Contact Numbers for Sphere Packings
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.
KeywordsSphere packings Kissing number Contact numbers Totally separable sphere packings Minimal rigidity Rigidity Erdős-type distance problems Colloidal matter
MSC (2010)(Primary) 52C17 52C15 (Secondary) 52C10
The first author is partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. The second author is supported by a Vanier Canada Graduate Scholarship (NSERC), an Izaak Walton Killam Memorial Scholarship and Alberta Innovates Technology Futures (AITF). The authors would like to thank the anonymous referee for careful reading and an interesting reference.
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