4. Generalizations.

4.1 Figures of constant width in many directions.

Instead of row, columns and diagonals, we now look for figures of constant width along a predetermined set of directions. Here I'll briefly sketch their construction.

A direction of the chessboard is determined by a vector d=(a,b) in x (we identify the infinite chessboard with x ). The lines of the chessboard along this direction are the sets

Definition. A figure of constant width w along the set of directions D is one such that every line with direction d in D intersects the figure in exactly w squares.

In order to build figures of constant width on a given set of directions I'll make use of the notions of composition of figures and of extended constant width . The reader is referred to [Her-Rob] and [Riv-Var-Zimm] for the definitions. The following proposition is easily verified.

Proposition 3. If A has constant width v along d and B has extended constant width w along d then A composed with B has constant width vw along d.

Let D be a given set of directions. We assume that any two directions in D are linearly independent. This implies that any two lines with different directions d and d' intersect in at most one square. For d in D let F(d) be the figure {kd, k=0,1,2,...,w-1}. We can always enclose this figure inside a finite chessboard of size N(d)xN(d) (N(d) chosen big enough), so that F(d) has extended constant width w along the direction d and extended constant width 1 along all the other directions. We now take F to be the composition of all figures F(d), d in D. By Proposition 3 the figure F has constant width w along all directions in D. Here is an example with w=2 and D={(0,1),(1,2),(1,-1)}:

Remarks.
1. The operation of composition is not associative. Thus, in order that F be well defined we need to specify the order in which the compositions are taken. No matter what order we choose, the resulting figure will have constant width.
2. Our construction of figures of constant width along the set of directions D is far from proving a theorem like Theorem 1. In particular, it is not clear whether there are figures of type (n,n,...,n,w). Unfortunately, the proof of Theorem 1 cannot be generalized to this case.

4.2 Figures of constant width on a toroidal chessboard.

As in the case of the n-queens problem, there are some arithmetical osbtructions to having a figure of type (n,k,w) on a toroidal chessboard.

4.3 Figures of constant width on an n-dimentional chessboard.

Nothing to say.