Regular Hexagon Research Articles

A New Family of Solids: The Infinite Kepler-Poinsot Polyhedra

The so-called Platonic solids have fascinated mathematicians and artists for over 2000 years. It is astonishing that there are only five cases of regular polyhedra, that is, of polyhedra in which regular polygons form the same spatial angles between them in each vertex. In 1619, Kepler added the small and great stellated dodecahedron to this list, but he allowed intersecting faces. Poinsot did so too, in 1809, and discovered the great dodecahedron and great icosahedron. In 20th century, Coxeter and Petrie added three more regular polyhedra, using infinitely repeating elements, based on the truncated tetrahedron, the cube and the truncated octahedron. The principle of intersecting faces, typical for the Kepler-Poinsot solids, can be combined with the Coxeter-Petrie generalization to the infinite case. Thus, a new regular polyhedron was discovered, based on the cubohemioctahedron but without its square faces. Placed side by side and on top of each other, identical regular hexagons meet in each vertex, always with the same spatial angle. There are 8 of them in each vertex, and so it is not a compound of twice two polyhedra with 4 hexagons in each vertex. The dual of this {6, 8} polyhedron of infinite Kepler-Poinsot type is indeed a {8, 6} polyhedron of infinite Kepler-Poinsot type, if two overlapping squares are considered as one 8/4 octagonal star. Kepler-Poinsot solids are difficult to interpret, with their intersecting faces, and this infinite case is even more difficult to grasp. The present paper tries to solve this using open faces so that one can see through the solids.

Numerical Evaluation of Wave Dissipation on a Breakwater Slope Covered by Precast Blocks with Different Geometrical Characteristics

Slopes suffer damage from waves in coastal environments. Precast blocks with well-designed geometrical characteristics can benefit the construction of revetments by mitigating the issue of wave overtopping and dissipating wave energy. In this study, we numerically studied the effect of the geometrical characteristics of precast blocks on wave overtopping by carrying out a numerical simulation of wave overtopping on a slope covered with precast blocks. A total of three different types of blocks were considered in this study to determine the optimal geometric shape using a validated numerical model. Our numerical investigation demonstrated that the roughness of the precast block plays an important role in lessening the height of the wave run-up. Concave and embedded regular hexagons could reduce the wave run-up height by 44.6% compared with smooth slopes within a 2 s wave period. Herein, we evaluate and discuss the influence of the geometrical characteristics of a given precast block, such as thickness, aperture, and wave dissipation notch, on wave run-up. We also present an empirical formula for predicting wave run-up on a slope covered by a concave and embedded regular hexagon-type prefabricated block. This study provides valuable insights into the design of prefabricated revetment blocks.

Dynamic analysis of multi-module floating photovoltaic platforms with composite mooring system by considering tidal variation and platform configuration

Floating Photovoltaics (FPV) are emerging as an innovative technology to utilizing solar energy which advances the exploration of renewable energy technologies in the deep sea. This investigation proposes a novel methodology to verify the feasibility of the designed models in time-domain analysis via the motion responses, the mooring analysis based on the finite element method (FEM) and the airgap analysis, considering a comparison of dynamic responses between the regular hexagon shape and rectangle shape semi-submersible floating photovoltaic platform (FPVP) both combinate the cables and the tensioners mooring system to adapt the tidal variation. Initially, the response amplitude operators (RAOs) of the two configurations of the FPVP numerical models established in AQWA are derived from the frequency-domain analysis. Then, the method of rectifying the RAOs by incorporating additional damping essential to compensating the fluid viscosity due to the limitation of the potential flow theory is verified through comparison with the Computational Fluid Dynamics (CFD) solver. Subsequently, the motion responses of the two conceptualized FPVPs with the composite mooring system under different depths of water are assessed in time-domain analysis through OrcaFlex which regards the cables and tensioners as Morison type based on the FEM theory. Simultaneously, the variations in the air gap of platforms with dissimilar stiffness tensioners are compared to assess the safety of FPV systems under different water depths. The long-term extreme forecast of effective tension force for various return periods derived from the Weibull distribution model is verified. Finally, considering the factors that influence the design and safety of multi-module floating photovoltaics in variable-depth water, this study provides essential guidance for the safe construction technology of semi-submersible FPVPs, serves as a valuable reference for promoting reliable and cost-effective FPV technologies for offshore environments.

볼록 육각형의 나폴레옹의 정리 확장에 대한 연구

This study was based on the research results conducted as a R&E project for the gifted students with a financial support from the Korea Foundation for the Advancement of Science and Creativity. In this study, the Napoleon hexagon ofhexagons was explored by expanding the Fermat-Torricelli point defined in a triangle and the Napoleon triangle into a convex hexagon. The results of this study are as follows. First, the equivalence condition for convex hexagons in which the Napoleon hexagon exists was discovered. Second, the area formula for Napoleon’s hexagon was shown. Third, we found various regular hexagons related to the Napoleon-containing hexagon and found that the centers of each regular hexagon were identical. Through this study, the concepts of the Fermat-Torricelli point and Napoleon triangle of triangles were expanded to convex hexagons, and it is expected that the expansion of mathematical concepts willcontribute to the development of mathematics. In addition, it is expected that additional research will be conducted in the future by exploring attractive regular hexagons whose centers coincide with the searched hexagons including Napoleon.

Ballistic Photons, Justification of their Structure and Movement Patterns

Aim: The work relates to the fundamentals of quantum physics and photonics, in particular to the formation of photons in two states: elementary particles and electromagnetic radiation, and the parameters of their structure and motion processes. The study of these problems is a relevant and important task, has not been fully studied. In general case, a photon is considered an elementary particle and wave, quantum of energy emitted by its electromagnetic field, which does not exist without moving at the speed of light. However, the presence of the material part of a photon is not always manifested, so a photon is considered a wave of electromagnetic radiation and the photon has no internal structure and dimensions, except for its wave parameters. The process of photon movement is also problematic. It is believed that it exists as a particle at the moment of its origin, and then it is only the movement of a wave, which, when meeting a target, again turns into an elementary particle. Moreover, ballistic photons is not, and their flight occurs by absorption and re-emission of waves by the environment. However, all these factors have contradictions, the elimination of which is the main goal of the work performed. Its scientific novelty is a new justification for the presence and parameters of a ballistic photon and the processes of its movement on the basis of reliable laws of physics. Methodology: The research methodology is based on the general principles of scientific knowledge, on the application of the laws of dialectics and reliable laws of physics. The author's method was also used – this access to the initial level of the material world, which is formed by physical fields and particles on the basis of natural physical constants: the speed of light c, Planck's constant h, gravitational constant G. New Results of the Work: The possibility of the existence of ballistic photons in the dual form of substance and electromagnetic waves is shown, within the framework of the general physical principles of the existence of elementary particles and the real principles of their movement in the material world. It was found that the disappearance of ballistic photons in flight after their birth and their appearance when meeting a target can be explained by their quantum structure and the process of their movement by quantum jumps based on the Heseisenberg uncertainty principle. Conclusions: The presence of the proposed quantum structure of the photon and the process of its movement is ensured by the absence of contradictions with the real laws of the material world that have a strict physical basis. For to confirm the reality of the existence of ballistic photons in the form of a real quantum mechanical particle, having the shape of a regular hexagon with a side of 1/ 6 of the wavelength λ of their radiation, as well as their movement in quantum jumps at 1/ 6 of the length of this wave, an experiment has been proposed. All laboratories with metrological equipment are invited to conduct the experiment.

CHURCHES ON MINING DAMAGE - A CASE STUDY

ABSTRACT: Aim: The research objective is to analyze selected three churches by architect Professor Adam Lisik realized on mining damage. The research conducted on the basis of the creative output of the prominent Polish architect of religious buildings is presented. The research is intended to show how difficult design conditions and locations on mining damage affect the functional and solid form of the churches. Methods: The following research methods were used to conduct the study: in-situ research of the objects in question, literature analysis, analysis of conceptual sketches and design documentation of the objects, interview with the author of the projects. The research of the objects takes the form of an expert assessment by the author of the article. Results: All three examples of churches built on high categories of mining damage show different projection solutions: square, regular hexagon, ellipse. The shapes used make it possible to obtain a compact projection with interesting aesthetics of shaping the interior space. The architect also showed a lot of freedom in shaping the various masses of the buildings. Limitations of the decision of the OUG (District Mining Office) for the architect not only did not constitute an obstacle, but on the contrary inspired the search for new solutions, giving satisfaction to the creator of the sacred temple. Conclusion: The results of the study allow us to conclude that the limitations of high mining damage categories (III and IV) did not limit creative freedom. It can be summarized by saying: large-scale religious buildings can be successfully implemented on mining damage. Much depends on the knowledge, skills, creative workshop and vision of the architect. Originality: Limitations of mining damage can pose creative challenges to architectural and structural designers. By complying with the principles of structural safety, one can break stereotypes and create architecturally interesting masses and spaces and interiors. Constraint can become an inspiration for creating new forms of architectural expression. Keywords: architecture, churches, design, construction, construction, mining damage, symbolism.

Surface stress redistribution and thermal performance of tempered vacuum glazing with printing sintering support pillars

The support stress distribution of the printing sintering support pillars (PSSP), which became rounded due to the shrinkage that occurred in sintering formation, was different from that of the conventional cylindrical metal support pillars (CMSP). In this paper, ANSYS software was used to establish the mechanical simulation model of tempered vacuum glazing with PSSP. The influence of PSSP arrangement (spacing D, and three arrangement styles of square, regular triangle, and regular hexagon) on the mechanical properties of tempered vacuum glazing were analyzed, including the maximum stress σ1max on tempered glass (T-glass) surface, the maximum deformation ω1max of T-glass, and the maximum stress σ2max on the PSSP. The simulation results showed that when the PSSP were arranged in a square with D of 50 mm, σ1max, ω1max, and σ2max were 26.39 MPa, 7.82 μm, and 546.72 MPa, respectively, which met the requirements of the evaluation indexes of mechanical properties of tempered vacuum glazing, and had the least number of PSSP. The mechanical properties of tempered vacuum glazing with regular triangle arrangement of PSSP were better than those of square and regular hexagon arrangements, and the σ1max, ω1max and σ2max of which were 16.24 MPa, 7.25 μm and 489.66 MPa respectively, but with a too large number of PSSP. Optimization of the triangle layout angle and side length could reduce PSSP numbers. When the PSSP were arranged in an isosceles triangle with a 65° base angle and 50 mm bottom length, the mechanical properties of tempered vacuum glazing were the best. The triangular arrangement of PSSP had an optimization effect on the surface stress redistribution of tempered vacuum glazing, and the number of stress peaks was twice that of the square arrangement. Meanwhile, tempered vacuum glazing samples were made and the surface stress, thermal conductivity tests were conducted. The results showed that the measured values of peak and valley stress had an error of ±2.8 MPa compared to the simulated values. The simulation results were reliable and the surface stress redistribution of tempered vacuum glazing was present. The measured heat transfer coefficient of tempered vacuum glazing with PSSP was 0.7402 W m−2 K−1, slightly higher than that of tempered vacuum glazing with CMSP. The difference between them was not significant.

Optimization of an architected composite with tailored graded properties

The aim of the present study is to design a solid material with specific and tailored mechanical properties through a suitably defined design framework and to evaluate the effectiveness of different microstructure geometries in an engineering perspective. To these ends, topology optimization algorithms are applied on a 2D homogenized equivalent model of a periodic structure. The design framework, developed in a previous work for 2D lattices made of regular hexagons, is here expanded and validated also in the cases of circular and square unit cells. The proposed approach involves optimizing porosity distribution of a homogenized equivalent solid, obtained through a Bloch–Floquet-based analysis, within a 2D lattice of regular unit cells forming the core element of a sandwich panel. Finite-element analyses on homogenized and fine structural models are carried out in order to validate the procedure, beyond the particular choice of the unit cell geometry and to detect its effectiveness and limits.

Optimization of cross-sectional dimensions of castellated beams with hexagonal openings

The object of this study is a castellated beam, in which the web openings have the shape of a regular hexagon. The beam is examined to find optimal cross-sectional dimensions. The optimization task is stated as the task of finding the optimal profile numbers for top and bottom Tees of the beam and the optimal width of the web opening while ensuring the required load-carrying capacity of the beam. Minimization of the volume of the beam material was considered as an optimality criterion. The stated optimization problem was solved using the exhaustive search method. For an assortment of normal I-beams with parallel flanges, castellated beams were obtained with optimal cross-sectional dimensions depending on the steel grade, the beam span, and the magnitude of transverse uniformly distributed load. The optimization calculations proved that it was possible to increase the elastic section modulus of the beam to 35.48...50 % through the use of a castellated web. Castellated beams with optimal cross-sectional dimensions at the same load-carrying capacity are characterized by lower steel consumption (up to 23.19 %) compared to I-beams with a solid web. Analysis of the results has made it possible to devise recommendations for the optimal distribution of material in the cross-sections of such beams. The results are valid only for the assortment of normal I-beam profiles and only for the case of uniformly distributed load acting on the beam when the compressed beam flange is laterally restrained from the bending plane and the beam web has openings in the form of regular hexagons. It is under such conditions that the reported results can be implemented in practice both at the stage of selecting cross-sections of the studied class of structures, and at the development of effective assortments of castellated beams

Eigenvalues of thick regular near hexagons

Eigenvalues of thick regular near hexagons

Core Optimization for Extending the Graphite Irradiation Lifespan in a Small Modular Thorium-Based Molten Salt Reactor

The lifespan of core graphite under neutron irradiation in a commercial molten salt reactor (MSR) has an important influence on its economy. Flattening the fast neutron flux (≥0.05 MeV) distribution in the core is the main method to extend the graphite irradiation lifespan. In this paper, the effects of the key parameters of MSRs on fast neutron flux distribution, including volume fraction (VF) of fuel salt, pitch of hexagonal fuel assembly, core zoning, and layout of control rod assemblies, were studied. The fast neutron flux distribution in a regular hexagon fuel assembly was first analyzed by varying VF and pitch. It was demonstrated that changing VF is more effective in reducing the fast neutron flux in both global and local graphite blocks. Flattening the fast neutron flux distribution of a commercial MSR core was then carried out by zoning the core into two regions under different VFs. Considering both the fast neutron flux distribution and burnup depth, an optimized core was obtained. The fast neutron flux distribution of the optimized core was further flattened by the rational arrangement of control rod channels. The calculation results show that the final optimized core could reduce the maximum fast neutron flux of the graphite blocks by about 30% and result in a more negative temperature reactivity coefficient, while slightly decreasing the burnup and maintaining a fully acceptable core temperature distribution.

Modelling and characterization of novel honeycomb structures with mass gradient produced by additive manufacturing

The dissemination of additive manufacturing methods has facilitated the design and production of complex structures which have a high strength-to-weight ratio. Cellular materials such as honeycombs have low-weight and high capacity to absorb energy which makes them desirable for the aerospace and automotive industries. The present work covers the study and comparison of metal-based regular honeycombs and functionally graded honeycombs. The latter encompass radial and linear/longitudinal gradients. Three repeating unit cells were studied: regular hexagons, Plateau and lotus. The structures were produced in aluminium using the laser powder bed fusion technique. Selected samples were submitted to a stress-relieving heat treatment. Numerical and experimental methods were used to assess the in-plane compressive properties. Finite element analysis was used to obtain the simulated force–displacement curves of each structure, allowing for the calculation of specific stiffness, absorbed energy and yield strength. The experimental method consisted of the compression of three specimens of three types of regular structures with and without stress-relieving heat treatment. The heat treatment reduced the yield strength and stiffness whilst increasing the ductility of the samples. The mechanical behaviour of the structures was found to depend upon a combined effect of the type of gradient, relative density, and unit cell structure. The results showed that an increase in the relative density would enhance the specific mechanical properties. The lotus configuration displayed the highest specific mechanical properties, as its geometry reduces the stress concentrations. The numerical results showed a reasonable match with the experimental results.

Design and Application of DG-FEM Basis Functions for Neutron Transport on Two-Dimensional and Three-Dimensional Hexagonal Meshes

Reactor design requires safety studies to ensure that the reactors will behave appropriately under incidental or accidental situations. Safety studies often involve multiphysics simulations where several branches of reactor physics are necessary to model a given phenomenon. In those situations, it has been observed that the neutron transport part is still a bottleneck in terms of computational times, with more than 80% of the total time. In the case of hexagonal lattice reactors, transport solvers usually invert the discretised Boltzmann equation by discretising the regular hexagon into lozenges or triangles. In this work, we seek to reduce the computational burden of the neutron transport solver by designing a numerical spatial discretisation scheme that would be more appropriate for honeycomb meshes. In our past research efforts, we have set up interesting discretisation schemes in the finite element setting in 2D, and we wish to extend them to 3D geometries that are prisms with a hexagonal base. In 3D, a rigorous method was derived to shrink the tensor product between 2D and 1D bases to minimum terms. We have applied these functions successfully on a reactor benchmark—Takeda Model 4—to compare and contrast the numerical results in a physical setting.

Auxetic System Based on Rotating Hexagons and Triangles

An auxetic metamaterial that exhibits isotropic in‐plane deformation is introduced herein in the form of rotating rigid units, consisting of regular hexagons and equilateral triangles, that are connected at their vertices. By inspection, the in‐plane Poisson's ratio is at a constant value of ν = −1. The size descriptor of this metamaterial can be expressed as a function of rotating unit sizes and their internal angle by means of geometrical construction. By energy method, the effective in‐plane Young's modulus of this metamaterial has been established and shown to be governed by the rotational stiffness at the connecting hinges in addition to the relative sizes of the rotating units and their internal angle. Results indicate that the in‐plane Young's modulus is linearly proportional to the rotational stiffness at the connecting hinges and inversely proportional to the rigid unit size. In addition, the Young's modulus is governed by the ratio of the hexagon‐to‐triangle size for small internal angle, and rapidly approaches extreme value as the internal angle approaches right angle. The characteristics of the proposed metamaterial would allow its use as a sieve that not only controls the filtering of particles by size but also by the aspect ratio of rod‐like particles.

Topology Design of Modular Unit and Cable Network of On‐Orbit Modular‐Assembled Antenna

With the increasing demands of ultra‐large space antennas, on‐orbit assembly stands out as a promising solution. However, this achievement necessitates the reconfiguration design of the whole antenna structure to maintain structural integrity and stability. In light of this challenge, a novel rapid design method is proposed for the unit configuration of modular‐assembled antennas in this paper. The method utilizes an inscribed regular hexagon, which is positioned within a horizontal topological circle. The hexagon is subsequently projected onto a spinning paraboloid, resulting in the creation of unit nodes that conform to the shape of the paraboloid. The configuration of the unit topology is then determined based on the arrangement of these nodes. By using the method, two units of space‐assembled antennas are designed. The topology design of the cable network is carried out for the modular unit. The pretensions of the cables are calculated. The results of the case study reveal that the developed modular units fulfill the requirements of the on‐orbit‐assembly antenna structure and can solve the engineering application difficulties of this type of antenna.

Computation of neighbourhood degree based topological indices for circumcoronene series of benzenoid

This article delves into the Circumcoronene series, a subset within the Benzenoid family of molecular structures known for their distinctive arrangement of benzene rings forming regular hexagons along the periphery. We compute various topological indices that are derived from open and closed neighborhood degrees, as well as reduced reverse indices, for the Circumcoronene series of Benzenoids. Furthermore, we conduct an analysis and comparative assessment of the distribution of these indices with the aid of MATLAB.

Some examples of quasiperiodic tilings obtained with a simple grid method

A grid method using tiling by fundamental domain of simple bi-dimensional lattices is presented. It refers to a previous work done by Stampfli in 1986 using two grids by regular hexagons, one rotated by relatively to the other. This leads to a quasiperiodic structure with a twelvefold symmetry made of regular triangles, squares and rhombuses. The tessellation of the plane by the overlap domains of two hexagons, each belonging to one of the two grids is considered. Vertices of the quasiperiodic tiling are the mid-point of the centers of the two overlapping hexagons. Edges of the quasiperiodic tiling are obtained by a Delaunay triangulation of the set of reference points. This method is extended to two other types of quasiperiodic tilings with other fundamental domains. A first example uses two square grids leading to the octagonal Ammann-Beenker quasiperiodic tiling. The second example is also based on the hexagonal lattice, but with grids defined by the lattice fundamental rhombic cell.

An Alternative Way to Produce High-Density Graphite from Carbonaceous Raw Materials

In this study, graphite, the most stable form of carbon, was examined for its hexagonal crystalline structure with specific dimensions (ao = 2.46 Ǻ; co = 6.70 Ǻ). Its framework comprises parallel carbon atom planes, forming regular hexagons (side length 1.415 Ǻ) and 120° angles between adjacent atoms. Two structural variations exist: hexagonal symmetry (1-2-1-2-1-2 planes) and rhomboidal symmetry (1-2-3-1-2-3 planes). The aim of this research was to produce high-density graphite utilizing carbonaceous raw materials. Graphite-based materials often exhibit high porosity, necessitating additional treatment. In this study, we successfully obtained mesophase tar pitch (yield: 45%), a pivotal raw material, and high-density graphite. The resulting graphite underwent characterization for physical properties (apparent and real density, porosity, and compression strength), demonstrating conformity with the existing literature data.

An energy-efficient irregular hexagonal tessellation-based approach for connected k-coverage in planar wireless sensor networks

In the design of planar wireless sensor networks (PWSNs), the preliminary tasks are to achieve both coverage and connectivity, which are essential for the correct operation of this type of network. A PWSN is said to be in perfect operational condition only when it is capable of guaranteeing both coverage of the field and connectivity among all the active sensor nodes. In order to achieve both coverage and connectivity in PWSNs, we intend to solve the problem of connected k-coverage in PWSNs, where each point in a field of interest is covered (or sensed) by at least k sensor nodes (k > 1) simultaneously and all the participating sensor nodes in the k-coverage process are connected to each other. In our study, we found an irregular hexagon, denoted by IrHx(rs/n), as the best polygon for tessellating a field of interest and allowing to deploy sensor nodes, where n > 1 is a natural number and rs is the radius of the sensing range of the sensor nodes. First, we construct our proposed irregular hexagon, IrHx(rs/n), which forms a tile, using the regular hexagon. Second, we compute the minimum sensor density required to k-cover a planar field of interest using our proposed IrHx(rs/n) tile. Third, we establish a relationship between the sensing and communication radii of the sensor nodes to ensure network connectivity in k-covered PWSNs. Finally, we substantiate our theory with various simulation results.

The Beauty of Regular Hexagons

The Beauty of Regular Hexagons