Study on 3D surfaces and tangles formed by Poncelet triangles.
problem Geometric and topological properties of 3D surfaces and tangles.
method Exploration of Poncelet triangles and associated points.
result Properties of 3D surfaces and tangles formed by Poncelet triangles.
This paper explores the geometric and topological properties of Poncelet porism for triangles.
problem Investigating the differential-geometric and topological properties of the Cayley condition in Poncelet porism for triangles.
method Demonstrated that the Cayley set is a smooth, connected, 9-dimensional complex manifold. Analyzed its structure through moduli spaces and fiber bundles.
result The Cayley set is a smooth, connected, 9-dimensional complex manifold.
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.
Study on Poncelet polygons' centers and circumcenters in various geometries.
problem Understanding Poncelet polygons' geometric centers in different geometries.
method Analyzing the Circumcenter of Mass and Center of Mass of Poncelet polygons, proving Dan Reznik's invariants, and exploring spherical geometry.
result Proof of Dan Reznik's invariants for billiard trajectories and insights into Poncelet polygons' centers in spherical geometry.
The paper explores conic-line arrangements via Poncelet's theorem and finds families of reducible curves.
problem Understanding the topology of conic-line arrangements using Poncelet's theorem.
method Study unramified double covers induced by Poncelet transverses.
result Existence of families of Zariski pairs of degree 2m+6 for m≥2. 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…
We deduce a recent theorem by R. Schwartz on the structure of the so-called Poncelet grid from complete integrability of the billiard in an ellipse
We give a simple proof of the Emch closing theorem by introducing a new invariant measure on the circle. Special cases of that measures are well-known and have been used in the literature to prove Poncelet's and Zigzag theorems. Some further generalizations are also obtained by applying the new measure.
Let D be any elliptic right cylinder. We prove that every type of knot can be realized as the trajectory of a ball in D. This proves a conjecture of Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use Jacobi's proof of Poncelet's theorem by means of elliptic functions.
Billiard motion in ellipses analyzed with canonical coordinates.
problem Understanding billiard motion in ellipses.
method Canonical coordinates and kinematic analysis.
result Explicit parametrization of billiard motions using Jacobian elliptic functions.
In the generalized Legendre approach, the equation describing an asymptotically locally Euclidean space of type Dn is found to admit an algebraic formulation in terms of the group law on a Weierstrass cubic. This curve has the structure of a Cayley cubic for a pencil generated by two transversal plane conics, that i…
We prove some recent experimental observations of D. Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the one-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric…
This paper connects billiards in ellipses to focal billiards in ellipsoids.
problem Proving the existence of isometric counterparts between billiards in ellipses and focal billiards in ellipsoids.
method Continuous transition via isometric focal billiards in a fixed ellipsoid.
result Established the connection between planar and spatial billiards.
We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a…
Napoleonic triangles don't exist in hyperbolic geometry.
problem The existence of Napoleonic triangles in hyperbolic geometry.
method Analyzing the construction of equilateral triangles on hyperbolic triangles.
result Hyperbolic triangles do not form Napoleonic triangles, except equilateral ones.
New bounds on inscribed triangles in arbitrary planar domains.
problem Finding inscribed triangles in arbitrary planar domains with specific angle constraints.
method Proving the existence of uniformly fat triangles and not-too-fat triangles in bounded open sets.
result Existence of a maximal number Θ (between 0 and 60) for inscribed triangles with angles ≥ Θ degrees.
Paper calculates eigenvalues of a specific triangle on a sphere.
problem Computing eigenvalues of a specific triangle on a sphere.
method Computed first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2) on the sphere.
result Computed the first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2).
Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.
problem Finding the minimum of the spectral determinant on isosceles triangles.
method Analyzing the determinant of the Laplacian on Euclidean isosceles triangle envelopes of fixed area.
result Equilateral triangle envelope minimizes the determinant of the Laplacian.
New method shows any triangle group generating pair is related to special coverings.
problem Understanding generating pairs of triangle groups.
method Special almost orbifold coverings.
result Any generating pair of a triangle group is represented by a special covering.
Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups t…
Defines band maps in unoriented link Floer homology forming a skein exact triangle.
problem Understanding unoriented link Floer homology through band maps.
method Defines and analyzes band maps in unoriented link Floer homology.
result Band maps form an unoriented skein exact triangle.
Proves a new skein exact triangle for real monopole Floer homology.
problem None explicitly stated; focuses on proving a new mathematical structure.
method Introduces a new exact triangle for real monopole Floer homology.
result Proves an unoriented skein exact triangle for real monopole Floer homology.
New surgery exact triangles in Heegaard Floer homology for rational slopes.
problem Constructing new surgery exact triangles in Heegaard Floer homology.
method Combining combinatorial triangle and quadrilateral counting in genus 1 Heegaard diagrams.
result Solving the combinatorial problem for rational slopes, including tricky cases.
We answer the question "Does the Y-triangle move preserve intrinsic knottedness?" in the negative by giving an example of a graph that is obtained from the intrinsically knotted graph K_7 by triangle-Y and Y-triangle moves but is not intrinsically knotted.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
We show that the triangle with angles Pi/12, Pi/3 and 7*Pi/12 has the lattice property and compute this triangle's Veech group.
We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metr…
Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise disjoint. We conjecture that it can be algorithmically checked if an (ordered or unorde…
New method proves mateability of triangle groups with Blaschke products.
problem Proving mateability of triangle groups with Blaschke products.
method Associating two piecewise analytic circle maps to the triangle group, mating these with Blaschke products, and constructing a common lift.
result Proves mateability of all cusped triangle groups with suitable Blaschke products.
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…
In Lorentzian geometry, limited definition of angles restricts the use of angle bisectors in study of triangles. This paper redefines angle bisectors so that they can be used to study attributes of triangles. Using the new definition, this paper investigates the existence of the incenter and the isogonal conjugate of a…
A formula for Rademacher symbols in triangle groups is provided.
problem No specific problem stated; focuses on a mathematical formula.
method Presentation of an explicit formula for Rademacher symbols.
result Generalizes Ghys' proof of modular knot linking numbers.
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
Criterion for stopping conjugacy class enumeration in triangle groups.
problem Enumerating all conjugacy classes in cocompact triangle groups.
method Encoding by P. Dehornoy and T. Pinsky; stopping criterion based on geometric length.
result Stopping criterion for the generation of conjugacy classes in cocompact triangle groups.
New skein exact triangles for link Floer homology.
problem Understanding link Floer homology through skein relations.
method Construction of skein triples for rational tangles.
result Established a framework for potential further skein exact triangles.
New theorem disproves Angle Defect for super triangles.
problem Angle Defect Theorem for N=1 super hyperbolic geometry.
method Action of OSp(1|2) on real super Minkowski space and brute-force computation.
result Disproves Angle Defect Theorem and provides novel additive function.
The study examines Lipschitz normally embedded Hölder triangles in 4D space.
problem Comparing ambient and outer Lipschitz geometry of Hölder triangles.
method Analyzes Lipschitz normally embedded Hölder triangles in \(\mathbb{R}^4\).
result Infinitely many equivalence classes of microknots.
Triangle Artin groups split as graphs of free groups under specific conditions.
problem Conditions for triangle Artin groups to split as graphs of free groups.
method Graph of free groups analysis and poly-free property proof.
result Triangle Artin groups split as graphs of free groups if and only if labels are greater than 5 and even.
The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.
problem Finding extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
method Characterizations obtained under suitable geometric constraints.
result Characterizations of extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
Triangle groups uniquely identified by their finite quotients.
problem Identifying triangle groups among finitely generated residually finite groups.
method Character varieties method to distinguish profinite completions.
result Certain Fuchsian triangle groups are profinitely rigid.
Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.
problem Understanding periodic billiard paths in right triangles and orbit closures in strata of Abelian and quadratic differentials.
method Classifying orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials.
result Computed orbit closures and deduced asymptotic number of periodic billiard trajectories in right triangles.
Paper shows any link can be diagrammed with only triangles and quadrilaterals.
problem Representing links with only triangles and quadrilaterals.
method Examined link diagrams as 4-valent graphs on a sphere.
result Any link can be represented with only triangles and quadrilaterals.
Given a closed polygon P having n edges, embedded in R^d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface in R^d having P as its geometric boundary. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert suface constru…
The paper extends Menelaus' and Ceva's theorems to translation triangles in various Thurston geometries.
problem Extending classical theorems to non-Euclidean geometries.
method Using projective models of Thurston geometries and defining a ``surface of a translation-like triangle".
result Generalization of Menelaus' and Ceva's theorems to non-constant curvature Thurston geometries.
We study groups generated by three half-turns in the Lobachevsky 3-space and their quotient orbifolds. These generalized triangle groups are closely related to the arbitrary 2-generator Kleinian groups. Our main result is a classification of the singular sets of the generalized triangle orbifolds. We also present a m…
Study on Teichmüller space of acute triangles using explicit calculations.
problem Analogue of Thurston's metric on acute triangles.
method Direct calculation of distance function and Finsler structure.
result Explicit expressions and properties of the metric on acute triangles.
Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).
problem Classifying Anosov representations of hyperbolic triangle groups into SL(3,R).
method Proving representations are Anosov if they lie in the Hitchin component or the Barbot component, with specific conditions for eigenvalues.
result Anosov representations in SL(3,R) have non-convex boundary maps.