由尺寸为1、2、3和4的正方形构成的单侧等可迁铺砌

中文摘要 AI 辅助

本文研究了Grünbaum与Shephard经典铺砌理论著作《Tilings and Patterns》中提出的问题：寻找所有由尺寸为1、2、3、…、n（n为自然数）的正方形构成的单侧等可迁铺砌。单侧性条件要求此类铺砌中相同尺寸的两个正方形不能沿整条边相交；等可迁性要求此类铺砌中相同尺寸的两个 tile（铺砌单元）必须属于铺砌对称群下的同一可迁类。这两个条件使正方形铺砌呈现出惊人有趣且复杂的结构。本研究针对n=4的情况给出了该问题的答案，同时开发了可扩展至n≥5情况的方法与计算机算法。

英文摘要

In this article we examine a problem stated in Grübaum and Shephard's classic book on tiling theory, Tilings and Patterns, which asks the reader to find all unilateral and equitransitive tilings by squares of sizes $1, 2, 3, \ldots, n$ where $n \in \mathbb{N}$. The unilaterality condition requires that two squares of the same size in such a tiling may not intersect along an entire side, and the equitransitivity requires that two tiles of the same size in such a tiling must be in the same transitivity class with respect to the symmetry group of the tiling. These two conditions on tilings by squares allows for surprisingly interesting and complex tilings by squares. In this work we provide the answer to Grübaum and Shephard's question for the case $n = 4$, as well as developing methods and computer algorithms that may be extended to the cases where $n \geq 5$.

发表机构

• University of Washington Bothell（华盛顿大学博塞尔分校）
• University of Wisconsin–Oshkosh（威斯康星大学奥什科什分校）
• University of Nebraska-Lincoln（内布拉斯加大学林肯分校）
• Tufts University（塔夫茨大学）

机构由 AI 辅助整理，请以论文原文为准。