论文标题
hitomezashi模式中的循环和区域
Loops and Regions in Hitomezashi Patterns
论文作者
论文摘要
Hitomezashi图案源自传统的日本刺绣，是单位长度线段的复杂布置称为针迹。针迹连接以形成hitomezashi链和hitomezashi循环，它们将平面分为区域。我们研究了这些模式的更深的数学特性，这些模式在角渗透研究中也具有突出的特性。以前知道，hitomezashi模式中的每个循环都具有奇怪的宽度和奇异的高度。我们还证明，这样的循环具有一致的$ 4 $ MODULO $ 8 $，而面积为$ 1 $ MODULO $ 4 $。尽管这些结果易于说明，但它们的证明要求我们了解可以应用于Hitomezashi模式的切片操作的精致拓扑和组合特性。我们还表明，随机$ m \ times n $ hitomezashi模式中的预期区域数（根据自然随机模型选择）渐近$ \ left（\ frac {π^2-9} {12} {12}+o（1）\ right）mn $。
Hitomezashi patterns, which originate from traditional Japanese embroidery, are intricate arrangements of unit-length line segments called stitches. The stitches connect to form hitomezashi strands and hitomezashi loops, which divide the plane into regions. We investigate the deeper mathematical properties of these patterns, which also feature prominently in the study of corner percolation. It was previously known that every loop in a hitomezashi pattern has odd width and odd height. We additionally prove that such a loop has length congruent to $4$ modulo $8$ and area congruent to $1$ modulo $4$. Although these results are simple to state, their proofs require us to understand the delicate topological and combinatorial properties of slicing operations that can be applied to hitomezashi patterns. We also show that the expected number of regions in a random $m\times n$ hitomezashi pattern (chosen according to a natural random model) is asymptotically $\left(\frac{π^2-9}{12}+o(1)\right)mn$.
