Hexagonal Tilings and Locally C6 Graphs

多边形平铺的1-2-3猜想

The 1-2-3 conjecture for polygonal tilings

研究基于多边形平铺的1-2-3猜想，通过利用非周期性平铺的额外结构，找到显式解，并为完全周期性平铺提供算法。

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 …

柔软镶嵌

Softening locally polyhedral tilings

本文通过构造新的边弯曲算法，证明了三维局部多面体镶嵌可完全柔软化。并给出了Domokos等人关于平面平衡多边形镶嵌平均尖刺数的结论简证。

A Local Characterization of Combinatorial Multihedrality in Tilings

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.

六边形网格图的独立复形

Independence Complexes of Hexagonal Grid Graphs

本文研究了六边形网格图的独立复形拓扑结构，确定了不同参数下的同伦类型，并给出了递归描述。

完美t-嵌入与菱形密铺

Perfect t-embeddings and Lozenge Tilings

本文首次为六角格正六边形构造完美t-嵌入，利用逆Kasteleyn矩阵与对称性推导精确公式，证明折纸映射收敛到Minkowski空间极大曲面，并给出高度涨落收敛到高斯自由场的新证明。

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。