Iwan Praton
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.
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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°.
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.
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。
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…
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.
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.
Can the entire plane be paved with a single tile that forces aperiodicity? This is known as the ein Stein problem (in German, ein Stein means one tile). This paper presents a monotile that delivers aperiodic tiling by design. It is based on the monotile developed by Taylor and Socolar (whose aperiodicity is forced by means of a non-connected tile that is mainly hexagonal) and motif-based hexagonal tilings that followed this major discovery. Here instead, a single substitution rule makes its shape, and when applying it, forces the tiling to be aperiodic. The proposed monotile, called HexSeed, i…
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。
We prove that for every $N\ne 4$ there is only one right triangle that tiles the regular $N$-gon.
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…
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…
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.
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。
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…
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 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…
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…
We consider polygons with the following ``pairing property'': for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon $K$ tiles the plane multiply when translated at the locations $Λ$, where $Λ$ is a multiset in the plane. The pairing property of $K$ makes this question particularly amenable to Fourier Analysis. After establishing a necessary and sufficient condition for $K$ to tile with a given lattice $Λ$ (which was first found by Bolle for the case of convex polygons-notice that all convex polygons that tile, necessarily hav…
We show it is possible to tile three-dimensional space using only tetrahedra with acute dihedral angles. We present several constructions to achieve this, including one in which all dihedral angles are less than $77.08^\circ$, and another which tiles a slab in space.
The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of quadrilateral subdivisions of the octahedron with $24$ tiles together with a flip modification for a special parameter, and a sequence of $3$-layer earth map tilings with $8n$ $(n\ge2)$ tiles together with two flip modifications for odd $n$. We also describe the moduli and calculate the geometric data.
On the Undecidability of Tiling the $3$-dimensional Space with a Set of $3$ Polycubes
本文针对单块平移平铺不可判定性的相关猜想，证明了用3个多立方体集合对3维空间进行平移平铺是不可判定的，为该猜想的证明提供了低维少瓦片的支撑结果。
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 appearing in the geometrization conjecture.
Periodic double tilings of the plane
针对由两个不同面积重复瓷砖组成的平面周期性铺砌，分类固定与任意晶格下的等周构型，依据面积比发现三种含不同圆形边缘数量且对应不同最优晶格的构型，计算等周轮廓并提出开放问题。
On monohedral tilings of a regular polygon
本文连接了正方形和圆盘的三块铺嵌结果，刻画了任意正n边形（n≥5）被至多三个拓扑圆盘单面铺嵌的情况。
We define rules for cellular automata played on quasiperiodic tilings of the plane arising from the multigrid method in such a way that these cellular automata are isomorphic to Conway's Game of Life. Although these tilings are nonperiodic, determining the next state of each tile is a local computation, requiring only knowledge of the local structure of the tiling and the states of finitely many nearby tiles. As an example, we show a version of a "glider" moving through a region of a Penrose tiling. This constitutes a potential theoretical framework for a method of executing computations in no…
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