Edges Of Polyhedron Research Articles

Realization of 3D random Voronoi mosaic cell in hydrate phases of clathrate compositions

It is shown that the cages in water frameworks of clathrate structures of crystalline gas hydrates are characterized by averaged values of the numbers NV (polyhedron vertices), NE (polyhedron edges), NF (polyhedron faces) and NVF (vertices per face), close to the values calculated for the unit cell of a 3D random Voronoi mosaic.

Weighted-Norm First-Order System Least-Squares (FOSLS) for Div/Curl Systems with Three Dimensional Edge Singularities

A weighted-norm first-order system least-squares (FOSLS) method for div/curl problems with edge singularities is presented. Traditional finite element methods, including least-squares methods, often suffer from a global loss of accuracy due to the influence of a nonsmooth solution near polyhedral edges. By minimizing a modified least-squares functional, optimal accuracy in weighted and nonweighted norms is recovered. Error estimates with and without mesh refinements are presented, and numerical results are given to confirm the theory.

Epsilon-Unfolding Orthogonal Polyhedra

An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, may need to be as thin as ɛ = 1/2Ω(n).

Computer modeling of the structure of large molecules: II. Cluster C1500

The structure of a C1500 cluster modeling a “planar” polymeric buckminsterfullerene(VIII) is determined by the PM3 quantum-chemical method. The cluster consists of 25 oblate octa-, hexa-, and tetra-valent C60 polyhedra and has D 2h symmetry. The calculated parameters of a rectangular unit cell are as follows: 9.016 A along the bridging bonds in the polymer plane and 9.084 A along the bridging bonds over and under the polymer plane. Hybrid σ orbitals of three-coordinate atoms are directed at an angle of 9.7°–9.8° to the polyhedron edges and those of four-coordinate atoms, at an angle of 8.3°–8.6° to the rectangular bridge edges. The s contribution into π orbitals is negligibly small. The GAMESS program with a new algorithm for diagonalization of large dense symmetrical matrices and a new VALENCY program were used for calculations.

A "Slice-and-Dice" Approach to Area Equivalence in Polyhedral Map Projections

For centuries, people have been intrigued by map projections upon polyhedra—the surface of the Earth laid out on the faces of polyhedral bodies (beautiful in themselves)—which are then unfolded to make flat maps of the world. This paper describes a simple "slice-and-dice" method to obtain area equivalence, which is applied to polyhedral globes. The implementations of this method include a projection that is equivalent to Snyder's equal-area projection for Platonic bodies, an equal-area projection with far smaller cusps, and an equal-area projection that obtains constant scale along the polyhedron's edges. The latter two are presented in terms of exact equations and are compared with Snyder's projection.

An Unprecedented Uranyl Phosphate Framework in the Structure of [(UO2)3(PO4)O(OH) (H2O)2](H2O).

AbstractFor Abstract see ChemInform Abstract in Full Text.

An unprecedented uranyl phosphate framework in the structure of [(UO2)3(PO4)O(OH)(H2O)2](H2O).

The new uranyl phosphate [(UO2)3(PO4)O(OH)(H2O)2](H2O) (1) with an unprecedented framework structure has been synthesized at 150 and 185 degrees C. The structure (tetragonal, P4(2)/mbc, a = 14.015(1) A, c = 13.083(2) A, V = 2575.6(4) A(3), Z = 8) contains uranyl phosphate chains composed of uranyl pentagonal and hexagonal bipyramids and phosphate tetrahedra linked by sharing of polyhedral edges. The uranyl phosphate chains are aligned both along [100] and [010] and are linked into a novel framework structure involving channels along [001]. Topologically identical chains occur linked into sheets in more than a dozen uranyl phosphate minerals, but these chains have never been observed in opposing orientations and linked into a framework as in 1.

Boundary Value Problems for the Laplace Operator in an Infinite Plane Sector

The starting point of the present paper (and some previous papers) is the theory of boundary value problems for elliptic pseudodifferential equations in domains with smooth boundary, constructed in [1, 2] in the mid-sixties. Nearly at the same time, the paper [3] was published, where the normal solvability was analyzed for general boundary value problems for differential operators in domains with conical (or corner) points (see also [4]). More complicated singularities like the vertex of a polyhedral angle and edges of various dimensions were also considered (V.G. Maz’ya, B.A. Plamenevskii, and A.I. Komech). Obviously, these papers do not exhaust the entire variety of problems appearing in nonsmooth situations. Very recently, the papers [5, 6] have appeared, where boundary value problems were considered for the Laplacian in a plane sector with the third boundary condition on the boundary. More complicated multidimensional statements can be found in [7, 8]. Interesting (and very close) results were obtained in [9], but only for differential operators. The author [10, 11] suggested a new approach to the theory of boundary value problems for elliptic pseudodifferential equations, which generalizes the Vishik–Eskin method to the case of nonsmooth domains and is based on a special factorization of the elliptic symbol of a pseudodifferential operator. On the other hand, this approach involves also the ideas due to Kondrat’ev and Eskin, where one applies the Mellin transform to the boundary value problem at the very beginning. This approach has the advantage that one can write out the general solution of a pseudodifferential operator in an infinite plane sector and use it to state a boundary value problem, which can be solved in closed form in a number of cases. This will be illustrated in the present paper for the simplest example of the Laplace operator.

Advances in Understanding of the Crystal Chemistry of Hexavalent Uranium

ABSTRACTResearch concerning the crystal chemistry of hexavalent U by the Environmental Mineralogy and Crystal Structures research group at Notre Dame has resulted in the description of more than 110 new structures of uranyl compounds (including 36 minerals). New insights into the crystal chemistry of U6+ are presented, with emphasis on recently discovered novel structural connectivities. The structural hierarchy of uranyl minerals and compounds, which was first established for 180 structures in 1996, has been extended to include 145 new structures. The hierarchy is based upon polymerization of polyhedra containing higher-valence cations, and consists of five distinct classes: structures containing isolated polyhedra (7), finite clusters of polyhedra (41), chains of polyhedra (52), sheets of polyhedra (184), and frameworks of polyhedra (41). The dominance of sheets in uranyl compounds (57% of known structures) arises from the unequal distribution of bond-valences within the uranyl polyhedra. Topological relations of the sheets in uranyl compounds are best understood by analysis of the topological distribution of anions within sheets in which sharing of polyhedral edges dominates, and by graphical representation of the connectivity of polyhedra in cases where sharing of vertices of polyhedra dominates the sheet.

Vertex-unfoldings of triangulated polyhedra

For a long time there has been a conjecture that any convex polyhedron could unfold to a non-overlapping, connected planar layout by cutting along polyhedron edges. The layout is composed of faces ...

Structure of conduction channels in superionic conductors: Crystallochemical modeling

The experimental and theoretical studies of the structure of conduction channels of superionic conductors are analyzed. A crystallochemical method of modeling the conduction channels is proposed in which they are treated as sets of polyhedron Vorono-Dirichlet sections outside of impenetrable spheres of the rigid sublattice. The allowance for both weak flexibility of the spheres and their mean-square displacements relative to the sublattice points yields a channel structure that “wraps” around the Vorono-Dirichlet polyhedron edges. Modeling the interionic potential for α-AgI with the help of equipotential surfaces suggests that the mobile-ion motion slips along the conduction channel walls. For α-AgI, the set of the equipotential surfaces specifies “the minimum energy trajectory” of transition into a superionic state, while the crystallochemical “wrapping” structure corresponds to a saddle point of a multidimensional potential surface. Symmetry selection rules are used for predicting mechanical trajectories as allowed oscillation modes for the tetrahedral and octahedral fragments of α-CuI.

A RE-EVALUATION OF THE STRUCTURE OF WEEKSITE, A URANYL SILICATE FRAMEWORK MINERAL

The structure of weeksite, K 1.26Ba0.25Ca0.12((UO2)2(Si5O13))H2O, orthorhombic, a 14.209(2), b 14.248(2), c 35.869(4) A, V 7262(2) A 3 , space group Cmmb, has been solved by direct methods using data collected with Mo KX-radiation and a CCD- based detector and refined by full-matrix least-squares techniques, on the basis of F 2 for all unique reflections, to an agreement factor (R1) of 7.0% and a goodness-of-fit ( S) of 1.04, calculated using the 1565 unique observed reflections ( Fo ≤ 4� F). The structure contains four unique U 6+ positions, each of which is part of a nearly linear (UO 2) 2+ uranyl ion. The uranyl ions ( Ur) are further coordinated by five atoms of oxygen arranged at the equatorial corners of UrO5 pentagonal bipyramids. There are ten silicon atoms, each of which is tetrahedrally coordinated by oxygen atoms. The uranyl polyhedra share equatorial edges to form chains, which in turn share polyhedron edges with silicate tetrahedra. The uranyl silicate chains are linked to crankshaft-like chains of vertex-sharing silicate tetrahedra, resulting in layers that are connected by the sharing of vertices between silicat e tetrahedra to form an open framework. Potassium, barium, calcium and H 2O are located in the channels within the uranyl silicate framework. Displacement of some atoms from their corresponding special positions indicates that the results represent an averag e structure.

A Qualitative Control Systems Synthesis by Changing the form of Phase Polyhedron Edges

In this paper, a new approach to high-dimensional control systems design is proposed. This approach allows to take into account the constraints onto system's state and controls. Moreover, the system can be under uncertainties, information about which is constrained as well. The proposed approach is based on phase constraints, being formed in real time, constitute the phase polyhedron, borders and tops of which can be to some extent deformated. Then, the criteria of solving synthesis problem will be placement of the polyhedron's tops, for which the rate-vectors of the system are directed towards the internal part of the polyhedron. Using this geometrial criterion and taking into account the properties, we can obtain the algebraic criterion for solving this problem, which is in form of inequalities, depending on finite number of tops. The conditions for solving these inequalities depending on tops deformation are provided. Conditions for choosing the directions of tops deformation to provide the geometric criterion are obtained. The illustrative example is provided.

3-D polyhedral face computation from two perspective views with the aid of a calibration plate

The 3-D reconstruction of visible polyhedral faces from a pair of general perspective views with the aid of a calibration plate is addressed. A polyhedron is placed on a planar calibration plate and two side views of both the polyhedron and the calibration plate are taken. Through proper arrangements we may assume that in the two views a number of polyhedral edges lying on the calibration plate and the whole calibration plate boundary are visible. We present an online camera calibration technique with the aid of the calibration plate and a two-stage process to find the vertex/edge correspondences without encountering the ambiguity problem of the conventional epipolar line technique. We then give a closed form solution to the 3-D polyhedral vertices visible in both images. We also describe other advantages of using our method for the 3-D polyhedron reconstruction. Experimental results show that the obtained 3-D polyhedral vertex coordinates are rather accurate.

A method to extract three-dimensional objects from three-dimensional NMR image data.

A method to extract shapes of bubble polyhedra from three-dimensional nuclear magnetic resonance image data of foam has been developed. The method consists of four steps: first, to compute series of maximum intensity projection images from the three-dimensional NMR image data of foam to visualize the network structure of the bubble edges; second, to measure the three-dimensional coordinates of vertices of the bubble polyhedra interactively on a CRT using the series of MIP images to obtain polyhedral edges; third, to construct polyhedral faces from the assembly of the edge data; and fourth, to construct bubble polyhedra from the assembly of the face data. The algorithm for construction of polygons and polyhedra is based on the selection of polygons or polyhedra from an enormous number of possible combinations of edges or faces under some simple selection rules. Since the computation time for the selection increased exponentially with the number of the edge data, a 'partial volume sweeping' method in which a cubic region which could accommodate any bubble polyhedron was swept over the three-dimensional image area, was developed to reduce the computation time. By using this method, it was demonstrated that the shapes of many bubble polyhedra could be extracted automatically from the assembly of the edge data within a practical computation time.

Citation for Mary Douglas, 1994 Bernal Prize Recipient

The Lord permitted the Israelites to eat chicken, whereas the ostrich, the pelican, the stork, the hoopoe, and the bat were forbidden. Members of other universities do not wear academic dress within the precincts of the University of Oxford, with the exception of members of the University of Cambridge who wear their robes when preaching the University sermon at Oxford. The Lele and the Kasai, neighboring tribes living on either side of the Kasai River in Zaire, celebrate their hot and cold seasons at opposite points in the calendar. Through the nineteenth century mathematicians disagreed in assessing the soundness of Euler's theorem about the relationship between the faces, vertices, and edges of polyhedra. Just to list these items together, to bring them into juxtaposition, is to identify this year's recipient of the Society's Bernal Prize. For almost thirty years Professor Mary Douglas has been telling academics why the consideration of such things might properly belong within the same inquiry, and why it is important that they be thought about together. And for those whose most viscerally located instincts bridle at any such suggestion, she has offered, and continually refined, a set of resources for the comparative study of culture, its forms, biases, and uses. Born in Italy in 1921, Mary Douglas took her first and doctoral degrees at Oxford where she studied with Edward Evans-Pritchard. Her fieldwork was done among the Lele of the then Belgian Congo, and the results of this research inspired much of her subsequent writing. Most of her academic career was spent progressing from lecturer to professor at University College, London, which she left in 1977 to become Director of Research on Culture at the Russell Sage Foundation in New York, then Avalon Foundation Professor in the Humanities at Northwestern, and Visiting Professor in the Department of Religion at Princeton, before retiring from formal academic

Betpakdalite Unmasked, and a Comment on Bond Valences

A new representation of the crystal structure of complex betpakdalite reported in 1984 (R ≈ 0.08) reveals a remarkable case of . hccchhccch 10-layer anion (O2-)/void ( ) close-packing. The transformation matrix from earlier C12/m 1 cell to new F 2/m 11 cell is [010/100/102], new a = 11.10, b = 19.44, c = 22. 93 �, α)/10 = 2.29 Ǻ. The packing efficiency is Ve = 15.45 � α3 for 10 layers of 32(O2-+ ) per layer. For Pauling r(O2-) = 1.40 �, Ve = 15.52 �3. Oxide anions populate only 46.25% of the close-packed slots; the rest are voids. The difference, Δ, between real structure and ideal (perfect) structure is (0.13 �) for O2-, (0.20 �) for cations and (0.15 A) for all atoms in the close-packed frame, range 0.00-0.43 �. The cell formula for x = 2 is H8 [K(H2O)6]4 [Ca(H20)6]8 [MO6+32 As5+8O148]28-.8H2O. It is suggested that H8 bonds with 8O(10) in the cell, the only anion bonded to only one Mo6+. The cell formula gives D(calc) = 2.90 g cm-3, close to reported D( obs ) = 2.98, 3.05; and 2.92 g cm-3. Bond strength/bond valence calculations for assorted Mp+6 � q- (M = octahedral cation linked to generalized anion �) often show extreme deviations. This is shown to be a consequence of Pauling's third rule of cation-cation repulsions across shared polyhedral edges and faces.

Graph‐Theory Derived Models for the Skeletal Chemical Bonding in Organometallic Metal Carbonyl Clusters

AbstractIdeas derived from topology and graph theory can be used to model the skeletal chemical bonding in metal carbonyl clusters as well as boranes and carboranes. In such molecules delocalized skeletal bonding occurs when there is a mismatch between the number of internal orbitals provided by a vertex atom and the number of polyhedral edges meeting at that vertex. Such global delocalization in deltahedral metal clusters may be regarded as a three‐dimensional analogue of the aromaticity in planar polygonal molecules such as benzene. Elementary graph‐theoretical considerations provide a basis for the 2n+2 skeletal electrons normally observed in n vertex delocalized deltahedral metal clusters lacking tetrahedral chambers. Electron‐rich n‐vertex metal clusters have more than 2n + 2 skeletal electrons and form polyhedra having at least one face with at least 4 edges. Electron‐poor n‐vertex metal clusters have less than 2n + 2 skeletal electrons and form deltahedra having at least one tetrahedral chamber. Ideas derived from group theory and graph theory can be used to analyze molecular orbital energy parameters for the most symmetrical deltahedral borane anions BnHn2− (particularly n = 6 and 12) computed using either semiempirical extended Hückel methods or ab initio methods based on Gaussian orbitals. Some of the fundamental aspects of the graph‐theory derived models for skeletal bonding in delocalized deltahedral metal clusters are closely related to the tensor surface harmonic theory of Stone.

A “vertex electron pair” scheme (VEPS) for describing the skeletal electron distribution in borane-type clusters

Abstract A predominantly localized electron pair scheme is outlined for describing the electron distribution and bonding in closo borane anions B n H n 2− and related electron deficient deltahedral clusters, in which a skeletal electron pair is assigned to each vertex, one pair being regarded as delocalized just inside the roughly spherical surface on which the skeletal atoms lie. The scheme gives a clearer picture of the electron distribution than is conveyed by resonating 2- and 3-centre bonds in the polyhedron edges and faces, and allows the bond orders of the polyhedron edge links to be calculated readily. The consequence of formal removal of BH 2+ units from closo species B n H n 2− to generate nido species B n −1 H n −1 4− and arachno species B n −2 H n −2 6− is explored, and seen to allow rationalization of two features of such deltahedral-fragment clusters: (i) why a high-connectivity vertex is left vacant and (ii) why the frontier orbitals of such species concentrate electronic charge around their open faces. Moreover, in the case of D 4 `h B 4 H 4 6− (cf. C 4 H 4 2− ) and D 5 h B 5 H 5 6− (cf. C 5 H 5 − ), the approach leads directly to the familiar picture for aromatic ring systems in which the highest filled, doubly degenerate π-bonding molecular orbital concentrates electronic charge in rings above and below the polygon on which the skeletal nuclei lie. It also leads to the expectation that arachno clusters with non-adjacent vacant vertices will be more stable than those with adjacent vacant vertices.

Molecular orbital (SCF-X?-SW) theory of metal-metal charge transfer processes in minerals

A molecular orbital description, based on Xα-Scattered wave calculations on a (FeTiO10)14− cluster, is given for Fe2+ → Ti4+ charge transfer transitions in minerals. The calculated energy for the lowest Fe2+ → Ti4+ metal-metal charge transfer transition is 18040 cm−1 in reasonable agreement with energies observed in the optical spectra of Fe-Ti oxides and silicates. As in the case of Fe2+ → Fe3+ charge transfer in mixed-valence iron oxides and silicates, Fe2+ → Ti4+ charge transfer is associated with Fe-Ti bonding across shared polyhedral edges. Such bonding results from the overlap of the Fe(t 2g ) and Ti(t 2g ) 3d orbitals.