As honeybees build their nests in preexisting tree cavities, they must deal with the presence of geometric constraints, resulting in nonregular hexagons and topological defects in the comb. In this work, we study how bees adapt to their environment in order to regulate the comb structure. Specifically, we identify the irregularities in honeycomb structure in the presence of various geometric frustrations. We 3D-print experimental frames with a variety of constraints imposed on the imprinted foundations. The combs constructed by the bees show clear evidence of recurring patterns in response to specific geometric frustrations on these starter frames. Furthermore, using an experimental-modeling framework, we demonstrate that these patterns can be successfully modeled and replicated through a simulated annealing process, in which the minimized potential is a variation of the Lennard-Jones potential that considers only first-neighbor interactions according to a Delaunay triangulation. Our simulation results not only confirm the connection between honeycomb structures and other crystal systems such as graphene, but also show that irregularities in the honeycomb structure can be explained as the result of analogous interactions between cells and their immediate surroundings, leading to emergent global order. Additionally, our computational model can be used as a first step to describe specific strategies that bees use to effectively solve geometric mismatches while minimizing cost of comb building.

中文翻译：

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几何挫折下蜂窝结构的结晶学

当蜜蜂在预先存在的树洞中筑巢时，它们必须处理几何约束的存在，从而导致蜂巢中出现不规则的六边形和拓扑缺陷。在这项工作中，我们研究了蜜蜂如何适应环境以调节蜂巢结构。具体来说，我们在存在各种几何挫折的情况下识别蜂窝结构中的不规则性。我们 3D 打印实验框架，对压印基础施加各种约束。蜜蜂建造的蜂巢清楚地显示出重复出现的图案，以响应这些起始框架上的特定几何挫折。此外，使用实验建模框架，我们证明可以通过模拟退火过程成功地建模和复制这些模式，其中最小化势是 Lennard-Jones 势的变体，它根据 Delaunay 三角剖分仅考虑第一邻域相互作用。我们的模拟结果不仅证实了蜂窝结构与其他晶体系统（如石墨烯）之间的联系，而且表明蜂窝结构中的不规则性可以解释为细胞与其周围环境之间类似相互作用的结果，从而导致出现全局秩序。此外，我们的计算模型可以作为第一步来描述蜜蜂用来有效解决几何不匹配同时最小化梳子构建成本的特定策略。我们的模拟结果不仅证实了蜂窝结构与其他晶体系统（如石墨烯）之间的联系，而且表明蜂窝结构中的不规则性可以解释为细胞与其直接环境之间的类似相互作用的结果，从而导致出现全局秩序。此外，我们的计算模型可以用作描述蜜蜂用来有效解决几何不匹配同时最小化梳子构建成本的特定策略的第一步。我们的模拟结果不仅证实了蜂窝结构与其他晶体系统（如石墨烯）之间的联系，而且表明蜂窝结构中的不规则性可以解释为细胞与其周围环境之间类似相互作用的结果，从而导致出现全局秩序。此外，我们的计算模型可以作为第一步来描述蜜蜂用来有效解决几何不匹配同时最小化梳子构建成本的特定策略。