Tilings

Tiling the plane with hexagons: improved separations for $k$-colourings

It has been common knowledge since 1950 that seven colours can be assigned to tiles of an infinite honeycomb with cells of unit diameter such that no two tiles of the same colour are closer than $d(7)=\frac{\sqrt{7}}{2}$ apart. Various authors have described tilings using $k>7$ colours, giving corresponding values for $d(k)$, but it is generally unknown whether these are the largest possible for a given $k$. Here, for many $k$, we describe tilings with larger values of $d(k)$ than previously reported.

用5个多联骨牌平铺平面的不可判定性

Undecidability of Tiling the Plane with a Set of 5 Polyominoes

本文证明五个多联骨牌能否平移平铺平面是不可判定的，通过新边标记法并专用一个多联骨牌实现匹配判定。

全等六边形对平坦环面的铺砌

Tilings of Flat Tori by Congruent Hexagons

本文在所有顶点度数为3的条件下，分类三类一般六边形及两种特殊情形的平面铺砌，并据此刻画平坦环面的六边形铺砌与模空间。

平面半正镶嵌的完美精确着色

Perfect precise colorings of plane semiregular tilings

本文研究平面半正镶嵌的完美精确着色问题，为$k\leq 6$的某些$k$价半正镶嵌族构造了满足对称群置换与顶点邻接颜色约束的着色方案。

具有奇数n的2n重旋转对称性的替换离散平面平铺

Substitution discrete plane tilings with $2n$-fold rotational symmetry for odd n

本文研究替换离散平面平铺，证明Sub Rosa平铺不是离散平面，并提出新的Planar Rosa替换平铺，证明其满足离散平面条件，提供了10重情况的显式构造和一般奇数n的构造方法。

$SL_k$-Tiling of the Plane

We study properties of (bi-infinite) arrays having all adjacent $k\times k$ adjacent minors equal to one. If we further add the condition that all adjacent $(k-1)\times (k-1)$ minors be nonzero, then these arrays are necessarily of rank $k$. It follows that we can explicit construct all of them. Several nice properties are made apparent. In particular, we revisit, with this perspective, the notion of frieze patterns of Coxeter. This shed new light on their properties. A connexion is also established with the notion of $T$-systems of Statistical Physics.

网格着色上的边覆盖平面填充曲线：一种直观方法

Edge-covering plane-filling curves on grid colorings: a pedestrian approach

本文研究正多边形平面铺嵌上的边覆盖平面填充曲线，通过将铺嵌对称群的边类划分为对应于有限子群的精化着色，构造出称为曲线集的平面填充曲线族，并以图示示例直观展示。

Perfect k-colored matchings and (k+2)-gonal tilings

We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.

Escher degree of non-periodic L-tilings by 2 prototiles

For a given tiling of the euclidean plane ${\bf E}^2$, we call the degree of freedom of perturbed edges of prototiles {\it escher degree}. In this paper we consider non-periodic L-tilings by 2 prototiles and obtain the escher degree of them.

Semi-regular tilings of the hyperbolic plane

A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.

Tilings of the plane with unit area triangles of bounded diameter

There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar.

Pair Correlation and Gap Distributions for Substitution Tilings and Generalized Ulam Sets in the Plane

We study empirical statistical and gap distributions of several important tilings of the plane. In particular, we consider the slope distributions, the angle distributions, pair correlation, squared-distance pair correlation, angle gap distributions, and slope gap distributions for the Ammann Chair tiling, the recently discovered fifteenth pentagonal tiling, and a few pertinent tilings related to these famous examples. We also consider the spatial statistics of generalized Ulam sets in two dimensions. Additionally, we carefully prove a tight asymptotic formula for the time steps in which Ulam …

平面上的花边图案与非周期平铺

Frieze patterns and aperiodic tilings of the plane

研究平面非周期花边图案，基于菱形彭罗斯平铺和戈德雷什 - 兰松 - 比拉尔平铺给出例子，通过用正整数装饰顶点使其满足菱形规则，类似康威和考克斯特定义的图案。

带侵入的六边形菱形铺砌 I：广义侵入

Lozenge Tilings of Hexagons with Intrusions I: Generalized Intrusion

本文通过从六边形区域移除广义侵入结构构造新区域，证明其菱形铺砌数量及带权生成函数具有简单乘积公式，从而推广 MacMahon 定理并给出 $q$-模拟。

使用正交凸多联骨牌对平面进行平移铺砌的不可判定性

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。

Counting perfect colourings of plane regular tilings

A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of the uncoloured pattern induces a global permutation of the colours. Two cases are distinguished: Either perfect colourings with respect to all symmetries, or with respect to orientation preserving symmetries only (no reflections). For the important class of colourings of regular tilings (and some Laves tilings) of the Euclidean or hyperbolic plane, this mainl…

Enumeration of lozenge tilings of hexagons with cut off corners

Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with ``maximal staircases'' removed from some of its vertices. The case of one vertex corresponds to Proctor's problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose t…

Construction of the discrete hull for the combinatorics of a regular pentagonal tiling of the plane

The article 'A "regular" pentagonal tiling of the plane' by P. L. Bowers and K. Stephenson defines a conformal pentagonal tiling. This is a tiling of the plane with remarkable combinatorial and geometric properties. However, it doesn't have finite local complexity in any usual sense, and therefore we cannot study it with the usual tiling theory. The appeal of the tiling is that all the tiles are conformally regular pentagons. But conformal maps are not allowable under finite local complexity. On the other hand, the tiling can be described completely by its combinatorial data, which rather auto…

Planar Rosa：一类具有 $2n$ 阶旋转对称性的准周期替换离散平面平铺

Planar Rosa : a family of quasiperiodic substitution discrete plane tilings with $2n$-fold rotational symmetry

提出 Planar Rosa 替换平铺族，证明其具有 $2n$ 阶旋转对称性且满足离散平面条件，并指出 Sub Rosa 平铺不满足该条件，完善了奇偶 $n$ 的结果。

Hexagon tilings of the plane that are not edge-to-edge

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…

三角网格中的三聚体覆盖：二十个大多未解决的问题

Trimer covers in the triangular grid: twenty mostly open problems

本文将三角网格铺砌研究从二聚体推广到三聚体，引入苯环区域并提出二十个大多未解决的精确计数问题。

用固定数量的Wang条铺满平面的不可判定性

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。

normed平面中的蜂窝猜想及Dowker定理的一个α-凸变种

The Honeycomb Conjecture in normed planes and an alpha-convex variant of a theorem of Dowker

本文研究了normed平面中的蜂窝猜想，证明了正凸平铺的平均平方周长最小，并探讨了与Dowker定理α-凸变种相关的联系，进一步给出了关于平铺单元等周比的问题解答。

用四个瓦片进行平面平移平铺的不可判定性

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。

平面的六边形铺砌

Hexagonal Tiling of the Plane

本文在弱于凸性的假设下证明了平面中恰好存在三种能铺砌的凸六边形。

Tegula -- exploring a galaxy of two-dimensional periodic tilings

Periodic tilings play a role in the decorative arts, in construction and in crystal structures. Combinatorial tiling theory allows the systematic generation, visualization and exploration of such tilings of the plane, sphere and hyperbolic plane, using advanced algorithms and software.Here we present a "galaxy" of tilings that consists of the set of all 2.4 billion different types of periodic tilings that have Dress complexity up to 24. We make these available in a database and provide a new program called Tegula that can be used to search and visualize such tilings. Availability: All tilings …

A Note on Bootstrap Percolation Thresholds in Plane Tilings using Regular Polygons

In \emph{$k$-bootstrap percolation}, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected) face has at least $k$ infected neighbors, then it becomes infected. For fixed $G$ and $p$, the \emph{percolation threshold} is the largest $k$ such that eventually all faces become infected, with probability at least $1/2$. For a large class of infinite graphs, we show that this threshold is independent of $p$. We consider bootstrap percolation in tilings o…

Ollinger 猜想的证明：用一组 8 个多联骨牌铺砌平面的不可判定性

A proof of Ollinger's conjecture: undecidability of tiling the plane with a set of $8$ polyominoes

本文证明 Ollinger 猜想，通过改变多联骨牌模拟 Wang tiles 的朝向并提出颜色编码新方法，确立用 8 个多联骨牌平移铺砌平面的不可判定性。

Periodic structure of translational multi-tilings in the plane

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…

A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings

We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at $-x_1,...,-x_n$, if $M$ is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if $M$ is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus tilings of a hexagon, which has an equivalent formulation in terms of plane partitions. A simila…

共形铺砌、组合曲率与类型问题

Conformal Tilings, Combinatorial Curvature, and the Type Problem

本文研究共形铺砌，证明对n≥7不存在复平面的边对边共形正则n边形铺砌，并揭示组合曲率与底层黎曼曲面万有覆盖类型的关系，基于Stone和Oh的工作给出新证明与应用。

由右黄金三角形构成的非周期平面铺砌家族

A family of non-periodic tilings of the plane by right golden triangles

本文研究了一种由两种大小右三角形构成的替换铺砌家族，证明其可通过局部规则用装饰瓷砖生成，即为sofic铺砌。

A factorization theorem for lozenge tilings of a hexagon with triangular holes

In this paper we present a combinatorial generalization of the fact that the number of plane partitions that fit in a $2a\times b\times b$ box is equal to the number of such plane partitions that are symmetric, times the number of such plane partitions for which the transpose is the same as the complement. We use the equivalent phrasing of this identity in terms of symmetry classes of lozenge tilings of a hexagon on the triangular lattice. Our generalization consists of allowing the hexagon have certain symmetrically placed holes along its horizontal symmetry axis. The special case when there …

Deciding multiple tiling by polygons in polynomial time

Suppose $P$ is a symmetric convex polygon in the plane. We give a polynomial time algorithm that decides if $P$ can tile the plane by transations at some level (not necessarily at level one; this is multiple tiling). The main technical contribution is a polynomial time algorithm that selects, if this is possible, for each $j=1,2,\ldots,n$ one of two given vectors $e_j$ or $τ_j$ so that the selection spans a discrete additive subgroup.

A class of maps on the torus and their vertex orbits

A tiling (edge-to-edge) of the plane is a family of tiles that cover the plane without gaps or overlaps. Vertex figure of a vertex in a tiling to be the union of all edges incident to that vertex. A tiling is $k$-vertex-homogeneous if any two vertices with congruent vertex figures are symmetric with each other and the vertices form precisely $k$ transitivity classes with respect to the group of all symmetries of the tiling. In this article, we discuss that if a map is the quotient of a plane's $k$-vertex-homogeneous lattice ($k \ge 4$) then what would be the sharp bounds of the number of verte…

范畴平铺理论：通过边反转构造有向平面平铺

Categorical Tiling Theory: Constructing Directed Planar Tilings via Edge Reversal

本文提出范畴平铺理论，通过构造有限范畴和预层模型，将平铺对齐问题形式化，并利用边反转构造反射对齐，生成具有对称性的对齐族。

Low-Complexity Tilings of the Plane

A two-dimensional configuration is a coloring of the infinite grid Z^2 with finitely many colors. For a finite subset D of Z^2, the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. This extended abstract is a short review of an algebraic method to study periodicity of such low complexity configurations.

Quotient Representations of Uniform Tilings

Given a flag in each of the vertex-transitive tessellations of the Euclidean plane by regular polygons, we determine the flag stabilizer under the action of the automorphism group of a regular cover. In so doing we give a presentation of these tilings as quotients of regular (infinite) polyhedra.

What percent of the plane can be properly 5- and 6-colored?

We present a tiling of more than 99.985698% of the Euclidean plane with six colors, reducing the previous record for uncovered fraction of the plane by about 12.8%. We also present a tiling of more than 95.99% of the plane with five colors. It is thus shown that any unit-distance graph of order at most 6992 and 24 in the plane can be properly 6-colored and 5-colored, respectively.

Deltoid tangents with evenly distributed orientations and random tilings

We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile rearrangements in the substitution rules associated to edge flips.

由 Ammann A2 非周期镶嵌导出的图的平均度

Average Degree of Graphs Derived From The Ammann A2 Aperiodic Tiling

本文研究由 Ammann A2 非周期镶嵌导出的图，将瓦片角点视为顶点、边视为边，并给出该图平均度的闭式公式。

关于平面中高阶 Brillouin 镶嵌及相关铺砌的角度

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

研究平面中局部有限集合的高阶 Brillouin 镶嵌及相关铺砌的角度单调性，证明平稳 Poisson 点过程下各类镶嵌的角度分布独立于阶数。

7个多边形的翻译非周期集

Translational Aperiodic Sets of 7 Polyominoes

本文证明了用7个多边形进行平面翻译铺砌是不可判定的，并构建了大小为7的非周期集，打破了Ammann的30年记录。

Three interactions of holes in two dimensional dimer systems

Consider the unit triangular lattice in the plane with origin $O$, drawn so that one of the sets of lattice lines is vertical. Let $l$ and $l'$ denote respectively the vertical and horizontal lines that intersect $O$. Suppose the plane contains a pair of triangular holes of side length two, distributed symmetrically with respect to $l$ and $l'$, and oriented so that both holes point toward $O$. Unit rhombus tilings of three different regions of the plane are considered, namely: tilings of the entire plane; tilings of the half plane that lies to the left of $l$ (where $l$ is considered a free b…

An aperiodic monotile that forces nonperiodicity through dendrites

We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar--Taylor monotile, but can be realised by shape alone. The second is a local growth rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forc…