On the structure of multiple translational tilings by polygonal regions

用一组4个四维多立方体对四维空间进行平移铺砌的不可判定性

Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes

本文证明用一组5个三维多立方体铺砌三维空间不可判定，并通过提升技术进一步证明用4个四维瓦片铺砌四维空间不可判定，为固定维数下单一瓦片平移铺砌不可判定猜想提供新证据。

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…

关于用3个多立方体集合平铺3维空间的不可判定性

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。

三个平铺块的平移平铺的不可判定性

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。

Convex pentagons and concave octagons that can form rotationally symmetric tilings

In this study, the properties of convex pentagons that can form rotationally symmetric edge-to-edge tilings are discussed. Because the rotationally symmetric tilings are formed by concave octagons that are generated by two convex pentagons connected through a line symmetry, they are considered to be equivalent to rotationally symmetric tilings with concave octagons. In addition, under certain circumstances, tiling-like patterns with a regular polygonal hole at the center can be formed using these convex pentagons.

Convex pentagons and convex hexagons that can form rotationally symmetric tilings

In this study, the properties of convex hexagons that can form rotationally symmetric edge-to-edge tilings are discussed. Because the convex hexagons are equilateral convex parallelohexagons, convex pentagons generated by bisecting the hexagons can form rotationally symmetric non-edge-to-edge tilings. In addition, under certain circumstances, tiling-like patterns with an equilateral convex polygonal hole at the center can be formed using these convex hexagons or pentagons.

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。

双铺砌问题是不可判定的

Two Tiling is Undecidable

该研究证明双多边形原型砖经旋转平移铺砌平面的问题不可判定，改进了此前三块砖的不可判定结果，还首次证明带边对边局部匹配约束的单铺砌问题不可判定。

边界镶嵌：密集的数字图像具有大的连通组件

Tiling with Boundaries: Dense digital images have large connected components

研究了数字图像中连通组件的密集排列问题，通过结构论和等周不等式确定了最大像素密度限制，并展示了结果的紧致性。

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。

单个多立方体的平移平铺三维空间是不可判定的

Tiling 3D by Translates of a Single Polycube is Undecidable

该研究证明单个连通多立方体平移平铺三维欧氏空间的问题不可判定，通过归约Wang平铺问题完成证明，且三维是该类问题可判定性的最优维度。

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…

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。

Classification of tile digit sets as product-forms

Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathc…