arXiv research

Classifies tilings of hyperbolic plane by regular polygons.

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the Spherical and Euclidean polygons with fixed perimeter the regular one…

Researchers calculate complexity of billiard paths in regular polygons.

In this article we investigate a family of nonlinear evolutions of polygons in the plane called the $β$-polygon flow and obtain some results analogous to results for the smooth curve shortening flow: (1) any planar polygon shrinks to a point and (2) a regular polygon with five or more vertices is asymptotically stable …

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…

Formula for Laplacian determinants on polygonal domains with slits.

The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.

We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex polygons, for instance their inscriptibility in a circle and their regularity i…

A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, …

This note characterizes monohedral tilings of regular polygons with up to three tiles.

Study on connection points on double regular polygons, providing coordinates and proving non-connection points.

We explain why numbers occurring in the classification of polygon spaces coincide with numbers of self-dual equivalence classes of threshold functions, or of regular Boolean functions, or of decisive weighted majority games.

Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles R^2 admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation ph…

Paper finds optimal shapes for minimizing average lengths of billiard trajectories in specific polygons.

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $L/k$, where $L$ is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show that every doubled regular $n$-gon admits a $1/2n$-geodesic. For the doubled regu…

A regular $n$-gon inscribing a knot is a sequence of $n$ points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular $n$-gon for any $n$.

This paper proves that for large n, the regular polygon minimizes the first eigenvalue of the Laplacian.

Extends boundary estimates for Monge-Ampère equations in polygonal domains.

New dynamical approach defines symmedian as hyperbolic barycenter.

Given a flag in each of the vertex-transitive tessellations of the Euclidean plane by regular polygons, we determine the flag stabilizer under the action of the automorphism group of a regular cover. In so doing we give a presentation of these tilings as quotients of regular (infinite) polyhedra.

Study the space of simple polygons and their moduli.

The Phi- relationship also known as Phi-factor appears in a number of lattice structures, mostly considering the lines within several separate circles or polygons. The paper considers a regular hexagonal tessellation as a lattice with the highest specific mechanical stiffness.

Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.

The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as $e^{-Z'(0)}$, where $Z(s)$, the zeta function, is the sum $\sum_n^{\infty} λ_n^{-s}$ analytically continued to $s$ around the origin. In this paper $Z'(0)$ is calculated for the Laplace operator with Dirichlet boundary…

The paper calculates the growth rates of billiard languages in hyperbolic polygons.

We study pairs of curves with Poncelet's porism properties and compute their vertex curves.

New elastic energy for irregular curves defined through polygonal approximations.

Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.

Study proves spectral determination of triangles and quadrilaterals, with restrictions on higher-order polygons.

We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the exis…

Study intersections of curves on translation surfaces, focusing on regular polygons and their Teichmüller disks.

Formula connects length and correlation functions via ghost polygons and Poisson bracket.

New methods classify convex lattice polygons for affine dimers.

The pentagram map's limit point is related to infinitesimal perturbations of polygons.

An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon…

We investigate a discrete version of the Möbius energy, that is of geometric interest in its own right and is defined on equilateral polygons with $n$ segments. We show that the $Γ$-limit regarding $L^{q}$ or $W^{1,q}$ convergence, $q\in [1,\infty]$ of these energies as $n\to\infty$ is the smooth Möbius energy. This re…

General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while s…

Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.

New property: polygons have a fixed dimension regardless of ambient space dimensions.

The paper classifies vertices in planar polygons formed by convex domains.

Any two equivalent discrete curves must have the same invariants at the corresponding points under an affine transformation. In this paper, we construct the moving frame and invariants for the discrete centroaffine curves, which could be used to discriminate the same discrete curves from different graphics, and estimat…

We study closed smooth convex plane curves $Γ$ enjoying the following property: a pair of points $x,y$ can traverse $Γ$ so that the distances between $x$ and $y$ along the curve and in the ambient plane do not change; such curves are called {\it bicycle curves}. Motivation for this study comes from the problem how to d…

In this paper, we discuss centroaffine geometry of polygons in $3$-space. For a polygon $X$ that is locally convex with respect to an origin together with a transversal vector field $U$, we define the centroaffine dual pair $(Y,V)$ similarly to [6]. We prove that vertices of $(X,U)$ correspond to flattening points for …

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. This map is known to interact nicely with Poncelet polygons, i.e. polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of R. Sc…

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexe…

Optimal Reeb graphs identified for polygon decomposition.

Study on Poncelet polygons' centers and circumcenters in various geometries.

New formula for spherical polygon area via prequantization.