Regular Convex Polygons 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.

A Novel Design Method of Deployable Semi-Regular Tessellated Surfaces with Plano-Spherical Units

The design of movable systems gives an opportunity to create transformable designs which respond to the environmental, functional, cultural, and aesthetical needs of today's architecture. This paper proposes a method for designing a family of deployable structures which can be applied to semi-regular tessellated planar surfaces such as roofs, walls, and shading devices. The generated modular approach and adaptability provides a wide usage area and various combinations for these designs. The regular convex polygon modules are designed as a network of the triangular units. The triangular unit is designed using Bennett's overconstrained plano-spherical linkage topology. The polygonal modules are assembled to each other in one-uniform semi-regular tessellations. The assembly of adjacent regular convex polygons in each tessellation is examined to find a proper solution for no collision during deployment and to properly fit a surface without any gaps or overlaps in the deployed position. The assembly method for creating 1-DoF deployable surfaces and mobility calculations for a unit, the polygonal modules, and the assemblies are computed, and motion studies are demonstrated with CAD models and exemplified for a square module for motion tests in a prototype.

Comparison of Quantitative Morphology of Layered and Arbitrary Patterns: Contrary to Visual Perception, Binary Arbitrary Patterns Are Layered from a Structural Point of View

Patterns found among both living systems, such as fish scales, bones, and tree rings, and non-living systems, such as terrestrial and extraterrestrial dunes, microstructures of alloys, and geological seismic profiles, are comprised of anisotropic layers of different thicknesses and lengths. These layered patterns form a record of internal and external factors that regulate pattern formation in their various systems, making it potentially possible to recognize events in the formation history of these systems. In our previous work, we developed an empirical model (EM) of anisotropic layered patterns using an N-partite graph, denoted as G(N), and a Boolean function to formalize the layer structure. The concept of isotropic and anisotropic layers was presented and described in terms of the G(N) and Boolean function. The central element of the present work is the justification that arbitrary binary patterns are made up of such layers. It has been shown that within the frame of the proposed model, it is the isotropic and anisotropic layers themselves that are the building blocks of binary layered and arbitrary patterns; pixels play no role. This is why the EM can be used to describe the morphological characteristics of such patterns. We present the parameters disorder of layer structure, disorder of layer size, and pattern complexity to describe the degree of deviation of the structure and size of an arbitrary anisotropic pattern being studied from the structure and size of a layered isotropic analog. Experiments with arbitrary patterns, such as regular geometric figures, convex and concave polygons, contour maps, the shape of island coastlines, river meanders, historic texts, and artistic drawings are presented to illustrate the spectrum of problems that it may be possible to solve by applying the EM. The differences and similarities between the proposed and existing morphological characteristics of patterns has been discussed, as well as the pros and cons of the suggested method.

Eigenfunctions and the integrated density of states on Archimedean tilings

We study existence and absence of \ell^2 -eigenfunctions of the combinatorial Laplacian on the 11 Archimedean tilings of the Euclidean plane by regular convex polygons. We show that exactly two of these tilings (namely the (3.6)^2 “kagome” tiling and the (3.12^2) tiling) have \ell^2 -eigenfunctions. These eigenfunctions are infinitely degenerate and are constituted of explicitly described eigenfunctions which are supported on a finite number of vertices of the underlying graph (namely the hexagons and 12-gons in the tilings, respectively). Furthermore, we provide an explicit expression for the Integrated Density of States (IDS) of the Laplacian on Archimedean tilings in terms of eigenvalues of Floquet matrices and deduce integral formulas for the IDS of the Laplacian on the (4^4) , (3^6) , (6^3) , (3.6)^2 , and (3.12^2) tilings. Our method of proof can be applied to other \mathbb Z^d -periodic graphs as well.

Hereditary polyhedra with planar regular faces

A skeletal polyhedron in Euclidean 3-space is called hereditary if the symmetries of each face extend to symmetries of the entire polyhedron. In this paper we describe the finite hereditary skeletal polyhedra which have regular convex polygons or regular star-polygons as faces.

An Adaptive Complex Collocation Method for Solving Linear Elliptic PDEs in Regular Convex Polygons Based on the Unified Transform

An Adaptive Complex Collocation Method for Solving Linear Elliptic PDEs in Regular Convex Polygons Based on the Unified Transform

Influence of particle shape in the additive manufacturing process for ceramics

Additive manufacturing (AM) of ceramics through a vat photopolymerization (VP) process is a promising technique due to the high intrinsic resolution and the evanescence of stresses introduced during the layerwise additive manufacturing process. Compared to the regular vat photopolymerization process, the addition of ceramic powder increases physical complexity. Hence, large scale adoption and optimization of AM for ceramics requires a thorough understanding of the underlying physics. This work focuses on the light scattering introduced through a complex interplay between optical properties of different constituents and its effect on the photopolymerization. The effect of the shape of the ceramic particle is investigated for isolated and multiple particle cases using a parametric analysis with regular convex polygons. Although the analysis of two-dimensional isolated particle shapes indicates a preference for one shape over the other, this difference vanishes in a multiple-scattering situation. A three-dimensional analysis furthermore shows that a two-dimensional approximation of the three-dimensional scattering problem is relevant and can be used to further the process understanding of ceramic VP.

Packing measure of super separated iterated function systems

Let J be the limit set of an iterated function system in $$\mathbb {R}^d$$ satisfying the open set condition. It is well known that the h-dimensional packing measure of J is positive and finite when h is given by Hutchinson’s formula. However, it may be hard to find a formula for the h-dimensional packing measure of J. We introduce the super separation condition and use it to reduce the problem of computing the packing measure to checking densities of a finite number of balls around each point in the limit set. We then use this fact to find formulas for the packing measure of a class of Cantor sets in $$\mathbb {R}$$ , a class of fractals based on regular convex polygons in $$\mathbb {R}^2$$ , and a class of fractals based on regular simplexes in $$\mathbb {R}^d$$ for $$d \ge 3$$ .

The Electronic Structure of Graphene Nanoislands: A CAS-SCF and NEVPT2 Study

This paper presents a tight binding and ab initio study of finite graphene nanostructures. The attention is focused on three types of regular convex polygons: triangles, rhombuses, and hexagons, which are the most simple high-symmetry convex structures that can be ideally cut out of a graphene layer. Three different behaviors are evidenced for these three classes of compounds: closed-shells for hexagons; low-spin open-shells for rhombuses; high-spin open-shells for triangles.

Pillow Distortion Analysis for a Space Mesh Reflector Antenna

Pillow distortion has become one significant limiting factor to the development of space mesh reflector antennas toward large-scale extension and high surface accuracy. However, the current state of the art about pillow distortion have great limitations in practical application, because they are mostly based on the assumptions that the stress applied on the mesh membrane is uniformly distributed and that the reflector is segmented into regular convex polygons by cable net. To improve the applicability and simplicity of the current state of the art, an alternative approach has been developed for the pillow disstortion analysis of space mesh reflector antennas without limiting assumptions in this paper. First of all, the triangular and quadrilateral cable-membrane elements are selected as conventional base units for space mesh reflector antennas. Second, equations and the calculation method were derived to investigate the pillow distortion of space mesh reflector antennas composed of triangular cable-membrane elements, based on the differential equations of membranes and cables, and then extended for that composed of other polygonal elements. To verify the validity and robustness of the proposed analytical method, some cable-membrane elements and a space mesh reflector antenna were taken as numerical examples. Eventually, the surface error distribution, root mean square error, and far-field electrical performance of an antenna influenced by the pillow distortion were conducted.

Numerical optimization method for packing regular convex polygons

An algorithm is presented for the approximate solution of the problem of packing regular convex polygons in a given closed bounded domain G so as to maximize the total area of the packed figures. On G a grid is constructed whose nodes generate a finite set W on G, and the centers of the figures to be packed can be placed only at some points of W. The problem of packing these figures with centers in W is reduced to a 0-1 linear programming problem. A two-stage algorithm for solving the resulting problems is proposed. The algorithm finds packings of the indicated figures in an arbitrary closed bounded domain on the plane. Numerical results are presented that demonstrate the effectiveness of the method.

On the diminishing process of Bálint Tóth

Let K K and K 0 K_0 be convex bodies in R d \mathbb {R}^d , such that K K contains the origin, and define the process ( K n , p n ) (K_n, p_n) , n ≥ 0 n \geq 0 , as follows: let p n + 1 p_{n+1} be a uniform random point in K n K_n , and set K n + 1 = K n ∩ ( p n + 1 + K ) K_{n+1} = K_n \cap (p_{n+1} + K) . Clearly, ( K n ) (K_n) is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in R d \mathbb {R}^d . We study this process for K K being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices. We also derive some new results in one dimension for non-uniform distributions.

Circle Numbers of Regular Convex Polygons

The circle number function is extended here to regular convex polygons. To this end, the radius of the polygonal circle is defined as the Minkowski functional of the circumscribed polygonal disc, and the arc-length of the polygonal circle is measured in a generalized Minkowski space having the rotated polar body as the unit disc.

A Characterization Theorem for the $$L^{2}$$ L 2 -Discrepancy of Integer Points in Dilated Polygons

Let C be a convex d-dimensional body. If \(\rho \) is a large positive number, then the dilated body \(\rho C\) contains \(\rho ^{d}\left| C\right| +\mathcal {O}\left( \rho ^{d-1}\right) \) integer points, where \(\left| C\right| \) denotes the volume of C. The above error estimate \(\mathcal {O}\left( \rho ^{d-1}\right) \) can be improved in several cases. We are interested in the \(L^{2}\)-discrepancy \(D_{C}(\rho )\) of a copy of \(\rho C\) thrown at random in \(\mathbb {R}^{d}\). More precisely, we consider Open image in new window where \(\mathbb {T}^{d}=\) \(\mathbb {R}^{d}/\mathbb {Z}^{d}\) is the d-dimensional flat torus and \(SO\left( d\right) \) is the special orthogonal group of real orthogonal matrices of determinant 1. An argument of Kendall shows that \(D_{C}(\rho )\le c\ \rho ^{(d-1)/2}\). If C also satisfies the reverse inequality \(\ D_{C}(\rho )\ge c_{1} \ \rho ^{(d-1)/2}\), we say that C is \(L^{2}\) -regular. Parnovski and Sobolev proved that, if \(d>1\), a d-dimensional unit ball is \(L^{2} \)-regular if and only if \(d\not \equiv 1\ ({\text {mod}}4)\). In this paper we characterize the \(L^{2}\)-regular convex polygons. More precisely, we prove that a convex polygon is not \(L^{2}\)-regular if and only if it can be inscribed in a circle and it is symmetric about the centre.

The zipper foldings of the diamond

A zipper folding of a polygon P given a source point x ∈ ∂P is the polyhedron generated by identifying all points in ∂P equidistant from x, measured along the perimeter of P , in essence “zipping” the boundary of the polygon. A theorem of Alexandrov shows that as long as every glued point has nonnegative curvature, then any zipper folding of a convex polygon leads to unique convex polyhedron (where a doubly covered polygon is considered a “flat” polyhedron). Alexandrov’s theorem is existential, but a more recent constructive proof by Bobenko and Izmestiev allows for the explicit construction of the polyhedron by solving a certain differential equation [2]. An implementation of the constructive algorithm has been coded by Stefan Sechelmann, which will output the folded convex polyhedron given a input triangulation of the polygon with gluing instructions. However, it is difficult to extract the creases and adjacencies from the initial polygon in their final output polyhedron. We seek a more combinatorial approach to computing this information. Previous work has also looked at determining all the combinatorially different polyhedra obtained via foldings, primarily for regular convex polygons as well as a few other shapes such as the Latin cross [3]. In this paper, we classify and compute the convex foldings of a diamond shape which are obtained via zipper foldings. Our primary goal was to seek a simpler combinatorial approach to testing for the correct set of folds, or crease pattern. As was observed by Alexandrov and noted in [1], there are a finite number of possible crease patterns. However, in our experience, verifying or discounting a crease pattern has been surprisingly difficult in more

JPD-Coloring of the Monohedral Tiling for the Plane

We introduce a definition of coloring by using joint probability distribution “JPD-coloring” for the plane which is equipped by tilingI. We investigate the JPD-coloring of ther-monohedral tiling for the plane by mutually congruent regular convex polygons which are equilateral triangles atr= 3 or squares atr= 4 or regular hexagons atr= 6. Moreover we present some computations for determining the corresponding probability values which are used to color in the three studied cases by MAPLE-Package.

Interpolating solutions of the Poisson equation in the disk based on Radon projections

We consider an algebraic method for reconstruction of a function satisfying the Poisson equation with a polynomial right-hand side in the unit disk. The given data, besides the right-hand side, is assumed to be in the form of a finite number of values of Radon projections of the unknown function. We first homogenize the problem by finding a polynomial which satisfies the given Poisson equation. This leads to an interpolation problem for a harmonic function, which we solve in the space of harmonic polynomials using a previously established method. For the special case where the Radon projections are taken along chords that form a regular convex polygon, we extend the error estimates from the harmonic case to this Poisson problem. Finally we give some numerical examples.

An Efficient Convex Hull Algorithm Using Affine Transformation in Planar Point Set

When trying to find the convex hull (CH) of a point set, humans can neglect most non-vertex points by an initial estimation of the boundary of the point set easily. The proposed CH algorithm imitates this characteristic of visual attention, starts by constructing an initial convex polygon (ICP), and measures the width and length of ICP through a shape estimation step. It then maps the point set into the new one by an affine transformation and makes most of the new points exist in a new initial convex polygon (NICP) which approximated to a regular convex polygon. Next, it discards the interior points in NICP by an inscribed circle and processes the remaining points outside NICP by Quickhull. Finally, the algorithm outputs the vertex set of CH. Two theorems are also proposed to solve an unconstrained optimization problem instead of the iteration method. Compared with four popular CH algorithms, the proposed algorithm can generate CH much faster than them and achieve a better performance.

Interpolation of harmonic functions based on Radon projections

We consider an algebraic method for reconstruction of a harmonic function in the unit disk via a finite number of values of its Radon projections. The approach is to seek a harmonic polynomial which matches given values of Radon projections along some chords of the unit circle. We prove an analogue of the famous Marr's formula for computing the Radon projection of the basis orthogonal polynomials in our setting of harmonic polynomials. Using this result, we show unique solvability for a family of schemes where all chords are chosen at equal distance to the origin. For the special case of chords forming a regular convex polygon, we prove error estimates on the unit circle and in the unit disk. We present an efficient reconstruction algorithm which is robust with respect to noise in the input data and provide numerical examples.

Visualizing Mutually Nondominating Solution Sets in Many-Objective Optimization

As many-objective optimization algorithms mature, the problem owner is faced with visualizing and understanding a set of mutually nondominating solutions in a high dimensional space. We review existing methods and present new techniques to address this problem. We address a common problem with the well-known heatmap visualization, since the often arbitrary ordering of rows and columns renders the heatmap unclear, by using spectral seriation to rearrange the solutions and objectives and thus enhance the clarity of the heatmap. A multiobjective evolutionary optimizer is used to further enhance the simultaneous visualization of solutions in objective and parameter space. Two methods for visualizing multiobjective solutions in the plane are introduced. First, we use RadViz and exploit interpretations of barycentric coordinates for convex polygons and simplices to map a mutually nondominating set to the interior of a regular convex polygon in the plane, providing an intuitive representation of the solutions and objectives. Second, we introduce a new measure of the similarity of solutions - the dominance distance - which captures the order relations between solutions. This metric provides an embedding in Euclidean space, which is shown to yield coherent visualizations in two dimensions. The methods are illustrated on standard test problems and data from a benchmark many-objective problem.