Rectilinear polygon

[1] A rectilinear polygon that is also simple is also called hole-free because it has no holes - only a single continuous boundary.

It is possible to distinguish several types of squares/rectangles contained in a certain rectilinear polygon P:[1] A maximal square in a polygon P is a square in P which is not contained in any other square in P. Similarly, a maximal rectangle is a rectangle not contained in any other rectangle in P. A square s is maximal in P if each pair of adjacent edges of s intersects the boundary of P. The proof of both sides is by contradiction: The first direction is also true for rectangles, i.e.: If a rectangle s is maximal, then each pair of adjacent edges of s intersects the boundary of P. The second direction is not necessarily true: a rectangle can intersect the boundary of P in even 3 adjacent sides and still not be maximal as it can be stretched in the 4th side.

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).

In general polygons, there may be squares that are neither continuators nor separators, but in simple polygons this cannot happen:[1] 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.

A golygon is a rectilinear polygon whose side lengths in sequence are consecutive integers.

A T-square is a fractal generated from a sequence of rectilinear polygons with interesting properties.