In Chapter 1 we were concerned with the way translates of a single cube fit together to tile space. In this chapter we examine tilings by translates of a finite collection of cubes, which we will call “clusters.” Chapters 3 and 4 will treat a special family of clusters that exists in all dimensions. Before we can state the main results of this chapter, we need some definitions.

As in Chapter 1, we assume a fixed coordinate system. We continue to identify each unit cube whose edges are parallel to the axes with its vertex that has the smallest coordinates. An n-dimensional cluster C is the finite union of unit cubes whose edges are parallel to the axes and which have integer coordinates. A cluster is not necessarily connected

Let C be a fixed cluster in n-space and assume that L is a set of vectors in n-space such that the set of translates {ν + C:ν ∈ L} tile n-space. (For a given cluster there may be no such lattice.) If all the coordinates of all the vectors in L are integers (rational numbers), we speak of an integer tiling (rational tiling) by C, or simply a Z-tiling (Q-tiling). If L is a lattice we speak of lattice tiling by C. Combining the two notions, we speak of a Z-lattice tiling and a Q-lattice tiling.

We will prove the following theorems, all of which concern tilings by translates of a cluster.