Hexagonal Tessellation Research Articles

2D hexagonal tessellations sustained by Br···Br/H contacts: From regular to semiregular to k-uniform tilings

2D hexagonal tessellations sustained by Br···Br/H contacts: From regular to semiregular to k-uniform tilings

Tessellation-Based Construction of Air Route for Wireless Sensor Networks Employing UAV.

In this paper, we consider a wireless sensor network consisting of an unmanned aerial vehicle (UAV) acting as a sink node and a number of sensor nodes scattered uncertainly on the ground. In the network, the UAV flies to a spatial point called point of interest and hovers to collect environmental data from neighboring sensor nodes. Then, the UAV proceeds to the next point of interest. The UAV must gather data from all the sensor nodes. On the other hand, a shorter round-trip air route of the UAV is more preferred since a battery-operated UAV needs regular recharging. To satisfy the requirement and to adhere to the recommendation as well, especially in the situation where only vague locational information about sensor nodes is available, we propose a scheme that follows three steps. First, it covers the sensor field of the wireless sensor network with three categories of hexagonal tessellations. Secondly, it establishes a point of interest at the centroid of each tile. Thirdly, it constructs an air route of the UAV, which visits every point of interest along a Hamiltonian cycle on the induced graph. Next, we develop a closed-form expression for the exact flight distance attained by the proposed scheme. For comparative evaluation, we discover some optimal schemes that minimize the flight distance by completely inspecting all patterns and corroborating the property of Hamiltonicity. The flight distance along the air route constructed by the proposed scheme is found to be only slightly longer than the flight distance yielded by an optimal scheme. Furthermore, the proposed scheme is proven to be practically valid when a common multicopter is employed as the sink node.

Spatial Diversity Changes in Land Use and Land Cover Mix in Central European Capitals and Their Commuting Zones from 2006 to 2018

This article investigates the land use and land cover (LULC) mosaic as a function of distance to the city centre. The research area is four Functional Urban Areas (FUAs) of the capitals of Central European countries: Czechia (Prague), Poland (Warsaw), Slovakia (Bratislava), and Hungary (Budapest). The article presents LULC mix changes in these FUAs in the context of transformations in urban cores and commuting zones of capital city metropolises, which have large populations and prominent positions in the country or region. The study makes use of Urban Atlas LULC data for 2006, 2012, and 2018. LULC change was analysed using a hexagonal tessellation with the hexagon as the basic spatial unit. Spatial entropy (ENT) and Simpson’s Diversity Index (SIDI) were employed. The change in ENT and SIDI were determined as a function of distance to urban cores, as well as changes on the first level of Urban Atlas nomenclature for 2006–2012 and 2012–2018, as well as on the third level of nomenclature for 2012–2018. The research shows that changes from 2006 to 2012 were more considerable than from 2012 to 2018. It also revealed that, if LULC classes are considered on the first level of nomenclature, diversity in urban cores grows and then declines in commuting zones. An analysis of diversity on the third level of nomenclature demonstrated its decline with the growing distance to the urban core. It has also been demonstrated that the mean values of ENT and SIDI are approximately twice as high in urban cores as in commuting zones, indicating a plateau of the mean value of ENT and SIDI over the study period in urban cores compared to an increase in commuting zones around them. The conducted research will be helpful to urban planners and decision-makers in directing the further, inevitable development of metropolitan areas in accordance with sustainable environmental management.

Programming quadric metasurfaces via infinitesimal origami maps of monohedral hexagonal tessellations: Part II

In Part I of this study, it was shown that all the three known types of monohedral hexagonal tessellations of the plane, those composed of equal irregular hexagons, have just a single deformation mode when tiles are considered as rigid bodies hinged to each other along the edges. A gallery of tessellated plates was simulated numerically to demonstrate the range of achievable deformed shapes. In Part II, the displacement field was first derived and a continuous interpolant for each type of tessellated plate. It turns out that all corresponding metasurfaces are described by quadrics. Afterwards, a parametric analysis was carried out to determine the effect of varying angles and edge lengths on the curvature, and the values of the geometric Poisson ratio of the plates. Finally, a method of fabrication is proposed based on the additive manufacturing of stiff tiles of negligible deformability and flexible connectors. Using this modular technique, it is possible to join together different monohedral tessellated plates able to deform into piece-wise quadrics. The nodal positions in the deformed configuration of the realized plates are measured after enforcing one principal curvature to assume a chosen value. The estimate of the other principal curvature confirms the analytical predictions. The presented tessellated plates permit to realize doubly curved shape-morphing metasurfaces with assorted shapes, which also can feature a certain surface roughness, and they can be employed in all applications demanding high surface accuracy and few actuators or just one.

Programming quadric metasurfaces via infinitesimal origami maps of monohedral hexagonal tessellations: Part I

The control of the shape of complex metasurfaces is a challenging task often addressed in the literature. This work presents a class of tessellated plates able to deform into surfaces of preprogrammed shape upon activation by any flexural load and that can be controlled by a single actuator. Quadric metasurfaces are obtained from infinitesimal origami maps of monohedral hexagonal tessellations of the plane, that is pavings in which all tiles are congruent to each other. Monohedral tessellated portions can be joined together to obtain more complex shapes, which can be locally synclastic or anticlastic and can have a certain roughness. We broaden previous work by providing a complete characterization of all the three known types of monohedral tessellations composed by irregular hexagons. The proposed two-dimensional structures may have applications in prosthetics, tissue engineering, wearable devices, energy harvesting devices, tunable focus mirrors and adaptive facades. The study is divided in two parts. In Part I, after introducing the discrete kinematics of tessellated plates, it is proved analytically that essentially each type of monohedral hexagonal tessellation possesses only one deformation mode. Afterwards, several numerical examples are provided to demonstrate the variety of achievable surface shapes. In Part II, first the metasurfaces corresponding to assigned tile geometries are given a continuum description, which establishes that the continuous interpolant is always a quadric. Then, experimental results on fabrication, assembly and surface accuracy are reported.

Strategic Deconfliction of Unmanned Aircraft Based on Hexagonal Tessellation and Integer Programming

Unmanned aircraft systems service suppliers adhere to interoperability standards that require unmanned aircraft operators to submit an operational intent, which describes the planned flight path in four-dimensional space. To ensure fairness, the central database follows a first-come, first-served approach, accepting new operational intents as long as they do not conflict with any active ones. However, creating a viable operational intent is challenging due to moving obstacles. This paper introduces an innovative optimization-based procedure to automate the intent filing process. It utilizes a stacked hexagonal tessellation to model the airspace, offering adjustable granularity. Path finding is accomplished using integer programming on the hex grid. The integer program is solved on a grid canvas that includes only necessary cells, striking a balance between computational efficiency and optimality. Simulation experiments demonstrate the procedure’s effectiveness in generating feasible trajectories, even in scenarios with dense, omnidirectional air traffic. This procedure has the potential to become the foundational software core for low-altitude air traffic management systems, providing strategic deconfliction and constraint management services.

Impact of grid size on spatiotemporal prediction of fine particulate matter

Incorporating high spatiotemporal resolution in prediction models can significantly improve the accuracy of results. The current study focuses on identifying the most suitable spatial resolution using hexagonal tessellations of different sizes over the study area. Hexagonal grids are selected after comparing different types of grids. Spatial interpolation methods, including inverse distance weighting and ordinary Kriging method, are used for creating augmented datasets for each grid. A hybrid deep-learning-based prediction model is developed combining Convolutional Neural Network (CNN) and Long Short-term Memory (LSTM) for multivariate time series regression to predict PM2.5. Three scenarios are considered based on the size of the hexagonal grids. The results from each model are compared using evaluation indices of Mean Absolute Error (MAE) and Root Mean Square Error (RMSE). The R2 values range from 0.8 to 0.95, which shows that the hybrid model gives accurate predictions. Analysis of models for different-sized grids shows that for 1, 3, and 5 km grid sizes, 17%, 24%, and 6% of the cells have RMSE values less than 20, respectively. Similarly, 39%, 50%, and 46% of the cells have MAE values less than 16 for 1, 3, and 5 km grid sizes, respectively. The comparison showed that the 3 km grid benefits data aggregation per prediction accuracy and computational time. The predictive capability of models is not dependent on the spatial relationship between data from different static monitors. In the future, mobile monitoring data can be included to increase the data points and models’ accuracy.

Visualizing 2018 lombok earthquake in Indonesia using crowdsouring data

Along with the development of science and technology, using big data, map makers can take advantage of crowdsourcing social media data on Twitter to obtain user location when uploading tweets, which can be called geolocated tweets. Earthquakes that occur very often in Indonesia often grab people's attention, especially netizens who use social media like Twitter. One of the major earthquakes that occurred in Indonesia in 2018 was the Lombok earthquake, which occurred twice in a row from July to August 2018. Using Twitter data, information and social responses related to the 2018 Lombok earthquake can be obtained, which can be used as evaluation material for public handling and responding. The information is then visualized in various forms, and one of the best visualization methods is selected.This study uses Twint package in Python as a way of obtaining location data from Twitter. The method used to collect Twitter data is a case study on the social impact of the Lombok earthquake in Indonesia in 2018. The data observation method used is a simulation of several types of map visualization and survey methods in selecting the best type of visualization. The method of analysis used is by mapping the data on the number of tweets as the main object using various types of maps, as well as calculating survey results by scoring each group of questions.The results of spatial data extraction from Twitter in this study obtained 2032 tweets that had been selected and cleaned from 11,584 tweets. Map visualization with the theme of the social impact of the Lombok earthquake in 2018 was compiled using five types of visualization, namely choropleth maps, proportional symbol maps, dot maps, hexagonal tessellation maps, and heat maps. Based on the results of the survey on selecting the best visualization, it was found that the choropleth map is the best visualization method according to respondents with a cartography background and respondents who are unfamiliar with cartography because the information displayed is easier to read and understand.

Precise and unbiased biomass estimation from GEDI data and the US Forest Inventory

Atmospheric CO2 concentrations are dependent on land-atmosphere carbon fluxes resultant from forest dynamics and land-use changes. These fluxes are not well-constrained, in part because reliable baseline estimates of forest carbon stocks and the associated uncertainties are lacking. NASA's Global Ecosystem Dynamics Investigation (GEDI) produces estimates of aboveground biomass density (AGBD) that are unique because GEDI's hybrid estimation framework enables formal uncertainty calculations that accompany the biomass estimates. However, GEDI's estimates are not without issue; a recent validation using design-based AGBD estimates from the US Forest Inventory and Analysis (FIA) program revealed systematic differences between GEDI and FIA estimates within a hexagon tessellation of the continental United States. Here, we explored these differences and identified two issues impacting GEDI's estimation process: incomplete filtering of low quality GEDI observations and regional biases in GEDI's footprint-level biomass models. We developed a solution to each, in the form of improved data filtering and GEDI-FIA fusion AGBD models, developed in a scale-invariant small area estimation framework, that were compatible with hybrid estimation. We calibrated 10 regional Fay-Herriot models at the hexagon scale for application at the unit scale of GEDI footprints, for which we provide a mathematical justification and empirical testing of the models' scale-invariance. These models predicted realistic distributions of unit level AGBD, with equal or improved performance relative to GEDI's L4A models for all regions. We then produced GEDI-FIA fusion estimates that were more precise than the FIA estimates and resulted in a bias reduction of 86.7% relative to the original GEDI estimates: 19.3% due to improved data filtering and 67.5% due to the new AGBD models. Our findings indicate that (1) small area estimation models trained in a scale-invariant framework can produce realistic predictions of AGBD, and (2) there is substantial spatial variability in the relationship between GEDI forest structure metrics and AGBD. This work is a step toward achieving reliable baseline forest carbon stocks, provides a viable methodology for training remote sensing biomass models, and may serve as a reference for other investigations of GEDI AGBD estimates.

A Proposal of a simulation-optimization methodology for allocation of agencies with human resources on hexagonal tessellation

The human resource assignment usually considers the search, qualification, selection, and allocation of human resources commonly based on their experience and professional development into different areas of a company. Rarely, the residence location and other characteristics of the personnel are considered.This paper takes a real case allocation problem from a public university social service program in Mexico City where the assignment process is restarted every six months. The lack of consideration of the personnel physical and functional attributes, led to an inefficient allocation process, increasing desertion of the program. To solve this problem, we propose a new methodology built on hexagonal tessellation capable of allocating personnel based on distance, schedules, and skills, which is later improved using simulation, reducing total travel distance in 585 km, improving matching schedules in 26%, matching skills in 37% and reduced the time dedicated to the allocation process from 5 days to 90 minutes.

Defects and frustration in the packing of soft balls

This work introduces the Hookean-Voronoi energy, a minimal model for the packing of soft, deformable balls. This is motivated by recent studies of quasi-periodic equilibria arising from dense packings of diblock and star polymers. Restricting to the planar case, we investigate the equilibrium packings of identical, deformable objects whose shapes are determined by an N-site Voronoi tessellation of a periodic rectangle. We derive a reduced formulation of the system showing at equilibria each site must reside at the “max-center” of its associated Voronoi region and construct a family of ordered “single-string” minimizers whose cardinality is O(N2). We identify sharp conditions under which the system admits a regular hexagonal tessellation and establish that in all cases the average energy per site is bounded below by that of a regular hexagon of unit size. However, numerical investigation of gradient flow of random initial data, reveals that for modest values of N the system preponderantly equilibrates to quasi-ordered states with low energy and large basins of attraction. For larger N the distribution of equilibria energies appears to approach a δ-function limit, whose energy is significantly higher than the ground state hexagon. This limit is possibly shaped by two mechanisms: a proliferation of moderate-energy disordered equilibria that block access of the gradient flow to lower energy quasi-ordered states and a rigid threshold on the maximum energy of stable states.

An analysis of three-dimensional patterns of experimental detonation cells

The notions of regularity and characteristic width of detonation cells are revisited based on crossed analyses of experimental front-view and longitudinal recordings obtained with the soot-plate technique. Tubes with cross-sections of different shapes, namely round, triangular and square, but the same surface area of 16 cm2, are used to detonate the stable mixture 2H2 + O2 + 2Ar with the initial pressure p0 varying from 15 kPa to 100 kPa and the initial temperature T0=294 K. The longitudinal recordings show the well-known regular arrangements for this mixture for all cross-section shapes and p0, but the front-view recordings show irregular patterns except for the square shape and low-enough p0. There are fewer cells in the round tube, more in the square one, and their average widths and relative differences decrease with increasing p0. All front-view cell patterns become irregular and independent of the cross-section shapes with increasing p0. The cellular dynamics at the walls of a tube is thus not representative of that on the whole detonation front, and longitudinal soot traces alone are not sufficient for describing the cellular structure. An analysis based on graph theory proposes that a tessellation of regular hexagons can model a large set of irregular front-view cells. A cell count on an experimental front-view recording thus defines an average cell width and an intrinsic but high minimum error for this width. Its resulting large sensitivity to the parameters of a simple Arrhenius rate of chemical progress indicates that detailed schemes of chemical kinetics and more advanced conceptual tools than a single length are necessary for characterizing the three-dimensional structures of detonation cells. The Voronoi tessellation supplemented with a physical criterion for the surface density of randomly-distributed point sources could represent a step forward.

Generic parallel data structures and algorithms to GPU superpixel image segmentation

We propose parallel implementation on GPU (graphics processing unit) system for some generic algorithms applied to superpixel image segmentation problem. The aim is to provide standard algorithms based on generic decentralized data structures that could be easily improved and customized on many optimization problems on parallel platforms. Note that superpixel segmentation are clustering algorithms applied to image processing. Two types of algorithms are presented and implemented on GPU based on common parallel data structures. Firstly, we present a parallel implementation of the well-known k-means algorithm with application to 3D data. It is based on a cellular grid subdivision of space that allows closest point findings in constant optimal time for bounded distributions. Secondly, we present an application of the parallel Boruvka minimum spanning forest algorithm to compute watershed segmentation. Both techniques are fully executed on GPU and share the same data structures that embed disjoint-set-trees and distributed-link-lists. We evaluate our k-means approach with regards to state-of-the-art methods, that are, the well known SLIC algorithm, and the adaptive segmentation approach SPASM. We argue that our implementation has the shortest execution time among the tested methods, with a near real time performance and quasi linear acceleration factor, while it provides more regular shape superpixel segmentation based on hexagonal tessellation. Watershed minimum spanning forest method is presented and evaluated accordingly to the same experimental framework.

Ecological network for species dependent on ancient broadleaf trees using Osmoderma barnabita as a model species: a new approach

Abstract Organisms that depend on rare and endangered microhabitats, such as old, hollow trees, have a high conservation value. To conserve these species, it is necessary to apply methods that identify habitats suitable for their survival. Until now, a comprehensive method has not been developed for predicting the occurrence of species associated with old, hollow trees. Here, we developed criteria to identify habitats for organisms associated with old, hollow trees, using Osmoderma barnabita as a model species. We used hexagonal tessellation to identify suitable habitats for hermit beetle. Criteria for predicting the occurrence of a hermit beetle were applied in practise, and as result 47.56% of the known hermit beetle localities coincided with the hexagons where the probability of occurrence of the species was the highest. Based on these results, an ecological network was created for species dependency on old, hollow trees, using the hermit beetle as a model species. The proposed approach makes it possible to plan and implement environmental protection measures for species living in territories spanning several countries. In this work, we propose a method to identify species occurrence in core areas and to determine ecological corridors and gaps. This method could tailor conservation planning efforts.

Integrable bootstrap for AdS3/CFT2 correlation functions

We propose an integrable bootstrap framework for the computation of correlation functions for superstrings in AdS3 × S3 × T4 backgrounds supported by an arbitrary mixture or Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. The framework extends the “hexagon tessellation” approach which was originally proposed for AdS5 × S5 and for the first time it demonstrates its applicability to other (less supersymmetric) setups. We work out the hexagon form factor for two-particle states, including its dressing factors which follow from those of the spectral problem, and we show that it satisfies non-trivial consistency conditions. We propose a bootstrap principle, slightly different from that of AdS5 × S5, which allows to extend the form factor to arbitrarily many particles. Finally, we compare its predictions with some correlation functions of protected operators. Possible applications of this construction include the study of wrapping corrections, of higher-point correlation functions, and of non-planar corrections.

Self-organized optimal packing of kinesin-5-driven microtubule asters scales with cell size.

Radial microtubule (MT) arrays or asters determine cell geometry in animal cells. Multiple asters interacting with motors, such as those in syncytia, form intracellular patterns, but the mechanical principles behind this are not clear. Here, we report that oocytes of the marine ascidian Phallusia mammillata treated with the drug BI-D1870 spontaneously form cytoplasmic MT asters, or cytasters. These asters form steady state segregation patterns in a shell just under the membrane. Cytaster centers tessellate the oocyte cytoplasm, that is divide it into polygonal structures, dominated by hexagons, in a kinesin-5-dependent manner, while inter-aster MTs form 'mini-spindles'. A computational model of multiple asters interacting with kinesin-5 can reproduce both tessellation patterns and mini-spindles in a manner specific to the number of MTs per aster, MT lengths and kinesin-5 density. Simulations predict that the hexagonal tessellation patterns scale with increasing cell size, when the packing fraction of asters in cells is ∼1.6. This self-organized in vivo tessellation by cytasters is comparable to the 'circle packing problem', suggesting that there is an intrinsic mechanical pattern-forming module that is potentially relevant to understanding the role of collective mechanics of cytoskeletal elements in embryogenesis. This article has an associated First Person interview with the first author of the paper.

Associations of atmospheric teleconnections with wintertime extratropical cyclones over East Asia and Northwest Pacific

Extratropical cyclones (ETCs) over East Asia and Northwest Pacific are identified and tracked by applying an objective algorithm to the 850-hPa relative vorticity fields from the ERA-Interim reanalysis. A total of 2866 ETCs originating at the western side of the date line have been identified in the extended November–March winters from 1979 to 2018. The ETC tracks are counted and visualized using a hexagonal tessellation rather than the regular longitude–latitude grids. Two generalized linear models (GLMs), Poisson regression model and Gamma regression model, are firstly applied to investigate the associations of wintertime ETCs with three atmospheric teleconnection patterns. The West Pacific (WP) pattern and the Pacific/North American (PNA) pattern are more responsible for the meridional variability of ETC activities in the North Pacific, while the influence of the Polar/Eurasia pattern on ETC activities is negligible. Results of composite analysis are qualitatively consistent with that of regression analysis. Composite maps of differences of jet stream, thermal gradient and mid-tropospheric baroclinicity in the positive and negative phases of teleconnection patterns also support the close associations of ETC activities with WP and PNA teleconnection patterns.

High-degeneracy Potts coarsening

I examine the fate of a kinetic Potts ferromagnet with a high ground-state degeneracy that undergoes a deep quench to zero temperature. I consider single spin-flip dynamics on triangular lattices of linear dimension 8≤L≤128 and set the number of spin states q equal to the number of lattice sites L×L. The ground state is the most abundant final state, and is reached with probability ≈0.71. Three-hexagon states occur with probability ≈0.26, and hexagonal tessellations with more than three clusters form with probabilities of O(10^{-3}) or less. Spanning stripe states-where the domain walls run along one of the three lattice directions-appear with probability ≈0.03. "Blinker" configurations, which contain perpetually flippable spins, also emerge, but with a probability that is vanishingly small with the system size.

A Novel Area Coverage Technique for Maximizing the Wireless Sensor Network Lifetime

This paper presents a mathematical model to geometrically optimize the density of active sensor nodes in a wireless sensor network (WSN) using concentric hexagonal tessellations and the concept of coverage contribution area for randomly deployed nodes in the field of interest (FOI). Some of the WSN applications, such as environmental monitoring, security, surveillance and health care require the target area to be covered a number of times. This number is denoted by a variable $$k$$ and is known as the ‘degree of coverage.’ The problem of achieving required degree of coverage is formulated as $$k$$ -coverage problem. An algorithm has been proposed to generate maximum number of disjoint-independent subsets of sensor nodes as an optimized solution to the $$k$$ -coverage problem, along with maximizing the WSN lifetime. Superiority and efficacy of the technique have been verified by mathematical analysis as well as simulations carried out using MATLAB.

Targeting Fatal Traffic Collision Risk from Prior Non-Fatal Collisions in Toronto

Research questionHow accurately can all locations of 44 fatal collisions in 1 year be forecasted across 1403 micro-areas in Toronto, based upon locations of all 1482 non-fatal collisions in the preceding 4 years?DataAll 1482 non-fatal traffic collisions from 2008 through 2011 and all 44 fatal traffic collisions in 2012 in the City of Toronto, Ontario, were geocoded from public records to 1403 micro-areas called ‘hexagonal tessellations’.MethodsThe total number of non-fatal traffic collisions in Period 1 (2008 through 2011) was summed within each micro-area. The areas were then classified into seven categories of frequency of non-fatal collisions: 0, 1, 2, 3, 4, 5, and 6 or more. We then divided the number of micro-areas in each category in Period 1 into the total number of fatal traffic collisions in each category in Period 2 (2012). The sensitivity and specificity of forecasting fatal collision risk based on prior non-fatal collisions were then calculated for five different targeting strategies.FindingsThe micro-locations of 70.5% of fatal collisions in Period 2 had experienced at least 1 non-fatal collision in Period 1. In micro-areas that had zero non-fatal collisions during Period 1, only 1.7% had a fatal collision in Period 2. Across all areas, the probability of a fatal collision in the area during Period 2 increased with the number of non-fatal collisions in Period 1, with 6 or more non-fatal collisions in Period 1 yielding a risk of fatal collision in Period 2 that was 8.7 times higher than in areas with no non-fatal collisions. This pattern is evidence that targeting 25% of micro-areas effectively could cut total traffic fatalities in a given year by up to 50%.ConclusionHighly elevated risks of traffic fatalities can be forecasted based on prior non-fatal collisions, targeting a smaller portion of the city for more concentrated investment in saving lives.