This work investigates dense packings of congruent hard infinitesimally thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle $\theta \in \left(\pi ,2\pi \right]$. Differently than those denotable as minor whose subtended angle $\theta \in \left[0,\pi \right]$, it is impossible for two hard infinitesimally thin circular arcs with $\theta \in \left(\pi ,2\pi \right]$ to arbitrarily closely approach once they are arranged in a configuration, e.g., on top of one another, replicable ad infinitum without introducing any overlap. This makes these hard concave particles, in spite of being infinitesimally thin, most densely pack with a finite number density. This raises the question as to what are these densest packings and what is the number density that they achieve. Supported by Monte Carlo numerical simulations, this work shows that one can analytically construct compact closed circular groups of hard major circular arcs in which a specific, $\theta$-dependent, number of them (counter) clockwise intertwine. These compact closed circular groups then arrange on a triangular lattice. These analytically constructed densest-known packings are compared to corresponding results of Monte Carlo numerical simulations to assess whether they can spontaneously turn up.

中文翻译：

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硬圆弧的密堆积

这项工作研究了二维欧几里得空间中全等硬的无限薄的圆弧的密集堆积。它侧重于那些被认为是专业的，其对向角度$\theta \in \left(\pi ，2\pi \right]$。不同于那些被称为对角的未成年人$\theta \in \left[0，\pi \right]$，不可能有两个坚硬的无限细的圆弧 $\theta \in \left(\pi ，2\pi \right]$一旦将它们以一种配置（例如彼此叠置）的方式布置，就可以无限紧密地接近，而不会引入任何重叠。这使这些坚硬的凹形颗粒尽管无限薄，却以有限的数密度密集地堆积。这就提出了关于这些最密堆积物是什么以及它们达到多少数量密度的问题。在蒙特卡洛（Monte Carlo）数值模拟的支持下，这项工作表明，人们可以解析地构造硬主要圆弧的密闭圆组，其中特定的，$\theta$取决于它们的数量（计数器）顺时针交织。然后，这些紧凑的闭合圆形组排列在三角形格子上。将这些分析构造最密集的填料与相应的蒙特卡洛数值模拟结果进行比较，以评估它们是否可以自发地翘起。