Polygonal Packing Research Articles

Densest plane group packings of regular polygons.

Packings of regular convex polygons (n-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly regarding densest lattice or double-lattice configurations. Here we consider all two-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the two-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometries. We formulate the plane group packing problem as a nonlinear constrained optimization problem. By means of the Entropic Trust Region Packing Algorithm that approximately solves this problem, we examine some known and unknown densest packings of various n-gons in all 17 plane groups and state conjectures about common symmetries of the densest plane group packings for every n-gon.

Atlas refinement with bounded packing efficiency

We present a novel algorithm to refine an input atlas with bounded packing efficiency. Central to this method is the use of the axis-aligned structure that converts the general polygon packing problem to a rectangle packing problem, which is easier to achieve high packing efficiency. Given a parameterized mesh with no flipped triangles, we propose a new angle-driven deformation strategy to transform it into a set of axis-aligned charts, which can be decomposed into rectangles by the motorcycle graph algorithm. Since motorcycle graphs are not unique, we select the one balancing the trade-off between the packing efficiency and chart boundary length, while maintaining bounded packing efficiency. The axis-aligned chart often contains greater distortion than the input, so we try to reduce the distortion while bounding the packing efficiency and retaining bijection. We demonstrate the efficacy of our method on a data set containing over five thousand complex models. For all models, our method is able to produce packed atlases with bounded packing efficiency; for example, when the packing efficiency bound is set to 80%, we elongate the boundary length by an average of 78.7% and increase the distortion by an average of 0.0533%. Compared to state-of-the-art methods, our method is much faster and achieves greater packing efficiency.

Jamming of Deformable Polygons.

We introduce the deformable particle (DP) model for cells, foams, emulsions, and other soft particulate materials, which adds to the benefits and eliminates deficiencies of existing models. The DP model combines the ability to model individual soft particles with the shape-energy function of the vertex model, and adds arbitrary particle deformations. We focus on 2D deformable polygons with a shape-energy function that is minimized for area a_{0} and perimeter p_{0} and repulsive interparticle forces. We study the onset of jamming versus particle asphericity, A=p_{0}^{2}/4πa_{0}, and find that the packing fraction grows with A until reaching A^{*}=1.16 of the underlying Voronoi cells at confluence. We find that DP packings above and below A^{*} are solidlike, which helps explain the solid-to-fluid transition at A^{*} in the vertex model as a transition from tension- to compression-dominated regimes.

Precise algorithm to generate random sequential adsorption of hard polygons at saturation.

Random sequential adsorption (RSA) is a time-dependent packing process, in which particles of certain shapes are randomly and sequentially placed into an empty space without overlap. In the infinite-time limit, the density approaches a "saturation" limit. Although this limit has attracted particular research interest, the majority of past studies could only probe this limit by extrapolation. We have previously found an algorithm to reach this limit using finite computational time for spherical particles and could thus determine the saturation density of spheres with high accuracy. In this paper, we generalize this algorithm to generate saturated RSA packings of two-dimensional polygons. We also calculate the saturation density for regular polygons of three to ten sides and obtain results that are consistent with previous, extrapolation-based studies.

Crowdsourced Quantification and Visualization of Urban Mobility Space Inequality

Most cities are car-centric, allocating a privileged amount of urban space to cars at the expense of sustainable mobility like cycling. Simultaneously, privately owned vehicles are vastly underused, wasting valuable opportunities for accommodating more people in a livable urban environment by occupying spacious parking areas. Since a data-driven quantification and visualization of such urban mobility space inequality is lacking, here we explore how crowdsourced data can help to advance its understanding. In particular, we describe how the open-source online platform What the Street!? uses massive user-generated data from OpenStreetMap for the interactive exploration of city-wide mobility spaces. Using polygon packing and graph algorithms, the platform rearranges all parking and mobility spaces of cars, rails, and bicycles of a city to be directly comparable, making mobility space inequality accessible to a broad public. This crowdsourced method confirms a prevalent imbalance between modal share and space allocation in 23 cities worldwide, typically discriminating bicycles. Analyzing the guesses of the platform’s visitors about mobility space distributions, we find that this discrimination is consistently underestimated in the public opinion. Finally, we discuss a visualized scenario in which extensive parking areas are regained through fleets of shared, autonomous vehicles. We outline how such accessible visualization platforms can facilitate urban planners and policy makers to reclaim road and parking space for pushing forward sustainable transport solutions.

A flexible labour division approach to the polygon packing problem based on space allocation

This paper deals with the two-dimensional satellite module polygon packing problem. Based on the duality of material and space, it regards the polygon packing problem as a space allocation problem, which involves allocating the container space to the given polygons reasonably and efficiently. Ant colony’s labour division is essentially a kind of task allocation. Using this task allocation to achieve the space allocation in polygon packing problems, a flexible labour division approach (FLD) is proposed based on the response threshold model. According to the characteristics of space allocation in polygon packing problems, FLD designs three actions for polygons to occupy the container space. With the interaction between environmental stimulus and response threshold, each polygon takes an appropriate action to complete the space allocation and a layout that meets the requirements of satellite module layout is obtained. The results of standard test instances demonstrate the effectiveness of FLD when compared with self-organisation emergence algorithm. Moreover, experiments on the general polygon packing problem also show that FLD is competitive with other existing algorithms.

Structural characterization of the packings of granular regular polygons.

By using a recently developed method for discrete modeling of nonspherical particles, we simulate the random packings of granular regular polygons with three to 11 edges under gravity. The effects of shape and friction on the packing structures are investigated by various structural parameters, including packing fraction, the radial distribution function, coordination number, Voronoi tessellation, and bond-orientational order. We find that packing fraction is generally higher for geometrically nonfrustrated regular polygons, and can be increased by the increase of edge number and decrease of friction. The changes of packing fraction are linked with those of the microstructures, such as the variations of the translational and orientational orders and local configurations. In particular, the free areas of Voronoi tessellations (which are related to local packing fractions) can be described by log-normal distributions for all polygons. The quantitative analyses establish a clearer picture for the packings of regular polygons.

Internal friction and absence of dilatancy of packings of frictionless polygons.

By means of numerical simulations, we show that assemblies of frictionless rigid pentagons in slow shear flow possess an internal friction coefficient (equal to 0.183±0.008 with our choice of moderately polydisperse grains) but no macroscopic dilatancy. In other words, despite side-side contacts tending to hinder relative particle rotations, the solid fraction under quasistatic shear coincides with that of isotropic random close packings of pentagonal particles. Properties of polygonal grains are thus similar to those of disks in that respect. We argue that continuous reshuffling of the force-bearing network leads to frequent collapsing events at the microscale, thereby causing the macroscopic dilatancy to vanish. Despite such rearrangements, the shear flow favors an anisotropic structure that is at the origin of the ability of the system to sustain shear stress.

Random packing of regular polygons and star polygons on a flat two-dimensional surface.

Random packing of unoriented regular polygons and star polygons on a two-dimensional flat continuous surface is studied numerically using random sequential adsorption algorithm. Obtained results are analyzed to determine the saturated random packing ratio as well as its density autocorrelation function. Additionally, the kinetics of packing growth and available surface function are measured. In general, stars give lower packing ratios than polygons, but when the number of vertexes is large enough, both shapes approach disks and, therefore, properties of their packing reproduce already known results for disks.

Simulation of cellular packing in non-proliferative epithelia

The physical laws governing the morphogenesis of biological tissues remain largely misunderstood. In particular, the role of the mechanical interactions occurring in this process needs to be better understood and studied. Inner follicular cells surrounding the oocytes of Ciona intestinalis form an epithelial monolayer resulting from an accretion process (without mitosis or apoptosis). This epithelium is elementary and useful for morphogenesis studies: the cells exhibit polygon packing with a specific but non-systematically repeatable topology (i.e. the distribution of pentagons, hexagons and heptagons changes). To understand the role of mechanical forces in tissue formation, we propose an innovative “2D spherical” model based on the physics of divided media. This approach simulates the cellular mechanical behavior and epithelium structuration by allowing cells to adopt a large variety of shapes and to self-organize in response to mechanical interactions. The numerical parameters considered in the model are derived from experimental data in order to perform pertinent and realistic simulations. The results obtained are compared to biological observations using the same counting method to characterize epithelium topology. Numerical and experimental data appear close enough to validate the model. It is then used for exploratory studies dealing with “Tissue Development Speed” variation, which is not easily attainable by experimentation. We show that the formation speed of the tissue influences its topology and hence its packing organization.

Postscheduling buffer management trade-offs in streaming software synthesis

Streaming applications, which are abundant in many disciplines such as multimedia, networking, and signal processing, require efficient processing of a seemingly infinite sequence of input data. In the context of streaming software synthesis from data flow graphs, we study the inherent trade-off between memory requirement and compilation runtime, under a given task firing schedule. We utilize postscheduling analysis granularity to control the amount of details in characterization of buffer's spatio-temporal footprints. Subsequently, we transform the buffer allocation problem to two-dimensional packing of polygons, where complexity of the packing problem (e.g., polygon shapes) is determined by the analysis granularity. We develop an evolutionary packing optimization algorithm which readily yields buffer allocations. Experimental results highlight the trade-off between complexity of the analysis and the total buffer size of generated implementations. In addition, they show dramatic improvements in total buffer size, if one is willing to pay the additional cost in optimization runtime.

Spider Silk Violin Strings with a Unique Packing Structure Generate a Soft and Profound Timbre

We overcome the difficulties in pulling long draglines from spiders, twist bundles of dragline filaments, and succeed in preparing violin strings. The twisting is found to change the cross section shapes of filaments from circular to polygonal and to optimize the packing structure with no openings among filaments providing mechanically strong and elastic strings. The spider string signal peaks of overtones for the violin are relatively large at high frequencies, generating a soft and profound timbre. Such a preferable timbre is considered to be due to the unique polygonal packing structure which provides valuable knowledge for developing new types of materials.

A global optimization point of view to handle non-standard object packing problems

This work originates from research carried out in support to the cargo accommodation of space vehicles/modules. The goal of this activity is to maximize the loaded cargo, taking into account the given accommodation requirements. Items can often be modelled as parallelepipeds, but it is even more frequent that real-world issues make this approximation no longer acceptable. These aspects and the presence of additional overall conditions, such as balancing, give rise to very challenging non-standard packing problems, not only in the frame of space engineering, but also in different application areas. This article considers first the orthogonal packing of tetris-like items, within a convex domain and subsequently the packing of polygons with (continuous) rotations in a convex domain. The proposed approach is based on mixed integer linear/non-linear programming (MIP, MINLP), from a global optimization point of view. The tetris-like formulation is exploited to provide the MINLP solution process with an approximated starting solution.

Steady state of tapped granular polygons

The steady state packing fraction of a tapped granular bed is studied for different grainshapes via a discrete element method. Grains are monosized regular polygons, fromtriangles to icosagons. Comparisons with disc packings show that the steady state packingfraction as a function of the tapping intensity presents the same general trends inpolygon packings. However, better packing fractions are obtained, as expected, forshapes that can tessellate the plane (triangles, squares and hexagons). In addition,we find a sharp transition for packings of polygons with more than 13 verticessignaled by a discontinuity in the packing fraction at a particular tapping intensity.Density fluctuations for most shapes are consistent with recent experimentalfindings in disc packing; however, a peculiar behavior is found for triangles andsquares.

A heuristic method for solving triangle packing problem

Given a set of triangles and a rectangle container, the triangle packing problem is to determine if these triangles can be placed into the container without overlapping. Triangle packing problem is a special case of polygon packing problem and also NP-hard, so it is unlikely that an efficient and exact algorithm can be developed to solve this problem. In this paper, a new concept of rigid placement is proposed, based on which a discrete solution space called rigid solution space is constructed. Each solution in the rigid solution space can be built by continuously applying legal rigid placements one by one until all the triangles are placed into the rectangle container without overlapping. The proposed Least-Destruction-First (LDF) strategy determines which rigid placement has the privilege to go into the rectangle container. Based on this, a heuristic algorithm is proposed to solve the problem. Combining Least-Destruction-First strategy with backtracking, the corresponding backtracking algorithm is proposed. Computational results show that our proposed algorithms are efficient and robust. With slight modification, these techniques can be conveniently used for solving polygon packing problem.

Role of anisotropy in the elastoplastic response of a polygonal packing

We study the effect of the anisotropy induced by loading on the elastoplastic response of a two dimensional discrete element model granular material. The anisotropy of the contact network leads to a breakdown of the linear isotropic elasticity. We report on a linear dependence of the Young moduli and Poisson ratios on the fabric coefficients, measuring the anisotropy of the contact network. The resulting nonassociated plastic flow rule and the linear relationship between dilatancy and stress ratio are discussed in terms of several existing models. We propose a paradigm for understanding soil plasticity, based on the correlation between the plastic flow rule and the induced anisotropy on the subnetwork of sliding contacts.

An algorithm for polygon placement using a bottom-left strategy

This paper describes a fast and efficient implementation of a bottom-left (BL) placement algorithm for polygon packing. The algorithm allows pieces to be nested within the partial layout produced by previously placed pieces, and produces an optimal BL layout in the sense that the positions considered are guaranteed to contain the bottom-left position of the infinite set of possibilities. Full details of the way in which these positions are calculated are given. Computational experiments comparing the results of different orderings on a variety of datasets from the literature are reported, and these illustrate that problems having in excess of 100 pieces of several piece types can be solved within one minute on a modern desktop PC. The procedure can easily be incorporated into algorithms that apply more sophisticated piece selection procedures.

Calculation of the incremental stress-strain relation of a polygonal packing.

The constitutive relation of the quasistatic deformation on two-dimensional packed samples of polygons is calculated using molecular dynamics simulations. The stress values at which the system remains stable are bounded by a failure surface, which shows a power law dependence on the pressure. Below the failure surface, nonlinear elasticity and plastic deformation are obtained, which are evaluated in the framework of the incremental linear theory. The results show that the stiffness tensor can be directly related to the microcontact rearrangements. The plasticity obeys a nonassociated flow rule with a plastic limit surface that does not agree with the failure surface.

An Efficient Algorithm for the Regular W 1 Packing of Polygons in the Infinite Plane

An Efficient Algorithm for the Regular W 1 Packing of Polygons in the Infinite Plane

An efficient algorithm for the regular W1 packing of polygons in the infinite plane

This paper describes a new algorithm, PLANEPACK, which determines an optimal or near optimal solution for the W1 packing of identical shapes in the infinite plane. Restricted to polygons for computational convenience, it is based on the no-fit polygon/configuration space obstacle approach. The algorithm was tested on a modest set of fourteen polygons (thirteen non-interlocking and one interlocking) and yielded a feasible solution for each. The solutions were optimal for four of the non-interlocking polygons and near optimal for the other nine. As expected though, the solution for the one interlocking polygon was sub-optimal and enhancements to the algorithm would be required for such cases.