标题： 球体堆积的接触数 显示英文标题

标题： Contact numbers for sphere packings

摘要： 在离散几何中，给定有限数量的非重叠球体的接触数被引入作为牛顿接吻数的推广。 这一概念不仅推动了有趣的数学发展，而且在自组装材料的科学领域（如胶体物质）也找到了应用。 由于几何学家、化学家、物理学家和材料科学家都在研究这一课题，因此有必要报告关于接触数问题的最新进展。 本文探讨了该问题的一般情况，并强调了一些重要的特殊情况，包括最小刚性和完全可分离球体堆积的接触数。 我们还讨论了在固定维度下识别接触图的复杂性。 此外，我们列出了一些猜想和开放问题。 显示英文摘要

摘要： In discrete geometry, the contact number of a given finite number of non-overlapping spheres was introduced as a generalization of Newton's kissing number. This notion has not only led to interesting mathematics, but has also found applications in the science of self-assembling materials, such as colloidal matter. With geometers, chemists, physicists and materials scientists researching the topic, there is a need to inform on the state of the art of the contact number problem. In this paper, we investigate the problem in general and emphasize important special cases including contact numbers of minimally rigid and totally separable sphere packings. We also discuss the complexity of recognizing contact graphs in a fixed dimension. Moreover, we list some conjectures and open problems.