Ratio of In-Sphere Volume to Polyhedron Volume of the Great Pyramid Compared to Selected Convex Polyhedral Solids

The architecture of the Great Pyramid at Giza is based on fascinating golden mean geometry. Recently the ratio of the in-sphere volume to the pyramid volume was calculated. One yields as result RV = π ⋅ φ5, where is the golden mean. It is important that the number φ5 is a fundamental constant of nature describing phase transition from microscopic to cosmic scale. In this contribution the relatively small volume ratio of the Great Pyramid was compared to that of selected convex polyhedral solids such as the Platonic solids respectively the face-rich truncated icosahedron (bucky ball) as one of Archimedes’ solids leading to effective filling of the polyhedron by its in-sphere and therefore the highest volume ratio of the selected examples. The smallest ratio was found for the Great Pyramid. A regression analysis delivers the highly reliable volume ratio relation , where nF represents the number of polyhedron faces and b approximates the silver mean. For less-symmetrical solids with a unique axis (tetragonal pyramids) the in-sphere can be replaced by a biaxial ellipsoid of maximum volume to adjust the RV relation more reliably.

Synthesis and Luminescent Property of Polycarbazole/Polyhedral Oligomeric Silsesquioxane Nanocomposites

Polyvinylcarbazole（PVK） composites containing organic-inorganic hybrid polyhedral oligomeric silse-squioxane（POSS） PVK-POSS were prepared by free radical polymerization. POSS monomers reacted with vinylcarbazole and were completely dispersed at molecular level in PVK matrix and PVK-POSS nanocomposites display higher glass transition temperature（Tg） in comparison with neat PVK. Optical properties of PVK/POSS nanocomposites were investigated by UV-spectrum and PL-spectrum and the results show that the PVK-POSS nanoparticles have a good interface effect and improve color purity effectively. The maximum absorption wavelength bathochromically shifts gradually with the increasing of the content of POSS. The luminescent intensity becomes higher and higher with the increase of POSS content, and reaches its maximum luminescent intensity when the POSS content is 3% （mass fraction）, while some POSS-rich nanoparticles are present in matrix when contents of POSS are beyond 5%.

Delay-dependent robust H_∞control of convex polyhedral uncertain fuzzy systems

The robust H∞ control problem for a class of uncertain Takagi-Sugeno fuzzy systems with timevarying state delays is studied. The uncertain parameters are supposed to reside in a polytope. Based on the delay-dependent Lyapunov functional method, a new delay-dependent robust H∞ fuzzy controller, which depends on the size of the delays and the derivative of the delays, is presented in term of linear matrix inequalities （LMIs）. For all admissible uncertainties and delays, the controller guarantees not only the asymptotic stability of the system but also the prescribed H∞ attenuation level. In addition, the effectiveness of the proposed design method is demonstrated by a numerical example.

Gregory Solid Construction for Polyhedral Volume Parameterization by Sparse Optimization

In isogeometric analysis,it is frequently required to handle the geometric models enclosed by four-sided or non-four-sided boundary patches,such as trimmed surfaces.In this paper,we develop a Gregory solid based method to parameterize those models.First,we extend the Gregory patch representation to the trivariate Gregory solid representation.Second,the trivariate Gregory solid representation is employed to interpolate the boundary patches of a geometric model,thus generating the polyhedral volume parametrization.To improve the regularity of the polyhedral volume parametrization,we formulate the construction of the trivariate Gregory solid as a sparse optimization problem,where the optimization objective function is a linear combination of some terms,including a sparse term aiming to reduce the negative Jacobian area of the Gregory solid.Then,the alternating direction method of multipliers(ADMM)is used to solve the sparse optimization problem.Lots of experimental examples illustrated in this paper demonstrate the effectiveness and efficiency of the developed method.

Study of Flame Retardation of Polycarbonate Modified by Polyhedral Oligomeric Silsesquioxane and Caged Bicyclic Phosphate

The flame-retarded polycarbonate（PC） has been made with octaphenyl polyhedral silsesquioxane（OPS） and/or caged bicyclic phosphate（Trimer）.Thermal gravimetric analysis（TGA）,Fourier-transform infrared（FTIR）,TGA-FTIR,limiting oxygen index（LOI）,and mechanical tests have been employed to characterize the modified PC.The additives of OPS and Trimer in PC have been proved to be effective flame-retardants because of the synergistic interaction between the elements of P and Si.The role of OPS and Trimer in PC degradation are different：OPS participates in the charring of PC,while Trimer makes PC degrade in advance.In addition,OPS and Trimer induced obvious differences in the mechanical properties of PC.

Synthesis and characterization of poly(methyl methacrylate-co-polyhedral oligomeric silsesquioxane)hybrid nanocomposites

A novel poly（methyl methacrylate-co-polyhedral oligomeric silsesquioxane） hybrid nanocomposite was synthesized by free radical polymerization and characterized by 1H NMR, 29Si NMR, and TGA technologies. Compared with PMMA homopolymer, the nanocomposite has better thermal stability.

Modeling calving process of glacier with dilated polyhedral discrete element method

Mass loss caused by glacier calving is one of the direct contributors to global sea level rise.Reliable calving laws are required for accurate modelling of ice sheet mass balance.Both continuous and discontinuous methods have been used for glacial calving simulations.In this study,the discrete element method(DEM)based on dilated polyhedral elements is introduced to simulate the calving process of a tidewater glacier.Dilated polyhedrons can be obtained from the Minkowski sum of a sphere and a core polyhedron.These elements can be utilized to generate a continuum ice material,where the interaction force between adjacent elements is modeled by constructing bonds at the joints of the common faces.A hybrid fracture model considering fracture energy is introduced.The viscous creep behavior of glaciers on long-term scales is not considered.By applying buoyancy and gravity to the modelled glacier,DEM results show that the calving process is caused by cracks which are initialized at the top of the glacier and spread to the bottom.The results demonstrate the feasibility of using the dilated polyhedral DEM method in glacier simulations,additionally allowing the fragment size of the breaking fragments to be counted.The relationship between crack propagation and internal stress in the glacier is analyzed during calving process.Through the analysis of the Mises stress and the normal stress between the elements,it is found that geometric changes caused by the glacier calving lead to the redistribution of the stress.The tensile stress between the elements is the main influencing factor of glacier ice failure.In addition,the element shape,glacier base friction and buoyancy are studied,the results show that the glacier model based on the dilated polyhedral DEM is sensitive to the above conditions.

Synthesis and Characterization of a New Titanium-silicate Material Containing Cage-like Polyhedral Oligosilsesquioxane

A new titanium-silicate material was synthesized with cubic cage-like tetramethylammonium octasilicate and TiCl4 ethanol solution as precursors. The product was characterized by FTIR, ^29Si NMR, UV-Vis, and XRD. Struc-tural and ingrediental analyses suggest that this material has a layered structure with cubic cage-like polyhedral oligo-silsesquioxane as building blocks and titanium as bridging atoms.

Polyhedral symmetry and quantum mechanics

A thorough study of regular and quasi-regular polyhedra shows that the symmetries of these polyhedra identically describe the quantization of orbital angular momentum, of spin, and of total angular momentum, a fact which permits one to assign quantum states at the vertices of these polyhedra assumed as the average particle positions. Furthermore, if the particles are fermions, their wave function is anti-symmetric and its maxima are identically the same as those of repulsive particles, e.g., on a sphere like the spherical shape of closed shells, which implies equilibrium of these particles having average positions at the aforementioned maxima. Such equilibria on a sphere are solely satisfied at the vertices of regular and quasi-regular polyhedra which can be associated with the most probable forms of shells both in Nuclear Physics and in Atomic Cluster Physics when the constituent atoms possess half integer spins. If the average sizes of the constituent particles are known, then the average sizes of the resulting shells become known as well. This association of Symmetry with Quantum Mechanics leads to many applications and excellent results.

A Geometric Method for Generating Discrete Trace Transition System of a Polyhedral Invariant Hybrid Automaton

Supervisory control and fault diagnosis of hybrid systems need to have complete information about the discrete states transitions of the underling system. From this point of view, the hybrid system should be abstracted to a Discrete Trace Transition System (DTTS) and represented by a discrete mode transition graph. In this paper an effective method is proposed for generating discrete mode transition graph of a hybrid system. This method can be used for a general class of industrial hybrid plants which are defined by Polyhedral Invariant Hybrid Automata (PIHA). In these automata there are no resetting maps, while invariant sets are defined by linear inequalities. Therefore, based on the continuity property of the state trajectories in a PIHA, the problem is reduced to finding possible transitions between all two adjacent discrete modes. In the presented method, the possibility and the direction of such transitions are detected only by computing the angle between the vector field and the normal vector of the switching surfaces. Thus, unlike the most other reachability methods, there is no need to solve differential equations and to do mapping computations. In addition, the proposed method, with some modifications can be applied for extracting Stochastic or Timed Discrete Trace Transition Systems.

Architecture of Platonic and Archimedean polyhedral links

A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic and Archimedean solids by using our method of the ‘three-cross-curve and dou- ble-twist-line covering’. There are five classes of polyhedral links that can be explored: the tetrahedral and truncated tetrahedral links; the hexahedral and truncated hexahedral links; the dodecahedral and truncated dodecahedral links; the truncated octahedral and icosahedral links. Our results show that the tetrahedral and truncated tetrahedral links have T symmetry; the hexahedral and truncated hexahedral links, as well as the truncated octahedral links, O symmetry; the dodecahedral and truncated dodeca- hedral links, as well as the truncated icosahedral links, I symmetry, respectively. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA and protein catenanes.

Ultrasmall Nanoparticle ROS Scavengers Based on Polyhedral Oligomeric Silsesquioxanes

Although tremendous efforts have been devoted to the structural and functional tailoring of natural polyphenol-functionalized nan oparticles,preparing ultrasmall sized(<6 nm)particles with precisely-defined structures has remained a grand challenge.In this work,we reported the preparation of ultra-small and precisely structured polyhedral oligomeric silsesquioxanes(POSS)-based polyphenol nanopartides(T8^-,T10^-,and T12^-GAPOSS)by accurately functionalizing the POSS surface with plant polyphenol gallic acid units via thiol-Michael"click"reactions.Those polyphenol nanoparticles exhibited strong free radical scavenging capacity,good biocompatibility and ability to resist cell oxidative damage,which dem on strated great potentials in inhibiting oxidative stress in duced pathologies.

A Polyhedral Theory on Graphs

This paper provides a polyhedral theory on graphs from which the criteria of Whit- ney and MacLane for the planarity of graphs are unified, and a brief proof of the Gauss crossing conjecture is obtained.

A novel method of determining the optimal polyhedral orientation for discrete global grid systems applicable to regionalscale areas of interest

The polyhedral discrete global grid system(DGGS)is a multi-resolution discrete earth reference model supporting the fusion and processing of multi-source geospatial information.The orientation of the polyhedron relative to the earth is one of its key design choices,used when constructing the grid system,as the efficiency of indexing will decrease if local areas of interest extend over multiple faces of the spherical polyhedron.To date,most research has focused on global-scale applications while almost no rigorous mathematical models have been established for determining orientation parameters.In this paper,we propose a method for determining the optimal polyhedral orientation of DGGSs for areas of interest on a regional scale.The proposed method avoids splitting local or regional target areas across multiple polyhedral faces.At the same time,it effectively handles geospatial data at a global scale because of the inherent characteristics of DGGSs.Results show that the orientation determined by this method successfully guarantees that target areas are located at the center of a single polyhedral face.The orientation process determined by this novel method reduces distortions and is more adaptable to different geographical areas,scales,and base polyhedrons than those employed by existing procedures.

Manifold Construction Over Polyhedral Mesh

We present a smooth parametric surface construction method over polyhe-dral mesh with arbitrary topology based on manifold construction theory.The surface is automatically generated with any required smoothness,and it has an explicit form.As prior methods that build manifolds from meshes need some preprocess to get poly-hedral meshes with specialtypes of connectivity,such as quad mesh and triangle mesh,the preprocess will result in more charts.By a skillful use of a kind of bivariate spline function which defines on arbitrary shape of 2D polygon,we introduce an approach that directly works on the input mesh without such preprocess.Fornon-closedpolyhe-dral mesh,we apply a global parameterization and directly divide it into several charts.As for closed polyhedral mesh,we propose to segment the mesh into a sequence of quadrilateral patches without any overlaps.As each patch is an non-closed polyhedral mesh,the non-closed surface construction method can be applied.And all the patches are smoothly stitched with a special process on the boundary charts which define on the boundary vertex of each patch.Thus,the final constructed surface can also achieve any required smoothness.

Synthesis of peptide dendrimers with polyhedral oligomeric silsesquioxane cores via click chemistry

Inorganic polyhedral oligomeric silsesquioxane （POSS） was used as the core for the synthesis of poly（t- lysine） peptide dendrimer via copper-catalyzed azide-alkyne click chemistry. The inorganic/organic composite dendrimer was characterized by MS, 1H NMR, FTIR, GPC and DLS.

Bridged polyhedral oligomeric silsesquioxane(POSS):A potential member of silsesquioxanes

Two novel and well-defined polyhedral oligomeric silsesquioxanes（POSS） with two same Si_8O_（12） cores and a reactive NH group, namely bridged-POSS（2a and 2b）,have been prepared by the traditional ＇corner-capping＇ reaction.X-ray diffraction demonstrates that those two POSS have the similar T_8 structure.From the thermo-gravimetric analysis,bridged-POSS shows the belter thermal degradation stability than the contrastive POSS.

TRACES OF NONSMOOTH FUNCTIONS ON POLYHEDRAL DOMAINS

Introduced in this paper are the definitions of the traces for a class of nonsmooth func-tions on polyhedral domains. By analyzing their properties we get the structures of thcsetraces.

Positive Luminescent Sensor for Aerobic Conditions Based on Polyhedral Oligomeric Silsesquioxane Networks

There are numerous numbers of hypoxia-selective luminescent probes based on oxygen quenching of phosphorescence.We show a unique design for luminescent probes to detect hyperoxia utilizing hybrid networks consisting of aggregation-induced emission(AIE)-active dyes and disulfide linkers.At the initial state,emission from the AIE-active dyes is inducible by suppressing energy-consumable intramolecular motions in the hybrid matrices,while the decrease in intensity was detected by releasing molecular motions corresponded to bond scission at the disulfide linkers.Particularly,it was shown that this process selectively proceeds in hypoxia.As a result,positive luminescent signals were obtained in hyperoxia.