Classifies tilings of hyperbolic plane by regular polygons.
problem Decidability of tiling patterns in hyperbolic plane.
method Finite set of local and inductive combinatorial constraints.
result First known weakly aperiodic protosets of regular polygons in hyperbolic plane.
The article studies polygon flows and their asymptotic behavior.
problem Investigating the behavior of polygon flows over time.
method Analogous to curve shortening flow, the article analyzes the β-polygon flow. result Regular polygons with five or more vertices are asymptotically stable.
The paper explores geodesics on doubled polygons and their existence.
problem Existence and behavior of 1/k-geodesics on doubled polygons.
method Investigation of 1/k-geodesics on doubled polygons, focusing on regular polygons.
result Every doubled regular n-gon admits a 1/2n-geodesic, and for odd primes p, conjectures k=2p as minimum for 1/k-geodesics.
Research determines criteria for semi-regular tilings in hyperbolic space.
problem Finding combinatorial criteria for semi-regular tilings in hyperbolic geometry.
method Combinatorial analysis of vertex-types and geodesic polygons.
result Determined criteria for existence and uniqueness of semi-regular tilings.
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.
problem Calculating the complexity of billiard paths in regular polygons.
method Counting saddle connections on lattice surfaces, focusing on combinatorial length.
result They answered a question about billiard language complexity in regular polygons.
Foliation of star-shaped polygons with fixed perimeter and area.
problem Characterizing star-shaped polygons with fixed perimeter and area.
method Analyzing families of star-shaped n-polygons in the Euclidean plane.
result Existence and properties of foliations on the space of star-shaped n-polygons.
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.
problem Determining the ζ-regularized determinant of the Laplacian on polygonal domains with slits. method Patchwork method for heat trace asymptotics, comparison formula for smooth conformal metrics.
result Polyakov-Alvarez type formula for Laplacian determinants on polygonal domains with slits.
The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.
problem Finding the minimum perimeter for polygons with a fixed area in hyperbolic geometry.
method Proving analogues of the discrete isoperimetric inequality for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons.
result Established two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with certain area or perimeter restrictions.
This note characterizes monohedral tilings of regular polygons with up to three tiles.
problem Characterizing monohedral tilings of regular polygons with up to three tiles.
method Connecting the results for squares and circles to generalize for any regular n-gon. result Characterization of monohedral tilings of any regular n-gon with up to three tiles. Characterizes extreme points in polygon limit sets.
problem Identifying boundary points in polygon limit sets.
method Characterization through affine dilations and polygon vertices.
result Characterizes which points lie on the boundary of convex hull.
Study on connection points on double regular polygons, providing coordinates and proving non-connection points.
problem Identifying connection points on double regular polygons.
method Examined coordinates in trace field, provided constructive proof for prime n. result For n=7, conjectured all remaining points are connection points; for n≥7 prime, provided explicit separatrix. 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.
Paper finds optimal shapes for minimizing average lengths of billiard trajectories in specific polygons.
problem Finding optimal shapes to minimize the average length of billiard trajectories.
method Used techniques from Teichmüller theory.
result Optimal shapes minimize average lengths of billiard trajectories in specific polygons.
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…
The paper calculates the Euler characteristic of regular spherical polygon spaces.
problem Determining the Euler characteristic of regular spherical polygon spaces.
method Constructing a manifold Xn and a function μ:XnoR such that μ−1(a)=Mn(a), determining the index of critical points, and using Morse surgeries. result Calculating the Euler characteristic χ(Mn(a)) for all a and odd n. The study of polygon areas with fixed perimeter.
problem Finding the minimum number of critical points for polygon areas.
method Analysis of the configuration space and critical points of the area function.
result Computed indices of critical points (regular stars) on the configuration space.
Study of hyperbolic polyhedral surfaces with regular faces.
problem Understanding the properties of hyperbolic polyhedral surfaces with regular faces.
method Combinatorial and geometric analysis of hyperbolic polyhedral surfaces with regular faces.
result There is a gap between the areas of non-smooth hyperbolic polyhedral surfaces and smooth hyperbolic surfaces.
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.
problem Finding the polygon with the smallest first eigenvalue of the Laplacian for a given area.
method Constructing polygonal manifolds and using spectral theory, tensor calculus, and symmetrization techniques.
result For large n, the regular polygon minimizes the first eigenvalue of the Laplacian.
Extends boundary estimates for Monge-Ampère equations in polygonal domains.
problem Boundary regularity for Monge-Ampère equations on convex polytopes with specific boundary conditions.
method Schauder-type techniques, inspired by Donaldson's work on the Abreu equation.
result Establishes boundary regularity result for Hölder continuous right-hand sides.
New dynamical approach defines symmedian as hyperbolic barycenter.
problem Understanding symmedian properties in hyperbolic geometry.
method Developed a new dynamical coordinatization.
result Symmedian point acts as hyperbolic barycenter.
Study on Vietoris-Rips complexes of regular polygons, revealing complex homotopy types.
problem Understanding the homotopy types and persistent homology of Vietoris-Rips complexes of regular polygons.
method Use of persistent homology, cyclic graphs, and winding fractions.
result Characterization of homotopy types and persistent homology of Vietoris-Rips complexes of Pn up to a scale parameter. 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.
problem Understanding the space of simple polygons and their moduli.
method Local and global descriptions of S(n) and M(n) using Morse theory and asymptotic analysis.
result Completely described S(4) and M(4) using a topological Morse function.
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.
problem Existence and non-existence of specific geodesic nets on flat spheres.
method The theorem of Gauss-Bonnet is applied to demonstrate results.
result Existence and non-existence of geodesic nets on regular doubled polygons.
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 ∑n∞λn−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.
problem Computing the exponential growth rates of billiard languages in polygons.
method New methods relating to minimal tiling paths.
result Explicit computation of exponential growth rates for q even, and bounds for q odd. Study of 2D Teichmüller spaces of surfaces with boundary.
problem Characterizing geometry of Teichmüller spaces with boundary.
method Examined pentagon and punctured triangle Teichmüller spaces, showing they are exhausted by regular geodesic polygons.
result Geodesics in these spaces diverge at most linearly.
We study pairs of curves with Poncelet's porism properties and compute their vertex curves.
problem Understanding pairs of curves with Poncelet's porism properties.
method Developed formulas to compute vertex curves for given envelope curves and vice versa, for all sufficiently regular pairs of Poncelet curves.
result Formulas produce all possible sufficiently regular pairs of Poncelet curves, including sets of curves analogous to pencils of conic sections.
New elastic energy for irregular curves defined through polygonal approximations.
problem Defining elastic energy for irregular curves in any space dimension.
method Relaxation process with p-rotation of inscribed polygonals, focusing on geometric curvature distribution. result Energy finite if and only if curve's arc-length parameterization has second order summability.
Study proves spectral determination of triangles and quadrilaterals, with restrictions on higher-order polygons.
problem Determining the geometry of convex polygons from their Steklov spectra.
method Analysis of characteristic polynomial and spectral properties of Steklov spectrum.
result Almost all triangles and certain quadrilaterals are uniquely determined by their Steklov spectra.
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.
problem Maximizing algebraic intersection between curves of given lengths.
method Investigate the quantity KVol defined for any closed orientable surface, focusing on regular n-gons for even n ≥ 8.
result Maximize algebraic intersection between curves of given lengths.
Formula connects length and correlation functions via ghost polygons and Poisson bracket.
problem Understanding functions on moduli spaces of Anosov representations.
method Introduced ghost polygons and ghost algebra to compute Poisson bracket.
result Stability of length and correlation functions under Poisson bracket.
The paper finds minimal-area metrics on polygons with multiple crossing geodesics.
problem Finding minimal-area metrics on polygons with specific geodesic conditions.
method Applied convex programs to polygons with length conditions on curves.
result The extremal metric coincides with the conformal extremal metric on regular polygons.
New methods classify convex lattice polygons for affine dimers.
problem Not all convex lattice polygons are characteristic polygons of affine dimers.
method General constructions and algorithm for finding affine dimers with prescribed polygons.
result All lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.
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 Lq or W1,q convergence, q∈[1,∞] of these energies as n→∞ is the smooth Möbius energy. This re…
The pentagram map's limit point is related to infinitesimal perturbations of polygons.
problem Understanding the limit point of the pentagram map and its relation to polygon perturbations.
method Interpreting Glick's operator as the infinitesimal monodromy of a polygon.
result Glick's operator measures the extent to which a perturbed polygon does not close up.
The paper explores centroaffine geometry of polygons and their duals.
problem Understanding centroaffine dual pairs of spatial polygons.
method Defining centroaffine dual pairs and proving properties of polygon duals.
result Constant curvature polygons are dual to planar polygons.
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…
The pentagram map preserves Poncelet polygons in convex cases.
problem Characterizing Poncelet polygons using the pentagram map.
method Theory of commuting difference operators, properties of real elliptic curves, and theta functions.
result A convex polygon is Poncelet if and only if it is projectively equivalent to its pentagram image.
Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.
problem Inverse Steklov problem on convex polygons.
method Analysis of Steklov eigenvalues and isoperimetric bounds.
result For almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum.
Study of polygon spaces, characterizing critical points of area function.
problem Characterizing critical points of area function in polygon spaces.
method Geometric characterization of critical points and calculation of Morse indices.
result Generalization of isoperimetric theorems for polygons in the plane.
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…