Crystallographic terminology associated with the honeycomb structure I'm trying to quickly learn basic crystallographic principles in 2D under pressure (it's me whose under pressure, not the lattice) so I'm checking many sources to find those that "speak to me" the best (including this excellent answer). Many sources (books, papers, researcher's websites) cite the "honeycomb lattice" as an example of a lattice which is non-Bravais in order to reinforce their explanation of what a Bravais lattice is.


some online examples where honeycomb is a "lattice" but not Bravais: slide 3 and here and slide 7
some online examples where the "lattice" designator is avoided: here and slide 20
ambiguous or noncommittal as to being a "lattice" slide 6


Briefly (I don't have the various texts with me right now to transcribe the paragraphs) it seems to me that a 2D Bravais lattice any set of points $m \vec a + n \vec b$ for all integers $m, n$.

The vertices of a honeycomb don't satisfy this because if you try to use that expression, 1/3 of the points are missing. If you want to use a Bravais lattice, you have to define a unit cell with two "atoms" in it.

Question: Can the vertices of a honeycomb or "honeycomb lattice" really be called a proper lattice (but just not a Bravais lattice), or should we really call it either a net† or a regular complex apeirogons or something else, but not refer to it as a lattice?



From http://www-personal.umich.edu/~sunkai/teaching/Winter_2018/01302018.pdf (click for full size) 





†The Five nets

From Elizabeth A. Wood's classic Vocabulary of Surface Crystallography Journal of Applied Physics 35, 1306 (1964); https://doi.org/10.1063/1.1713610


  In triperiodic structures the equivalent points form
  a three-dimensional lattice in which the space units are
  unit cells. In diperiodic structures the equivalent points
  form a two-dimensional net in which the area units are
  unit meshes. The fourteen Bravais space lattices are
  replaced, in the diperiodic case, by five nets, described
  in Table I and diagrammed in Fig. 2.
the concept of lattice is very (too!) often misinterpreted outside crystallography. By definition, the lattice points are all identical, that is each lattice point has identical surroundings to all others. In a honeycomb pattern you can distinguish 2 different sets of points, the red and the blue ones in the figure.
So honeycomb pattern is actually a 2-dimensional structure (not a lattice), resulting from the convolution of a basis (in this casis constituted of one blu + one red point) with a 2D lattice (the hexagonal one in the figure representing the five nets)