As we noted before, each semi-square can be characterized by yi + xi, the diagonal on which the hypotenuse lies. A clique C is now represented by two ordered lists: H(C), a list of lists of semi-squares for each diagonal on which at least one semi-square has its hypotenuse, and B(C) a list of lists of semi-squares for each y-coordinate on which at least one basis of any semi-square in the clique is positioned. A clique at a given x-position is now an assembly of ordered hypotenuses and bases as depicted in Fig. 5. A new semi-square X can intersect the set of bases and hypotenuses in the following ways: