Dirk Frettlöh, April L. D. Say-awen, M. L. A. N. de las Peñas
It is shown that there are primitive substitution tilings with dense tile orientations invariant under n-fold rotation for n=2,3,4,5,6,8. The proof for dense tile orientations uses a general result about irrationality of angles in certain parallelograms.
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A square tiling of the unit square is said to have the minimal tile property if the smallest tile can tile all the other tiles. We show that in such a tiling, the smallest tile cannot be too small.
A tiling of the unit square is an MTP tiling if the smallest tile can tile all the other tiles. We look at the function $f(n)=\max \sum s_i$, where $s_i$ is the side length of the $i$th tile and the sum is taken over all MTP tilings with $n$ tiles. If $n=k^2+3$, it was conjectured that $f(k^2+3)=k+1/k$. We show that any tiling that violates the conjecture must consist of at least three tile sizes and has exactly one minimal tile.
Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles
本文利用Type 5族凸五边形单调瓷砖，通过两种透视法导出四种凹多边形瓷砖，证明其对应非周期平铺的簇，并探讨配对瓷砖集与ASPmr的关联。
How many different tiles are needed at the minimum to create aperiodicity? Several tilings made of two tiles were discovered, the first one being by Penrose in the seventies. Since then, scientists discovered other aperiodic tilings made of two tiles, including the square-triangle one, a tiling that has been particularly useful for the study of dodecagonal quasicrystals and soft matters. An open problem still exists: Can one tile be sufficient to create aperiodicity? This is known as the ein stein problem. We present in this paper an aperiodic tiling made of one single tile: an isosceles right…
A convex pentagonal tile is a convex pentagon that admits a monohedral tiling. We show that a convex pentagonal tile that admits a periodic tiling has a property in which the sum of three internal angles of the pentagon is equal to 360°.
Converting non-periodic tilings with Tile(1, 1) into tilings with three types of pentagons, I
本文提出一种转换方法，将 Smith 等人利用替换法构造的 Tile(1, 1) 非周期铺砌转化为三种五边形铺砌，并指出排除任意替换后仅有两种图案，且转换中不细分菱形。
In [B.Gruenbaum, G.C. Shephard, Spherical tilings with transitivity properties, in: The geometric vein, Springer, New York, 1981, pp. 65-98], they proved "for every spherical normal tiling by congruent tiles, if it is isohedral, then the graph is a Platonic solid, an Archimedean dual, an n-gonal bipyramid (n>2), or an n-gonal trapezohedron (i.e., the pseudo-double wheel of 2n faces)". In the classification of spherical monohedral tilings, one naturally asks an "inverse problem" of their result: For a spherical monohedral tiling of the above mentioned topologies, when is the tiling isohedral? W…
We show that every tiling of a convex set in the Euclidean plane $\mathbb{R}^2$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of tilings of the full plane $\mathbb{R}^2$, which is based on a surprising connection to a random walk on a directed graph.
Triangle Tiling V: Tilings by a tile with integer sides
本文研究三角形平铺问题，证明若存在整数边长且含120度角的瓦片的N-平铺，则N至少为96，并据此排除多种N值，如7、11、14等，同时给出各形状ABC的最小未解N值。
An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile'". The tile may or may not be similar to ABC . This paper is the first of several papers, which together seek a complete characterization of the triples (ABC,N,T) such that ABC can be N -tiled by T . In this paper, we consider the case in which the tile is similar to ABC, the case in which the tile is a right triangle, and the case when ABC is equilateral. We use (only) techniques from linear algebra and elementary field …
Aperiodic tiling is a well-know area of research. First developed by mathematicians for the mathematical challenge they represent and the beauty of their resulting patterns, they became a growing field of interest when their practical use started to emerge. This was mainly in the eighties when a link was established with quasi-periodic materials. Several aperiodic tilings made of two tiles were discovered, the first one being by Penrose in the seventies. Since then, scientists discovered other aperiodic tilings including the square-triangle one, a tiling that has been particularly useful for t…
This paper shows that a multiple translative tile in the plane must be a multiple lattice tile.
The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters fo…
An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC . We wish to understand possible tilings by completely characterizing the triples (ABC, T, N) such that ABC can be N -tiled by T. In particular, this understanding should enable us to specify for which N there exists a tile T and a triangle ABC that is N-tiled by T; or given N, determine which tiles and triangles can be used for N-tilings; or given ABC, to determine which tiles …
Let $K$ be a convex body. It is known that, in general, if $K$ is a $k$-fold translative tile (for some positive integer $k$), then $K$ may not be a (onefold) translative tile. However, in this paper I will show that for every convex body $K$, $K$ is a twofold translative tile if and only if $K$ is a translative tile.
Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for all substitution tilings showing tiles in infinitely many orientations.
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) whose vertices are supported by the $2n$-th cyclotomic field $\mathbb{Q}\left(ζ_{2n}\right)$ is extended to cases with Dense Tile Orientations (DTO). It is shown that every CAST with DTO has an inflation multiplier $η$ with irrational argument so that $\frac{kπ}{2n}\neq\arg\left(η\right)\notin\mathbb{πQ}$. The minimal inflation multiplier $η_{min.irr.}$ is discussed for $n\geqq2$. Examples of CASTs with DTO, minimal inflation multiplier $η_{min.irr.}$ and individual dihedral symmetry $D_{2n}$ are introduced for $n\in\left\{ 2,3,4,5,…
Tiling of Hyperbolic Surface by Multiple Tiles
研究双曲面负欧拉特征数情况下用n边形(n≥7)铺砌的有限问题，提出固定瓷砖数的算法，并计算小亏格表面用两个瓷砖的铺砌数，讨论多瓷砖铺砌中不同边长的数量。
General substitution rules for non-periodic rhomb tilings are derived. From the requirement that all substitution tiles consist of a discrete number of prototiles, it follows that a substitution tile with angle s*pi/n must be built out of pairs of prototiles with angles (s+t)*pi/n and (s-t)*pi/n, except if t=0. In addition, we require that a discrete number of prototile edges must fit between the beginning and endpoint of the substitution tile edge. By comparing the total area of the discrete number of prototiles constituting the substitution tile and the total area derived from the assumed ed…
Tilings of an Isosceles Triangle
本文研究等腰三角形ABC用三角形T铺砌的问题，探讨了直角、角倍数关系及120度角三种瓷砖形式，证明了直角瓷砖N必须为偶数，其他两种情况需满足边长比例有理数及N的必要条件。
In this paper we show that the function defined by a primitive, star shaped substitu- tion tiling of the plane, can be realized as a Jacobian of a biLipschitz homeomorphism of R^2. In particular it holds for any Penrose tiling.
In this paper we prove the existence of quasiperiodic rhombic substitution tilings with 2n-fold rotational symmetry, for any n. The tilings are edge-to-edge and use [n/2] rhombic prototiles with unit length sides. We explicitly describe the substitution rule for the edges of the rhombuses, and prove the existence of the corresponding tile substitutions by proving that the interior can be tiled consistently with the given edge substitutions.
We say that a triangle $T$ tiles a polygon $A$, if $A$ can be dissected into finitely many nonoverlapping triangles similar to $T$. We show that if $N>42$, then there are at most three nonsimilar triangles $T$ such that the angles of $T$ are rational multiples of $π$ and $T$ tiles the regular $N$-gon. A tiling into similar triangles is called regular, if the pieces have two angles, $\al$ and $\be$, such that at each vertex of the tiling the number of angles $\al$ is the same as that of $\be$. Otherwise the tiling is irregular. It is known that for every regular polygon $A$ there are infinitely…
Fractals Generated by Modifying Aperiodic Substitution Tilings
本研究提出利用非周期替换密铺（如Ammann Chair密铺）结合Sierpinski地毯概念生成无限多分形的方法，计算并证明其相似维数稠密，并用Java实现开源的图像生成器。
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry Dn or D2n, i.e. the tiling contains an infinite number of patches of any size with dihedral symmetry Dn or D2n only by iteration of substitution rules on a single tile.
In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. It is known that there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane, but there is a centrally symmetric convex decagon which can form a five-fold lattice tiling. This paper characterizes all the convex domains which can form a five-fold lattice tiling of the Euclidean plane.
A set of tiles for covering a surface is composed of two types of tiles. The base shape of each one of them is a diamond or rhombus, both with angles 60 and 120 degrees. They are distinguished by labeling one as an acute diamond with a base angle of 60 degrees, the other one as an obtuse diamond with a base angle of 120 degrees. The two types of tiles can be marked with arrows, notches, or colored lines to keep them distinct. Notches can be used as matching rules such that the acute diamonds can form a star with 6-fold rotational symmetry among other matches. Similarly, three obtuse diamonds c…
In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. It is known that there is no other convex domain which can form two-, three- or four-fold translative tiling in the Euclidean plane, but there are centrally symmetric convex octagons and decagons which can form five-fold translative tilings. This paper characterizes all the convex domains which can form five-fold translative tilings of the Euclidean plane.
A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with 11-fold symmetry appearing in the literature.
The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular $k$-gonal tile with 120-degree angles is isoperimetric. For area $π/3$, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls th…
There are exactly eight edge-to-edge tilings of the sphere by congruent equilateral pentagons: three pentagonal subdivision tilings with 12, 24, 60 tiles; four earth map tilings with 16, 20, 24, 24 tiles; and one flip modification of the earth map tiling with 20 tiles.
NP-completeness of Tiling Finite Simply Connected Regions with a Fixed Set of Wang Tiles
该研究证明用23个Wang瓦片的固定集平铺有限单连通区域是NP完全问题，还推导出111个矩形的对应问题为NP完全，减少了所需瓦片数，改进了前人结果。
Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing geometric and combinatorial, rather than topological, features of the original. Our method is easy to apply to particular substitution tilings, permits experimentation, and can be used to construct bord…
In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ 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 the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer and Guy in 1994.
Icosahedron and dodecahedron can be dissected into tetrahedral tiles projected from 3D-facets of the Delone polytopes representing the deep and shallow holes of the root lattice D_6. The six fundamental tiles of tetrahedra of edge lengths 1 and τare assembled into four composite tiles whose faces are normal to the 5-fold axes of the icosahedral group. The 3D Euclidean space is tiled face-to-face by the composite tiles with an inflation factor τgenerated by an inflation matrix. The aperiodic tiling is a generalization of the Tubingen triangular tiling in 2-dimensions for the faces of the tiles …
This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is exhibited that can fill the Euclidean plane only with a tiling that contains k distinct isohedral sets of tiles, where k can be made arbitrarily large. It is shown that the construction cannot work for a simply connected 2D tile with matching rules for adjacent tiles enforced by shape alone. It is also shown that any of the following modifications allows the construction to work: (1) coloring the ed…
Two results about equidistribution of tile orientations in primitive substitution tilings are stated, one for finitely many, one for infinitely many orientations. Furthermore, consequences for the associated diffraction spectra and the dynamical systems are discussed.
A Recognizable Substitution for the Anti-Penrose Tiling
本文提出一种具有10重旋转对称性的菱形铺砌的替换规则，通过标记原型瓷砖和显式替换规则，证明其可识别性，从而能从局部配置中唯一恢复铺砌的层次结构。
Forcing nonperiodic tilings with one tile using a seed
本文探讨“爱因斯坦问题”的弱化形式，通过使用种子瓦片强制非周期平铺，并列举与讨论匹配规则的类型。
Undecidability of Translational Tiling of the Plane with Four Tiles
本文研究平面平移平铺的不可判定性，将所需多联骨牌数量从5个降至4个，证明了用4个不连通多联骨牌进行平面平移平铺不可判定。
Undecidability of Translational Tiling of the Plane with Orthogonally Convex Polyominoes
针对仅由几何形状决定的平面平移铺砌问题，本文摆脱以往极凹铺砌块构造，证明用一组 7 个正交凸多联骨牌铺砌平面仍不可判定。
We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar--Taylor tilings. Howe…
Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of \cite{AL} to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has a quasicrystalline structure.
We show that any primitive substitution tiling of the plane creates a separated net which is biLipschitz to the integer lattice. Then we show that if H is a primitive Pisot substitution in an Euclidean space, for every separated net Y, that corresponds to some tiling of the tiling space, there exists a bijection F between Y and the integer lattice that translate every element of Y a bounded distance. As a corollary we get that we have such an F for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.
We introduce a new general framework for constructing tilings of Euclidean space, which we call multiscale substitution tilings. These tilings are generated by substitution schemes on a finite set of prototiles, in which multiple distinct scaling constants are allowed. This is in contrast to the standard case of the well-studied substitution tilings which includes examples such as the Penrose and the pinwheel tilings. Under an additional irrationality assumption on the scaling constants, our construction defines a new class of tilings and tiling spaces, which are intrinsically different from t…
In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets associated with tilings of Euclidean space. First we consider substitution tilings, and exhibit a substitution matrix associated with two distinct substitution rules. The first rule generates onl…
Substitution tilings with transcendental inflation factor
本文针对任意λ>2，在无限字母表上构造替换，生成具有超越膨胀因子的替换平铺，证明相关动力系统严格遍历，并提供显式构造。
Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes
本文证明了用6个多立方体集合对三维整数格进行平移平铺的不可判定性，扩展了已知的二维结果，并推进了该领域的研究。
A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in an earlier paper, which characterizes the case of combinatorial tile-transitivity.
We describe a computer algorithm that searches for substitution rules on a set of triangles, the angles of which are all integer multiples of π/n. We find new substitution rules admitting 7-fold rotational symmetry at many different inflation factors.
Undecidability of tiling the plane with a fixed number of Wang bars
本文证明了使用29个Wang条铺满平面的问题是不可判定的，并进一步推导出颜色缺失25的Wang瓷砖集的铺砌问题同样不可判定。
Undecidability of Translational Tiling with Three Tiles
本文证明用三个连通平铺块对4维空间进行平移平铺是不可判定的，为固定维度下单调平铺不可判定猜想提供了新证据。
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.
An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer $k\geq 3$, there exists a normal tiling of the Euclidean plane by convex hexagons of unit area with exactly $k$ irregular vertices. Using the same approach we show that there are normal edge-to-edge tilings of the plane by hexagons of unit area and exactly $k$ many $n$-gons ($n>6$) of unit area. A result of Akopyan yields an upper bound for $k$ depending on the maximal diameter and minimum area of the tiles. Our result complement…
We prove that for the uniquely ergodic ${\bf R}^d$ action associated with a primitive substitution tiling of finite local complexity, every measurable eigenfunction coincides with a continuous function almost everywhere. Thus, topological weak-mixing is equivalent to measure-theoretic weak-mixing for such actions. If the expansion map for the substitution is a pure dilation by $θ>1$ and the substitution has a fixed point, then failure of weak-mixing is equivalent to $θ$ being a Pisot number.
Suppose $f\in L^1(\mathbb{R}^d)$, $Λ\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+Λ$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity. Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficien…
The $k$-tiling problem for a convex polytope $P$ is the problem of covering $\mathbb R^d$ with translates of $P$ using a discrete multiset $Λ$ of translation vectors, such that every point in $\mathbb R^d$ is covered exactly $k$ times, except possibly for the boundary of $P$ and its translates. A classical result in the study of tiling problems is a theorem of McMullen that a convex polytope $P$ that 1-tiles $\mathbb R^d$ with a discrete multiset $Λ$ can, in fact, 1-tile $\mathbb R^d$ with a lattice $\mathcal L$. A generalization of McMullen's theorem for $k$-tiling was conjectured by Gravin, …
We consider substitution tilings in R^d that give rise to point sets that are not bounded displacement (BD) equivalent to a lattice and study the cardinality of BD(X), the set of distinct BD class representatives in the corresponding tiling space X. We prove a sufficient condition under which the tiling space contains continuously many distinct BD classes and present such an example in the plane. In particular, we show here for the first time that this cardinality can be greater than one.
We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but CAN be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable nu…
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $β$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion $β$, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized $β$-transformation.
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