Note on ordering and complexity of Platonic and Archimedean polyhedra based on solid angles

The missing values for the solid angles of the two snub semiregular polyhedra have been calculated, and integrated into the whole series of Platonic and Archimedean polyhedra. This is the only criterion which so far gives an unambiguous answer (without any degeneracy leading to posets) on how to order these polyhedra according to their increasing complexity.     

Modulating Orientational Order to Organize Polyhedral Nanoparticles into Plastic Crystals and Uniform Metacrystals

In nanoparticle self-assembly, the current lack of strategy to modulate orientational order creates challenges in isolating large-area plastic crystals. Here, we achieve two orientationally distinct supercrystals using one nanoparticle shape, including plastic crystals and uniform metacrystals. Our approach integrates multi-faceted Archimedean polyhedra with molecular-level surface polymeric interactions to tune nanoparticle orientational order during self-assembly. Experiments and simulations show that coiled surface polymer chains limit interparticle interactions, creating various geometrical configurations among Archimedean polyhedra to form plastic crystals. In contrast, brush-like polymer chains enable molecular interdigitation between neighboring particles, favoring consistent particle configurations and result in uniform metacrystals. Our strategy enhances supercrystal diversity for polyhedra comprising multiple nondegenerate facets.     

Multicomponent self-assembly of a nested co(24) @co(48) metal-organic polyhedral framework

A tale of two polyhedra: Two nested Archimedean metal-organic polyhedra, a rhombicuboctahedron (Co(48) cage) and a cuboctahedron (Co(24) cage), have been assembled from two types of cobalt dimers and two complementary ligands. Within the 3D covalent cubic array of outer Co(48) cages and framework lie encapsulated inner Co(24) cages that are linked into a separate "hidden" 3D framework.     

Low-level self-assembly of open framework based on three different polyhedra: metal-organic analogue of face-centered cubic dodecaboride

Dinuclear paddlewheels linked by 5-methylisophthalate and dabco act as 5-connecting nodes to form a highly symmetric open framework based on three different Archimedean polyhedra.     

Codes in platonic, Archimedean, Catalan, and related polyhedra: a model for maximum addition patterns in chemical cages

The notion of d-code is extended to general polyhedra by defining maximum sets of vertices with pairwise separation > or =d. Codes are enumerated and classified by symmetry for all regular and semiregular polyhedra and their duals. Partial results are also given for the series of medials of Archimedean polyhedra. In chemistry, d-codes give a model for maximal addition to or substitution in polyhedral frameworks by bulky groups. Some illustrative applications from the chemistry of fullerenes and boranes are described.     

Copper Keplerates: High‐Symmetry Magnetic Molecules

Keplerates are molecules that contain metal polyhedra that describe both Platonic and Archimedean solids; new copper keplerates are reported, with physical studies indicating that even where very high molecular symmetry is found, the low‐temperature physics does not necessarily reflect this symmetry.     

Nonconvex polyhedra by repeated truncation of semiregular polyhedra

Some of the semiregular (Archimedean) polyhedra (1–13 in Table 1) afford on truncation polyhedra that contain vertices where the sum of planar degrees for the faces which meet at those vertices is equal to (for 17, 18, and 23 in Table 3) or higher than 360° (21, 22, 24–26 in Table 3). Therefore such polyhedra are nonconvex.     

Molecules and crystals with both icosahedral and cubic symmetry

Notwithstanding the apparent incompatibility between octahedral and icosahedral symmetries, fragments with the two types of symmetry coexist in many molecules and crystals, as evidenced by continuous shape and symmetry measures. A geometric analysis of Platonic and Archimedean polyhedra and of a variety of molecular and crystal structures strongly suggests that octahedral symmetry is latent in icosahedral polyhedra and vice versa. In this Feature Article, new concepts and structural data from the literature combine to offer a perspective view of complex molecular and extended structures. Its influence on the common cubic packing of icosahedral molecules is discussed for a variety of examples, including water clathrates, dodecahedrane, Buckminsterfullerene, the Pd145 and Mo132 clusters and several intermetallic phases.     

Partition of <Emphasis Type="Italic">π</Emphasis>-electrons between faces of polyhedral carbon aggregates

We report for the trivalent regular and semiregular polyhedra (three Platonic and seven Archimedean carbon polyhedra) the π-electron partition between rings of various sizes based on considering all their resonance structures. It was found that small odd-membered (3-and 5-membered) faces are assigned a lower share of π-electrons than that corresponding to equipartition (i.e., 1/3 of an electron for carbon atoms shared between three rings). In contrast, 4-membered rings obtain a larger share of π-electrons than that corresponding to equipartition.     

Structural preferences of single-walled silica nanostructures: nanospheres and chemically stable nanotubes

Structural preferences of single-walled and coordinatively saturated spherical and tubular nanostructures of silica have been determined by ab initio calculations. Two families of spherical (SiO2)n clusters derived from Platonic solids and Archimedean polyhedra are depicted, with n ranging from 4-120. The analogue of a truncated icosidodecahedron, Ih-symmetric Si120O240, is favored in energy, closely followed by the Ih-symmetric Si60O120-truncated icosahedron. The silica nanotubes derived from spherical clusters are capped by Si2O2 rings, whereas the tubular section consists of single oxygen bridges. Periodic studies performed with open-ended silica nanotubes and the alpha-quartz polymorph of silica, along with a comparisons to fullerenes and carbon nanotubes, suggest that tubes with diameters of approximately 1 nm should be chemically stable.     

Molecular structures of alumina nanoballs and nanotubes: a theoretical study

Molecular structures of alumina nanoballs and nanotubes have been determined. Tetrahedral, octahedral, and icosahedral alumina nanostructures were derived from Platonic solids and Archimedean polyhedra and were optimized by quantum chemical methods. I(h)-symmetric balls, resembling their isovalence electronic analogues, fullerenes, are preferred. The nanoballs consist of adjacent Al(5)O(5) and Al(6)O(6) rings, similar to C(5)- and C(6)-rings of fullerenes. The structural characteristics of alumina nanoballs are dominated by pi-electron donation from oxygen to aluminum. Alumina nanotubes can be derived from icosahedral nanoballs. The tubes alternate between D(5d)- and D(5h)-symmetries and are capped by halves of the icosahedral balls.     

Molecular structures of magnesium dichloride sheets and nanoballs

The structures and relative stabilities of (MgCl(2))(n)() sheetlike clusters and nanoballs were studied by quantum chemical methods. The sheets as discrete molecules were studied up to Mg(100)Cl(200). Their stabilities increase systematically as a function of the size of the sheet. Periodic ab initio calculations were performed for (001) monolayer sheets of alpha- and beta-MgCl(2), beta-sheet being slightly favored. Nanoballs were constructed from Archimedean polyhedra, producing tetrahedral, octahedral, and icosahedral symmetries, and were studied up to Mg(60)Cl(120). Nanoballs prefer to take the shape of truncated cuboctahedron (Mg(48)Cl(96)). Comparisons to sheetlike clusters and periodic calculations suggest that magnesium dichloride nanoballs are stable.     

Structural Preferences of Single‐Walled Silica Nanostructures: Nanospheres and Chemically Stable Nanotubes

Structural preferences of single‐walled and coordinatively saturated spherical and tubular nanostructures of silica have been determined by ab initio calculations. Two families of spherical (SiO₂)_n clusters derived from Platonic solids and Archimedean polyhedra are depicted, with n ranging from 4–120. The analogue of a truncated icosidodecahedron, I_h‐symmetric Si₁₂₀O₂₄₀, is favored in energy, closely followed by the I_h‐symmetric Si₆₀O₁₂₀‐truncated icosahedron. The silica nanotubes derived from spherical clusters are capped by Si₂O₂ rings, whereas the tubular section consists of single oxygen bridges. Periodic studies performed with open‐ended silica nanotubes and the α‐quartz polymorph of silica, along with a comparisons to fullerenes and carbon nanotubes, suggest that tubes with diameters of approximately 1 nm should be chemically stable.     

Complex nanoscale cage clusters built from uranyl polyhedra and phosphate tetrahedra

Five cage clusters that self-assemble in alkaline aqueous solution have been isolated and characterized. Each is built from uranyl hexagonal bipyramids with two or three equatorial edges occupied by peroxide, and three also contain phosphate tetrahedra. These clusters contain 30 uranyl polyhedra; 30 uranyl polyhedra and six pyrophosphate groups; 30 uranyl polyhedra, 12 pyrophosphate groups, and one phosphate tetrahedron; 42 uranyl polyhedra; and 40 uranyl polyhedra and three pyrophosphate groups. These clusters present complex topologies as well as a range of compositions, sizes, and charges. Two adopt fullerene topologies, and the others contain combinations of topological squares, pentagons, and hexagons. An analysis of possible topologies further indicates that higher-symmetry topologies are favored.     

Ytterbium Coordination Polymer with Four Different Coordination Numbers：The First Structural Characterization of Lanthanide Phthalate Complex

The novel ytterbium coordination polymer is a t4wo-dimensional framework in which the central metal ions have four different coordination numbers and form four kinds of coordination poly-hedra,The four kinds of coordination polyhedra connect into infinite chains by sharing oxygen atoms.     

Onion-shell metal-organic polyhedra (MOPs): a general approach to decorate the exteriors of MOPs using principles of supramolecular chemistry

Metal-organic polyhedra with surface-exposed organic groups have been designed. The polyhedra are based on concentric shells of alternating negative-positive-negative charges and have been used to design homochiral hosts.     

General theoretical analysis of four-connected polyhedral molecules

Molecular orbital calculations on four-connected polyhedral molecules have resulted in the following generalisations. Spherical four-connected transition metal carbonyl polyhedra are characterised by 14n + 2 electrons and their Main Group analogues by 4n + 2 electrons. Non-spherical four-connected polyhedra have variable electron counts of 14n to 14n + 4 (or 4n to 4n + 4). These generalisations have been analysed in terms of Stone's Tensor Surface Harmonic Theory. The development of electron counting rules for condensed polyhedra derived from four-connected polyhedral fragments is also described.     

On topological form in structures

This paper begins with a review of the Euler relation for the polyhedra and presents the corresponding Schläfli relation in n, the polygonality, and p, the connectivity of the polyhedra. The use of ordered pairs as given by (n, p), the Schläfli symbols, to organize the mapping of the polyhedra and its extension into the two-dimensional (2D) and three-dimensional (3D) networks is described. The topological form index, represented by l, is introduced and is defined as the ratio of the polygonality, n, to the connectivity, p, in a structure, it is given by l = n/p. Next a discussion is given of establishing a conventional metric of length in order to compare topological properties of the polyhedra and networks in 2D and 3D. A fundamental structural metric is assumed for the polyhedra. The metric for the polyhedra is, in turn, used to establish a metric for tilings in the Euclidean plane. The metrics for the polyhedra and 2D plane are used to establish a metric for networks in 3D. Once the metrics have been established, a conjecture is introduced, based upon the metrics assumed, that the area of the elementary polygonal circuit in the polyhedra and 2D and 3D networks is proportional to a function of the topological form index, l, for these structures. Data of the form indexes and the corresponding elementary polygonal circuit areas, for a selection of polyhedra and 2D and 3D networks is tabulated, and the results of a least squares regression analysis of the data plotted in a Cartesian space are reported. From the regression analysis it is seen that a quadratic in l, the form index, successfully correlates with the corresponding elementary polygonal circuit area data of the polyhedra and 2D and 3D networks. A brief discussion of the evident rigorousness of the Schläfli indexes (n, p) over all the polyhedra and 2D and 3D networks, based upon the correlation of the topological form index with elementary polygonal circuit area in these structures, and the suggestion that an Euler–Schläfli relation for the 2D and 3D networks, is possible, in terms of the Schläfli indexes, concludes the paper.