Polyhedral Shape Research Articles

Analyzing the Polyhedral Shape of Mineral Inclusions in Diamond Crystals Using Goniometric Data

Analyzing the Polyhedral Shape of Mineral Inclusions in Diamond Crystals Using Goniometric Data

Controlling the Formation of Polyhedral Block Copolymer Nanoparticles: Insights from Process Variables and Dynamic Modeling.

This study delves into the formation of nanoscale polyhedral block copolymer particles (PBCPs) exhibiting cubic, octahedral, and variant geometries. These structures represent a pioneering class that has never been fabricated previously. PBCP features distinct variations in curvature on the outer surface, aligning with the edges and corners of polyhedral shapes. This characteristic sharply contrasts with previous block copolymers (BCPs), which displayed a smooth spherical surface. The emergence of these cornered morphologies presents an intriguing and counterintuitive phenomenon and is linked to process parameters, such as evaporation rates and initial concentration, while keeping other variables constant. Using a system of coupled Cahn-Hillard (CCH) equations, we uncover the mechanisms driving polyhedral particle formation, emphasizing the importance of controlling relaxation parameters for shape variable u and microphase separation v. This unconventional approach, differing from traditional steepest descent method, allows for precise control and diverse polyhedral particle generation. Accelerating the shape variable u proves crucial for expediting precipitation and aligns with experimental observations. Employing the above theoretical model, we achieve shape predictions for particles and the microphase separation within them, which overcomes the limitations of ab initio computations. Additionally, a numerical stability analysis discerns the transient nature versus local minimizer characteristics. Overall, our findings contribute to understanding the complex interplay between process variables and the morphology of polyhedral BCP nanoparticles.

Elastic properties and compressive mechanical behaviour of closed-cell porous materials: Effect of microstructural morphology

This paper studies the effect of different combinations of morphological parameters of porous closed-cell materials on their elastic properties and mechanical behaviour. Three-dimensional representative volumes of porous media with different polyhedral void shapes are investigated, considering variation of statistical realisations of morphological features, such as shape, size and distribution of pores. Description of morphology of each investigated representative volume is formalized using second- and fourth-order correlation functions. The effective properties of representative volumes are calculated with finite-element analysis. Additionally, samples of the studied models were additively manufactured with polystyrene, using fused filament fabrication (FFF/FDM) 3D-printing technique, and subjected to compression. Experimental results for elastic mechanical properties and distributions of strain fields on the surface of the samples, obtained with micro-DIC (digital image correlation), are compared with results of numerical finite-element computations. This is accompanied by analysis of internal-stress distributions to assess the mechanical state of the structures. Patterns and trends observed in mechanical responses of porous materials depending on their different morphological parameters are outlined and discussed.

The influence of olivine settling on the formation of basaltic cumulates revealed by micro-tomography and numerical simulations

The speed at which crystals settle in magmatic reservoirs affects the solidification rate of magmas and their differentiation. Despite extensive prior work on the subject, most of our quantitative understanding of the process is still restricted to treating crystals as spherical particles and does not address the geometric complexities of natural crystals. Here, we use three-dimensional (3D) X-ray microcomputed tomography (X-μCT) observations on olivine crystals from Kīlauea Volcano (Hawai‘i) to document their highly intricate and variable geometries, and the textural growth relationships between the olivine crystals and their inclusions (melt, spinel, and fluid/vapor bubbles). Olivine crystals generally have clustered polyhedral or skeletal shapes, which reflect variable magmatic conditions (or growth rates) during their formation. The cumulative presence of spinel, melt, fluid, and vapor inclusions affects the density of the host crystals by up to 6% relative, and thus plays a limited role on modifying crystal settling rate. In contrast, the overall crystal shape plays a major role. We performed numerical simulations employing a finite element method to investigate the effect of crystal morphology on settling rate and the evolution of the particle volume fraction in a magmatic convective layer. We show that for all olivine geometries investigated, the settling velocity is highest when the long axis of the crystal is aligned with the flow direction of the melt. Increasing the aspect ratio of the olivine tends to decrease its settling velocity and results in an increase in the influence of its orientation on the terminal velocity (UT). Extrapolation of the simulation results to variable particle volume fractions (Φ = 0–0.5) indicates high crystal settling rates (UT = ∼9.7 × 10−6–1.4 × 10−5 m/s) that are used to estimate the timescales for the formation of olivine cumulates in natural melt-dominated basaltic systems. The formation of olivine cumulates is therefore rapid, potentially leading to the accumulation of a crystal layer at the bottom, where the frictional contacts between the crystals exert a rheological lock-up acting against further convection. Crystal accumulation in the locked layer (parametrized with the solid/liquid volume ratio in the reservoir) is a function of the reservoir size and crystal fraction, and takes a few years in small reservoirs (<1 km thick) and a few decades in larger reservoirs (several kms thick). We propose a calibration of olivine suspension timescales for mafic magma reservoirs (based on the knowledge of the particle volume fraction, reservoir height, and olivine morphology). This calibration is used to estimate the rate of cumulate build-up, and can help interpret crystal size distributions in the framework of crystal suspension times.

Systematic schemes for buckling analyses of a subsea bio-inspired non-circular FGM polyhedral liner with an arch invert

This study develops theoretical frameworks for the buckling failure of a non-circular thin-walled liner encased in a cracked pipeline with an arch invert. The buckling performance of the pipeline-liner system is improved by combining the polyhedral profiles and the Functionally Graded Materials (FGM). The liner is close-fitting to the inner surface of the pipeline. The energy function is obtained by assembling the thin-walled shell principle and the theory of minimum potential energy. Then, the nonlinear equilibrium equations and the buckling equations are derived to predict the buckling performances of the FGM polyhedral pipeline-liner system with an arch invert. Moreover, the present buckling behaviors of the system are compared with other available results, and good agreements are accomplished when the FGM polyhedral liner degenerates into an FGM circular liner or a homogeneous polyhedral liner. In addition, an enhancement coefficient is defined as the ratio of the buckling pressure of a polyhedral liner to a circular one. Finally, several key geometric and material parameters are examined to quantify their effects on the buckling behavior of the FGM polyhedral pipeline-liner system with an arch invert. These parameters are volume fraction exponents, polyhedral shapes, and thickness-to-radius ratios, respectively.

ZnxFe3-xO4–ZnO heterojunction interfaced with poly(L-DOPA) electron transfer mediator layer for enhanced visible light photocatalytic activity

Z-scheme heterojunctions comprising ZnxFe3-xO4@poly(L-DOPA)@ZnO (ZF@PD@ZnO) were synthesized using solvothermal and precipitation techniques. The study aimed to investigate the impact of the poly(L-DOPA) electron transfer mediator on the photocatalytic performance of ZnxFe3-xO4@ZnO (ZF@ZnO) nanocomposites under visible light irradiation. X-ray diffraction (XRD) and HRTEM analysis confirmed the formation of heterostructures, identifying the crystalline phases of both ZnO and zinc ferrite. BFTEM images revealed a polyhedral shape for all samples, with a tendency toward mild agglomeration. The partial inversion of the zinc ferrite cubic spinel structure was evidenced by FT-IR, XPS, and VSM analyses. Zinc ferrite was found to be ferromagnetic, with cation distributions between tetrahedral (A) and octahedral (B) positions as [Zn0.62+Fe0.43+]A[Fe0.42+Fe1.63+]BO4. The samples exhibited notable photocatalytic activity against Rhodamine B, serving as a model contaminant. Furthermore, the stability and reusability of the samples were examined. Additionally, the study elucidated the photocatalytic mechanism produced by the ternary Z-scheme ZF@PD@ZnO, involving spin-polarized charge carriers. Energy band alignment was evaluated by combining ultraviolet photoelectron spectroscopy (UPS) with UV-Vis spectroscopy. The interplay between band alignment and reactive oxygen species (ROS) generation was discussed in correlation with EPR results. Enhanced ROS generation under visible light illumination, observed in the ZF@PD@ZnO nanocomposite, is mainly due to the addition of the conductive poly(L-DOPA) layer, acting as an electron transfer mediator in the Z-scheme heterojunction, ensuring enhanced charge separation.

Evaluation of magnetic and electrochemical performance of copper oxide nanoparticles using Myristica fragrans (mace) extract

Abstract The synthesis of metal oxide nanoparticles using natural extract encourages the futuristic design of an environmentally friendly system by getting rid of the dangerous, toxic substances. The food industry in India is paying a lot of attention to Myristica fragrans, often known as mace, due to its rich medicinal significance. In the current study, M. fragrans (Mace) aqueous extract was used to prepare copper oxide (CuO) nanoparticles. Phytochemical screening confirms the presence of bioactive substances such as alkaloids, sterols, glycosides, and flavonoids in the extract. XRD and SEM measurements show that the nanoparticles have a monoclinic structure with polyhedral shape. Using the Debye-Scherrer formula, the material’s average crystallite size was found to be 85 nm. Based on the Tauc plot, an optical band gap of the prepared CuO NPs was calculated as 2.6 eV. At room temperature, the material’s magnetic property was investigated using VSM analysis. Congo red was used to examine the photocatalytic properties of the materials with various timings. CuO nanoparticles’ antibacterial activity was evaluated at various doses against Staphylococcus aureus and Klebsiella pneumoniae. Almost, CuO NPs exhibit better response against both the bacteria. Moreover, research investigations using cyclic voltammetry was carried out to assess the produced nanoparticles’ pseudocapacitive qualities. At a scan rate of 10 mV s−1, the material produced a good specific capacitance of 233.8 F/g with 1 M of KOH as an electrolyte.

The protective effects of &lt;em&gt;Zingiber officinale&lt;/em&gt; essential oils extract against Atenolol induced adrenal gland in rats (&lt;em&gt;Rattus norvigicus&lt;/em&gt;)

Zingiber officinale Roscoe is a perennial herb belonging to the Zingiberaceae family. The chemical compounds were identified by Gas chromatography‒mass spectrometry, the results showed that 38 components found in Z. officinale, beta-Sesquiphellandrene had the highest percentage was (28.94%), followed by 2- Butanone, 4-(2,6,6-trimethyl-2-cyclohexen-1-yl)-, (R)- (16.82%), alpha.-Farnesene (14.45%). The rest of the compounds constituted a smaller percentage, they were represented by Zingiberene, beta.-Phellandrene, Camphene, 1,8-Cineole, Alpha-Cedrene, Elemol, and Shogaol. Histological sections of the adrenal gland treated with atenolol drug showed that the cortex composed of atrophied zona granulosa, vascularized and inflamed zona fasciculate and sever inflammation of zona reticularis, the medulla appear with mild hemorrhage extended between the chromaffin cells, the parenchyma of the medulla consist of loosely, disorganized into clusters and cords resemble the epithelial, the cells large is size, polyhedral shape and appear empty in sections separated by sinusoidal capillaries while the deep layer is zona reticularis formed by pink stained cells and irregular arrangement. While in adrenal gland treated with Z. officinale showed more normal histological sections, so that treated with (atenolol and Z. officinale) showed mild moderate changes in different adrenal zona, the zona glomerulosa showed normal structure, small cluster beneath the capsule with crowded nuclei, mild hemorrhage area and the cells appeared resemble the lipoblasts.

Sequencing, Analysis and Organization of the Complete Genome of a Novel Baculovirus Calliteara abietis Nucleopolyhedrovirus (CaabNPV).

Baculoviridae, a virus family characterized by a single large double stranded DNA, encompasses the majority of viral bioinsecticides, representing a highly promising and environmentally friendly pesticide approach to insect control. This study focuses on the characterization of a baculovirus isolated from larvae of Calliteara abietis (Erebidae, Lymantriidae) collected in Mongolian pinaceae forests. This new isolate was called Calliteara abietis nucleopolyhedrovirus (CaabNPV). CaabNPV exhibits an irregular polyhedron shape, and significant variation in the diameter of its occlusion bodies (OBs) was observed. Nucleotide distance calculations confirmed CaabNPV as a novel baculovirus. The CaabNPV genome spans 177,161 bp with a G+C content of 45.12% and harbors 150 potential open reading frames (ORFs), including 38 core genes. A comprehensive genomic analysis categorizes CaabNPV within Group II alphabaculovirus, revealing a close phylogenetic relationship with Alphabaculovirus orleucostigmae (OrleNPV). Additionally, repeat sequence analysis identified three highly repetitive sequences consisting of 112 bp repeat units, known as homologous regions (hrs). This research contributes valuable insights into CaabNPV's phylogenetic placement, genomic structure, and its potential applications in insect biocontrol.

Antimicrobial, antibiofilm, and antivirulence properties of Eisenia bicyclis-extracts and Eisenia bicyclis-gold nanoparticles towards microbial pathogens

Nanomaterials derived from seaweed have developed as an alternative option for fighting infections caused by biofilm-forming microbial pathogens. This research aimed to discover potential seaweed-derived nanomaterials with antimicrobial and antibiofilm action against bacterial and fungal pathogens. Among seven algal species, the extract from Eisenia bicyclis inhibited biofilms of Klebsiella pneumoniae, Staphylococcus aureus, and Listeria monocytogenes most effectively at sub-MIC levels. As a result, in the present study, E. bicyclis was chosen as a prospective seaweed for producing E. bicyclis-gold nanoparticles (EB-AuNPs). Furthermore, the mass spectra of E. bicyclis reveal the presence of a number of potentially beneficial chemicals. The polyhedral shape of the synthesized EB-AuNP with a size value of 154.74 ± 33.46 nm was extensively described. The lowest inhibitory concentration of EB-AuNPs against bacterial pathogens (e.g., L.monocytogenes, S. aureus, Pseudomonas aeruginosa, and K. pneumoniae) and fungal pathogens (Candida albicans) ranges from 512 to >2048 μg/mL. Sub-MIC of EB-AuNPs reduces biofilm formation in P. aeruginosa, K. pneumoniae, L. monocytogenes, and S. aureus by 57.22 %, 58.60 %, 33.80 %, and 91.13 %, respectively. EB-AuNPs eliminate the mature biofilm of K. pneumoniae at > MIC, MIC, and sub-MIC concentrations. Furthermore, EB-AuNPs at the sub-MIC level suppress key virulence factors generated by P. aeruginosa, including motility, protease activity, pyoverdine, and pyocyanin, whereas it also suppresses the production of staphyloxanthin virulence factor from S. aureus. The current research reveals that seaweed extracts and a biocompatible seaweed-AuNP have substantial antibacterial, antibiofilm, and antivirulence actions against bacterial and fungal pathogens.

An Improved Shape Annealing Algorithm for the Generation of Coated Deoxyribonucleic Acid Origami Nanostructures

Abstract In recent years, the field of structural DNA nanotechnology has advanced rapidly due to transformative design tools. Although these tools have been revolutionary, they still bear one overall limitation of requiring users to fully conceptualize their designs before designing. Recently, a simple computational casting technique was developed using generative optimization strategies to automate the DNA origami nanostructure design. This approach employs a shape annealing algorithm, which creates a formal language of honeycomb nanostructures with shape grammars and drives designs from the language toward a desired configuration using simulated annealing. This initial demonstration of the approach can generate novel scaffold routing schemes for creating solid or hollow structures constrained by the boundaries of polyhedral meshes. The results from the initial approach, particularly from the hollow structures, reveal a challenging design space. This simple technique generates novel scaffold routing schemes that do not replicate the overall polyhedral mesh shape and are limited in their ability to control scaffold path exploration in the design space. This paper demonstrates an approach for achieving different levels of consistent effective wall thicknesses and improving the quality of mesh coverage for hollow structures that can be tuned and optimized by introducing a more refined computational casting technique. We achieve these improvements through changes in the simulated annealing algorithm by adding a Hustin move set algorithm that dynamically adjusts the performance of the overall design and redefining how these hollow designs are articulated. This work illustrates how the technique can navigate a challenging design space to generate effective hollow designs.

Correlation between surface-to-volume ratio of the particle shape and elastic properties of the particulate composites

This work is motivated by real-world particle shapes, observed using a scanning electron microscopy. The focus of the presented studies was to understand the influence of the particle shapes on the effective elastic properties of the two-phase composites. For this, particles with polyhedral, undulated and other shapes were numerically modeled using analytical functions. Creation of some shapes, like polyhedral, are known from the literature but Laplace’s spherical harmonics, as well as the Goursat’s surface and some others, were used for the first time to create novel particle shapes. Elastic properties of the composites with different particle shapes were calculated using the finite element analysis. The obtained results show good agreement with mean-field homogenization methods such like Mori-Tanaka and Lielens as well as other numerical results available in the literature. Further, the dependence of the effective Young’s moduli of the composite on the shape and the corresponding surface-to-volume ratio of the particles was studied. It was observed that the effective Young’s moduli increase with the surface-to-volume ratio of the particles in the case where particles are stiffer in comparison to the matrix. It was also remarked that, in the case of particles of similar shapes, the particle surface-to-volume ratio and the effective Young’s moduli differ significantly with the surface curvature and the edge sharpness of the particles.

Edge-sharing water prisms.

Regular-shaped water clusters and nanostructures can be of particular interest for the self-assembly of complex structures and functional materials involving water molecules. Polyhedral water clusters are the most well studied. The low-energy structures of water hexamer, octamer and decamer are shaped like right prisms. The cavities of gas hydrates also have a polyhedral shape. Ice nanotubes are of no less interest both as configurations of global energy minima and in terms of possible applications. This article presents the results of the first systematic study of the structure and properties of a special class of water clusters. It reveals different combinations of three right prisms sharing an edge. According to modern calculations, the configurations of global energy minima of water clusters with the number of molecules being eighteen and twenty have this structure precisely. The topological characteristics of the edge-sharing water prisms are studied and the formulas for estimating the residual entropy are obtained. For these clusters (3-prisms), the usefulness of using a discrete model of intermolecular interaction [strong-weak-effective-bond model], previously developed for polyhedral water clusters, is shown. Calculations of the binding energy were performed using the non-additive Amoeba potential. To understand the relative stability of 3-prisms among other cluster forms, we use the structures recently reported in a published database containing over 3 × 106 unique water cluster networks (H2O)N of size N = 3-25 [Rakshit et al., J. Chem. Phys., 2019, 151(21), 214307, DOI: 10.1063/1.5128378].

Shape-controlled synthesis of micro-/nanosized Cu particles with spherical and polyhedral shapes using the polyol process.

This study reports the synthesis of Cu micro-/nanosized particles through the polyol process. Cu particles were synthesized by reducing copper(ii) chloride in ethylene glycol (EG), polyvinylpyrrolidone (PVP), and potassium bromide (KBr) at low temperatures with or without the use of sodium borohydride (NaBH4).

Mesoscale simulations of diffusion and sedimentation in shape-anisotropic nanoparticle suspensions.

We determine the long-time self-diffusion coefficient and sedimentation coefficient for suspensions of nanoparticles with anisotropic shapes (octahedra, cubes, tetrahedra, and spherocylinders) as a function of nanoparticle concentration using mesoscale simulations. We use a discrete particle model for the nanoparticles, and we account for solvent-mediated hydrodynamic interactions between nanoparticles using the multiparticle collision dynamics method. Our simulations are compared to theoretical predictions and experimental data from existing literature, demonstrating good agreement in the majority of cases. Further, we find that the self-diffusion coefficient of the regular polyhedral shapes can be estimated from that of a sphere whose diameter is the average of their inscribed and circumscribed sphere diameters.

Clustering characteristics of Fe-rich intermetallics in high vacuum die cast AlSiMgMn alloys with high resolution μ-CT inspection

The AlSiMgMn alloys with different Fe and Mn contents were prepared with high vacuum die casting (HVDC) process. The Fe-rich intermetallics in the surface layer and center of the alloys were inspected with high-resolution 3D X-ray micro-computed tomography (μ-CT), and the growth mechanism of the primary Fe-rich phase was analyzed through their different 3D morphographies. By combining Bisecting K-means and DBSCAN (Density-Based Spatial Clustering of Applications with Noise) methods, the intermetallic clusters in the surface layer and center were quantified. The results show that the primary Fe-rich intermetallics were characterized as hexahedron and octagonal dendrite shapes with large size while the eutectic Fe-rich intermetallics were polyhedron and Chinese-script shapes. When Fe and Mn contents of the alloys were lower, the intermetallic clusters in the surface layer were with less cluster number and larger intermetallic distances showing uniform distribution whereas the clusters in the center were more and had shorter distances. In the case of alloys with higher Fe and Mn contents, the intermetallics showed clustering distribution in both the surface layer and center. The mechanism for the intermetallic cluster formation in the surface layer and center was further discussed.

Spreading ceramic stereolithography pastes: Insights from shear- and orthogonal-rheology

We study the shear rheological behavior of a commercial stereolithography paste containing ≈50 vol. % of zirconia particles (diameter ≈ 100 nm) with the aim to clarify physical mechanisms occurring during the “scraping” step of this yield stress fluid. Beyond a flow curve characterized by a high zero-shear viscosity accompanied with an overall shear-thinning behavior, we investigate in a systematic way the transient regime through start-up experiments. We demonstrate that a structural transition occurs between 10−2 and 10−1 s−1, resulting in an apparent interruption of the shear-thinning. The corresponding transient response presents a pronounced extra-growth of the shear stress before to stabilize at high strain amplitude and a negative first normal stress difference peak, both effects become stronger at higher shear rates. These observations are rationalized based on the high interparticle friction owing to the polyhedral shape and the roughness of the particles. In addition, relaxation tests following the start-up experiments reveal that the samples submitted to shear rates higher than 10−1 s−1 cannot relax the shear stress to the same level as in low shear rate experiments, suggesting a durable structural modification likely to impact the quality of the parts prior to their debinding and densification. Finally, we utilize orthogonal superposition rheology to illustrate how the application of an oscillatory deformation during the scraping procedure could help to reduce the shear-thinning interruption and improve the stereolithography processing as already observed empirically during scraping.

Fe doped LiNi0.8Co0.15Al0.05 O2 (NCA) cathode materials: Synthesis, structural and morphological studies

As a result of its substantial energy density, layered LiNi0.8Co0.15Al0.05O2 cathode materials are thought to be the most promising of the next-generation cathode materials for electric vehicles (EVs). In this research, self propagating combustion (SPC) is used to prepare cathode materials made of pristine LiNi0.8Co0.15Al0.05O2 (NCA) and NCA doped with 1% Fe to produce LiNi0.8Co0.14Fe0.01Al0.05O2. Based on X-Ray Diffraction (XRD) results, pristine and doped NCA cathode materials were pure and single phase. Reference Intensity Ratio (RIR) values for pristine and doped NCA are 0.87 and 1.12 respectively indicating that the amount of cation mixing was reduced with Fe doping. Through Rietveld refinement, it was discovered that the addition of Fe to NCA resulted in a decrease in cation mixing from 13.56% to 4.07%.It was found that both pristine and doped NCA possessed polyhedral like shape morphology. It can be seen that doping with 1% Fe does not change much in the crystallite size of the materials. By having less cation mixing, Fe doped was found to greatly improve the structural integrity of NCA cathode materials, which in turn improved the materials’ electrochemical performance.

The Growth Mechanism, Luminescence, and Lasing of Polyhedral ZnO Microcrystals with Whispering-Gallery Modes

This work studies the features of the formation of isometric polyhedral ZnO microcrystals that provide stimulated emission and whispering-gallery-mode (WGM) lasing in the near-UV range. For this purpose, the growth stages of such crystals in the process of gas-transport synthesis and the luminescent properties of the structures obtained at each stage were investigated. It was shown that the growth of laser microcrystals begins with the formation of microspheroids with thin ZnO shells. Such spheroids exhibit mainly white luminescence with a small contribution of near-UV emission. Increasing the synthesis duration results in thickening and faceting of the spheroid shells, as well as a decrease in the contribution of the yellow–red component to the luminescence spectrum. At the same time, ZnO microcrystallites nucleate and grow inside the spheroids, using as a material the remains of a liquid zinc drop and oxygen entering the spheroids through their shells. Such growth conditions allow them to take on an equilibrium polyhedral shape. Eventually, upon destruction of the spheroid shell, a polyhedral ZnO microcrystal supporting WGMs is observed.

Integrating geometries of ReLU feedforward neural networks.

This paper investigates the integration of multiple geometries present within a ReLU-based neural network. A ReLU neural network determines a piecewise affine linear continuous map, M, from an input space ℝm to an output space ℝn. The piecewise behavior corresponds to a polyhedral decomposition of ℝm. Each polyhedron in the decomposition can be labeled with a binary vector (whose length equals the number of ReLU nodes in the network) and with an affine linear function (which agrees with M when restricted to points in the polyhedron). We develop a toolbox that calculates the binary vector for a polyhedra containing a given data point with respect to a given ReLU FFNN. We utilize this binary vector to derive bounding facets for the corresponding polyhedron, extraction of "active" bits within the binary vector, enumeration of neighboring binary vectors, and visualization of the polyhedral decomposition (Python code is available at https://github.com/cglrtrgy/GoL_Toolbox). Polyhedra in the polyhedral decomposition of ℝm are neighbors if they share a facet. Binary vectors for neighboring polyhedra differ in exactly 1 bit. Using the toolbox, we analyze the Hamming distance between the binary vectors for polyhedra containing points from adversarial/nonadversarial datasets revealing distinct geometric properties. A bisection method is employed to identify sample points with a Hamming distance of 1 along the shortest Euclidean distance path, facilitating the analysis of local geometric interplay between Euclidean geometry and the polyhedral decomposition along the path. Additionally, we study the distribution of Chebyshev centers and related radii across different polyhedra, shedding light on the polyhedral shape, size, clustering, and aiding in the understanding of decision boundaries.