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Concept: Icosidodecahedron


The structural description of disordered systems has been a longstanding challenge in physical science. We propose an atomic cluster alignment method to reveal the development of three-dimensional topological ordering in a metallic liquid as it undercools to form a glass. By analyzing molecular dynamic (MD) simulation trajectories of a Cu(64.5)Zr(35.5) alloy, we show that medium-range order (MRO) develops in the liquid as it approaches the glass transition. Specifically, around Cu sites, we observe “Bergman triacontahedron” packing (icosahedron, dodecahedron and icosahedron) that extends out to the fourth shell, forming an interpenetrating backbone network in the glass. The discovery of Bergman-type MRO from our order-mining technique provides unique insights into the topological ordering near the glass transition and the relationship between metallic glasses and quasicrystals.

Concepts: Physics of glass, Scientific method, Icosidodecahedron, Iron, Titanium, Glass transition, Solid, Glass


Dense particle packing in a confining volume remains a rich, largely unexplored problem, despite applications in blood clotting, plasmonics, industrial packaging and transport, colloidal molecule design, and information storage. Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to N=60 constituent polyhedral particles. We examine the interplay between anisotropic particle shape and isotropic 3D confinement. Densest clusters exhibit a wide variety of symmetry point groups and form in up to three layers at higher N. For many N values, icosahedra and dodecahedra form clusters that resemble sphere clusters. These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. We also investigate cluster density as a function of N for each particle shape. We find that, in contrast to what happens in bulk, polyhedra often pack less densely than spheres. We also find especially dense clusters at so-called magic numbers of constituent particles. Our results showcase the structural diversity and experimental utility of families of solutions to the packing in confinement problem.

Concepts: Icosidodecahedron, Stellation, Polyhedral compound, Platonic solid, Icosahedron, Dual polyhedron, Polyhedron, Dodecahedron


The irreversible transformation from an icosahedral quasicrystal (i-QC) CaAu4.39Al1.61 to its cubic 2/1 crystalline approximant (CA) Ca13Au56.31(3)Al21.69 (CaAu4.33(1)Al1.67, Pa ̅ 3(No. 205); Pearson symbol: cP728; a = 23.8934(4) Å), starting at ~570 ○C and complete by ~650 ○C, is discovered from in-situ, high-energy, variable-temperature powder X-ray diffraction (PXRD), thereby providing direct experimental evidence for the relationship between QCs and their associated CAs. The new cubic phase crystallizes in a Tsai-type approximant structure under the broader classification of polar intermetallic compounds, in which atoms of different electronegativities, viz., electronegative Au + Al vs. electropositive Ca, are arranged in concentric shells. From a structural chemical perspective, the outermost shell of this cubic approximant may be described as interpenetrating and edge-sharing icosahedra, a perspective that is obtained by splitting the traditional structural description of this shell as a 92-atom rhombic triacontahedron into an 80-vertex cage of primarily Au [Au59.86(2)Al17.14□3.00] and an icosahedral shell of only Al [Al10.5□1.5]. Following the proposal that the cubic 2/1 CA approximates the structure of the i-QC and based on the observed transformation, an atomic site analysis of the 2/1 CA, which shows a preference to maximize the number of heteroatomic Au-Al nearest neighbor contacts over homoatomic Al-Al contacts, implies a similar outcome for the i-QC structure. Analysis of the most intense reflections in the diffraction pattern of the cubic 2/1 CA that changed during the phase transformation shows correlations with icosahedral symmetry and the stability of this cubic phase is assessed using valence electron counts. According to electronic structure calculations, a cubic 1/1 CA, “Ca24Au88Al64” (CaAu3.67Al2.67) is proposed.

Concepts: Icosidodecahedron, Diffraction, Powder diffraction, Fundamental physics concepts, X-ray, Atom, Crystal, Crystallography


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 Ag180 nanocage 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 C60 can be described mathematically as a curling up of a 6.6.6 Platonic tiling, the Ag180 cage can be described as a curling up of a Archimedean tiling. High-resolution electrospray ionization mass spectrometry reveals that {Ag3}n subunits coexist with the Ag180 species 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 Ag180 to the Silver-Trigon Assembly Road (STAR), an assembly path that might be further employed to fabricate larger, elegant silver cages.

Concepts: Carbon, Icosahedron, Icosidodecahedron, Platonic solid, Mass spectrometry, Electrospray ionization, Truncated icosahedron, Polyhedron


The creation of a perfect hollow nanoscopic sphere of metal centres is clearly an unrealisable synthetic challenge. It is however an inspirational challenge, from the viewpoint of chemical architecture and also as finite molecular species may provide unique microscopic insight into the origin and onset of phenomena such as topological spin-frustration effects found in infinite 2D and 3D systems. Herein, we report a series of high symmetry gadolinium(III) (S = 7/2) polyhedra, Gd20, Gd32, Gd50 and Gd60, to test an approach based on assembling polymetallic fragments that contain different polygons. Structural analysis reveals the Gd20 cage resembles a dodecahedron; the vertices of the Gd32 polyhedron exactly reveal symmetry Oh; Gd50 displays an unprecedented polyhedron in which an icosidodecahedron Gd30 core is encapsulated by an outer Gd20 dodecahedral shell with approximate Ih symmetry; and the Gd60 shows a truncated octahedron geometry. Experimental and theoretical magnetic studies show that this series produces the expected antiferromagnetic interaction that can be modelled based on classical spins at the Gd sites. From the magnetization analyses we can roughly correlate the derivative bands to the Gd-O-Gd angles. Such a magneto-structural correlation may be used as “fingerprints” to identify these cages.

Concepts: Magnetism, Polygon, Icosidodecahedron, Icosahedron, Dodecahedron, M. C. Escher, Geometry, Polyhedron


Bimetallic nanoclusters Au19Cu30 with chemical composition of [Au19Cu30(CºCR)22(Ph3P)6Cl2](NO3)3 (where RCºC is from 3-ethynylthiophene (H3C4S-3-CºCH) or ethynylbenzene (PhCºCH)) has been synthesized. Single X-ray structural analysis reveals that Au19Cu30 has a multishelled core structure of Au@Au12@Cu30@Au6, comprising a centered icosahedron Au13 (Au@Au12) surrounded by an ico-sidodecahedral Cu30 shell and an outmost shell of a chair-like hexagon Au6. The alkynyl carbon is bound to the hollow sites on the Au19Cu30 nanocluster surface, which is a novel interfacial binding mode in alkynyl-protected alloy nanoclusters. The Cu30 icosidodecahedron is unprecedented and Au19Cu30 represents the first alkynyl-protected Au-Cu alloy nanocluster.

Concepts: Nanoparticle, Alkyne, Icosidodecahedron, Structure, The Hollow, Truncated icosahedron, Atom, Chemical element


Quasicrystals (QCs) are well-ordered but aperiodic crystals with classically forbidden symmetries (such as 5-fold). High-dimensional (HD) crystallography is a standard method to locate atom positions explicitly. However, in practice, it is still challenging because of its complexity. Here, we report a new simple approach to three-dimensional (3D) atomic modeling derived from X-ray diffraction data, and apply it to the icosahedral QC Al0.63Cu0.25Fe0.12. Electron density maps were calculated directly from 3D diffraction data indexed with noninteger (fractional) numbers as measured, with proper phases; each of 2(5) = 32 possible phase assignments for the five strongest reflections was used for Fourier synthesis. This resulted in an initial phasing model based on chemically sensible electron density maps. The following procedure was exactly the same as that used to determine ordinary crystal structures, except that fractional indices were assigned to the reciprocal vectors relative to the three orthogonal 2-fold axes in icosahedral (Ih) symmetry to which the observed diffraction data conformed. Finally, ∼30 000 atoms were located within a sphere of a ∼48 Å radius. Structural motifs or basic repeating units with a hierarchical nature can be found. Isolated icosahedral clusters are surrounded by a concentric dodecahedron, beyond which there is a concentric truncated icosahedron. These are strikingly similar to those obtained via HD crystallography, but show very clear real-space relationships between the clusters.

Concepts: Icosidodecahedron, Dodecahedron, Icosahedron, X-ray, Atom, Truncated icosahedron, Crystallography, Crystal


We present the formation of the largest titanium-oxo cluster [Ti42(μ3-O)60(OiPr)42(OH)12)]6- with the first fullerene-like Ti-O shell structure. The {Ti42O60} core of this compound exemplifies icosahedral (Ih) symmetry as C60, the highest possible symmetry for molecules. According to the coordination environments, the Ti centers in this cluster can be arranged into a Platonic {Ti12} icosahedron and an Archimedean {Ti30} icosidodecahedron. The solution stability of this cluster has been confirmed by ESI-MS study. The spherical body of the {Ti42O60} core has an inside diameter of 1.05 nm and an outside diameter of 1.53 nm, which can be even directly visualized by high-resolution TEM. Our results demonstrate that titanium oxide can also form fullerene-like shell structures.

Concepts: Truncated icosahedron, Chemistry, Polyhedron, Truncated dodecahedron, Platonic solid, Icosidodecahedron, Dodecahedron, Icosahedron


A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A3), W(H2) × W(A1) and W(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B4 onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B5 lattice is used to describe both fivefold and tenfold symmetries. The lattice B6 can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

Concepts: M. C. Escher, Platonic solid, Dodecahedron, Icosidodecahedron, Rhombic dodecahedron, Group theory, Polyhedron, Group


Despite significant progress in the structural characterization of the quasicrystalline state, the chemical origins of long- and short-range icosahedral order remain mysterious and a subject of debate. In this Article, we present the crystal structure of a new complex intermetallic phase, Ca10Cd27Cu2 (mC234.24), whose geometrical features offer clues to the driving forces underlying the icosahedral clusters which occur in Bergman-type quasicrystals. Ca10Cd27Cu2 adopts a C-centered monoclinic superstructure of the 1/1 Bergman approximant structure, in which [110] layers of Bergman clusters in the 1/1 structure are separated through the insertion of additional atoms (accompanied by substantial positional disorder). An examination of the coordination environments of the Ca and Cu (in the ordered regions) reveals that the structure can be viewed as a combination of coordination polyhedra present in the nearest binary phases in the Ca-Cd-Cu compositional space. A notable feature is the separation of Ca-Cd and Cu-Cd interactions, with Bergman clusters emerging as Ca-Cd Friauf polyhedra, which are derived from the MgZn2-type CaCd2 phase, encapsulate a Cu-Cd icosahedron similar to those appearing in Cu2Cd5. DFT-chemical pressure calculations on nearby binary phases point to the importance of this segregation of Ca-Cd and Cu-Cd interactions. The mismatch in atomic size between Cu and Cd leads to an inability to simultaneously satisfy Ca-Cu and Ca-Cd interactions in the Friauf polyhedra of the nearby Laves phase CaCd2. The relegation of the Cu atoms to icosahedra prevents this frustration, while nucleating the formation of Bergman clusters.

Concepts: Platonic solid, Dual polyhedron, Polyhedron, Polyhedral compound, Icosidodecahedron, Dodecahedron, Crystal, Icosahedron