Low-Dimensional Lattices VII: Coordination Sequences

J. H. Conway, 
Mathematics Department 
Princeton University,
Princeton, NJ 08540 

N. J. A. Sloane 
Information Sciences Research 
AT&T Research
Murray Hill, NJ 07974 

Present address:
Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971

July 9, 1996 

ABSTRACT

The coordination sequence {S(n)}  of a lattice or net gives the number
of nodes that are n bonds away from a given node.
S(1) is the familiar coordination number.

Extending work of O'Keeffe and others, we give explicit
formulae for the coordination sequences of the root lattices
A_d, D_d, E_6, E_7, E_8 and their duals.

Proofs are given for many of the formulae, and for the fact that in every case
S(n) is a polynomial in n, although some of the individual
formulae are conjectural.

In the majority of cases the set of nodes that are at most n bonds
away from a given node form a polytopal cluster whose shape
is the same as that of the contact polytope for the lattice.

It is also shown that among all the Barlow packings in three dimensions
the hexagonal close packing has the greatest coordination sequence, and
the face-centered cubic lattice the smallest, as conjectured by O'Keeffe.


This paper was published (in a somewhat different form) in
Proc. Royal Soc. London, Series A, Vol. 453 (1997), 2369-2389.


For the full version see
http://www.research.att.com/~njas/doc/ldl7.pdf (pdf) or
http://www.research.att.com/~njas/doc/ldl7.ps (ps)