Physical systems are regularly studied as spatial point sets, and so understanding the structure in such sets is a very natural problem. However, aside from several special cases, describing the manner in which a set of points can be arranged in space has been historically challenging. In the first part of this talk, I will show how consideration of the configuration space of local arrangements of neighbors, and a few simple results in metric geometry, can shed light on essential challenges of this problem, and in the classification of data more generally. In the second part of the talk I will introduce basic ideas from Voronoi cell topology and show how it can be used to define crystals, defects, and order more generally in a somewhat precise manner.