Regular Tetrahedron Research Articles

A necessary and sufficient condition for the existence of simple closed geodesics on regular tetrahedra in spherical space

Доказано необходимое и достаточное условие существования простой замкнутой геодезической типа $(p,q)$ на правильном тетраэдре в сферическом пространстве. Библиография: 6 названий.

A Maximum Localization Rate Algorithm for 3D Large-Scale UWSNs

Underwater wireless sensor networks (UWSNs) play an increasingly important role in monitoring the marine environment. The research of sensor node localization is one of the biggest concerns in UWSNs, but there are still many formidable obstacles to overcome, including localization accuracy, localization rate, and other issues. In contrast to localization accuracy, however, few studies focus on increasing the node localization rate. Therefore, this paper proposes a maximum localization rate (MLR) algorithm for 3D large-scale UWSNs. In view of the number of anchor nodes around the unknown node, the MLR algorithm designs different positioning strategies, such as the triangular cosine method and the dynamic position-assisted localization method. The MLR algorithm also adopts the regular tetrahedron network topology and the collinear judgment mechanism to guarantee localization accuracy. The simulation results demonstrate that excluding isolated nodes, the MLR algorithm can achieve the maximum localization rate while maintaining a certain level of positioning accuracy at the same time.

Numerical analysis and experimental study of an ocean wave tetrahedral triboelectric nanogenerator

Ocean wave energy is increasingly regarded as promising blue energy for large-scale applications. Due to the low-frequency profile of the ocean wave energy, Triboelectric Nanogenerator provides an efficient harvesting way and convert into electrical energy. Contact area is an important factor for improving the output performance of triboelectric nanogenerators. However, in the popular structures (like nested rolling-spheres) of triboelectric nanogenerator, the contact area is small due to the point contact mode, which limits the energy harvest efficiency. In this paper, a contact-separation triboelectric nanogenerator based on a tetrahedron structure was designed and tested. The structure contains a smaller tetrahedron suspending inside a larger tetrahedron. Two regular tetrahedrons would contact and separate in the face-face mode when the structure is excited by external oscillation. The first order oscillation frequency of the structure is designed to match the major frequency of the ocean wave.The side length ratio between the internal and external regular tetrahedrons, and the posture of the structure were optimized to improve the power output. Experimental data showed that peak-to-peak voltage output and short-circuit current were at up to 200 V and 30 μA, respectively. Meanwhile, the peak power output reached 14.7 mW with 10 MΩ resistive load. The proposed TENG was able to light up to twenty commonly-used LED lamps (working voltage 3.0–3.4 V). A general thermohygrograph was succesfully powered by the proposed structure. This novel design coule be potentially used in the applications like the self-powered sensor node in the ocean by harvesting the free water wave energy.

Structure and rotational dynamics of water around hydrogen peroxide

The density functional theory-based molecular dynamics simulations were induced to study various properties of water around different solvation pockets around hydrogen peroxide(H2O2). Our study aims to understand water behavior in the vicinity of H2O2 molecule through analyzing structure, hydrogen bond statistics, orientational and non-diffusive jump profile. A water-like peroxide molecule can form four hydrogen bonds with nearby water molecules: two as a proton donor and two as a proton acceptor, which must disturb the local structure and dynamics of the surrounding water. The presence of two different peak maxima in radial distribution function (RDF) allows us to resolve the solvation environment into three solvation sites: solvation shell I, solvation shell II, and bulk. The probability distribution of the average distance of central water molecule from 4 nearest water molecules shows a shift in the peak for solvation shell I water molecules towards a ∼0.3Å higher value compared to two other solvation shells. This further affects tetrahedral order parameters as water molecules shift from the regular tetrahedron. For the first solvation layer, the water molecules lose their tetrahedrality to a significant extent as qtet probability distribution peaks at 0.5 and broaden in the higher values of qtet without any shoulder. For the second solvation layer, we find the qtet peak at 0.87 with a minor shoulder appearing at qtet = 0.55, a sign of slight loss of tetrahedrality in the unstructured water compared to undisturbed water. Loss of tetrahedrality hints at the change of orientation of H2O molecules in the vicinity of H2O2 and affects the water-water hydrogen bond (HB) formation. A noteworthy reduction in the HB donor/acceptor account and per water HB count shows interference of H2O2. The water molecules prefer H2O2 compared to neighboring water molecules for HB formation. The orientation autocorrelation function provides the rotational pace of water molecules in the different solvation layers. The reorientation time constant difference in the solvation shells I and II fluctuate by mere 0.20 ps, showcasing the trivial nature of hydrogen peroxide and water HBs. Using the HB descriptors, we also report the non-diffusive jump profile involving vicinal water molecules of H2O2. The jump angle profile of solvation shell water molecules is in line with bulk water.

Investigation of defect, mechanical, thermal properties and refractive index on an Er:LuYSGG mixed laser crystal

We investigate the defect, mechanical, thermal properties, and refractive index of Er:LuYSGG mixed crystal grown by the Czochralski method. The chemical etching pit patterns present a regular tetrahedron, which consists well with the corresponding atomic arrangement schematics. The Mohs hardness is obtained to be 6.60. The thermal expansion coefficient and the thermal conductivity along <111> crystallographic orientation are achieved to be 7.91 × 10−6 K−1 and 3.423 W m−1 K−1, respectively, indicating the Er:LuYSGG has a relatively high damage threshold. Based on the measured values of refractive index, the Sellmeier equation is fitted and the refractive index curve in the range of 400–3000 nm is displayed. All these results are of great significance for the further research and laser application of Er:LuYSGG crystal.

Design of Broadband and Wide-Angle Hexagonal Metamaterial Absorber Based on Optimal Tiling of Rhombus Carbon Pixels and Implantation of Copper Cylinders

A design method for a broadband and wide-angle metamaterial absorber is proposed based on optimal tiling of rhombus carbon pixels on and implantation of metal cylinders inside an acrylic substrate for which the backside is blocked by the perfect conductor. First, an intermediate carbon metapattern is achieved via optimal tiling of rhombus carbon pixels based on the genetic algorithm (GA), which can minimize the reflectances of both of the transverse electric (TE) and transverse magnetic (TM) polarized electromagnetic (EM) waves for the incident angles 0∘ and 60∘ simultaneously. Then, copper cylinders are implanted inside the substrate to boost the absorptions of both of the TE and TM polarizations for the 60∘ oblique incidences. To extend the absorption bandwidth, the design is finalized by evolving the intermediate metapattern using the GA. Based on the finalized carbon metapattern, the 90% absorption bandwidth is confirmed in the frequency range 8.8 to 11.6 GHz, for which the fractional bandwidth is 27.5% for both of the two polarizations with the incident angles from 0∘ to 60∘. The proposed method could open a way to design a broadband and wide-angle EM metamaterial absorber that can be applied to the edges of three-dimensional structures such as a regular tetrahedron or square pyramid that have interior angles of 60∘ that cannot be covered by conventional square or rectangular metamaterial absorbers.

Crystal structure and semiconductor properties of copper(II) complex incorporating chiral (R)-(+)-α-Ethylbenzylammonium cations:[(R)-C9H14N]3[CuBr4].Br.

The molecular organic-inorganic hybrid halogenometallate [(R)-C9H14N]3[CuBr4].Br has been synthesized by a slow evaporation method. The single-crystal X-ray diffraction experiment evidences that [(R)-C9H14N]3[CuBr4].Br crystallizes in the monoclinic system with the non-centrosymmetric space group P21 at T ​= ​150K. The compound displays a zero-dimensional (0D) structure which consists in three chiral [(R)-C9H14N]+ cations, one anionic [CuBr4]2- and free Br− ion. The structure analysis reveals that the Cu(II) has an intermediate geometry between regular tetrahedron (Td) and square planar (D4h). In the crystal structure, the cations and anions are arranged in alternating stacks which are interconnected via hydrogen bonding contacts N–H⋯Br. This hybrid compound presents good thermal stability up to 370 ​K. The DSC and electric measurements show that no phase transition occurs in the compound over the temperature range 223–423 ​K. Optical absorption measurements suggest that [(R)-C9H14N]3[CuBr4].Br has a narrow direct optical band gap (Eg) of ∼2.15 ​eV which makes it a promising material in optoelectronic devices.

Constellation-shaped three-dimension 14QAM mapping scheme based on regular tetrahedron cells

An innovative constellation-shaped three-dimensional 14-quadrature amplitude modulation QAM (3D-14QAM) mapping scheme based on regular tetrahedron is proposed. The non-uniform 14QAM signals are designed by Huffman code and mapped onto the designed 3D constellation based on regular tetrahedron. The simulation demonstration is carried out successfully by VPI software in a 4 × 100 Gb / s wavelength division multiplexing (WDM) polarization division multiplexed coherent optical communication system with a transmission distance of 300 km. For WDM system, the simulation results show that the proposed scheme is capable of achieving about 1.8 dB OSNR gain over the traditional 3D-16QAM scheme at the hard-decision forward error correction (HD-FEC) limit of 3.8 × 10 − 3. In addition, the proposed 3D-14QAM can obtain 1.1 dB OSNR gain compared with the traditional 3D-16QAM for single-channel scenario at the HD-FEC. The outstanding performance indicates that the proposed scheme can provide a performance improvement for both single carrier system and WDM system. And benefited from the ability of suppress nonlinearity, the proposed scheme can obtain a larger gain in the WDM system than a single carrier system.

Magnetic Field Reconstruction for a Realistic Multi-Point, Multi-Scale Spacecraft Observatory

Future in situ space plasma investigations will likely involve spatially distributed observatories comprised of multiple spacecraft, beyond the four and five spacecraft configurations currently in operation. Inferring the magnetic field structure across the observatory, and not simply at the observation points, is a necessary step towards characterizing fundamental plasma processes using these unique multi-point, multi-scale data sets. We propose improvements upon the classic first-order reconstruction method, as well as a second-order method, utilizing magnetometer measurements from a realistic nine-spacecraft observatory. The improved first-order method, which averages over select ensembles of four spacecraft, reconstructs the magnetic field associated with simple current sheets and numerical simulations of turbulence accurately over larger volumes compared to second-order methods or first-order methods using a single regular tetrahedron. Using this averaging method on data sets with fewer than nine measurement points, the volume of accurate reconstruction compared to a known magnetic vector field improves approximately linearly with the number of measurement points.

Molecular structure and DFT calculations of aqua(5,10,15,20-tetrakis[4-(benzoyloxy)phenyl] porphyrinato)magnesium-dioxane

In this work, complex [Mg(TPBP)(H2O)], where TPBP is the 5,10,15,20-tetrakis[4-(benzoyloxy)phenyl]porphyrin ligand, was obtained unintentionally as a by-product of the reaction of the [Mg(TPBP)] complex with an excess of dioxane in dichloromethane. The crystal structure of five coordinated Mg(II) [Mg(TPBP)(H2O)] (dioxane) (1), was determined using single crystal X-ray diffraction (SCXRD) analysis and studied using Hirshfeld surfaces computational method. The metal cation is surrounded by four porphyrin nitrogen atoms and one axially bonded water molecule, forming a regular tetrahedron. In this complex, the molecular structure and the three-dimensional framework were stabilized by inter-and intramolecular O–H⋯O and C–H⋯O hydrogen bonds, and by weak C–H⋯Cg π interactions. The results of the infrared and proton nuclear magnetic resonance spectroscopies are in good agreement with the transformation of the porphyrinic Mg(II) complex [Mg(TPBP)] to the one involving a bonded water molecule. Density Functional Theory (DFT) calculations on [Mg(TPBP)(H2O)] (dioxane) (1) complex revealed its enhanced thermodynamic stabilization which is attributed to the existence of hydrogen bonding interactions and the kinetic lability will be attributed to the small difference between the energy levels.

SY9-1. Predicting and preventing long-term cardiovascular disease after preeclampsia

We investigate the defect, mechanical, thermal properties, and refractive index of Er:LuYSGG mixed crystal grown by the Czochralski method. The chemical etching pit patterns present a regular tetrahedron, which consists well with the corresponding atomic arrangement schematics. The Mohs hardness is obtained to be 6.60. The thermal expansion coefficient and the thermal conductivity along <111> crystallographic orientation are achieved to be 7.91 × 10−6 K−1 and 3.423 W m−1 K−1, respectively, indicating the Er:LuYSGG has a relatively high damage threshold. Based on the measured values of refractive index, the Sellmeier equation is fitted and the refractive index curve in the range of 400–3000 nm is displayed. All these results are of great significance for the further research and laser application of Er:LuYSGG crystal.

SIMPLICIUS ON THE FILLING OF SPACE, IN CAELO 655.9–656.5: A DELIBERATE MISTAKE?

In In cael. 655.9–656.5 Simplicius reports an argument in which an apparent justification is offered for the false claim by Aristotle that ‘pyramids’ (regular tetrahedra) can completely fill space. This argument was analysed by Ian Mueller in an Appendix to his translation of In caelo, and the outline of an alternative has been presented in Myrto Hatzimichali's study of Potamo of Alexandria. In this article I contest Mueller's interpretation, and expand on the one reported by Hatzimichali. I also contest Mueller's claim that a version of his interpretation can be found in the partial commentary by Peter of Auvergne. It is suggested here that the ‘justification’ reported by Simplicius is a deliberate slip in logic, which is accompanied by a carefully constructed cover-up involving some quite tricky geometry. Simplicius makes frequent reference to Alexander of Aphrodisias, but it is argued here that he has been very selective with these citations.

The ground state structures and spectra of Ag20 clusters and the adsorption to carbon monoxide

The most stable structures and spectra of neutral, anionic and cationic Ag20 clusters have been studied by using density functional theory in conjunction with an unbiased CALYPSO structure searching method. Structural searches indicate that the regular tetrahedron which was widely regarded as the ground-state of neutral Ag20 cluster is a low-lying isomer. Several new isomers with lower energy than the regular tetrahedron were found in geometry optimizations. The ground state structures of neutral and anionic Ag20 cluster are confirmed by the optical absorption and photoelectron spectra. On the other hand, the investigation of CO adsorption on Ag20 clusters shows that CO adsorption on anionic Ag20 cluster is stronger than that on neutral and cationic Ag20 clusters. The lowest energy structure of anionic Ag20 cluster can bind CO molecular number up to 19. The adsorption of CO on Ag20 clusters significantly changes the Infrared spectra of neutral and cationic Ag20 clusters and the Raman spectrum of anionic Ag20 cluster. Raman scattering intensity of CO adsorbed on Ag20 cluster was enormously enhanced.

Design and analysis of three-dimensional printing of a porous titanium scaffold

ObjectiveMechanic strength, pore morphology and size are key factors for the three-dimensional (3D) printing of porous titanium scaffolds, therefore, developing optimal structure for the 3D printed titanium scaffold to fill bone defects in knee joints is instructive and important.MethodsStructural models of titanium scaffolds with fifteen different pore unit were designed with 3D printing computer software; five different scaffold shapes were designed: imitation diamond-60°, imitation diamond-90°, imitation diamond-120°, regular tetrahedron and regular hexahedron. Each structural shape was evaluated with three pore sizes (400, 600 and 800 μm), and fifteen types of cylindrical models (size: 20 mm; height: 20 mm). Autodesk Inventor software was used to determine the strength and safety of the models by simulating simple strength acting on the knee joints. We analyzed the data and found suitable models for the design of 3D printing of porous titanium scaffolds.ResultsFifteen different types of pore unit structural models were evaluated under positive pressure and lateral pressure; the compressive strength reduced when the pore size increased. Under torsional pressure, the strengths of the imitation diamond structure were similar when the pore size increased, and the strengths of the regular tetrahedron and regular hexahedron structures reduced when the pore size increased. In each case, the compressive strength of the regular hexahedron structure was highest, that of the regular tetrahedron was second highest, and that of the imitation diamond structure was relatively low. Fifteen types of cylindrical models under a set force were evaluated, and the sequence of comprehensive compressive strength, from strong to weak was: regular hexahedron > regular tetrahedron > imitation diamond-120° > imitation diamond-90° > imitation diamond-60°. The compressive strength of cylinder models was higher when the pore size was smaller.ConclusionThe pore size and pore morphology were important factors influencing the compressive strength. The strength of each structure reduced when the pore size (400, 600 and 800 μm) increased. The models of regular hexahedron, regular tetrahedron and imitation diamond-120°appeared to meet the conditions of large pore sizes and high compressive strength.

Correlation of Tunable CoSi4 Tetrahedron with the Superconducting Properties of LaCoSi.

It is known that as the FeAs4 tetrahedron in the Fe-based superconductor is close to the regular tetrahedron, critical temperature (Tc) can be greatly increased. Recently, a Co-based superconductor of LaCoSi (4 K) with "111" structure was found. In this work, we improve the Tc of LaCoSi through structural regulation. Tc can be increased by the chemical substitution of Co by Fe, while the superconductivity is suppressed by the Ni substitution. The combined analysis of neutron and synchrotron X-ray powder diffractions reveals that the change of the Si-Co-Si bond angles of the CoSi4 tetrahedron is possibly responsible for the determination of superconducting properties. The Fe chemical substitution is favorable for the formation of the regular tetrahedron of CoSi4. The present new Co-based superconductor of LaCoSi provides a possible method to enhance the superconductivity performance of the Co-based superconductors via controlling Co-based tetrahedra similar to those well established in the Fe-based superconductors.

Structural and Vibrational Study of CsSbGe3O9, a New Germanate in the ASbGe3O9 Family (A = Alkali Metal).

The germanate CsSbGe3O9, grown spontaneously via the high-temperature solution method from Cs2Mo4O13 used as a flux, crystallizes in the orthorhombic system with a = 12.3636(2), b = 13.8521(2), c = 31.4824(5) Å, and V = 5391.73(2) Å3. Its structure, determined from single-crystal X-ray diffraction data, is most probably noncentrosymmetric despite our choice to report it in the centrosymmetric maximal supergroup Pnma D2h16 (no. 62) in which agreement factors R1 = 0.0371 and wR2 = 0.0706 (all data) were obtained. The unit cell contains 24 formulas CsSbGe3O9. The three-dimensional network is built up with regular germanate tetrahedra at nine crystallographically independent Ge sites. The Sb atoms (four independent positions) adopt octahedral coordination with O atoms, and the Cs+ cations are located in the channels of the 3-D network. A more in-depth analysis of the structure of CsSbGe3O9 is carried out in the light of the structures previously determined for the compounds ASbGe3O9 (A = K, Rb) having the same chemical formula but significantly different atomic arrangements. The structural characteristics are discussed related to literature and the nature of the monovalent cation A+.

Electronic and crystal structures of LnFeAsO1−x H x (Ln = La, Sm) studied by x-ray absorption spectroscopy, x-ray emission spectroscopy, and x-ray diffraction: II pressure dependence

We examine electronic and crystal structures of iron-based superconductors LnFeAsO1−x H x (Ln = La, Sm) under pressure by means of x-ray absorption spectroscopy (XAS), x-ray emission spectroscopy (XES), and x-ray diffraction. In LaFeAsO the pre-edge peak on high-resolution XAS at the Fe-K absorption edge gains in intensity on the application of pressure up to 5.7 GPa and it saturates in the higher pressure region. We found integrated-absolute difference values on XES for Ln = La, corresponding to a spin state, decline on the application of pressure, and then it is minimized when the T c approaches the maximum at around 5 GPa. In contrast, such the optimum value was not detected for Ln = Sm. We reveal that the superconductivity is closely related to the lower spin state for Ln = La unlike Sm case. We observed that As height from the Fe basal plane and As–Fe–As angle on the FeAs4 tetrahedron for Ln = La deviate from the optimum values of the regular tetrahedron in superconducting (SC) phase, which has been widely accepted structural guide to SC thus far. In contrast, the structural parameters were held near the optimum values up to ∼15 GPa for Ln = Sm.

Tetrahedral Pt10– Cluster with Unique Beta Aromaticity and Superatomic Feature in Mimicking Methane

Utilizing a customized metal cluster source in tandem with a flow tube reactor and a reflectron time-of-flight mass spectrometer, we have obtained well-resolved pure metal clusters Ptn- and observed their gas-phase reactions with a few small gas molecules. Interestingly, the remarkable inertness of Pt10- was repeatedly observed in different reactions. Meanwhile, we have determined the structure of Pt10- within a regular tetrahedron. Considering that Pt possesses 5d96s1 electron configuration, the tetrahedral Pt10- exhibits unexpected stability at neither a magic number of valence electrons nor a shell closure of geometric structure. Comprehensive theoretical calculations unveil the stability of Pt10- is significantly associated with the all-metal aromaticity. In addition to the classical total aromaticity, which is mainly due to 6s electrons, there is unique beta-aromaticity ascribed to spin-polarized beta 5d electrons pertaining to singly occupied multicenter bonds. Further, we demonstrate the superatomic feature of such a transition metal cluster Pt10-, as Pt6@Pt4-, in mimicking methane.

Local crystalline order features in disordered packings of monodisperse spheres

A new tetrahedral structure model was developed for disordered sphere packings, including not only regular tetrahedron (T), but also quartoctahedron (Q) and bcc simplex (B), the tetrahedral building blocks in fcc, hcp and bcc crystal structures, respectively. Both geometric frustrated configurations and local configurations associated with crystalline order in disordered packings can be comprehensively characterized. It is found that with increasing packing fraction, the population of T, Q, and B increases. Moreover, the crystal-type local configurations formed by face-adjacent T, Q and B increases as packing fraction increases, which is more prevalent than the geometric frustrated ones formed by face-adjacent T. In addition, as packing fraction increases, the local density of irregular simplexes is found to increase more quickly than regular ones, playing more important roles in the density increase in disordered packings. These structure features are found to be intrinsic in the jammed random sphere packings with different friction coefficients, which is independent of interparticle friction and only determined by the packing fraction. Our findings provide new understanding for the structural nature of disordered packings and the underlying structural basis of the random close packing.

Isodiametry, Variance, and Regular Simplices from Particle Interactions

Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant $$d_\infty $$ metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on $${{\mathbf {R}}}^n$$ whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass $$1/(n+1)$$ to each vertex of a (unit-diameter) regular simplex.