Study on random triangles and quadrilaterals in plane geometry.
problem Probability of a random triangle being obtuse.
method Grassmann manifold correspondence and probability measure on plane polygons.
result Answered Lewis Carroll's and Sylvester's questions about random triangles and quadrilaterals.
The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.
problem Proving the Discrete Schwarz-Pick Lemma for circle packings with various inversive distances.
method Using a variational principle for circle packings with inversive distances, the paper extends the lemma to a broader range of packings.
result The Discrete Schwarz-Pick Lemma holds for circle packings with inversive distances in (−1,1], provided an additional condition on triangle weights. Study on least symmetric triangles in spherical geometry.
problem Finding the least symmetric triangle, both in planar and molecular contexts.
method Using Grassmannian correspondence and hyperoctahedral group action, compute the furthest point from the boundary in the Grassmannian.
result Exact computation of least symmetric triangles, including obtuse and acute types.
The paper extends circle pattern theory to include obtuse angles.
problem Existence and rigidity of circle patterns with non-obtuse exterior intersection angles.
method Topological degree theory, variational principle, Teichmüller theory, Sard's Theorem.
result The Circle Pattern Theorem is generalized to include obtuse angles.
Paper extends circle pattern theory to obtuse angles.
problem Circle patterns with obtuse angles not previously covered.
method Using topological degree theory, extends Koebe-Andreev-Thurston Theorem.
result Generalized Andreev's Theorem for obtuse dihedral angles.
Paper generalizes Andreev's theorem with obtuse angles.
problem Characterizing hyperbolic polyhedra with obtuse angles.
method Established discrete analog of weak solution/regularity theory.
result Generalized Andreev's Theorem to include obtuse angles.
New invariant found in spaces with curvature below, related to volume.
problem Geometric invariant of spaces with curvature below.
method Relations between new invariant and normalized volume, rigidity for maximal case.
result New invariant related to volume and rigidity for maximal case.
The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.
problem Understanding the combinatorial types of 3D polytopes in specific Riemannian manifolds.
method Analysis of mean curvature convex Riemannian polyhedra with non-obtuse dihedral angles in positive scalar curvature 3-manifolds.
result Determination of combinatorial types of 3D simple convex polytopes.
From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these d…
Given a combinatorial description C of a polyhedron having E edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize C is generally not a convex subset of RE \cite{DIAZ}. If C has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angle…
We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume b…
The paper studies circle patterns on surfaces with specific angles and curvature maps.
problem Investigating circle patterns with obtuse angles on surfaces of finite type.
method Characterizing curvature maps and establishing combinatorial Ricci flow conditions.
result Generalizations of circle pattern theorem and a computational method to find patterns.
Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's Method. Under the restriction that the dihedral angles are non-obtuse, Andreev's Theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyh…
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.
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.
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).
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.
Introduces Coxeter polyhedra in various geometries.
problem None explicitly stated, focuses on introducing concepts.
method Introduction of Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries.
result Fundamental theorems and descriptions of polyhedra and tessellations.
Shorter sides in geodesic triangles in hyperbolic plane.
problem Properties of geodesic triangles in hyperbolic surfaces.
method Analyzing lifts of a closed geodesic in hyperbolic 2-space.
result Sides of triangles formed by geodesics are shorter than the geodesic itself.
The study proves conditions for triangle comparison on surfaces of revolution.
problem Conditions for triangle comparison on Riemannian manifolds.
method Model surfaces of revolution; necessary and sufficient conditions for geodesic triangles.
result Necessary and sufficient conditions for a triangle comparison theorem.
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…
The paper explores isotopic triples of triangles in 3D space.
problem Determining if triples of triangles are combinatorially isotopic.
method Continuous motion of triangles with disjoint outlines, algorithmic checks, and elementary proofs.
result Different types of triples of disjoint triangles are not isotopic.
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.
Two proofs show the triangle inequality for Jaccard distance.
problem Triangle inequality for Jaccard distance
method Simple proofs using nonnegative, monotone, submodular functions
result Triangle inequality proven for Jaccard distance
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…
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…
Study of complex tori using twistor triangles and algebraic representations.
problem Understanding the geometry of complex tori through twistor triangles.
method Using representation theory of algebras to analyze the period domain of complex tori.
result Introduced pseudometric invariants to distinguish triangles up to G-equivalence. Classifies complex hyperbolic triangle groups by types.
problem Classifying complex hyperbolic triangle groups.
method By types defined by the ellipticity of two short words.
result Improves Schwartz conjecture.
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.
Triangle groups show rigidity in hyperbolic spaces.
problem Local rigidity of triangle groups generated by reflections.
method Geometric representation and diagonal embeddings in PGL(2,R) and PSp±(2n,R). result Triangle groups are locally rigid in hyperbolic spaces.
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.
The paper studies exact triangles in stable vector bundles on tori.
problem Understanding exact triangles in stable vector bundles on tori.
method Geometric interpretation via Fukaya category and homological mirror symmetry.
result Geometric interpretation of exact triangles in terms of Fukaya category.
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.
Counterexamples to Wise's theorem found using triangle groups and Ramanujan graphs.
problem A stronger form of Wise's theorem does not hold.
method Generalized triangle groups and Ramanujan graphs.
result A desired stronger form of Wise's malnormal special quotient theorem does not hold.
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.