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.

High-efficiency sodium storage of Co_(0.85)Se/WSe_(2) encapsulated in N-doped carbon polyhedron via vacancy and heterojunction engineering

With the advantage of fast charge transfer,heterojunction engineering is identified as a viable method to reinforce the anodes'sodium storage performance.Also,vacancies can effectively strengthen the Na+adsorption ability and provide extra active sites for Na+adsorption.However,their synchronous engineering is rarely reported.Herein,a hybrid of Co_(0.85)Se/WSe_(2) heterostructure with Se vacancies and N-doped carbon polyhedron(CoWSe/NCP)has been fabricated for the first time via a hydrothermal and subsequent selenization strategy.Spherical aberration-corrected transmission electron microscopy confirms the phase interface of the Co_(0.85)Se/WSe_(2) heterostructure and the existence of Se vacancies.Density functional theory simulations reveal the accelerated charge transfer and enhanced Na+adsorption ability,which are contributed by the Co_(0.85)Se/WSe_(2) heterostructure and Se vacancies,respectively.As expected,the CoWSe/NCP anode in sodium-ion battery achieves outstanding rate capability(339.6 mAh g^(−1) at 20 A g^(−1)),outperforming almost all Co/W-based selenides.

Facile synthesis of the Mn_(3)O_(4)polyhedron grown on N-doped honeycomb carbon as high-performance negative material for lithium-ion batteries

Because of their large volume variation and inferior electrical conductivity,Mn_(3)O_(4)-based oxide anode materials have short cyclic lives and poor rate capability,which obstructs their development.In this study,we successfully prepared a Mn_(3)O_(4)/N-doped honeycomb carbon composite using a smart and facile synthetic method.The Mn_(3)O_(4)nanopolyhedra are grown on N-doped honeycomb carbon,which evidently mitigates the volume change in the charging and discharging processes but also improves the electrochemical reaction kinetics.More importantly,the Mn-O-C bond in the Mn_(3)O_(4)/N-doped honeycomb carbon composite benefits electrochemical reversibility.These features of the Mn_(3)O_(4)/N-doped honeycomb carbon(NHC)composite are responsible for its superior electrochemical performance.When used for Li-ion batteries,the Mn_(3)O_(4)/N-doped honeycomb carbon anode exhibits a high reversible capacity of 598 mAh·g^(−1)after 350 cycles at 1 A·g^(−1).Even at 2 A·g^(−1),the Mn_(3)O_(4)/NHC anode still delivers a high capacity of 472 mAh·g^(−1).This work provides a new prospect for synthesizing and developing manganese-based oxide materials for energy storage.

Unique NiFe–NiCoO2 hollow polyhedron as bifunctional electrocatalysts for water splitting

It is of significance to design of stable and cost-effective electrocatalyst for water splitting with high efficiency in an alkaline medium.The major obstacles for practical application of water splitting devices are lack of high-efficiency and low-cost electrocatalysts with low overpotential for both HER and OER.In this paper,we report a NiFe alloy decorated NiCoO2 hollow polyhedron(denoted as Ni Fe–Ni Co O2)by using[NiFe(CN)6]- intercalated NiCo–LDH as precursor.As evidenced by the electrochemical active surface area,the resultant Ni Fe–Ni Co O2 composite shows unique hollow nanostructure,which can not only provide abundant mass transport channels,but also increase the contact area of the NiFe–Ni Co O2 material with the electrolyte.The overpotential(η)demand is 286 mV for OER and 102 mV for HER at the current density of 10 mA/cm2 in an alkaline medium of 1 M KOH for the NiFe/NiCoO2 composite.This work provides a new pathway for preparation of the highly efficient bifunctional electrocatalysts for water splitting.

Optimal fuzzy PID controller with adjustable factors based on flexible polyhedron search algorithm

A new kind of optimal fuzzy PID controller is proposed, which contains two parts. One is an on line fuzzy inference system, and the other is a conventional PID controller. In the fuzzy inference system, three adjustable factors x p, x i , and x d are introduced. Their functions are to further modify and optimize the result of the fuzzy inference so as to make the controller have the optimal control effect on a given object. The optimal values of these adjustable factors are determined based on the ITAE criterion and the Nelder and Mead′s flexible polyhedron search algorithm. This optimal fuzzy PID controller has been used to control the executive motor of the intelligent artificial leg designed by the authors. The result of computer simulation indicates that this controller is very effective and can be widely used to control different kinds of objects and processes.

Improved non-uniform subdivision scheme with modified Eigen-polyhedron

In this study,a systematic refinement method was developed for non-uniform Catmull-Clark subdivision surfaces to improve the quality of the surface at extraordinary points(EPs).The developed method modifies the eigenpolyhedron by designing the angles between two adjacent edges that contain an EP.Refinement rules are then formulated with the help of the modified eigenpolyhedron.Numerical experiments show that the method significantly improves the performance of the subdivision surface for non-uniform parameterization.

Fast,Exact and Robust Set Operations on Polyhedrons Using Localized Constructive Solid Geometry Trees

Regularized Boolean operations have been widely used in 3D modeling systems. However, evaluating Boolean operations may be quite numerically unstable and time consuming, especially for iterated set operations. A novel and unified technique is proposed in this paper for computing single and iterated set operations efficiently, robustly and exactly. An adaptive octree is combined with a nested constructive solid geometry （CSG） tree by this technique. The intersection handling is restricted to the cells in the octree where intersection actually occurs. Within those cells, a CSG tree template is instanced by the surfaces and the tree is converted to planebased binary space partitioning （BSP） for set evaluation; Moreover, the surface classification is restricted to the ceils in the octree where the surfaces only come from a model and are within the bounding-boxes of other polyhedrons. These two ways bring about the efficiency and scalability of the operations, in terms of runtime and memory. As all surfaces in such a cell have the same classification relation, they are classified as a whole. Robustness and exactness are achieved by integrating plane-based geometry representation with adaptive geometry predicate technique in intersection handling, and by applying divide-and-conquer arithmetic on surface classification. Experimental results demonstrate that the proposed approach can guarantee the robustness of Boolean computations and runs faster than other existing approaches.

Ferromagnetic Calculation of Terbium Dysprosium and Holmium with Polyhedron Electron Shell

Based on the regular polyhedron model of multi-electronic atom combined with the Bohr hypothesis, the following supposition is put forward: the electron momentum multiplied by the inscribed sphere radius of edges of each regular polyhedron is equal to the Planck constant. The relationship between saturation magnetization rates and Planck constants is determined, and the ferromagnetism of atoms is obtained from regular dodecahedron and regular hexahedron. Then, terbium, dysprosium, and holmium saturation magnetization rate are obtained from electronic regular polyhedron configuration. Derivation of matter wave formula is from thermodynamics, avoiding over speed of light.

The Atomic Regular Polyhedron Electronic Shell

The periodic table of elements is arranged based on a series of regular polyhedron. The stability of inert gas atoms can be explained by the distribution of electrons, as well as their motion and magnetic force structure. A magnetic force regular octahedron is proposed. It is a unique configuration that best satisfies the convergence of electrons moving in the same direction within regular polyhedra. In the case of an electrostatic force crust, the formal electron spin accounts for the crusts intrinsic magnetic moment exceeding the speed of light. If one is to consider that the electron has a magnetic outer layer and an electrostatic inner layer, then the question can be solved and abovementioned inference can provide the basis for magnetic force and momentum for the regular octahedron model. The electron periphery has twenty-petal adsorptive substances;the existence of adsorptive substance causes the magnetic force greater than the electrostatic force. Each electronic shell in the regular polyhedron is in accordance with the electron configuration of periodic table of elements;the kinetic track of each electron is a surface of regular polyhedron. The magnetic properties of iron, cobalt, and nickel can be explained by the regular dodecahedron electronic shell of an atom. The electron orbit converged from reverse direction can explain diamond. The adsorptive substances found in atomic nuclei and electrons are defined as magnetic particles called magnetons. The thermodynamic magneton theory can be better explained when it is analyzed using principles of thermodynamics, superconductivity, viscosity, and even in the creation of glass. The structure of the light is a helical line.

Solvothermal Synthesis of Polyhedron Cobalt Nanomaterials

Co nanostructure tetradecahedron shape like were prepared via a simple solvothermal route. The shape and size of Co nanopolyhedron were tuned by changing the volume composition of the solvents and the synthesis temperature. Phase purity was confirmed by X-ray diffraction （XRD） and crystal size was determined by scanning electron microscopy （SEM）. The magnetic hysteresis loops of the polyhedron wareneasured using a SQUID magnetometer at 5 and 300 K. The results show ferromagnetic characteristics with coercivities of 22 and 15 Oe, respectively.

Two Algorithms for Fast Polyhedron Ray-Tracing

This note presents two fast polyhedron ray-tracing algorithms that can be applied not only in ray-convex polyhedron intersection, but also in ray-concave polyhedron intersection (Algorithm 2 permits polygons to contain internal loops of not).In the basis of surface trian gulation, Algorithm 1 can accelerate the surface normal vector interpolation by the intersection point's parameters. And besides, Algorithm 2 does not need any pre_procession such as surface triangulation. Moreover, it requires a few memories with more difficult operations such as division, extraction of roots and transcendental functions avoided entirely. Their simplicity and efficiency permit easy software or hardware implementation.

Excess Activity Tuned by Distorted Tetrahedron in CoMoO_(4) for Oxygen Evolution

Oxygen vacancies enable modulating surface reconstruction of transition metal oxides containing metal-oxygen polyhedrons into metallic oxyhydroxide for oxygen evolution reaction(OER),while revealing reconstructing mechanism is stuck by the requirement to precisely control exact sites of these vacancies.Herein,oxygen vacancies are localized only within MoO_(4)tetrahedrons rather than CoO_(6)octahedrons in CoMoO_(4)catalyst,guaranteeing coherent reconstruction of CoO_(6)octahedrons into pure CoOOH with tunable activities for OER.Meanwhile,distorted tetrahedron accelerates the dissolution of Mo atoms into alkaline electrolyte,triggering spontaneous transition of partial CoMoO_(4)into Co(OH)_(2).CoO_(6)octahedrons in both CoMoO_(4)and Co(OH)_(2)can transform pure CoOOH completely at lower potential,resulting in excess intrinsic activity whose summit is identified by overpotential at 10 mA cm^(-2)with 22.9%reduction and Tafel slope with 65.3%reduction.Well-defined manipulation over the distorted polyhedrons offers one versatile knob to precisely modulate electronic structure of oxide catalysts with outstanding OER performance.

负离子配位多面体生长基元在晶体生长中的应用

It has been pointed out by the theoretic model of anion coordination polyhedra of growth units that the crystal form is dominated by the stability of combination of anion coordination polyhedra on the growth interface.The face has the fastest growth rate and disappears easily if the vertex of coordination polyhedron is pointed at.The face has the slowest growth rate and appears easily if the plane of coordination polyhedron is pointed at.The growth rate is in median for the face that the edge of coordination polyhedron is pointed at.A few of problems are discussed in the paper on the basis of this theory. (1)Several examples of the morphology of allomorph and homoeomorphism are given in the paper.There are three allomorphs of TiO 2 rutile,brookite and anatase.The morphology of these allomorphs are totally different because the stability of combination of octahedral [Ti O 6] 8- vary with the orientation and linked edges of coordination polyhedra.Another example is the homoeomorphism of cassiterite SnO 2.The orientation and combination of octahedral [Sn O 6] 8- in cassiterite are similar to that of octahedral [Ti O 6] 8- in rutile,so that cassiterite and rutile have similar morphology even for their form of twin and twin face.The combination of octahedral [Al O 6] 9- ,[Fe O 6] 9- and [Ti O 6] 8- respectively is similar at the orientation and linked form of coordination polyhedra in three crystals of corundum,hematite and ilmenite,so the same faces appear in these three crystals,resulted in similar morphology.

A NEW FORMULA WITHOUT BOUNDARY INTEGRALS ON A STEIN MANIFOLD

A new Koppelman-Leray-Norguet formula of (p,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the solution of -equation on this domain which does not involve integrals on boundary is given, so one can avoid complex estimates of boundary integrals.

面向函数的循环优化序列定制方法

讨论了一种可针对程序中的不同函数(routine)采用不同的循环优化序列的方法,该方法基于polyhedron模型,使用简化cache失效率方程分别对每一个函数作优化序列评估,以迭代编译方式为每个函数寻找一个独特的循环优化序列。该方法降低了变换实施的复杂度,降低了对编译器具体实现的依赖;考虑程序中不同函数的差异以获得更好的优化效果。对SPEC2006的实验数据表明,经过定制优化之后,较之Open64-O3,加速比为1.05～1.13。

Numerical Analysis of Ice Rubble with a Freeze-Bond Model in Dilated Polyhedral Discrete Element Method

Freezing in ice rubble is a common phenomenon in cold regions,which can consolidate loose blocks and change their mechanical properties.To model the cohesive effect in frozen ice rubble,and to describe the fragmentation behavior with a large external forces exerted,a freeze-bond model based on the dilated polyhedral discrete element method(DEM)is proposed.Herein,imaginary bonding is initialized at the contact points to transmit forces and moments,and the initiation of the damage is detected using the hybrid fracture model.The model is validated through the qualitative agreement between the simulation results and the analytical solution of two bonding particles.To study the effect of freeze-bond on the floating ice rubble,punch-through tests were simulated on the ice rubble under freezing and nonfreezing conditions.The deformation and resistance of the ice rubble are investigated during indenter penetration.The influence of the internal friction coefficient on the strength of the ice rubble is determined.The results indicate that the proposed model can properly describe the consolidated ice rubble,and the freeze-bond effect is of great significance to the ice rubble properties.

Modeling method and preliminary model of Asteroid Toutatis from Chang'E-2 optical images

Shape modeling is fundamental to the analysis of dynamic environment and motion around asteroid. Chang＇E- 2 successfully made a flyby of Asteroid 4179 Toutatis and obtained plenty of high-resolution images during the mis- sion. In this paper, the modeling method and preliminary model of Asteroid Toutatis are discussed. First, the optical images obtained by Chang＇E-2 are analyzed. Terrain and silhouette features in images are described. Then, the modeling method based on previous radar model and preliminary information from optical images is proposed. A preliminary polyhedron model of Asteroid Toutatis is established. Finally, the spherical harmonic coefficients of Asteroid Toutatis based on the polyhedron model are obtained. Some parameters of model are analyzed and compared. Although the model proposed in this paper is only a preliminary model, this work offers a valuable reference for future high-resolution models.

Mg_2(OH)_2CO_3·3H_2O Whiskers Growth Mechanism

From the perspective of growth units, the growth mechanism of Mg2（OH）2CO3.3H2O whisker is investigated in this paper. Results show that the growth morphology of Mg2（OH）2CO33H2O whisker is consistent with the model of anion coordination polyhedron growth units. The growth solution Raman shift of Mg2（OH）2CO;3H2O was monitored using Raman spectroscopy. The growth units are [Mg-（OH）4]2- and H2COv The growth process of Mg2（OH）2COf3H2O whisker is as follows： growth unit [Mg-（OH）4]2- first incorporates into the larger dimension [Mg-（OH）4]2-, then the [Mg-（OH）4]2-n combines with H2CO3 into a linear skeleton Mg2（OH）2CO3 in the same line. Mg2（OH）2CO3 combines with H2O by hydrogen bonds and ultimately transforms into Mg2（OH）2COf3H2O whisker. Magnesium carbonate whiskers have a layered structure, each of which is made of magnesium, carbon, oxygen, with H2O in between each layer. When skeletons are superimposed within the same plane as a parallelepiped one, they grow into solid cuboid-shaped whiskers. When the parallelepiped skeletons planes combine with each other through the cascading links, they grow into hollow cylindrical whiskers.

Growth Habit of the Basic Oxysulfate Magnesium Whisker

The growth habit of the basic magnesium oxysulfate whisker was investigated based on the theoreticalmodelof anion coordination polyhedron growth units.It is found that typicalbasic magnesium oxysulfate whisker growth is consistent with anion tetrahedralcoordination incorporation rules.The growth units of basic magnesium oxysulfate whiskers are [Mg-（OH）_4]^（2-） and HSO_4^-.[Mg-（OH）_4]^（2-） is the favorable growth unit and whisker growth is in the direction of the [Mg-（OH）_4]^（2-） combination.A plurality of [Mg-（OH）_4]^（2-） s combine and become a larger dimensionalgrowth unit in a one-dimensionaldirection.Then HSO_4^- and larger dimensionalgrowth units connect as basic magnesium sulfate whiskers,according to the structuralcharacteristics of the basic magnesium sulfate whisker,which can guide the synthesis of magnesium hydroxide whisker.

Ternary alloy and metal oxides embedded in yolk–shell polyhedrons as bifunctional oxygen electrocatalyst

To improve the efficiency of oxygen electrolysis,exploiting bifunctional electrocatalysts with excellent activity and stability is extremely important due to the four-electron transfer dynamics of oxygen evolution reaction(OER)and oxygen reduction reaction(ORR).Herein,a series of yolk-shell hollow polyhedrons(YHPs)embedded with NiCoFe ternary alloy and metal oxides,which are named YHP-x(x=1,2,3,4),were reported.By controlled etching multi-layered zeolitic imidazolate frameworks and following pyrolytic integration,YHPs are endowed with mass transfer tunnels,accessible inner active sites,and good electrical conductivity.Due to the synergetic effect of the alloy,metal oxides and the yolk-shell structure,YHP-1 exhibits excellent ORR performance with a half-wave potential of 0.79 V and YHP-2 displays superior OER performance with a low overpotential of 257 mV at 10 mA cm−2.The strategy described in this work can be extended to a number of hollow/yolk-shell electrocatalysts for water splitting and metal–air batteries.