Planar Geometry Research Articles

THE BISECTOR TRIANGLE WITH A 90° ANGLE

For the examination of the triangle discussed in the article, profound knowledge of plane geometry is required, and grasping the specific approach to solving it can be quite challenging without prior knowledge of what to rely on. The focus is on a triangle with a 120° angle, commonly known as the Shebarshin problem. The article presents both computational and purely geometric methods for solving a problem of elevated complexity. This particular problem appears in entrance exams for universities and in mathematical Olympiads at the highest levels. The theme of singular triangles remains constantly relevant in geometry. For instance, many properties and theorems are known about an orthocentric triangle, which have been discussed in detail and already proven. I propose to consider in detail the properties and theorems of an equally interesting triangle, the sides of which connect the bases of the bisectors of the interior angles of the triangle. The goal of my article is to alleviate the drawback of complexity while retaining accessibility and originality in the geometric problem involving a triangle with a 120° angle and an inscribed bisectral triangle with a corresponding 90° angle.

Synthesis of bimetallic palladacycles via C[sbnd]H bond activation: Crystal structure, DNA binding and in vitro cytotoxicity study

Herein, we report the synthesis, characterization and in vitro anticancer activity of two μ-Cl and μ-O bridged dimeric cyclometalated palladium(II) complexes having general formula [Pd2(L1)Cl2] (1) and [Pd2(L2)Cl2] (2). The complexes are synthesized via palladium assisted aromatic CH bond activation of coordinating ligands (HL1/HL2) and thoroughly characterized by spectroscopic techniques. X-ray crystal structure analysis has confirmed the square planar geometry of the complexes. DFT and TDDFT results have interpreted the electronic structure and solution spectra of the complexes. Cytotoxicity of the complexes was studied by in vitro model with human gastric cancer cell lines (AGS). Cytotoxicity investigation reveals that the complexes have promising anti-cancer activity, with IC50 values of 13.03 µM and 17.86 µM for 1 and 2, respectively. UV–vis and fluorescence techniques are used to examine how the complexes interact with CT DNA. Fluorescence-based competitive binding analysis with ethidium bromide (EB) indicates that the complexes effectively displace EB from the EB-DNA complex.

On the relation between maximal thermal performance and minimal fouling deposition rate in heat exchanger-like devices

The present article numerically studies the relationship between a heat exchanger’s thermal performance and the occurrence of fouling, here represented by the deposition of calcium sulfate (CaSO4) on its surface. For that, a 2D geometry denoting the internal flow of a CaSO4 saturated aqueous solution between two parallel plates is simulated. The flow, which can be laminar or turbulent (k-ω SST model), is promoted by a known pressure difference across the channel, while the CaSO4 fouling is thermally triggered by a heat source in the central part of the channel. The numerical model considers the solution of the conservative equations of mass, momentum, and energy, while the mass transfer rate is determined based on a correlation available in the literature. Two approaches were used to determine the CaSO4 fouling rate: (i) the CaSO4 concentration on the fouling’s surface was assumed to be identical to the concentration of the bulk flow, and (ii) the fouling concentration varies spatially, hence requiring the solution of the differential equation for conservation of species. The steady analysis showed that for both approaches, the optimal plate-to-plate spacing that maximizes the thermal performance is similar to the spacing that minimizes the fouling deposition. Finally, the transient analysis, which also considers the effect of the fouling removal rate through shear, shows that deviations might occur between the optimal spacings for maximal thermal performance and minimal deposition, especially for lower pressure drop values.

Bis(diphenyl)phosphinomethane Platinum(II) complex of 1-(benzo[d]thiazol-2-yl)-3-phenylthiourea: Crystal structure and DFT study

This study extensively describes synthesizing and characterizing a novel Pt(II) thiourea complex and its mixed ligand bisdiphenylphosphinomethane complex. The first complex [Pt(BtztH)2] is obtained by reacting K2PtCl4 and 1-(benzo[d]thiazol-2-yl)-3-phenylthiourea in 1:2 molar ratio. The second complex [Pt(Btzt)(dppm)] was synthesized by reacting dichlorobis(diphenyl)phosphinomethane platinum(II) complex with 1-(benzo[d]thiazol-2-yl)-3-phenylthioure in equimolar ratio with the presence of Et3N. Both complexes adopt square planar geometry, which were determined by FT-IR, 1H-nmr and X-ray crystallography. In comparison with the first complex, the second complex has doubly deprotonated, resulting in a dianion Pt(II) complex, which was determined by X-ray single crystal measurement. The asymmetric unit contained two unique molecules with distorted square planar geometry. A couple of non-covalent or intermolecular interactions stabilized the solid-state supramolecular assembly. Hirshfeld surface analysis was conducted in order to have a deeper understanding of the supramolecular assembly. Enrichment ratio calculations determined the pair of chemical species with the highest propensity for making interatomic contact. The [Pt(Btzt)(dppm)] compound was also investigated theoretically. The DFT study was performed by using B3LYP/Def2-TZVP method. Theoretical findings indicate that the HOMO-LUMO gaps range from 2.2 to 3.22 eV, indicating that the compounds possess favorable semiconductor properties. Within these complexes, the coordinated atoms exhibit strong electron donations to the Pt cation, while the Pt cation demonstrates significant back-donation to the ligand.

Strong Dimensional and Structural Dependencies of Hot Carrier Effects in InGaAs Nanowires: Implications for Photovoltaic Solar Cells.

III-V nanowire structures are among the promising material systems with applications in hot carrier solar cells. These nanostructures can meet the requirements for such photovoltaic devices, i.e., the suppression of thermalization loss, an efficient hot carrier transport, and enhanced photoabsorption thanks to their unique one-dimensional (1D) geometry and density-of-states. Here, we investigate the effects of spatial confinement of photogenerated hot carriers in InGaAs-InAlAs core-shell nanowires, which presents an ideal class of hot carrier solar cell materials due to its suitable electronic properties. Using steady-state photoluminescence spectroscopy, our study reveals that by increasing the degree of spatial confinement and Auger recombination, the effects of hot carriers increase, which is in good agreement with theoretical modeling. However, for thin nanowires, the temperature of hot carriers decreases as the effects of crystal disorder increase. This observation is confirmed by probing the extent of the disorder-induced Urbach tail and linked to the presence of a higher density of stacking defects in the limit of thin nanowires. These findings expand our knowledge of hot carrier thermalization in nanowires, which can be applied for designing efficient 1D hot carrier absorbers for advanced-concept photovoltaic solar cells.

Fractal penis and golden penis. The hidden dimensions of penile tissue

The penis is an organ with a unique and complex structure. It is formed by a fractal architecture, meaning its design is based on highly complex repetitive patterns. In addition, its dimensions follow what are known as "golden proportions", also known as "the divine proportion". These proportions are a mathematical relationship that is found in many natural structures and, in the case of the penis, provide stability and improve its performance. This design is not the result of chance, but rather the result of an underlying mathematical message. Fractal geometry, based on complex numbers, is a key tool in understanding the internal structure of the penis, which cannot be described using traditional Euclidean geometry. In addition to fractal geometry, the presence of golden proportions in the penis suggests that its design is influenced by mathematics. To fully understand this relationship, additional studies are required and the creation of innovative mathematical models that include tools such as artificial intelligence. In conclusion, the penis is an example of how mathematics and nature are closely related and how mathematical complexity can improve the functioning of natural structures.

Art in sacred geometry: a new perspective for multidisciplinary STEAM education

Blending sacred geometry with elementary geometry education not only adds depth and richness to the subject but also transcends traditional boundaries, emphasizing the interconnectedness of art, science, and architecture. The discovery that ancient consecrated structures based on sacred geometry store electromagnetic energy underscores the profound relationship between the arts, sciences, and architecture as perceived by ancient civilizations by fostering a multidisciplinary approach akin to STEAM education. This motivates us to trace the scientific grounds for integrating sacred geometry into STEAM curricula. We contemplate the foundation of sacred geometry through some elementary geometrical constructions and show how they can be organized in the most elementary style as a part of sufficiently flexible, rather than a rigid, core curriculum that would not only help students in grasping the significant interrelationships between art, science, geometry and architecture, but also contribute to a more profound and interconnected understanding of the modern world.

Solving olympiad geometry without human demonstrations

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1–4, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

The role of the areal parameters on turbulent flow over 2D Gaussian roughness

In the past decades, numerous efforts have been dedicated to establishing a direct correlation between a geometric parameter that represents wall roughness and the corresponding velocity reduction, known as the Roughness Function ΔU+. This reduction is influenced by various statistical measures of roughness height, including average roughness height, peak-to-valley roughness distance, roughness root mean square, and Effective Slope, among others. It has been demonstrated that a singular measure of roughness height cannot sufficiently predict the Roughness Function across all turbulent regimes. Consequently, many studies have concentrated on identifying a universal correlation between roughness geometry and the downward shift of the mean velocity profile. In this study, we investigated the correlation between various geometrical parameters and the Roughness Function using Large Eddy Simulation (LES) techniques in channel flows at a friction Reynolds number of Reτ=400. Given the complexity introduced by the random nature of irregular roughness, we explored specific aspects of the relationship between wall irregularities and the roughness function by studying 2D geometries. This approach allowed us to systematically investigate the impact of geometrical properties on the roughness function and isolate the effects of roughness density and coverage area. Several irregular rough surfaces, characterized by different average oscillations height, different distributions and different densities, were designed throughout 2D Gaussian functions. With the aim to find a universal correlation between the roughness geometry and the Roughness Function, new geometrical quantities were investigated, based on the global area occupied by the roughness A∗. The prediction of the roughness function can be thus obtained using apriori data. To predict the roughness function, we introduced a parameter called Effective Area (EA), which is derived from the correlation between the Effective Slope (ES) and the roughness area A∗. Our findings indicate that a single geometric parameter, whether ES or Area A, is insufficient to predict the roughness effect. Conversely, combining these two parameters enhances predictive accuracy, at least for the proposed roughness model. This improvement can be attributed to the ability of ES to interpret the roughness distribution and height, while the coverage area is effective in predicting roughness density over a flat plate.

On a Family of Polyhedral Singular Vertices

We provide a local description of the curves with minimal length based at singularities in a family of polyhedral surfaces. These singularities are accumulation points of vertices with conical angles equal to π and 4π (or 3π, in a variation). While a part of the minimizing curves behaves quite like the ones reaching conical vertices, the singularities present features such as being connected to points arbitrarily close to them by exactly two minimizing curves. The spaces containing such singularities are constructed as metric quotients of an euclidean half-disk by certain identification patterns along its edge. These patterns are examples of what is known as paper-folding schemes, and we provide the foundational aspects about them which are necessary for our analysis. The arguments are based on elementary metric geometry and calculus.

Anisotropic spin transport and photoresponse characteristics detected by tip movement in magnetic single-molecule junction

Orientation-dependent transport properties induced by anisotropic molecules are enticing in single-molecule junctions. Here, using the first-principles method, we theoretically investigate spin transport properties and photoresponse characteristics in trimesic acid magnetic single-molecule junctions with different molecular adsorption orientations and electrode contact sites. The transport calculations indicate that a single-molecule switch and a significant enhancement of spin transport and photoresponse can be achieved when the molecular adsorption orientation changes from planar geometry to upright geometry. The maximum spin polarization of current and photocurrent in upright molecular junctions exceeds 90%. Moreover, as the Ni tip electrode moves, the tunneling magnetoresistance of upright molecular junctions can be increased to 70%. The analysis of the spin-dependent PDOS elucidates that the spinterfaces between organic molecule and ferromagnetic electrodes are modulated by molecular adsorption orientation, where the molecule in upright molecular junctions yields higher spin polarization. Our theoretical work paves the way for designing spintronic devices and optoelectronic devices with anisotropic functionality base on anisotropic molecules.

Accelerated charging dynamics in shell-and-multi-tube latent heat storage systems for building applications

Thermal energy storage (TES) is an emerging technique to store the surplus energy generated during periods of excess supply and utilise it when demand arises. Phase change material (PCM) stands out as a highly efficient TES system. Enhancing heat transfer during phase change is a major challenge in creating effective PCM-based TES systems. This study focuses on enhancing the melting performance of a shell-and-tube latent heat thermal energy storage (LHTES) system. This improvement is achieved by conducting an extensive parametric study involving various geometric factors including tube position from the bottom, vertical tube spacing, tube diameter, different heat exchanger shell aspect ratios, PCM temperature, tube temperature, and PCM type. A three-dimensional model is simplified to a two-dimensional geometry for numerical simulation. The results indicate that the PCM melting time decreased by 20.6 % and 4.4 % with a 20 mm vertical tube distance compared to 15 mm and 25 mm. Among the considered scenarios, the configuration with a doubled number of tubes in width consistently exhibits a higher melt fraction throughout the entire duration. Regarding the aspect ratio, the square-shaped system demonstrates both quicker melting times and a higher energy storage rate. When comparing PCM types, it is evident that RT 35 exhibits a 3.2 % higher heat storage rate and a 44.7 % shorter melting time compared to RT 35 HC.

Parameter Identification Problem for Drying of Solid Propellants: Effective Diffusivity

Abstract In the manufacture of propellants, drying is a crucial step in which liquid solvents are extracted from the propellants' microstructure in order to impart the desired physical and chemical qualities. Therefore, the prediction of the effective diffusivity, a fundamental characteristic defining the rate of solvent transfer during vacuum drying of propellants, is particularly valuable for determining optimal drying conditions in the operation of drying systems and optimizing the drying process. This paper presents an elegant way to estimate the lumped value of the effective diffusivity for the mass transfer process in the course of drying propellants. Analytical solutions for the effective diffusivity at planar, cylindrical, and spherical geometries are obtained. These solutions are used to investigate effects of the propellant geometry on the drying process. Using experimental data for the transient moisture content, it becomes possible to determine geometries, which should be preferred for a given function of the mass flux.

Modeling of vehicle carbon emissions on horizontal curve road sections

The high carbon emissions of vehicles traveling on horizontal curve road sections cannot be ignored. Facing the difficulty of accurately quantifying the carbon emission of driving on horizontal curves and the unknown causes of high carbon emission, this study proposes to construct a carbon emission prediction model applicable to road sections with different planar geometries. The direct and indirect effects of horizontal curve alignment on vehicle carbon emissions are represented in the model in terms of travel stabilization and speed changes, respectively. A lateral force coefficient parameter was introduced into the model to integrate the carbon emission quantification problem for different planar geometry sections. Meanwhile, field tests were conducted to assess the reliability of the model and the research findings. The model reveals that the geometric parameters of horizontal curves that affect carbon emissions are the radius of the circular curve, the superelevation, and the length of the gentle curve. The root causes of high carbon emissions on horizontal curve road sections are curve driving resistance and speed fluctuations. Under the free-flow driving condition of the highway, the maximum curve radius affecting the carbon emissions of passenger cars and trucks is 400 m and 550 m, respectively. The research results can realize the carbon emission quantification of vehicles on the road sections with different plane geometries. Also, it is helpful to control the high carbon emission of vehicles traveling on horizontal curve road sections.

A combinatorial curvature flow in spherical background geometry

In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for any initial data the flow converges exponentially fast to a circle pattern with given total geodesic curvature on each circle. Our result could be regarded as a resolution of the problem in the spherical case. As far as we know, this is the first combinatorial curvature flow in spherical background geometry with fine properties, and it provides an algorithm to find the desired ideal circle pattern.

A Harnack inequality for a class of 1D nonlinear reaction–diffusion equations and applications to wave solutions

In this paper, a differential-geometric method is applied to build some Li–Yau–Hamilton-type Harnack inequalities for the positive solutions to a one spatial dimensional nonlinear reaction–diffusion equation in a plane geometry. The class of reaction–diffusion equation that is considered here contains several important equations some of which are Newel–Whitehead–Segel, Allen–Cahn and Fisher–KPP equations. The Harnack inequalities so derived are used to discuss some other important properties of positive solutions and in the characterization of positive wave solutions.

Synthesis, characterization and catalytic activities of palladium(II)-imidazopyridine

The ligand 2-phenylimidazo(1,2-α)pyridine (1) was synthesized by neat reaction 2-aminopyridine and 1-bromo-acetophenone; reaction of 1 with acetonitrile solution of PdCl2 leads to [Pd(1)Cl2(CH3CN)] (2). Refluxing [Pd(1)Cl2(CH3CN)] (2) in pyridine leads to [Pd(1)Cl2(py)] (3).The complexes were characterized by various spectroscopic techniques and the solid state structure of 2 was elucidated by single crystal X-ray diffraction (SCXRD) analyses. SCXRD analyses support the square planar geometry of 2. A series of DFT calculations were also performed to gain further insight into the respective structures of the complexes. From molecular hardness calculation it is observed slightly better reactivity of 2 compared to 3. Molecular Hirshfeld surface analyses support various non-covalent secondary interactions that observed in packing of solid state structure of 2. Complexes 2 and 3 were found to facilitate Suzuki coupling reactions under relatively mild conditions and slightly better yields were obtained in case of 2.

Graphene-embedded planar and cylindrical Oldenburger–Kolakoski aperiodic photonic crystals

In this paper, we examine the transmission and absorption spectra of light waves in both planar and cylindrical quasiperiodic structures known as Photonic Crystals. In these quasiperiodic multilayers, graphene sheets are inserted between planar and concentric layers of two different dielectric materials (namely TiO2 and SiO2). These structures are called Planar Photonic Crystals (PPCs) and Cylindrical Photonic Crystals (CPCs), respectively. The spatial arrangement of the dielectric layers follows the aperiodic Oldenburger–Kolakoski (O–K) sequence. The transfer matrix method is used to calculate the transmission and absorption spectra of obliquely incident TE and TM waves to investigate the effects of graphene, aperiodic disorder, and incidence angle on the physical behavior of the system. The numerical results reveal that the transmission spectra for cylindrical geometry differ significantly from those for planar geometry, even in the case of a normal incidence angle. Furthermore, the transmission of TM modes is higher than that of TE modes for incidence angles greater than 30°. Finally, we have demonstrated that a substantial number of peaks in the absorption spectra for the CPC case are higher than those for the PPC case, which is a result that may be useful for enhancing the efficiency of optical sensors.

Noncommutativity in Configuration Space Induced by a Conjugate Magnetic Field in Phase Space

An external magnetic field in configuration space coupled to quantum dynamics induces noncommutativity in its velocity momentum space. By phase space duality, an external vector potential in the conjugate momentum sector of the system induces noncommutativity in its configuration space. Such a rationale for noncommutativity is explored herein for an arbitrary configuration space of Euclidean geometry. Ordinary quantum mechanics with a commutative configuration space is revisited first. Through the introduction of an arbitrary positive definite ∗-product, a one-to-one correspondence between the Hilbert space of abstract quantum states and that of the enveloping algebra of the position quantum operators is identified. A parallel discussion is then presented when configuration space is noncommutative, and thoroughly analysed when the conjugate magnetic field is momentum independent and nondegenerate. Once again the space of quantum states may be identified with the enveloping algebra of the noncommutative position quantum operators. Furthermore, when the positive definite ∗-product is adapted to the conjugate magnetic field, the coordinate operators span a Fock algebra of which the coherent states are the analogues of the structureless points in a commutative configuration space. These results generalise and justify a posteriori within ordinary canonical quantisation the heuristic approach to quantum mechanics in the noncommutative Euclidean plane as constructed and developed by F. G. Scholtz and his collaborators.

Nanoscale Ion Transport Enhances Conductivity in Solid Polymer-Ceramic Lithium Electrolytes.

The predictive design of flexible and solvent-free polymer electrolytes for solid-state batteries requires an understanding of the fundamental principles governing the ion transport. In this work, we establish a correlation among the composite structures, polymer segmental dynamics, and lithium ion (Li+) transport in a ceramic-polymer composite. Elucidating this structure-property relationship will allow tailoring of the Li+ conductivity by optimizing the macroscopic electrochemical stability of the electrolyte. The ion dissociation from the slow polymer segmental dynamics was found to be enhanced by controlling the morphology and functionality of the polymer/ceramic interface. The chemical structure of the Li+ salt in the composite electrolyte was correlated with the size of the ionic cluster domains, the conductivity mechanism, and the electrochemical stability of the electrolyte. Polyethylene oxide (PEO) filled with lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) or lithium bis(fluorosulfonyl) imide (LiFSI) salts was used as a matrix. A garnet electrolyte, aluminum substituted lithium lanthanum zirconium oxide (Al-LLZO) with a planar geometry, was used for the ceramic nanoparticle moieties. The dynamics of the strongly bound and highly mobile Li+ were investigated using dielectric relaxation spectroscopy. The incorporation of the Al-LLZO platelets increased the number density of more mobile Li+. The structure of the nanoscale ion-agglomeration was investigated by small-angle X-ray scattering, while molecular dynamics (MD) simulation studies were conducted to obtain the fundamental mechanism of the decorrelation of the Li+ in the LiTFSI and LiFSI salts from the long PEO chain.