it can be stretched in the 4th side. Corollary: every maximal square/rectangle in P has at least two points, on two opposite edges, that intersect the boundary of P. A corner square is a maximal square s in a polygon P such that at least one corner of s overlaps a convex corner of P. For every convex corner, there is exactly one maximal (corner) square covering it, but a single maximal square may cover more than one corner. For every corner, there may by many different maximal rectangles covering it. A separator square in a polygon P is a square s in P such that P−s is not connected. Lemma: in a simple rectilinear polygon, a maximal square that does not contain a knob is a separator. A square containing a knob may or may not be a separator. The number of different separator squares may be infinite and even uncountable. For example, in a rectangle, every maximal square not touching one of the shorter sides is a separator. A continuator square is a square s in a polygon P such that the intersection between the boundary of s and the boundary of P is continuous. A maximal continuator is always a corner square. Moreover, a maximal continuator always contains a knob. Hence the number of continuators is always finite and bounded by the number of knobs. There are several different types of continuators, based on the number of knobs they contain and their internal structure (see figure). The balcony of a continuator is defined as its points that are not covered by any other maximal square (see figure). No square can be both a continuator and a separator. In general polygons, there may be squares that are neither continuators nor separators, but in simple polygons this cannot happen: In a simple rectilinear polygon, every maximal square is either a separator or a continuator. This is also true for rectangles: every maximal rectangle is either a separator or a continuator. In a simple rectilinear polygon which is not a square, there are at least two continuators. There is an interesting analogy between maximal squares in a simple polygon and nodes in a tree: a continuator is analogous to a leaf node and a separator is analogous to an internal node. Special cases The simplest rectilinear polygon is an axis-aligned rectangle - a rectangle with 2 sides parallel to the