Abstract. The triangular ratio metric is studied in subdomains of the complex plane and Euclidean n-space. Various inequalities are proven for this metric. The main results deal with the behavior of this metric under quasiconformal maps. We also study the smoothness of metric disks with small radii.

Proton conduction is vital for living systems to execute various physiological activities. The understanding of its mechanism is also essential for the development of state-of-the-art applications, including fuel-cell technology. We herein present a bottom-up strategy, that is, the self-assembly of Cage-1 and -2 with an identical chemical composition but distinct structural features to provide two different supramolecular conductors that are ideal for the mechanistic study. Cage-1 with a larger cavity size and more H-bonding anchors self-assembled into a crystalline phase with more proton hopping pathways formed by H-bonding networks, where the proton conduction proceeded via the Grotthuss mechanism. Small cavity-sized Cage-2 with less H-bonding anchors formed the crystalline phase with loose channels filled with discrete H-bonding clusters, therefore allowing for the translational diffusion of protons, that is, vehicle mechanism. As a result, the former exhibited a proton conductivity of 1.59 × 10 –4 S/cm at 303 K under a relative humidity of 48%, approximately 200-fold higher compared to that of the latter. Ab initio molecular dynamics simulations revealed distinct H-bonding dynamics in Cage-1 and -2 , which provided further insights into potential proton diffusion mechanisms. This work therefore provides valuable guidelines for the rational design and search of novel proton-conducting materials.

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The spontaneous resolution of racemates, from natural compounds to artificial structures, has long been pursued to shed light on the origin of homochirality in life. Even though diverse synthetic systems have been elegantly devised to elaborate the underlying principles of spontaneous symmetry breaking, their complexity is still unparalleled to the natural masterpieces including DNA helix and proteins, which convey remarkable coalescence at both molecular and supramolecular levels. Here, we report on the spontaneous resolution of a pair of homochiral entities from a racemic mixture of a triply interlocked cage-catenane comprising 720 possible stereoisomers. This cage-catenane comprises six methyldithiane ring-containing linkers (denoted rac-2). As each methyldithiane ring has two chiral centers, it exhibits four possible diastereomers. These otherwise equimolar diastereomers are preferentially differentiated with the equatorial conformers over their axial analogues, leading to the dominant formation of ( S , R )-2 and ( R , S )-2, i.e., diastereomeric enrichment at the molecular level. This diastereomeric enrichment is unbiasedly transferred from precursor rac-2 to cage-catenane rac-4, from which a pair of homochirals ( S , R ) 6 -4 and ( R , S ) 6 -4 is narcissistically self-sorted upon crystallization. This powerful symmetry breaking is attributed to a supramolecular synergy of directional π–π stacking with the multivalency of erstwhile weak S···S contacts (with an unusual distance of 3.09 Å) that are cooperatively arranged in a helical fashion. This work highlights the attainability of complex homochiral entities by resorting to coalesced covalent and noncovalent contributions and therefore provides additional clues to the symmetry breaking of sophisticated yet well-defined architectures.

A variety of organic cages with different geometries have been developed during the last decade, most of them exhibiting a single cavity. In contrast, the number of organic cages featuring a pair of cavities remains scarce. These structures may pave the way towards novel porous materials with emergent properties and functions.We herein report on rational design of a three-dimensional hexaformyl precursor 1, which exhibits two types of conformers, i.e. Conformer-1 and -2, with different cleft positions and sizes. Aided by molecular dynamics simulations, we select two triamino conformation capturers (denoted CC). Small-sized CC-1 selectively capture Conformer-1 by matching its cleft size, while the large-sized CC-2 is able to match and capture both conformers. This strategy allows the formation of three compounds with twin cavities, which we coin diphane. The self-assembly of diphane units results in superstructures with tunable proton conductivity, which reaches up to 1.37×10-5 S cm-1.