I am interested in properties of regular neighbourhood meshes, but I feel like
I'm missing *keywords* to investigate further.

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In the 2D world I, like both triangular neighbourhood and square neighbourhood, because:

* they are both **regular**, which means to me:
	+ there is always *same distance* between two close neighbours (say, 1)
	+ there is always the *same angle* between directions towards two closest neighbours (60° for the triangle, 90° for the square)

[![enter image description here](https://i.stack.imgur.com/9xPfm.jpg)](https://i.stack.imgur.com/9xPfm.jpg)[![enter image description here](https://i.stack.imgur.com/Uh0S8.gif)](https://i.stack.imgur.com/Uh0S8.gif)

**But** I like triangular best, because:

* it offers more neighbours than the square (6 against 4 only)
* 2 closest neighbours are also neighbours themselves (whereas in the square world, if your he-friend goes one step north and your she-friend goes one step east, they are now *more* than 1 step away)

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In the 3D world, I only know the square neighborhood well.

[![enter image description here](https://i.stack.imgur.com/r0suj.png)](https://i.stack.imgur.com/r0suj.png)

But I don't like how it only offers 6 neighbours and how this last "neighbours neighbouring" feature is not respected.

Is there another regular 3D neighbourhood with more than 6 neigbours per node and featuring "close-neighbours-neighbouring"?
['https://math.stackexchange.com/questions/2491023', 'https://math.stackexchange.com', 'https://math.stackexchange.com/users/165143/']
My interpretation of the close-neighbours-neighbouring property is as follows:

> 
> For each node, the minimum separation among all distinct pairs of its neighbours is the same as the distance between the node and its neighbours.
> 
> 
> 

Then the [fcc lattice](https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres) (right side of the diagram below) satisfies the two properties, having 12 neighbours per node.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Close_packing_box.svg/1125px-Close_packing_box.svg.png)
The spaces between the nodes then form the [alternated cubic/tetrahedral–octahedral honeycomb](https://en.m.wikipedia.org/wiki/Tetrahedral-octahedral_honeycomb).
![](https://upload.wikimedia.org/wikipedia/commons/6/6a/Alternated_cubic_tiling.png)
Consider that a rhombic dodecahedron tiles Euclidean 3 space.