The usual quantizer based on an n-dimensional lattice # maps a point x # R n to a closest lattice point. Suppose # is the intersection of lattices # 1 , . . . , # r . Then one may instead combine the information obtained by simultaneously quantizing x with respect to each of the # i . This corresponds to decomposing R n into a honeycomb of cells which are the intersections of the Voronoi cells for the # i , and identifying the cell to which x belongs. This paper shows how to write several standard lattices (the face-centered and body-centered cubic lattices, the root lattices D 4 , E # 6 , E 8 , the Coxeter-Todd, Barnes-Wall and Leech lattices, etc.) in a canonical way as intersections of a small number of simpler, decomposable, lattices. The cells of the honeycombs are given explicitly and the mean squared quantizing error calculated in the cases when the intersection lattice is the face-centered or body-centered cubic lattice or the lattice D 4 . 1.