Dense packings of hard circular arcs

TL;DR

This paper explores the densest arrangements of congruent, infinitesimally-thin circular arcs with angles greater than π, revealing new packing structures and densities through analytical constructions and Monte Carlo simulations.

Contribution

It introduces analytical methods to construct densest packings of major circular arcs and compares these with numerical simulations to understand their spontaneous formation.

Findings

Analytical densest packings form triangular lattice arrangements.

Constructed packings depend on the subtended angle θ.

Monte Carlo simulations support the analytical results.

Abstract

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 (anti-)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.