The identification and classification of the crystallographic space groups took much of the nineteenth century, and the confirmation of the completeness of the list was finished by the theorems of Evgraf Fedorov and Arthur Schoenflies.
The interpretation in science and engineering is that since a Euclidean graph representing a material extending through space must satisfy conditions (1), (2), and (3), non-crystalline substances from quasicrystals to glasses must violate (4).
The cycles of a crystal net are related[8] to another invariant, that of the coordination sequence (or shell map in topology[9]), which is defined as follows.
See[10] [11] for a mathematical aspect of topological density which is closely related to the large deviation property of simple random walks.
One of the major systematic crystal net enumeration algorithms extant[14] is based on the representation of tessellations by a generalization of the Schläfli symbol by Boris Delauney and Andreas Dress, by which any tessellation (of any dimension) may be represented by a finite structure,[15] which we may call a Dress–Delaney symbol.
The three-dimensional Dress–Delaney symbol enumerator of Delgado-Friedrichs et al. has predicted several novel crystal nets that were later synthesized.
The extension of the symmetry group to 3-space permits the characterization of a fundamental domain (or region) of 3-space, whose intersection with the net induces a subgraph which, in general position, will have one vertex from each orbit of vertices.
[19] Other programs have been developed that similarly generate copies of an initial fragment and glue them into a periodic graph[20]