré disc model of the hyperbolic plane. This allows us to pinpoint the role of the intrinsic geometry and probe the role of the Gaussian curvature of the film on the resulting assemblies. We find three distinct symmetric arrays of discs favoured for specific values of the average Gaussian curvature of the film, whose relative appearance as a function of the average film curvature is common to all three minimal surfaces, which strongly suggests these assemblies are not influenced by the extrinsic embedding of the film. In general, the assembled discs are located in regions with minimal (absolute) Gaussian curvature, likely in order to minimize the effective line tension within the system corresponding to the enthalpic cost of co-polymer assembly along the perimeters of the discs. This article is part of the theme issue ‘Geometry, materials and the imagination’.

Many biomolecules and their synthetic counterparts preferentially assemble into double-helices with a preferred handedness and parallelism, such as all-antiparallel, right-handed double-helical polymorphs of DNA and RNA favored under ambient conditions by Watson–Crick duplexing. We analyze knots formed by double-helices which are exclusively all-parallel (or exclusively all-antiparallel) and all right-handed (or all left-handed) . Most simpler knot enantiomers are unrealizable. Among the nearly 500 distinct knot enantiomers with up to 10 crossings, fewer than 10% can be formed by such double-helices. Simplest designs for the folding of admissible knot enantiomers via duplexing of complementary base pairs in single-stranded RNA (or DNA) are enumerated for those knots.

Nature offers a remarkable diversity of nanomaterials that have extraordinary functional and structural properties. Intrinsic to nature is the impressive ability to form complex ordered nanomaterials via self-organization. One particularly intriguing nanostructure is the gyroid, a network-like structure exhibiting high symmetry and complex topology. Although its existence in cells and tissues across many biological kingdoms is well documented, how and why it forms remains elusive and uncovering these formation mechanisms will undoubtedly inform bioinspired designs. A beautiful example is the smooth single gyroid that is found in the wing scales of several butterflies, where it behaves as a photonic crystal generating a vibrant green color. Here, we report that the gyroid structures of the Emerald-patched Cattleheart, Parides sesostris , develop as woven fibrillar structures, in contrast to the commonly held assumption that they form as smooth constructs. Ultramicroscopy of pupal tissue reveals that the gyroid geometry consists of helical weavings of fibers, akin to hyperbolic line patterns decorating the gyroid. Interestingly, despite their fibrillar nature, electron diffraction reveals the absence of crystalline order within this material. Similar fibrillar structures are also observed in the mature wing scales of P. sesostris specimens with surgically altered pupal development, leading to a blue coloration. Our findings not only introduce a variation of the gyroid in biology but also have significant implications for our understanding of its formation in nature.

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

Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.

The comprehension of form generally assumes a euclidean three-dimensional perspective. I argue here that non-euclidean geometry has much to offer in understand­ing structures of atomic crystals, molecular liquid crystals and related mesoporous inorganic materials and biominerals.

Contemporary scientific research is competitive, costly and coupled to the parallel universe of commerce. A Faustian bargain between scientists and politicians allows the funding to flow. There is another path: to slow down, think and experiment without the pressure of competition and frequent publication. That path will come at a cost: reduced funding for people and equipment. The article compares and contrasts the most creative musical and visual artists with the current scientific model. I suggest that science requires acceptance that true creativity can only come by decoupling from current commercial and political imperatives.

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.

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.

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-called H , K and L pseudoknotted 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.