发表机构
School of Physics, University of Bristol（布里斯托大学物理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为三维强非周期单调块建立最小面装饰要求，提出三色箭头匹配规则，并证明可放宽为仅基于方块颜色的规则，以促进物理实现。
AI 中文摘要
最近的一篇预印本[arXiv:2609.19214]提出了一种三维(3D)强非周期单调块：一种仅能以非周期方式铺满欧几里得空间且不允许无限阶对称性的形状。该证明采用了Lee和Moody先前识别的3D椅子块，并在其面上添加几何装饰以强制非周期性（没有这些装饰，椅子块也允许周期铺砌）。在此，我们建立关于面装饰的一般性要求以达到相同目的，从而促进物理实现的搜索。我们发现这些要求是最小的。我们提供了使用三种颜色箭头的匹配规则。这些规则与原规则不等价，但通过强制椅子块组合成线性尺寸加倍的超椅子块，从而强制相同的铺砌。在此过程中，匹配规则本身唯一地组合，这是先前证明的核心。进一步放宽此约束，我们发现仅使用基于方块颜色的匹配规则（无论方向如何）即可强制相同的结构。任何编码这些规则（几何或其他方式）的物理系统都将强制强非周期单调铺砌。我们提供了简单示例。
英文摘要
A recent pre-print [arXiv:2609.19214] proposed a three-dimensional (3D) strongly aperiodic monotile: a shape that tiles Euclidean space only non-periodically and which admits no symmetry of infinite order. The proof takes the 3D Chair tile identified previously by Lee and Moody, and adds geometric decorations to the faces so as to force non-periodicity (without these decorations The Chair also admits periodic tilings). Here we establish general requirements on face decorations to achieve the same end, in order to facilitate the search for physical realisations. We find that the requirements are minimal. We provide matching rules using three colours of arrow that are equivalent to the original rules, in that they force the same local and global configurations. These rules force Chairs to compose into `Superchairs' with doubled linear dimensions. In this process the matching rules themselves compose uniquely. We find that the same global structure can be forced using simpler rules based on the colours of squares, regardless of orientation. Any physical system encoding these rules (geometrically or otherwise) will force the same strongly aperiodic monotilings. We provide simple examples.
Comments6 pages, 4 figures. v2 updated to correct an error in Fig 2b, plus minor modifications elsewhere
