Self-Assembly of an [M8 L24 ]16+ Intertwined Cube and a Giant [M12 L16 ]24+ Orthobicupola

Through coordination-driven self-assembly, aesthetically captivating structures can be formed by tuning the length or flexibility of various components. The self-assembly of an elongated rigid terphenyl-based tetra-pyridyl ligand (L1) with a cis-Pd(II) acceptor produces an [M12 L16 ]24+ triangular orthobicupola structure (1). When flexibility is introduced into the ligand by the incorporation of a -CH2 - group between the dipyridylamine and terphenyl rings in the ligand (L2), anunique [M8 L24 ]16+ water-soluble 'intertwined cubic structure' (2) results. The inherent flexibility of ligand L2 might be the key factor behind the formation of the thermodynamically stable and 'intertwined cubic structure' in this scenario. This research showcases the ability to design and fabricate novel, topologically distinctive molecular structures by a straightforward and efficient approach.

Periodic Borromean rings, rods and chains

This article describes periodic polycatenane structures built from interlocked rings in which no two are directly linked. The 2-periodic vertex-, edge- and ring-transitive families of hexagonal Borromean rings are described in detail, and it is shown how these give rise to 1- and 3-periodic ring-transitive (isonemal) families. A second isonemal 2-periodic family is identified, as is a unique 3-periodic Borromean assembly of equilateral triangles. Also reported is a notable 2-periodic structure comprising chains of linked rings in which the chains are locked in place but no two chains are directly interlinked, being held in place as a novel `quasi-Borromean' set of four repeating components.

Symmetry groups of two-way twofold and three-way threefold fabrics

This work discusses the symmetry groups of two classes of woven fabrics, two-way twofold fabrics and three-way threefold fabrics. A method to arrive at a design of a fabric is presented, employing methods in color symmetry theory. Geometric representations of all possible layer group or diperiodic symmetry structures of the fabrics are derived. There are 50 layer symmetry groups corresponding to two-way twofold fabrics and 27 layer symmetry groups corresponding to three-way threefold fabrics.

Piecewise-linear embeddings of decussate extended θ graphs and tetrahedra

An nθ graph is an n-valent graph with two vertices. From symmetry considerations, it has vertex–edge transitivity 1 1. Here, they are considered extended with divalent vertices added to the edges to explore the simplest piecewise-linear tangled embeddings with straight, non-intersecting edges (sticks). The simplest tangles found are those with 3n sticks, transitivity 2 2, and with 2⌊(n − 1)/2⌋ ambient-anisotopic tangles. The simplest finite and 1-, 2- and 3-periodic decussate structures (links and tangles) are described. These include finite cubic and icosahedral and 1- and 3-periodic links, all with minimal transitivity. The paper also presents the simplest tangles of extended tetrahedra and their linkages to form periodic polycatenanes. A vertex- and edge-transitive embedding of a tangled srs net with tangled and polycatenated θ graphs and vertex-transitive tangled diamond (dia) nets are described.

Symmetric Tangling of Honeycomb Networks

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 Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space

We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. Embeddings of graphs are given as piecewise linear with finite, non-intersecting edges. We call attention to problems of conflicting terminology when disciplines such as materials chemistry and mathematics collide.

Tangled piecewise-linear embeddings of trivalent graphs

A method is described for generating and exploring tangled piecewise-linear embeddings of trivalent graphs under the constraints of point-group symmetry. It is shown that the possible vertex-transitive tangles are either graphs of vertex-transitive polyhedra or bipartite vertex-transitive nonplanar graphs. One tangle is found for 6 vertices, three for 8 vertices (tangled cubes), seven for 10 vertices, and 21 for 12 vertices. Also described are four isogonal embeddings of pairs of cubes and 12 triplets of tangled cubes (16 and 24 vertices, respectively). Vertex 2-transitive embeddings are obtained for tangled trivalent graphs with 6 vertices (two found) and 8 vertices (45 found). Symmetrical tangles of the 10-vertex Petersen graph and the 20-vertex Desargues graph are also described. Extensions to periodic tangles are indicated. These are all interesting and viable targets for molecular synthesis.

Reentrant tensegrity: A three-periodic, chiral, tensegrity structure that is auxetic

We present a three-periodic, chiral, tensegrity structure and demonstrate that it is auxetic. Our tensegrity structure is constructed using the chiral symmetry Π⁺ cylinder packing, transforming cylinders to elastic elements and cylinder contacts to incompressible rods. The resulting structure displays local reentrant geometry at its vertices and is shown to be auxetic when modeled as an equilibrium configuration of spatial constraints subject to a quasi-static deformation. When the structure is subsequently modeled as a lattice material with elastic elements, the auxetic behavior is again confirmed through finite element modeling. The cubic symmetry of the original structure means that the auxetic behavior is observed in both perpendicular directions and is close to isotropic in magnitude. This structure could be the simplest three-dimensional analog to the two-dimensional reentrant honeycomb. This, alongside the chirality of the structure, makes it an interesting design target for multifunctional materials.

Isotopy classes for 3-periodic net embeddings

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.

Crystallographic descriptions of regular 2-periodic weavings of threads, loops and nets

Piecewise linear descriptions are presented of weavings of threads, loops and 2-periodic nets. Crystallographic data are provided for regular structures, defined as those with one kind (symmetry-related) of vertex (corner) and edge (stick). These include infinite families of biaxial thread weaves, interwoven square lattices (sql), honeycomb (hcb) nets, and tetragonal and hexagonal polycatenanes.

The geometry of periodic knots, polycatenanes and weaving from a chemical perspective: a library for reticular chemistry

The geometry of the most regular polycatenanes and weavings, as an extended family of discrete knots and catenanes, is described in terms of sticks and corners in their optimal embeddings.

On the group-theoretical approach to the study of interpenetrating nets

Using group–subgroup and group–supergroup relations, a general theoretical framework is developed to describe and derive interpenetrating 3-periodic nets. The generation of interpenetration patterns is readily accomplished by replicating a single net with a supergroupGof its space groupHunder the condition that site symmetries of vertices and edges are the same in bothHandG. It is shown that interpenetrating nets cannot be mapped onto each other by mirror reflections because otherwise edge crossings would necessarily occur in the embedding. For the same reason any other rotation or roto-inversion axes fromG \ Hare not allowed to intersect vertices or edges of the nets. This property significantly narrows the set of supergroups to be included in the derivation of interpenetrating nets. A procedure is described based on the automorphism group of aHopf ring net[Alexandrovet al.(2012).Acta Cryst.A68, 484–493] to determine maximal symmetries compatible with interpenetration patterns. The proposed approach is illustrated by examples of twofold interpenetratedutp,diaandpcunets, as well as multiple copies of enantiomorphic quartz (qtz) networks. Some applications to polycatenated 2-periodic layers are also discussed.

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].