We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds ap…
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48 results for “aperiodic tilings”
We give a homological construction of aperiodic tiles for certain open Riemannian surfaces admitting actions of Grigorchuk groups of intermediate growth.
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 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 …
Paper bridges matching rules and height functions in aperiodic tilings.
problem Relationship between matching rules and height functions in aperiodic tilings.
method Cochain-first framework to establish equivalence between matching rules, Ammann bar continuity, cycle closure of 1-cochains, and height-function existence.
result Unified framework for aperiodic tilings including Penrose and canonical projection tilings.
A full Mealy automaton is associated with a graph and a square complex, which contains an anti-torus if and only if the automaton is bi-reversible and the graph is aperiodic.
problem Determining the existence of anti-tori in square complexes associated with Mealy automata
method Associating a graph and a square complex with a Mealy automaton and proving the equivalence between bi-reversibility and aperiodicity of the graph
result The square complex contains an anti-torus if and only if the automaton is bi-reversible and the graph is aperiodic
We show that uniformly finite homology of products of trees vanishes in all degrees except degree , where it is infinite dimensional. Our method is geometric and applies to several large scale homology theories, including almost equivariant homology and controlled coarse homology. As an application we determine …
We introduce a quantitative condition on orbits of dynamical systems which measures their aperiodicity. We show the existence of sequences in the Bernoulli-shift and geodesics on closed hyperbolic manifolds which are as aperiodic as possible with respect to this condition.
The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptio…
The study examines aperiodicity properties of automorphism groups of free products of groups.
problem Investigating aperiodicity properties of automorphism groups of free products of groups.
method Analyzing the subgroup of Out(G) preserving conjugacy classes and proving aperiodicity properties.
result The group IA(G, G, 3) is torsion-free and has notable aperiodicity properties.
Parrondo's effects with aperiodic protocolsphysics.soc-ph
Study shows aperiodic sequences enhance Parrondo's effect, with Thue-Morse outperforming others.
problem Enhancing Parrondo's effect through strategic switching protocols.
method Investigated Fibonacci, Thue-Morse, and Rudin-Shapiro sequences; analyzed capital correlation and persistence.
result Thue-Morse sequence outperforms other aperiodic sequences and benchmark games in capital gain.
ARISE models efficient markets without periodogram or Gaussianity assumptions.
problem Mimicking and learning long-term memory in efficient markets.
method ARISE process using aperiodic spectrum estimation and infinite-sum function of known processes.
result ARISE process has mean-square convergence, consistency, and asymptotic normality without periodogram and Gaussianity assumptions.
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 -gon.
result Characterization of monohedral tilings of any regular -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.
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-Tilesmath.GT
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.
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.
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.
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…
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.
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.
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.
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.
We consider the demixing problem of two (or more) structured high-dimensional vectors from a limited number of nonlinear observations where this nonlinearity is due to either a periodic or an aperiodic function. We study certain families of structured superposition models, and propose a method which provably recovers t…
Square-tiled torimath.GT
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})…
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…
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.
Soft tilingsmath.MG
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.
Three-dimensional Rep-tilesmath.GT
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 .
result A 3-manifold is a 3D rep-tile if and only if it is the exterior of a connected graph in .
In this note we prove that any monohedral tiling of the closed circular unit disc with topological discs as tiles has a -fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain t…
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.
We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two different viewpoints, one with constructive techniques to enumerate such tiling…
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.
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…
We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group has the fixed point property FW for walls (e.g. if it has property (T)), every aperiodic action of by diffeomorphisms that are of class with countably many singularities is con…
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to …
The groups and -gon tilingsmath.GT
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.
Arithmeticity of the Kontsevich--Zorich monodromies of certain families of square-tiled surfacesmath.DS
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.
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 …
Tileable Surfacesmath.DG
Constructs surfaces that can be tiled by a finite set of rigid motion congruence classes of tiles.
problem Creating surfaces that can be tiled by a finite set of rigid motion congruence classes of tiles.
method Constructs examples with various topologies and describes all monotilings by finite edge prototiles.
result Describes all monotilings by finite edge prototiles with three or less edges.
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these for…
Conformally correct tilingsmath.HO
We discuss the art and science of producing conformally correct euclidean and hyperbolic tilings of compact surfaces. As an example, we present a tiling of the Chmutov surface by hyperbolic (2, 4, 6) triangles.
Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfacesmath.GT
We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To a…