BUFFON’S COIN AND NEEDLE PROBLEMS FOR THE SNUB HEXAGONAL TILING

In this paper we consider the snub hexagonal tiling of the plane ($(3^4, 6)$ Archimedean tiling) and compute the probability that a random circle or a random segment intersects a side of the tiling.

Geometrical manipulation of complex supramolecular tessellations by hierarchical assembly of amphiphilic platinum(II) complexes

Here we report complex supramolecular tessellations achieved by the directed self-assembly of amphiphilic platinum(II) complexes. Despite the twofold symmetry, these geometrically simple molecules exhibit complicated structural hierarchy in a columnar manner. A possible key to such an order increase is the topological transition into circular trimers, which are noncovalently interlocked by metal···metal and π–π interactions, thereby allowing for cofacial stacking in a prismatic assembly. Another key to success is to use the immiscibility of the tailored hydrophobic and hydrophilic sidechains. Their phase separation leads to the formation of columnar crystalline nanostructures homogeneously oriented on the substrate, featuring an unusual geometry analogous to a rhombitrihexagonal Archimedean tiling. Furthermore, symmetry lowering of regular motifs by design results in an orthorhombic lattice obtained by the coassembly of two different platinum(II) amphiphiles. These findings illustrate the potentials of supramolecular engineering in creating complex self-assembled architectures of soft materials.

Assembly of silver Trigons into a buckyball-like Ag180nanocage

Buckminsterfullerene (C60) represents a perfect combination of geometry and molecular structural chemistry. It has inspired many creative ideas for building fullerene-like nanopolyhedra. These include other fullerenes, virus capsids, polyhedra based on DNA, and synthetic polynuclear metal clusters and cages. Indeed, the regular organization of large numbers of metal atoms into one highly complex structure remains one of the foremost challenges in supramolecular chemistry. Here we describe the design, synthesis, and characterization of a Ag180nanocage with 180 Ag atoms as 4-valent vertices (V), 360 edges (E), and 182 faces (F)––sixty 3-gons, ninety 4-gons, twelve 5-gons, and twenty 6-gons––in agreement with Euler’s rule V − E + F = 2. If each 3-gon (or silver Trigon) were replaced with a carbon atom linked by edges along the 4-gons, the result would be like C60, topologically a truncated icosahedron, an Archimedean solid with icosahedral (Ih) point-group symmetry. If C60can be described mathematically as a curling up of a 6.6.6 Platonic tiling, the Ag180cage can be described as a curling up of a 3.4.6.4 Archimedean tiling. High-resolution electrospray ionization mass spectrometry reveals that {Ag3}nsubunits coexist with the Ag180species in the assembly system before the final crystallization of Ag180, suggesting that the silver Trigon is the smallest building block in assembly of the final cage. Thus, we assign the underlying growth mechanism of Ag180to the Silver-Trigon Assembly Road (STAR), an assembly path that might be further employed to fabricate larger, elegant silver cages.

Archimedean tiling graphs with Gallai’s property

Gallai in 1966 raised the question about the existence of graphs with the property that every vertex is missed by some longest path. This property will be called Gallai’s property. In this paper we present some graphs embeddable into Archimedean tiling graphs, with both connectivity 1 and 2, satisfying Gallai’s property.

Irregular model DNA particles self-assemble into a regular structure

DNA nanoparticles with three-fold coordination have been observed to self-assemble in experiment into a network equivalent to the hexagonal (6.6.6) tiling, and a network equivalent to the 4.8.8 Archimedean tiling.

Concentration-dependent rhombitrihexagonal tiling patterns at the liquid/solid interface

A self-assembling Pd(ii) complex forms sophisticated concentration-dependent Archimedean tiling patterns composed of three types of polygons at the liquid/solid interface.