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.     

Novel crystalline carbon-cage structure synthesized from laser-driven shock wave loading of graphite

We report a novel crystalline carbon-cage structure synthesized from laser-driven shock wave loading of a graphite-copper mixture to about 14+/-2 GPa and 1000 +/- 200 K. Quite unexpectedly, it can be structurally related to an extremely compressed three-dimensional C60 polymer with random displacement of C atoms around average positions equivalent to those of distorted C60 cages. Thus, the present carbon-cage structure represents a structural crossing point between graphite interlayer bridging and C60 polymerization as the two ways of forming diamond from two-dimensional and molecular carbon.     

Lists of face-regular polyhedra

We introduce a new notion that connects the combinatorial concept of regularity with the geometrical notion of face transitivity. This new notion implies finiteness results in the case of bounded maximal face size. We give lists of structures for some classes and investigate polyhedra with constant vertex degrees and faces of only two sizes.     

Multimolecular polyhedra and QSPR

A multimolecular polyhedron (MP) is defined when studying molecular sets. When any molecular set is described using some chosen \(N\) -dimensional vector tags attached to every molecular object in the set, thus forming a tagged set, then the tag set also forms a MP. In the particular case when the molecular tags are density functions, then he quantum object set formed in this manner also defines some MP in \(\infty \) -dimensional space. In principle, MP are defined with respect an arbitrary origin, but once constructed one can shift it in two natural ways: (a) by each vertex at a time and (b) once for all by means of the polyhedron centroid. The quantum QSAR frames, which can be constructed following these two origin shift procedures, provide two not so different results when used for property evaluation. The differences between both shifting techniques are here revealed and discussed. A single density function for a given molecule is also studied from this point of view as a sequel of the previous study of MP.     

Reticular chemistry of metal-organic polyhedra

Metal-organic polyhedra (MOPs), are discrete metal-organic molecular assemblies. They are useful as host molecules that can provide tailorable internal volume in terms of metrics, functionality, and active metal sites. As a result, these materials are potentially useful for a variety of applications, such as highly selective guest inclusion and gas storage, and as nanoscale reaction vessels. This review identifies the nine most important polyhedra, and describes the design principles for the five polyhedra most likely to result from the assembly of secondary building units, and provides examples of these shapes that are known as metal-organic crystals.     

Surface tension-driven self-folding polyhedra

We discuss finite element simulations and experiments involving the surface tension-driven self-folding of patterned polyhedra. Two-dimensional (2D) photolithographically patterned templates folded spontaneously when solder hinges between adjacent faces were liquefied. Minimization of interfacial free energy of the molten solder with the surrounding fluidic medium caused the solder to ball up, resulting in a torque that rotated adjacent faces and drove folding. The simulations indicate that the folding process can be precisely controlled, has fault tolerance, and can be used to fold polyhedra composed of a variety of materials, ranging in size from the millimeter scale down to the nanometer scale. Experimentally, we have folded metallic, arbitrarily patterned polyhedra ranging in size from 2 mm to 15 microm.     

Topological aspects of chemically significant polyhedra

This paper considers both static and dynamic properties of chemically significant polyhedra. Static properties of polyhedra consider relationships between the numbers and degrees/sizes of polyhedral vertices, edges, and faces; polyhedral symmetries; and numbers of topologically distinct polyhedra of various types. Dynamic properties of polyhedra involve studies of polyhedral isomerizations from both macroscopic and microscopic points of view. Macroscopic aspects of polyhedral isomerization can be described by graphs called topological representations in which the vertices correspond to different permutational isomers and the edges to single degenerate polyhedral isomerization steps. Such topological representations are presented for isomerizations of polyhedra having five, six, and eight vertices. Microscopic aspects of polyhedral isomerizations arise from consideration of the details of polyhedral topology, such as the topological aspects of diamond-square-diamond processes. In this connection, Gale diagrams are useful for describing isomerizations of five- and six-vertex polyhedra, including the Berry pseudorotation of a trigonal bipyramid through a square pyramid intermediate and the Bailar or Ray and Dutt twists of an octahedron through a trigonal prism intermediate.     

Nesting of fullerenes and Frank-Kasper polyhedra

The duality relationship between fullerenes and Frank-Kasper polyhedra suggests that these two families of polyhedra appear nested in solid state structures. Magic numbers, described by simple mathematical relationships, identify four families of fullerenes with tetrahedral and icosahedral symmetries.     

The characterisation of coordination polyhedra by invariants

We have extended the use of the invariants of spherical harmonics for the characterisation of the angular distribution of coordinating atoms about a central atom. These invariants are suitable indices for use in clustering algorithms for automatic crystal chemistry. Invariants for coordination polyhedra in intermetallic compounds and Ca-O silicates are considered. A study of invariants in coordination polyhedra distorted by the Jahn-Teller effect is presented. In addition, Invariants from small clusters and the cuboctahedron-icosahedron transition are analyzed.     

Truncating and chamfering diagrams of regular polyhedra

Truncating the vertex and chamfering the edges of a polyhedron is the classic way to obtain Archimedean polyhedra taking a Platonic solid as the starting point. We have considered the set of polyhedra obtained by this method and have found singular diagrams that show a special relation between these Archimedean polyhedra.     

Eigenvalue spectra and geometric transformation of polyhedra

Starting from one fullerene, the three geometric transformations, cap, face dual and edge dual produce series of carbon clusters and deltahedra. The geometric relation between these polyhedra implies that their topological matrices and eigenvalue spectra must be relative. We have developed a matrix algebra method to research some polyhedra with high symmetry and one kind, two or three kinds of equivalent vertices such as C₆₀(I_h), resolve their exact eigenvalues, and proved this point.     

配位多面体的群重叠积分

本文讨论了配位多面体的群重叠积分计算问题, 给出了这种群重叠积分的一般形式以及它们满足的一些关系, 用第一类点群将群重叠积分的计算化简, 由此定义了有关的几何参数并研究了它们的性质, 以正六面体为例说明了这些参数的具体计算。     

A catalogue of growth transformations of fullerene polyhedra

Carbon insertion or extrusion mechanisms transforming one fullerene to another are presented as patch replacements on the fullerene surface. A systematic catalogue is constructed for the topologically distinct local insertion/extrusion transformations of fullerenes, classified by patch boundary and pentagon content. All pairs of patches with the same boundary but different numbers of atoms, i.e., growth patches, containing up to five pentagons, with an upper limit for the boundary length that depends on the number of pentagons, are listed. New transformations and infinite series of transformations are identified.     

Rapid self-assembly of uranyl polyhedra into crown clusters

Clusters built from 32 uranyl peroxide polyhedra self-assemble and crystallize within 15 min after combining uranyl nitrate, ammonium hydroxide, and hydrogen peroxide in aqueous solution under ambient conditions. These novel crown-shaped clusters are remarkable in that they form so quickly, have extraordinarily low aqueous solubility, form with at least two distinct peroxide to hydroxyl ratios, and form in very high yield. The clusters, which have outer diameters of 23 ?, topologically consist of eight pentagons and four hexagons. Their rapid formation and low solubility in aqueous systems may be useful properties at various stages in an advanced nuclear energy system.     

A catalogue of isomerization transformations of fullerene polyhedra

Representation of isomerization and carbon insertion or extrusion mechanisms as patch replacements on a fullerene surface allows construction of a catalogue of topologically distinct local transformations of fullerenes, classified by patch boundary and pentagon content. All isomerization patches and isomerization pairs containing up to five pentagons and with an upper limit for the boundary length depending on the number of pentagons are listed. Several infinite series of transformations are identified.     

Coordination polyhedra TeO n in crystal structures

The Voronoi-Dirichlet (VD) polyhedra and the intersecting spheres method have been used to analyze the coordination of Te(IV) and Te(VI) atoms by oxygen atoms in the crystal structures of 317 compounds. The quantitative estimation of the steric effect of the Te(IV) lone electron pair, giving rise to the asymmetry of the coordination sphere and manifesting itself as a substantial (on average, by 0.5(1) Å) displacement of Te(IV) atoms from the centroids of their VD polyhedra, is presented.