This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile R3. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to hig…
New tiles in higher dimensions are shown to be homeomorphic to balls.
problem Characterizing self-affine tiles in higher dimensions as balls.
method Using Brouwer's invariance of domain theorem and a horizontal distance tool.
result Necessary and sufficient conditions for tiles to be d-dimensional tame balls. Self-affine tiles homeomorphic to a ball proven for a specific digit set.
problem Topology of self-affine tiles with collinear digit sets.
method Proving homeomorphism to a ball using integral self-affine tiles with collinear digit sets.
result A large class of integral self-affine tiles with collinear digit sets is homeomorphic to a closed 3-dimensional ball.
Let M be a 3×3 integer matrix each of whose eigenvalues is greater than 1 in modulus and let D⊂Z3 be a set with ∣D∣=∣detM∣, called digit set. The set equation MT=T+D uniquely defines a nonempty compact set T⊂R3. If T has positive L…
We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on the dimensions and intersections of boundary sets. As an application, we descri…
Let T:=T(A,D) be a disk-like self-affine tile generated by an integral expanding matrix A and a consecutive collinear digit set D, and let f(x)=x2+px+q be the characteristic polynomial of A. In the paper, we identify the boundary ∂T with a sofic system by constructing a ne…
An iterated function system Φ consisting of contractive similarity mappings has a unique attractor F⊆Rd which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling T of the con…
Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
Self-affine arcs without inner weak separation are parabolic segments.
problem Characterizing self-affine Jordan arcs without parabolic segments.
method Analyzing the weak separation property and proving implications for arc types.
result Self-affine Jordan arcs without parabolic segments are attractors of multizippers.
We test for departures from normal and independent and identically distributed (NIID) returns, when returns under the alternative hypothesis are self-affine. Self-affine returns are either fractionally integrated and long-range dependent, or drawn randomly from an L-stable distribution with infinite higher-order moment…
The paper examines properties of self-affine Sierpiński sponges using metric invariants.
problem Investigating properties of self-affine Sierpiński sponges using metric invariants.
method Examined through maximal power law property and perfectly disconnectedness.
result Characterized self-affine Sierpiński sponges by their metric properties.
Let A be an expanding d×d matrix with integer entries and D⊂Zd be a finite digit set. Then the pair (A,D) defines a unique integral self-affine set K=A−1(K+D). In this paper, by replacing the Euclidean norm with a pseudo-norm w in terms of A, we…
Study finds a measure for sponge components of Lalley-Gatzouras type.
problem Understanding the distribution of δ-connected components in self-affine sponges.
method Generalized existing results to self-affine sponges of Lalley-Gatzouras type, proving a measure relationship.
result Existence of a Bernoulli measure for cylinder components with a specific asymptotic relation.
New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.
problem Designing aesthetic shapes in equiaffine geometry.
method Introducing a new symmetry (ESA) to characterize planar curves.
result The new class of curves includes the quadratic curve and logarithmic spiral.
In this paper, we consider the connectedness of planar self-affine set T(A,D) arising from an integral expanding matrix A with characteristic polynomial f(x)=x2+bx+c and a digit set D={0,1,…,m}v. The necessary and sufficient conditions only depending on b,c,m are given for the $T(A…
We study the connectedness of the planar self-affine sets T(A,D) generated by an integer expanding matrix A with ∣det(A)∣=3 and a non-collinear digit set D={0,v,kAv} where k∈Z∖{0} and v∈Z2 such that {v,Av} is linearly independent. By chec…
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.
In the paper, we focus on the connectedness of planar self-affine sets T(A,D) generated by an integer expanding matrix A with ∣det(A)∣=3 and a collinear digit set D={0,1,b}v, where b>1 and v∈R2 such that {v,Av} is linearly independent. We discuss the domain of…
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.
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.
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})…
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.
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.
Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.
problem Investigating the Hausdorff dimension of Anosov subgroup limit sets with self-affine complexity.
method Analyzing the Hausdorff dimension of projective limit sets Λ1(Γ) of Anosov subgroups Γ under specific assumptions about their affine complexity. result The Hausdorff dimension of Λ1(Γ) is determined by the critical exponent of the first simple root under partial quasi-self-similarity. 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. In this note we prove that any monohedral tiling of the closed circular unit disc with k≤3 topological discs as tiles has a k-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…
We propose a construction which transforms a self-similar zipper in Rn to a self-affine zipper Rn+1 whose attractor is a smooth curve.
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…
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 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.