The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-d…
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The study classifies tilings of the sphere by congruent quadrilaterals.
problem Classifying edge-to-edge tilings of the sphere by congruent quadrilaterals.
method Classification of tilings into three classes based on geometric data and parameters.
result Three classes of tilings are identified: 2-layer earth map tilings, quadrilateral subdivisions of the octahedron, and 3-layer earth map tilings.
The study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.
Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
problem Understanding the structure of decorated hyperbolic polygons.
method Combinatorial approach using pseudo-manifolds and shellability.
result Arc complexes of decorated hyperbolic polygons are closed piecewise linear balls.
The study finds the bounds of vertex orbits in maps derived from specific lattices.
problem Determining the bounds of vertex orbits in maps derived from -vertex-homogeneous lattices.
method Analyzing maps as quotients of -vertex-homogeneous lattices.
result Sharp bounds of the number of vertex orbits are identified.
New tilings of the 2-sphere from convex polyhedra in 3-sphere.
problem Finding tilings of the 2-sphere from convex polyhedra in 3-sphere.
method Using the Lie group and its Maurer-Cartan forms.
result Existence of two canonical tilings of the 2-sphere.
Paper tackles shape graph registration using neural networks.
problem Constrained registration of shape graphs with varying nodes and edges.
method Shape-Graph Matching Network (SGM-net) with an elastic shape metric loss function.
result State-of-the-art matching performance and reduced computational cost.
Study of straight-line flows on a unique infinite surface.
problem Understanding straight-line flows on a specific infinite surface.
method Geometric description and characterization of periodic and drift orbits; use of rigid symmetries and Veech group.
result Complete characterization of periodic directions and proof of density of periodic and ergodic directions.
Graph matching involves combinatorial optimization based on edge-to-edge affinity matrix, which can be generally formulated as Lawler's Quadratic Assignment Problem (QAP). This paper presents a QAP network directly learning with the affinity matrix (equivalently the association graph) whereby the matching problem is tr…