Calculate the number of nearest neighbours in a particular lattice in $\mathbb R^d$ I have been wondering about the following for a while now:
Can you construct a lattice in $L \subset \mathbb R^d$ such that the euclidean distance between any two nearest neighbours is $1$ and we have the property that $\textbf{0} \in L$. Also we require that the lattice is maximal in some sense - not sure how to write this maximality down but hopefully examples will help: 


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*In $d = 1$, clearly $\mathbb Z$ is such a lattice

*In $d=2$, I am thinking of a hexagonal lattice with all side lengths equal to 1.
Questions


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*Does such a lattice exist for $ d > 2$? Is there a name for it?

*Is there a closed formula for the number of nearest neighbours, $N(d)$, in such a lattice? ($N(1) = 2, N(2) = 6, ... $)

 A: We can give natural measure of lattice maximality as density — number of points lying in some (big) cube relative to its volume (or, equivalently, volume of $\Bbb R^n/\Lambda)$. Another (more used) convention is to draw actual balls of diameter $1$ around points and calculate filled volume — I'll use this one, but they differ by only dimension dependent multiplicative constant — volume ratio of ball and cube. There's also more general question — what if we do not require our points to be an additive subgroup of $\Bbb R^n$, but only require that distance between them is $\geq 1$?


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*$n=1$: obviously $\Bbb Z \subset \Bbb R$ gives what we want

*$n = 2$: hexagonal lattice is densest (Fejes-Toth theorem, 1940)

*$n = 3$: cubical lattice (equal to layer-alternating hexagonal lattice) is densest (Hayes, 1998)

*$n = 8, 24$: E8 lattice and Leech lattice are densest (Viazovska, 2016)
All these constructions are most dense over all packings, not only lattice ones.
Little is known apart from it. There are some asymptotic bounds for $D(n)$ := maximal possible density:


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*$D(n) > 2^{-n}$: obvious

*$D(N) > 2^{-n+1}$: Minkovski, 1901

*$D(n) < 2^{-Kn}, K \simeq 0.599$, Kabatianski, Levenstein, 1973

*$D(n) > C \cdot n \cdot log \,log\, n \, 2^{-n}$: Venkatesh, 2012 
There's somewhat outdated (in terms of records) but very interesting book "Sphere Packings, Lattices and Groups" by J. H. Conway about all this business.
But, actually, situation is worse: even without requirement on some regularity, we do not know how many balls can touch given one! It's called kissing number problem. Only in 6 dimensions definite number is known — 1, 2, 3, 4, 8 and 24, and except for 4 optimal configuration is already listed above.
