The properties which (as far as I know) are unique to the hexagonal lattice:
All of the cells have the same shape and there is perfect translatonal symmetry when shifting between the cells;
Each cell is completely surrounded by its closest neighbors (where we define 'closest' in terms of Euclidean distance between the centers of the cells) - i.e. there is no path from a cell which doesn't go through one of its closest neighbors first (we are including the boundaries of the cells of course).
Again, as far as I know only hexagonal lattice fits these properties. And I suppose an 1D lattice with equal segments, but this is a trivial case.
Square lattice ($d=2$) or cubic lattice ($d=3$) or any generalization for larger $d$ don't work, because there are ways to leave the cell without going through one of its closest neighbors first.
Truncated octahedral lattice in $d=3$ (which I consider a natural generalization of hexagonal lattice) doesn't work as well, since we have square faces of a truncated octahedron, and the disance in this direction is larger than for hexagonal faces.
Is there any other integer lattice for $d>1$ which has these two properties?
Edit
To clarify the second condition:
- Two cells which are not nearest neighbours are not allowed to touch (share boundary)
(Thanks, @Simon)
- Important! The lattice have to be space-filling as well. Euclidean space filling in case someone is wondering.