laminated lattice

A lattice (in the sense of def. 1) that is built up in layers. One takes an n-dimensional lattice, and in n+1-dimensional space place copies of it next to one another so as to build up an n+1-dimensional lattice. The vectors that determine how succeeding layers lie next to one another are called "glue vectors".

The lattice that is used to store cannonballs, (and to display fruit) is a laminated lattice built up out of successive layers of the hexagonal close packing.

--back to combinatorics--

Every (n+1)-dimensional lattice is built up of layers of n-dimensional lattices. An (n+1)-dimensional laminated lattice is built up of layers of n-dimensional laminated lattices packed as densely as possible. Thus, the definition of a laminated lattice is recursive, and what counts as a laminated lattice in n dimensions depends on what happens in all the lower dimensions. The process of "laminating" lattices can be viewed as a "greedy" way of constructing dense lattices. In some dimensions, this construction yields the densest possible lattices; in other dimensions, there are denser lattices that cannot be constructed in this way.