Art and science of conformally correct tilings of compact surfaces.
problem Producing conformally correct tilings of compact surfaces.
method Discussing and presenting examples of tilings.
result Presentation of a tiling of the Chmutov surface by hyperbolic triangles.
In this paper we describe the pentagonal tiling of the plane defined in the article "A regular pentagonal tiling of the plane" by P. L. Bowers and K. Stephenson as a conformal substitution tiling and summarize many of its properties given in the mentioned article. We show furthermore why such tiling is not FLC with res…
Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to three-dimensional cube tilings. The connections between discrete extremal length …
Framework verifies global correctness of neural networks for perception tasks.
problem Verifying robustness of neural networks is insufficient; global correctness needs to be ensured.
method Specified a state space and observation process to define the target input space. Tiled the spaces and compared ground truth and network output bounds to deliver error bounds.
result Framework can verify error bounds globally over the target input space and detect illegal inputs.
New tiling algorithm for hyperbolic 3-manifolds, characterizing cusp areas.
problem Characterizing and computing maximal cusp areas in hyperbolic 3-manifolds.
method Developed a new tiling algorithm and provided simpler expressions for distances.
result Completely characterized the space of cusp neighborhoods and found the Epstein-Penner decomposition.
Researchers found the maximum number of holes in polyominoes grows proportionally to the dimension.
problem Finding the maximum number of holes in polyominoes of varying dimensions.
method Used concepts from error-correcting codes and dynamical systems.
result Proved that fd(n)/no(d−1)/d as n goes to infinity for all d≥2. A hybrid scheme uses GAs and DL for tile panel reconstruction.
problem Reconstructing Portuguese tile panels with real-world effects.
method Enhanced GA-based puzzle solver with novel DLCM.
result 82% accuracy for tile reconstruction compared to 3.5% for best known method.
Study geometrically measures to decide if modular companions are conformally equivalent.
problem Deciding if two modular companions are conformally equivalent under a given group action.
method Construct a moduli space and equivariant tilings to measure conformal equivalence.
result Presented a geometric measure to decide conformal equivalence of modular companions.
The hyperbolic structure of equilateral pentagons is mapped to a tiling of the hyperbolic plane.
problem Mapping the realization space of equilateral pentagons to a hyperbolic plane.
method Combining combinatorial correspondence, Riemann mapping theorem, and normalization procedure.
result A full conformal parameterization of the space of equilateral pentagons.
New framework links fractal complexity to separation dimension.
problem Quantifying the complexity of fractal partitions.
method Introducing Separation Dimension ($\sepdim$) and Geometrically Regular Partitions (GRPs).
result Sharp upper bound for chromatic number of fractal partitions.
A new test evaluates risk estimation accuracy using probability integral transform.
problem Measuring the accuracy of financial market risk estimations.
method Probability Integral Transform (PIT) of ex post realized returns against ex ante probability distributions.
result The new test shows the importance of capturing the dynamic of financial markets.
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. The study of tiling homology on flat surfaces, proving impossibility of certain tilings.
problem Proving the non-existence of polyomino tilings on specific square-tiled surfaces.
method Study of homology groups for topological tilings, using coloring proofs.
result Several results about the non-existence of polyomino tilings on certain square-tiled surfaces.
Circular disc can be tiled with up to 3 congruent pieces, showing symmetry.
problem Tiling a circular disc with congruent pieces.
method Proving the existence of a k-fold rotational symmetry for k≤3. result First nontrivial estimate on minimum number of tiles for certain tiling configurations.
New tiles allow efficient knot mosaics for small knots.
problem Efficient representation of small knots on a grid.
method Introducing corner connection tiles for knot mosaics.
result Efficient knot mosaics for knots with crossing number 8 or less.
Rep-tiles fill cubes in any dimension.
problem Finding compact submanifolds that can tile cubes.
method Classifying and constructing rep-tiles for any finite CW complex.
result Every smooth compact submanifold with connected boundary is topologically isotopic to a rep-tile.
Shellable tilings on simplicial complexes help understand their structure.
problem Understanding the structure of simplicial complexes through tilings.
method Proving the existence of shellable h-tilings on finite simplicial complexes after stellar subdivisions.
result The h-vector of a tiling is determined by the critical vector, with palindromic properties for closed triangulated manifolds.
Study tiling spaces over irrational tori using diffeological classification.
problem Understanding the structure of tiling spaces over irrational tori.
method Diffeological classification of irrational tori and analysis of fiber bundle structures.
result Inherited diffeological equivalence of one-dimensional tiling spaces over irrational tori.
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.
In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic…
New method constructs tilings of the plane using directed edges and alignments.
problem Modeling tilings of the Euclidean or hyperbolic plane as presheaves over categories.
method Introducing finite categories for polygons with labeled directed edges, constructing reflective alignments.
result Characterizing alignments of tilings by comparing edge directions and generating families with elegant symmetry.
New tile types for knots and links reduce complexity.
problem Determining the minimum number of tiles needed for knot representations.
method Introduced new tile types and analyzed their impact on knot complexity.
result Corner tile number lies between tile number and 3 times tile number.
Cannon, Floyd and Parry have studied the modulus of finite subdivision rules extensively. We investigate the properties of the modulus of subdivision rules with linear and exponential growth at every vertex, using barycentric subdivision and a subdivision rule for the Borromean rings as examples. We show that the subdi…
Study on tilings of the plane with two types of tiles of varying areas.
problem Classifying tilings with minimal interface length.
method Analysis of isoperimetric configurations for different lattice types and tile areas.
result Three distinct tilings configurations found based on tile area ratio.
Classifies tilings of Euclidean and hyperbolic planes using topological methods.
problem Classifying crystallographic tilings of Euclidean and hyperbolic planes.
method Topological approaches, including enumeration and classification of tilings as decorations of orbifolds.
result Classification up to equivariant equivalence achieved.
New spectral sequences derived from shellable tilings.
problem Discrete Morse theory and shellable complexes.
method Introduced tilings and quivers to support spectral sequences.
result Spectral sequences converge to relative (co)homology.
Max-rank improves multiple testing in conformal prediction.
problem Simultaneous testing of multiple hypotheses in scientific inquiries.
method Introduces max-rank, a novel correction for positive dependencies in simultaneous testing.
result Max-rank efficiently controls family-wise error rate and improves predictive uncertainty estimates.
The main goal of this paper is to define a 1-1 correspondence between between substitution tilings constructed by inflation and the arithmetic of positional representation in the underlying real vector space. It introduces a generalization of inflationary tessellations to equivalence classes of tiles. Two tiles belong …
We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…
Mathematical tools for tiling hyperbolic surfaces are developed.
problem Exploring isotopy classes of tilings on hyperbolic surfaces.
method Extension of mapping class groups to orbifolds, combinatorial tiling theory.
result Complete enumeration of isotopically distinct tilings of hyperbolic surfaces.
Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.
problem Creating space-filling shapes without sharp corners.
method Edge bending algorithm to deform polyhedral tilings into soft tilings.
result Soft tilings derived from minimal surfaces can be continuously transformed into one another.
We determine the topology of the moduli space of periodic tilings of the plane by parallelograms. To each such tiling, we associate combinatorial data via the zone curves of the tiling. We show that all tilings with the same combinatorial data form an open subset in a suitable Euclidean space that is homotopy equivalen…
Softens tilings in 3D space, proving conjectures about polyhedral tilings.
problem Proving that all locally polyhedral tilings in 3D space can be softened.
method Developed a new edge-bending algorithm to prove the statement.
result Proved conjectures about polyhedral tilings in 3D space and the plane.
Paper proves corner connection tiles can represent knots with fewer tiles.
problem Finding the minimum number of tiles for knot representation.
method Developed corner connection tiles and proved their efficiency.
result Corner connection tiles can represent knots with fewer tiles than traditional tiles.
This paper classifies all 3D rep-tiles up to homeomorphism.
problem Identifying compact 3D shapes that can be tiled into smaller copies of themselves.
method Examined all 3D rep-tiles up to homeomorphism, showing equivalence to the exterior of a connected graph in S3. result A 3-manifold is a 3D rep-tile if and only if it is the exterior of a connected graph in S3. New method for counting distinct tilings with symmetrical surfaces.
problem Counting distinct tilings with symmetrical surfaces.
method Deriving representations of mapping class groups and describing tilings as decorations on orbifolds.
result Explicit enumeration of isotopically distinct tilings.
4-ball can be tiled with knotted surfaces.
problem Tiling the 4-ball with knotted surfaces.
method Using congruent knotted surfaces isotopic to the original surface.
result Tiling of the 4-ball with knotted surfaces.
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.
Study higher rank inner products and their tilings to describe tori degenerations.
problem Understanding metric degenerations of tori.
method Introduce higher rank inner products and their tilings, use to describe degenerations.
result Describe metric degenerations of polarized tori and Hausdorff limits of tilings.
The study calculates the number of convex tilings on a sphere using triangles, squares, or hexagons.
problem Counting convex tilings on a sphere by triangles, squares, or hexagons.
method Using Thurston's lattice correspondence, extending to squares and hexagons, computing relevant lattice Λ, integrating Siegel theta function, and determining formulas using modular forms.
result Explicit formulas for the weighted number of convex tilings with a given number of tiles.
The paper finds bounds and specific tile numbers for knot mosaics.
problem Determining the minimum number of tiles needed to represent knots.
method Analyzing the relationship between tile number and mosaic number of knots.
result Strict bounds and specific tile numbers for various knots are determined.
The paper explores different perspectives on rhombile tilings.
problem None explicitly stated, focuses on different viewpoints.
method Four ways of looking at rhombile tilings: cubes, groups, lines, and points.
result Different methods provide insights into rhombile tilings.
Shear moves connect square-tiled surfaces in quadratic differentials.
problem Connecting square-tiled surfaces via specific moves.
method Shear moves corresponding to diagonal flips preserving square-tiled properties.
result Connected components of reconfiguration problem are in bijection with moduli space of quadratic differentials.
The study finds arithmetic groups often in square-tiled surface monodromies.
problem Understanding arithmetic properties of square-tiled surfaces.
method Analyzing variations of Hodge structures and Kontsevich-Zorich monodromies.
result Arithmetic groups are frequent in low genus square-tiled surfaces.
This note connects tiling billiards dynamics to Novikov's problem via helicoidal construction.
problem Understanding dynamics of tiling billiards and topology of subsurface sections.
method Helicoidal construction by Ivan Dynnikov.
result Relationship between tiling billiards and Novikov's problem in higher genus.
The level set of an elliptic function is a doubly periodic point set in C. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in C^2 and its sections (``cuts'') by C. We give S a crystallographic isometry in C^2 by defining a fundamental surface element as a conformal ma…
Research examines tiling problems with colored cubes and bricks on a 3D board.
problem Tiling problems with colored cubes and bricks on a (2imes2imesn)-board. method Recursive approach considering (n−1)-long board to determine number of tilings. result Identifies identities for recursions using breakability.
3D self-affine tiles with specific digit sets have boundary homeomorphic to a 2-sphere.
problem Characterizing 3D self-affine tiles with collinear digit sets whose boundary is a sphere.
method Using lattice tiling combinatorics and topological properties of spheres, the paper characterizes such tiles.
result The boundary of these tiles is homeomorphic to a 2-sphere under certain conditions.