Regular Hexagon Research Articles

A web of confocal parabolas in a grid of hexagons

If one erects regular hexagons upon the sides of a triangle T, several surprising properties emerge, including: (i) the triangles which flank said hexagons have an isodynamic point common with T, (ii) the construction can be extended iteratively, forming an infinite grid of regular hexagons and flank triangles, (iii) a web of confocal parabolas with only three distinct foci interweaves the vertices of hexagons in the grid. Finally, (iv) said foci are the vertices of an equilateral triangle.

Normal-mode oscillations for the circular and dipolar states of a filled hexagonal magnetic dipole cluster.

We analyze the rotational dynamics of six magnetic dipoles of identical strength at the vertices of a regular hexagon with a variable-strength dipole in the center. The seven dipoles spin freely about fixed axes that are perpendicular to the plane of the hexagon, with their dipole moments directed parallel to the plane. Equilibrium dipole orientations are calculated as a function of the relative strength of the central dipole. Small-amplitude perturbations about these equilibrium states are calculated in the absence of friction and are compared with analytical results in the limit of zero and infinite central dipole strength. Normal modes and frequencies are presented. Bifurcations are seen at two critical values of the central dipole strength, with bistability between these values.

Enumerating the Nonisomorphic Configurations of a Modular Reconfigurable Robot

Abstract By introducing the modular and reconfigurable design, the limbs of a hexapod robot can be assembled into different configurations to meet various task requirements. Contrary to the fixed configuration system, the task capabilities of the modular reconfigurable robot vary with the configurations. This article addresses the problem of the configuration enumeration and configuration expression of a modular reconfigurable robot. First, by establishing the permutation group of the regular hexagon (the base shape of the modular reconfigurable robot) and adopting the Pólya enumeration theorem, the theoretical formula of the nonisomorphic configuration enumeration is derived. Then, considering the change in structural features, equivalence relations based on structural features are defined and the structural feature method (SFM) is proposed to enumerate the nonisomorphic configurations. In addition, to express configurations more intuitively, a senary vertex identification array with the same dimension as the number of modules is presented, which can be transformed from the decimal index of the configuration. Simulation analysis demonstrates that the result of the SFM is consistent with that of the theoretical calculation, which also confirms the effectiveness and accuracy of the two methods applied in the nonisomorphic configuration enumeration of the modular reconfigurable robot.

Behaviour of inradius, circumradius, and diameter in generalized Minkowski spaces

We provide a study of Blaschke–Santaló diagrams, i.e. complete systems of inequalities, for the inradius, diameter, and circumradius, measured with respect to different gauges. This contrasts previous works on those diagrams, which are all considered for the Euclidean measure. By proving several new inequalities and properties between these three functionals, we compute the intersection and the union over all possible gauges of those diagrams, showing that they coincide with the corresponding diagrams of a parallelotope and (in the planar case) a triangle, respectively. We also show that the planar spaces whose unit balls are regular pentagons or hexagons play an important role in the understanding of further extreme behaviours.

Quadratic Bezier Curve as the base for pot modelling

Pot is a place or container made of clay, earthenware, cement, or plastic to plant trees or plants. In addition to a place to plant plants, pots also function to decorate the appearance of the house, both on the outside (outdoor) or in the room (indoor). So that the selection of the right pot model can make the yard or corner of the room look more comfortable, beautiful, and attractive. Some pot models that are already on the market have a circular or rectangular base with a monotonous pot shape. Thus, the researcher conducted on the pot modelling with the initial data of regular hexagon polygons as the base of the pot and the Quadratic Bezier curve as the basis for modelling pots with the help of Maple software. This research method uses applied research methods, namely the application of the Quadratic Bezier Curve in making pot models, several pot models were produced, namely a pot with two tiers and three tiers, where each level has several potted models with twisted and symmetrical types. So that the pot models obtained are more attractive and have artistic value.

Building a dynamic model of profit maximization for a carsharing system accounting for the region’s geographical and economic features

This paper describes a dynamic model of profit maximization for a car-sharing system, taking into consideration the geographical and economic characteristics of a region. To solve the model construction task, a technique to cover the region with geometric shapes has been described. It was established that when modeling a car-sharing system, it is rational to cover a region with a grid of equal regular hexagons located side to side. For each subregion, quantitative parameters were calculated: the number of free cars in the subregions, the probability of a car traveling from one sub-region to another, the cost of maintenance and operation of the car, and the income from the trip. This takes into consideration the dynamic nature of the specified parameters. Based on these parameters, an objective function is constructed including constraints for the dynamic model. These constraints take into consideration the economic and geographical features of each subregion. A dynamic profit maximization model was built for the car-sharing system in the city of New York (USA) based on the TCL dataset. To calculate the parameters of the model, data on 776,285,070 trips over the period from January 2016 to July 2021 were used. Maps of the beginning and completion of trips in the region and a map of trips tied to hexagonal grid cells using the Kepler visualization service have been built. The frameworks H3 and S2 were analyzed in terms of determining the length of the route between the subregions. Modeling was carried out according to the built unidirectional dynamic model of profit maximization. It has been established that taking into consideration the average economic and geographical characteristics of a region makes it possible to increase the profit of the car-sharing system by 12.36 %. Accounting for the dynamics of economic and geographical features of the region of customers in the model makes it possible to increase profits by an additional 4.18 %

Bio-Inspired Bianisotropic Magneto-Sensitive Elastomers with Excellent Multimodal Transformation.

Magneto-sensitive soft materials that can accomplish fast, remote, and reversible shape morphing are highly desirable for practical applications including biomedical devices, soft robotics, and flexible electronics. In conventional magneto-sensitive elastomers (MSEs), there is a tradeoff between employing hard magnetic particles with costly magnetic programming and utilizing soft magnetic particle chains causing tedious and small deformation. Here, inspired by the shape and movement of mimosa, a novel soft magnetic particle doped shape material bianisotropic magneto-sensitive elastomer (SM bianisotropic MSE) with multimodal transformation and superior deformability is developed. The high-aspect-ratio shape anisotropy and the material anisotropy in which the magnetic particles are arranged in a chainlike structure together impart magnetic anisotropy to the SM bianisotropic MSE. A magneto-elastic analysis model is proposed, and it is elucidated that magnetic anisotropy leads to peculiar field-direction-dependent multimodal transformation. More importantly, a quadrilateral assembly and a regular hexagon assembly based on this SM bianisotropic MSE are designed, and they exhibit 2.4 and 1.7 times the deformation capacity of shape anisotropic samples, respectively. By exploiting the multidegree of freedom and excellent deformability of the SM bianisotropic MSE, flexible logic switches and ultrasoft magnetic manipulators are further demonstrated, which prove its potential applications in future intelligent flexible electronics and autonomous soft robotics.

A Generalization of Group-Graded Modules

In this article, we generalize the concept of group-graded modules by introducing the concept of G-weak graded R-modules, which are R-modules graded by a set G of left coset representatives, where R is a G-weak graded ring. Moreover, we prove some properties of these modules. Finally, results related to G-weak graded fields and their vector spaces are deduced. Many considerable examples are provided with more emphasis on the symmetric group S3 and the dihedral group D6, which gives the group of symmetries of a regular hexagon.

EKSPLORASI ETNOMATEMATIKA PADA BENTUK ANYAMAN ENDE

The purpose of this study is to explore the ethnomathematical concepts contained in the woven form of the Ende tribe. This type of research is exploratory research using an ethnographic approach. The subjects in this study were woven gedek and khidhe craftsmen, mothers who used to make plaits in Rewa Rangga Village, Ende Regency. The instruments in this research are field study interviews and documents. Data analysis used is a triangulation technique. The results showed that there were geometric shapes in woven shapes and motifs. Geometric shapes that are explored are building spaces: beams; flat shapes: circles, regular hexagons, rectangles, and rhombus.

The Impact of the Cut-Out Shape on the Dynamic Behavior of Composite Thin Circular Plates

This study measures the impact of the cut-out shape on the fundamental frequency and the harmonic response of thin composite circular plates under fixed boundary conditions. For this purpose, the Finite Element Method has been employed using the SHELL181 element of ANSYS Workbench 18.2 software. Nine different circular plates having central equilateral triangle, square, regular pentagon, regular hexagon, regular heptagon, regular octagon, regular nonagon, regular decagon, and circular cut-outs have been designed to understand the changes in the vibrational behavior of a structure as the cut-out geometry approximates to circular. Additionally, a solid circular plate has been modelled for comparison purposes. To investigate the effect of the cut-out shape more deeply, all structures have been designed as cross-ply and angle-ply composites. The free vibration analysis has been performed to obtain the fundamental frequency of each structure. The harmonic response analysis problem has been solved by employing the Mode Superposition Method and considering a constant damping ratio. The results of the study have been interpreted by considering the displacement response, stress, and phase shifts at a certain frequency range. The results indicate that the shape of the cut-out considerably affects the fundamental frequency and harmonic response of the circular structure regardless of its fiber orientation.

Continuous forcing spectrum of regular hexagonal polyhexes

For any perfect matching M of a graph G, the forcing number (resp. anti-forcing number) of M is the smallest cardinality of an edge subset S⊆M (resp. S⊆E(G)∖M) such that the graph G−V(S) (resp. G−S) has a unique perfect matching. The forcing spectrum of G is the set of forcing numbers of all perfect matchings of G. Afshani et al. [2] proved that any finite set of positive integers can be the forcing spectrum of a planar bipartite graph. A polyhex is a 2-connected plane bipartite graph whose interior faces are regular hexagons. In this paper, we give the minimum forcing number and anti-forcing number of a polyhex with a 3-divisible perfect matching. Further, we prove that the forcing spectrum of any regular hexagonal polyhex is continuous, i.e. an integer interval, by applying some changes on prolate triangle polyhexes to obtain a sequence of perfect matchings of the graph.

Resonance modes of periodically structuralized microwave magnetic elements

Here we consider a flower-like structure of a resonator consisting of six elliptical elements, referred to as petals, made from a magnetic material. The petals are positioned with their centres at the corners of a regular hexagon. Using numerical simulations (CST Studio) we examine the effect of different radial orientations of petals. We study resonance modes with a specific distribution of the electromagnetic field within the resonator as well as the effect of the rotation of petals on the field distribution. The mode character is crucial to understand the behaviour of the frequency spectrum. E.g., the rotation of petals influences significantly the frequency of the lowest mode only, while the other frequencies are almost unchanged and this effect is directly related to the profiles of modes. The system studied is a promising candidate for a tuneable component of an integrated detection system.

Hysteretic transition between states of a filled hexagonal magnetic dipole cluster

By minimizing the magnetostatic potential energy and by finding zeros in the sum of the squares of the torques, we find the equilibrium states of six dipoles of identical strength at the vertices of a regular hexagon and a variable-strength dipole at the center. The seven dipoles spin freely about fixed axes that are perpendicular to the plane of the hexagon, with their dipole moments directed parallel to the plane. When the central dipole is weak compared with the perimeter dipoles, a “circular” state applies in which the perimeter dipole moments circle around the central dipole, which points toward a perimeter dipole. When the central dipole is strong, a more symmetric “dipolar” state applies in which the perimeter dipole moments align approximately with the field of the central dipole. Over an intermediate range of dipole strengths bounded by two critical values, both states are locally stable and the state of the system depends upon its history. Iron filings are used to observe both states in experiments on small spherical neodymium magnets. A “misaligned” state that is barely unstable theoretically is also observed experimentally; this state resembles the circular state except that the central dipole moment points toward a point of contact between two perimeter magnets.

A new polyurethane-steel honeycomb composite pier anti-collision device: Concept and compressive behavior

A new type of pier anti-collision composite structure composed of honeycomb steel and polyurethane (PU) elastomer was proposed in this study. The impacts of the shape and filling materials of inner core cells on the failure mode, load–displacement cure, bearing capacity, structural stability, and energy absorption were studied by conducting uniaxial compression tests on device segments. Test results showed that the bearing capacity, structural stability, and energy absorption of honeycomb steel structure were significantly improved by PU elastomer filling. Besides, when compared with the square honeycomb structure and the regular hexagon honeycomb structure, the maximum values of average load, total energy absorption (TEA), and specific energy absorption (SEA), which were 69.6 kN, 1986.1 J, and 1300 J/kg, respectively, for the regular triangle honeycomb structure without PU filling, increased to 459.3%, 376.38%, and 212.5%, respectively, for the regular hexagonal core cell structure with PU filling, which was proved to be the most suitable core structure for pier anti-collision device.

A Novel Path Planning Model Based on Nested Regular Hexagons for Mobile Anchor-Assisted Localization in Wireless Sensor Networks

A Novel Path Planning Model Based on Nested Regular Hexagons for Mobile Anchor-Assisted Localization in Wireless Sensor Networks

Research on Optimization of Tube Structure of Phase Change Heat Storage Device

In view of the low heat transfer efficiency of the phase-change heat storage device and the inability to quickly respond to temperature changes, the fluid tube structure in the heat storage device was optimized, and four types of structures: circular structure, square structure, regular hexagon structure and regular triangle structure were compared and studied. Numerical simulation and optimization analysis are carried out on the pipes with the same thermal fluid circulation area but different sizes. The simulation results show that the triangular structure has the highest heat transfer efficiency in the initial stage of heat transfer due to the longest perimeter, and the heat transfer efficiency is positively correlated with the perimeter under the heat transfer mode in which heat conduction is the main body. Under the heat transfer mode dominated by convective heat transfer in the later stage of melting, the time required for the complete melting of phase change materials outside the square tube is the shortest.

Analysis of effect of variation of Honeycomb core cell size and sandwich panel width on the stiffness of a sandwich structure

A sandwich structure consists of three main parts i.e. the facing skins, the core and the adhesive. It acts in a way similar to that of the I- Beam. In this research, a sandwich structure has been designed with a regular hexagon honey-comb core made up of Kevlar® and face sheet of carbon fiber. The design has been modelled and the model has also been validated with the experimental and analytical method. Six different configurations of sandwich structures have been proposed. Out of these six, three configurations have the varying cell size i.e. 3.2 mm, 4 mm and 4.8 mm and the other three configurations have the varying panel width i.e. 40 mm, 45 mm and 50 mm keeping rest of the design parameters unchanged. Using ANSYS, analysis has been performed for all these six configurations and equivalent stiffness has been calculated. It has been observed that the honeycomb core cell size does not have a significant effect on the stiffness properties of a composite sandwich panel. The analysis also reveals that with the increased panel width the stiffness of composite panel increases significantly.

Hybrid auxetic materials implemented in crates & non-pneumatic wheels for shock absorption

In this study, new hybrid auxetic structures have been designed, and modifications of the pre-existing auxetic & hybrid auxetic structures have been made to analyze the shock absorption capability of the designed structures. The basic hybrid auxetic structure consists of a bowtie structure (exhibits auxetic property) which is partnered with a regular hexagon (for stability). The modifications are made to the hexagon along its vertices. On a few hybrid auxetic models, the bowtie is also significantly redesigned. These modifications result in the enhancement of the Negative Poisson's ratio as compared to the basic hybrid auxetic structure. In addition to that, they are also structurally stable than their predecessor. Depending upon their property & performance, it is determined whether to use them as support structures in the packaging (crates) or as the internal frame of non-pneumatic wheels.

Experimental observation of O-solitary waves in shallow water

Solitary waves have been well studied in various fields, including water waves, optical communications, plasmas, and biophysics. We present experimental results that a two-dimensional solitary wave, with its crest shaped like the letter “O,” occurs around an island terrain in shallow water. We name it the O-soliton (a cylindrical-type soliton) in this paper. This is the first experimental observation of the O-soliton including both outward and inward propagation in shallow water. O-solitons are observed with stable and periodic occurrences under typical coastal conditions. A water column at its center is higher than eight times the initial mean wave height measured by six wave gauges arranged in a regular hexagon. A one-dimension solitary wave incurs a phase shift when it collides with the O-soliton. However, no phase shift is observed in the O-soliton itself. Furthermore, when it collapses to its center point, it remains in this state for a short period. This is analogous to the pausing of the wave crest during a head-on collision of solitary waves or the reflection of a solitary wave at a vertical wall. These results also enrich the general understanding of solitary waves, whose repeated occurrence suggests that they may naturally occur in the ocean, and motivate analogous studies in other fields of physics.

Zero-curvature point of minimal graphs

Motivated by a classical result of Finn and Osserman (J Anal Math 343(12):351–364, 1964), who proved that the Jenkins–Serrin graph over the square inscribed in the unit disk is extremal for the Gaussian curvature of the point O (so-called centre) of the minimal graphs above the center 0 of unit unit disk, provided the tangent plane is horizontal, we ask and answer to the question concerned the extremal of "second derivative" of the Gaussian curvature of such graphs provided that its curvature at O is zero. We prove that the extremals are certain Jenkins–Serrin graph over the regular hexagon inscribed in the unit disk, provided that the Gaussian curvature vanishes and the tangent plane is horizontal at the centre.