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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.
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    So, to clarify the leaving-path-condition: You consider the edges of a cell as part of the cell, but not corners? Then you have to clarify this for higher dimensions, because a 3D-Polyhedron will have faces, edges, and corners (i.e. "0d/1d/2d"-boundaries). Which of these should be part of the cell and which don't? – 2017-02-22
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    @Simon, corners as well. Thanks. – 2017-02-22
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    @Simon, I kind of thought 'boundary' was a general term – 2017-02-22
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    yes, It is the general term. But in that case square lattice work fine (for all dimensions). If you leave a square (in d=2) you will have to go through one of the 4 nearest squares. At least though a corner of one of them. – 2017-02-22
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    @Simon, could you then help me with better wording of this intuitively clear concept? (I hope it is intuitively clear) – 2017-02-22
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    How about "two cells which are not nearest neighbours are not allowed to touch"? – 2017-02-22
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    @Simon, thank you. Though there was a popular question here about what 'touching' precisely means, and it was not resolved as fat as I remember. I think 'share a boundary' works – 2017-02-22
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    What do you mean by a "closest neighbour"? (I.e. how do you measure the distance between the cells.) Also: Please, calarify what do you mean by a "lattice": The standard definition does not mention any cells, it simply means "a discrete subgroup of $R^n$ isomorphic to $Z^n$." Do you mean Voronoi cells for such lattices? – 2017-02-28
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    What about [rhombic dodecahedron](https://en.wikipedia.org/wiki/Rhombic_dodecahedron)? – 2017-02-28
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    @SimonMarynissen, seems likely, thank you for the reference, I didn't know about this polyhedron. However, I can't be sure until I see how exactly they surround each other, so some 3d modelling is in order – 2017-03-01
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    You say *we define closeness in terms of Euclidean distance*, but as Moishe Cohen points out this is ambiguous. The ambiguity is critical to the meaning of what you are asking. E.g. if the distance between cells is defined as the length of a translation taking one cell to another, as mentioned in the first bullet, then diagonally adjacent squares in a chessboard pattern would not be nearest neighbours, but if the distance is defined as the infimum of the distances between a point of one cell and a point of the other, then they would. I think an explicit definition is needed. – 2017-03-03
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    @MartinRattigan, see the edit please – 2017-03-06

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