Conway and aperiodic tilings

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。

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…

通往非周期单瓦片之路

The Path to Aperiodic Monotiles

本文面向本科生介绍铺砌理论的关键问题，包括Heesch问题、等面数问题与非周期单瓦片的存在性，旨在激发更多学生研究该领域的兴趣。

乌龟、帽子与幽灵：菱形铺砌上的非周期结构

Turtles, Hats and Spectres: Aperiodic structures on a Rhombic tiling

本文借助菱形匹配规则、递归替换、Fibonacci 词切割投影及变形论证，给出 Turtle 与 Hat 非周期性的多种证明，并提出两条新的 Spectre 替换规则。

An aperiodic tile with edge-to-edge orientational matching rules

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…

Lots of Aperiodic Sets of Tiles

Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational applications, but its study and use has been hindered by a lack of easily generated examples. Through readily generalized, robust techniques for controlling hierarchical structure, we increase the catalogue of explicit constructions of aperiodic sets of tiles hundreds-fold, in lots, easily assembled and configured from atomic subsets of 211 tiles, enforcing 25,380 dis…

Substitutive structure of Jeandel-Rao aperiodic tilings

Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a nontrivial translation. We study herein the Wang shift $Ω_0$ made of all valid tilings using the set $\mathcal{T}_0$ of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift $X_0$ of $Ω_0$ such that every tiling in $X_0$ can be decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112…

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。

Local Rules for Computable Planar Tilings

Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question is to characterize, among a class of non-periodic tilings, the aperiodic ones. In this paper, we answer this question for the well-studied class of non-periodic tilings obtained by digitizing irrational vector spaces. Namely, we prove that such tilings are aperiodic if and only if the digitized vector spaces are computable.

All $SL_2$-tilings come from infinite triangulations

An $SL_2$-tiling is a bi-infinite matrix of positive integers such that each adjacent 2 by 2 submatrix has determinant 1. Such tilings are infinite analogues of Conway-Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory. We show that, by means of so-called Conway-Coxeter counting, every $SL_2$-tiling arises from a triangulation of the disc with two, three or four accumulation points. This improves earlier results which only discovered $SL_2$-tilings with infinitely many entries equal to 1. Indeed, our methods show that…

一种非周期单瓦片

An aperiodic monotile

本文解决了长期悬而未决的“爱因斯坦”问题，构造出一类连续的非周期单瓦片，并通过元瓦片替换规则、几何不可公度性论证及计算机辅助证明，证实其只能形成非周期铺砌。

一个手性非周期单调块

A chiral aperiodic monotile

研究仅用平移和旋转实现非周期铺砌的单形状问题，证明hat的等边近亲是弱手性非周期单调块，并修改其边缘得到严格手性非周期单调块族Spectres。

A note on reduction of tiling problems

We show that translational tiling problems in a quotient of $\mathbb{Z}^d$ can be effectively reduced or ``simulated'' by translational tiling problems in $\mathbb{Z}^d$. In particular, for any $d \in \mathbb{N}$, $k < d$ and $N_1,\ldots,N_k \in \mathbb{N}$ the existence of an aperiodic tile in $\mathbb{Z}^{d-k} \times (\mathbb{Z} / N_1\mathbb{Z} \times \ldots \times \mathbb{Z} / N_k \mathbb{Z})$ implies the existence of an aperiodic tile in $\mathbb{Z}^d$. Greenfeld and Tao have recently disproved the well-known periodic tiling conjecture in $\mathbb{Z}^d$ for sufficiently large $d \in \mathb…

Fixed point theorem and aperiodic tilings

We propose a new simple construction of an aperiodic tile set based on self-referential (fixed point) argument. People often say about some discovery that it appeared "ahead of time", meaning that it could be fully understood only in the context of ideas developed later. For the topic of this note, the construction of an aperiodic tile set based on the fixed-point (self-referential) approach, the situation is exactly the opposite. It should have been found in 1960s when the question about aperiodic tile sets was first asked: all the tools were quite standard and widely used at that time. Howev…

非周期结构永不崩溃：用于无损压缩的斐波那契层级

Aperiodic Structures Never Collapse: Fibonacci Hierarchies for Lossless Compression

研究非周期层级在无损压缩中的优势，证明斐波那契准晶体铺砌避免了周期性层级的有限深度崩溃，提升字典复用效率。

A metric characterisation of repulsive tilings

A tiling of $\mathbb{R}^d$ is repulsive if no $r$-patch can repeat arbitrarily close to itself, relative to $r$. This is a characteristic property of aperiodic order, for a non repulsive tiling has arbitrarily large local periodic patterns. We consider an aperiodic, repetitive tiling $T$ of $\mathbb{R}^d$, with finite local complexity. From a spectral triple built on the discrete hull $Ξ$ of $T$, and its Connes distance, we derive two metrics $d_{sup}$ and $d_{inf}$ on $Ξ$. We show that $T$ is repulsive if and only if $d_{sup}$ and $d_{inf}$ are Lipschitz equivalent. This generalises previous …

多重网格对偶铺砌上的多边形冠状极限

Polygonal corona limit on multigrid dual tilings

本文研究多重网格对偶铺砌中的逐细胞传播极限形状，通过定义多重网格与对偶铺砌的特征多边形，将 Penrose 铺砌中正十边形极限推广到任意正则情形。

An aperiodic hexagonal tile

We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The space--filling tiling that can be built from copies of the prototile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. There are two local isomorphism classes consistent with the matching rules and there is a nontrivial relation between these …

三骨牌平铺的五边形数定理

A pentagonal number theorem for tribone tilings

本文研究六边形网格中双参数近似六边形区域的 tribones 平铺问题，证明平铺存在当且仅当两个参数为成对五边形数。

非周期单形砖：从几何学到群论

Aperiodic monotiles: from geometry to groups

针对2023年两项分别基于几何与群论的非周期单形砖（爱因斯坦问题）成果，本文提出统一框架，通过群论平铺模拟满足条件的形砖系统，将Hat形砖转化为非周期群单形砖并分析其对称性。

Tiling Problems on Baumslag-Solitar groups

We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.

三维中的强非周期单瓦片

A Strongly Aperiodic Monotile in Three Dimensions

本研究提出三维强非周期单瓦片 Chair44，通过形状特征强制非周期性，证明其平铺无平移周期且对称群阶有限，并给出机器验证的严格证明。

An Integer Linear Programming Model for Tilings

In this paper, we propose an Integer Linear Model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how this model can be used to efficiently check the necessity of the Coven-Meyerowitz's $(T2)$ condition and also to define an iterative algorithm that finds all the possible tilings of the rhythm A. To conclude, we run several experiments to validate the time efficiency of this model.

非周期单铺块Tile(1,1)的同手性膨胀

Homochiral inflation for the aperiodic monotile Tile(1,1)

针对手性单铺块Tile(1,1)的准周期铺砌问题，提出同手性膨胀构造，固定各步骤手性，用两种簇分解得到12种元铺块并映射至三角晶格，实现对称要素分离并算出取向分布。

作为迭代函数系统的帽形多kite（hat polykite）

The hat polykite as an Iterated Function System

本研究将著名的非周期帽形铺砌对应到重叠迭代函数系统，通过研究分量一致收敛的无穷迭代函数系统序列，为帽形铺砌的非完美替换系统提供了理论建模。