Q:

Hexagonal lattice and geometry

![Hello, first of all thanks for all who take their time to read my question and try to put value out. My question is regarding the hexagonal lattice I posted as image. I extended it by my own but the numbers 11,12,13 & 14 seem to be not correct. That's the first problem that I don't know where and second what are the real distances for the lattice numbers 11, 12, 13 & 14 observed from point 0 regarding the geometry rules? 
Best regards.

A:

Take a look at the following representation that I just obtained :

One can see figures in the cells : they represent exact distances from the origin to the center of each cell (well, rounded to $10^{-2}$).
One can see arcs of circle centered at the origin numbered $1,2,3,...$ : their radii are the different distances one finds on the figure. 
Then your labelling follows this logic : a cell is labelled $n$ if the $n$th circle passes trough its center...
For example, the 12 cells at distance $2.65$ from the origin are labelled $4$ because circle n° 4 passes though their centers.
One finds back the first values given in your text :
$$1, \sqrt{3}=1.73, 2, \sqrt{7}=2.65, 3, 2\sqrt{3}=3.46...$$
About your numbering : I am afraid you have to reconsider it...

