Regular N-gon Research Articles

On Dirichlet eigenvalues of regular polygons

We prove that the first Dirichlet eigenvalue of a regular N-gon of area π has an asymptotic expansion of the form λ1(1+∑n≥3Cn(λ1)Nn) as N→∞, where λ1 is the first Dirichlet eigenvalue of the unit disk and Cn are polynomials whose coefficients belong to the space of multiple zeta values of weight n and conjecture that their coefficients lie in the space of single-valued multiple zeta values. We also explicitly compute these polynomials for all n≤14.

Regular n-gon rotating equilibria with logarithm gravitational interaction

Regular n-gon rotating equilibria with logarithm gravitational interaction

Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere

We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in MR2. An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.

Numerical instability and its treatment for steady state heat conduction problems in exchanger tubes using the dual boundary element method

In this paper, we employed the boundary element method (BEM) to determine the heat conduction shape factor (CSF) for the steady-state heat conduction problem in the heat exchanger tube. Various doubly-connected geometric shapes were considered. Numerical results of the conduction shape factor matched well with the theoretical data. Numerical instability was found for a special size. Degenerate scale occurred for all cases of regular N-gons in determining the conduction shape factor and matched the data as analytically predicted. Numerical results showed that a degenerate scale is only dependent on the outer boundary instead of the inner boundary. Three regularization techniques, the dual BEM, the CLEEF(UT), and the CLEEF(LM), were provided to avoid numerical instability. It was found that the larger number of sides for the N-gon shape is given, the lower the CSF is obtained. As N approaches infinity, it converges to the minimum value of the solution for an annulus.

On monohedral tilings of a regular polygon

A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J Comb Theory Ser A 66:40–52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa et al. (Mediterr J Math 17:156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular n-gon with at most three tiles for any n≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 5$$\\end{document}.

On the centered co-circular central configurations for the n-body problem

For the power-law potential n-body problem, we study a special kind of central configurations where all the masses lie on a circle and the center of mass coincides with the center of the circle. It is also called the centered co-circular central configuration. We get some symmetry results for such central configurations. We show that for positive numbers α>0 and integers n≥3 satisfying 1n∑j=1n−1cscα⁡jπn≤1+α4, the regular n-gon with equal masses is the unique centered co-circular central configuration for the n-body problem with power-law potential Uα. It quickly follows that for the Newtonian n-body problem (in the case α=1) and n≤6, the regular n-gon is the unique centered co-circular central configuration.

힐튼 피더슨 근사접기를 활용한 지오지브라로 정11각형 한 번에 그리기 소재 연구

This study is the result of research with R&E students at Chungnam Science High School. We questioned whether it would be possible to accurately divide paper into thirds, and studied how to fold lines into equal parts. By expanding this, I was curious about whether it was possible to fold an angle into equal parts, and it was found that it was possible to divide an angle into thirds that could not be constructed by Hilton Pederson approximation. From the relational expression for the size of an exterior angle of a regular polygon, it was inferred that folding an angle into n equal parts is related to the regular n-gon folding problem. Pierpont said that the smallest regular n-gon that cannot be origami is a regular 11 polygon, and he deduced that an 11 polygon could be folded with an approximate folding method that is easier to access than multi-folds. Therefore, the purpose of the study was set as follows. First, by using the Fujimoto approximation method and the Hilton Pederson approximation method, we investigate how to fold an regular 11 polygon approximately, and examine the relationship between the two methods. Second, we look at how to actually implement regular 11 polygon approximation using two methods using GeoGebra. The researcher observed the R&E students through the assignment research guidance and found the following implications. First, it has significance in repeatedly repeating the interaction of asking, answering, arranging, and verifying questions between teachers and students. Second, Students had the experience of solving and generalizing the problem of dividing angles into thirds with a new method called origami. Third, through the experience of actually dividing an angle into 11 parts with origami and GeoGebra, it aroused curiosity in creating coding that reduces errors and solves problems more easily. Fourth, mathematical reasoning ability and problem-solving ability could be improved by experiencing the process of proving and inferring two approximate folds mathematically.

Periodic points on the regular and double n-gon surfaces

Using the transfer principle of Lie theory, we classify the periodic points on the regular n-gon and double n-gon translation surfaces and deduce consequences for the finite blocking problem on rational triangles that unfold to these surfaces.

A ring of spikes in a Schnakenberg model

For the Schnakenberg model, we consider a highly symmetric configuration of N spikes whose locations are located at the vertices of a regular N-gon inside either a unit disk or an annulus. We call such configuration a ring of spikes. The ring radius is characterized in terms of the modified Green’s function. For a disk, we find that a ring of 9 or more spikes is always unstable with respect to small eigenvalues. Conversely, a ring of 8 or less spikes is stable inside a disk provided that the feed-rate A is sufficiently large. More generally, for sufficiently high feed-rate, a ring of N spikes can be stabilized provided that the annulus is thin enough. As A is decreased, we show that the ring is destabilized due to small eigenvalues first, and then due to large eigenvalues, although both of these thresholds are separated by an asymptotically small amount. For a ring of 8 spikes inside a disk, the instability appears to be supercritical, and deforms the ring into a square-like configuration. For less than 8 spikes, this instability is subcritical and results in spike death.

The area of reduced spherical polygons

We confirm two conjectures of Lassak on the area of reduced spherical polygons. The area of every reduced spherical nonregular n-gon is less than that of the regular spherical n-gon of the same thickness. Moreover, the area of every reduced spherical polygon is less than that of the regular spherical odd-gons of the same thickness and whose number of vertices tends to infinity.

On Smith normal forms of q-Varchenko matrices

In this paper, we investigate q-Varchenko matrices for some hyperplane arrangements with symmetry in two andthree dimensions, and prove that they have a Smith normal formover Z[q]. In particular, we examine the hyperplane arrangement forthe regular n-gon in the plane and the dihedral model in the spaceand Platonic polyhedra. In each case, we prove that the q-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over Z[q] and realize their congruent transformation matrices over Z[q] as well.

Nonlinear stability of regular vortex polygons in a Bose–Einstein condensate

The problem of stability of rotating regular vortex N-gons (Thomson's configurations) in a Bose–Einstein condensate in a harmonic trap is considered. A reduction procedure on the level set of the momentum integral is proposed. The dependence of the velocity of rotation ω of vortex polygon about the center of the trap is obtained as a function of the number of vortices N and the radius of the configuration, R. The analysis of the orbital linear and nonlinear stability of the motion of such configurations is carried out. For N ⩽ 6, regions of orbital stability of configurations in the parameter space are constructed. It is shown that vortex N-gons for N>6 are unstable for any parameters of the system. In this paper, we study the stability of rotating regular vortex N-gons in a Bose–Einstein condensate in a harmonic trap. The analysis of the orbital linear and nonlinear stability of motion is carried out. The dependence of the stability of regular vortex N-gons on the number of vortices N and the parameters of the system is given.

On the Existence of Symmetric Bicircular Central Configurations of the 3n-Body Problem

In this paper, we consider central configurations of the planar 3n-body problem consisting of n masses at the vertices of a regular n-gon inscribed in a circle of radius r and 2n masses at the vertices of a second (not necessarily regular) concentric 2n-gon inscribed in a circle of radius ar which are symmetric in the sense that the set of positions of the 3n masses and the set of the corresponding masses are invariant under the action of a finite subgroup of O(2). There are two different types of such configurations. In the first type, called regular bicircular central configurations of the 3n-body problem, the second 2n-gon is regular, n of the vertices of the second n-gon are aligned with the vertices of the first regular n-gon and the masses at the vertices of this 2n-gon alternate values. In the second type, called semiregular bicircular central configurations of the 3n-body problem, the second 2n-gon is semiregular and the masses at its vertices are all of them equal. A semiregular 2n-gon has n pair of vertices symmetric by a reflection of an angle \(\beta \) with respect to the axis of symmetry of the first regular n-gon. Our aim is to analyze the set of values of the parameter a for the regular 2n-gon and of the parameters \((a,\beta )\) for the semiregular 2n-gon providing symmetric bicircular central configurations. In particular, for all \(n\ge 2 \) we prove analytically the existence of symmetric bicircular central configurations with a (respectively \((a,\beta )\)) satisfying some particular conditions. Using either computer-assisted results or numerical results, we also describe the complete set of values of a (respectively \((a,\beta )\)) providing symmetric bicircular central configurations for \(n=2,3,4,5\) and we give numerical evidences that the pattern for \(n>5\) is the same as the one for \(n=5\).

Symmetries in porous flows: recursive solutions of the Brinkman equation in polygonal ducts

In this paper, we solve the Brinkman equation for unidirectional, steady, incompressible and Newtonian flow through a duct of uniform cross-section in the shape of a regular N-gon under no-slip boundary condition on the surface of the duct in the absence of applied forces. We calculate solutions for small values of N and by analysing them we devise a recursive scheme of generating solutions for all . We obtain approximate expressions for the vortex lines for all values of N by applying a trigonometric perturbation on circular vortex lines and graphically compare them with exact vortex lines. We also graphically compare both exact and approximate vortex lines with Schwarz-Christoffel transformation.

Commensurable diagonals in regular n-gons

In this paper, we will show a characterization for pairs of commensurable diagonals of regular n-gons. This result gives a positive answer to a recent question.

Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials

We say that a monic polynomial with integer coefficients is a polygomial if its each zero is obtained by squaring the edge or a diagonal of a regular n-gon with unit circumradius. We find connections of certain polygomials with Morgan-Voyce polynomials and further with Chebyshev polynomials of second kind.

ENUMERATING NECKLACES WITH TRANSITIONS

AbstractNecklaces are the equivalence classes of words under the action of the cyclic group. Let a transition in a word be any change between two adjacent letters modulo the word’s length. We present a closed-form solution for the enumeration of necklaces in n beads, k colours and t transitions. We show that our result provides a more general solution to the problem of counting alternating (proper) colourings of the vertices of a regular n-gon.

Irregular Tilings of Regular Polygons with Similar Triangles

We say that a triangle T tiles a polygon A, if A can be dissected into finitely many nonoverlapping triangles similar to T. We show that if N>42, then there are at most three nonsimilar triangles T such that the angles of T are rational multiples of pi and T tiles the regular N-gon. A tiling into similar triangles is called regular, if the pieces have two angles, alpha and beta , such that at each vertex of the tiling the number of angles alpha is the same as that of beta . Otherwise the tiling is irregular. It is known that for every regular polygon A there are infinitely many triangles that tile A regularly. We show that if N>10, then a triangle T tiles the regular N-gon irregularly only if the angles of T are rational multiples of pi . Therefore, the number of triangles tiling the regular N-gon irregularly is at most three for every N>42.

An Affine Model of a Riemann Surface Associated to a Schwarz–Christoffel Mapping

In this paper, we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz–Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated n-gon in the complex plane.

From the Lagrange polygon to the figure eight I

The present work studies the continuation class of the regular n-gon solution of the n-body problem. For odd numbers of bodies between $$n = 3$$ and $$n = 15$$ , we apply one parameter numerical continuation algorithms to the energy/frequency variable and find that the figure eight choreography can be reached starting from the regular n-gon. The continuation leaves the plane of the n-gon and passes through families of spatial choreographies with the topology of torus knots. Numerical continuation out of the n-gon solution is complicated by the fact that the kernel of the linearization there is high dimensional. Our work exploits a symmetrized version of the problem which admits dense sets of choreography solutions and which can be written as a delay differential equation in terms of one of the bodies. This symmetrized setup simplifies the problem in several ways. On the one hand, the direction of the kernel is determined automatically by the symmetry. On the other hand, the set of possible bifurcations is reduced and the n-gon continues to the eight after a single symmetry breaking bifurcation. Based on the calculations presented here, we conjecture that the n-gon and the eight are in the same continuation class for all odd numbers of bodies.