Chapter 3 - Tiling by the Semicross and Cross

Summary

When we hear the expression “convex body” we probably visualize such famous sets as the ball, the cube, or the tetrahedron. However, the expression “non-convex body” triggers no specific image, just the general sense of an object with dents. In this chapter and the next we explore two families of non-convex sets that may well be viewed as the prototypes of non-convex bodies.

These two particular families of clusters in n-space have drawn the attention of mathematicians for several reasons. First, their tiling, packing, and covering properties can be analyzed with the aid of existing algebraic and combinatorial tools. Second, they raise many new questions, even about structures as simple as finite cyclic groups. Third, they are a convenient source of examples and counterexamples for questions concerning bodies that are not convex. Finally, they also appear naturally in such a real-world application as coding theory. In this chapter we define these two families and examine the way they tile n-space. In the next chapter we look at their packings and coverings. At the end of this chapter we sketch their history.

1. Definitions

In this chapter we will restrict our attention to Z-tilings. Recall that if a cluster tiles Rⁿ then it tiles Zⁿ (Theorem 4 in Chapter 2). However a cluster may lattice tile Rⁿ but not lattice tile Zⁿ, as was shown in Chapter 2 by the cluster consisting of two squares separated by a square.