Symmetric tangled Platonic polyhedra

Fine-Tuning of the Sequential Self-Assembly of Entangled Polyhedra by Exploiting the Side-Chain Effect

The control of the sequential self-assembly processes of highly entangled (Ag3L2)n (n=2,4,6,8) and Ag21L12 coordination polyhedra using side-chain effects was studied via the introduction of linear or branched side chains into the tripodal ligands. In addition to changes in the intermediate polyhedral species affording the multi- pathway process, disruption of the kinetic control of the sequential self-assembly was observed, thus demonstrating the utility of steric control for the construction of 3D-entangled molecular materials on the 5 nm scale with high molecular complexity.

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.

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.

Amplification of weak chiral inductions for excellent control over the helical orientation of discrete topologically chiral (M3L2) n polyhedra

Superb control over the helical chirality of discrete (M₃L₂)_n polyhedra (n = 2,4,8, M = Cu^I or Ag^I) created from the self-assembly of propeller-shaped ligands (L) equipped with chiral side chains is demonstrated here. Almost perfect chiral induction (>99 : 1) of the helical orientation of the framework was achieved for the largest (M₃L₂)₈ cube with 48 small chiral side chains (diameter: ∼5 nm), while no or moderate chiral induction was observed for smaller polyhedra (n = 2, 4). Thus, amplification of the weak chiral inductions of each ligand unit is an efficient way to control the chirality of large discrete nanostructures with high structural complexity.

Superb control over the helical chirality of highly-entangled (M₃L₂)_n polyhedra (M = Cu(i), Ag(i); n = 2,4,8) was achieved via multiplication of weak chiral inductions by side chains accumulated on the huge polyhedral surfaces.