Any two polygons can be continuously "extended" or ....?

[paizhaulski]

I am looking for theorems/information related to the following statement: any polygon can be created by an infinite number of infinitely small "extensions" or "croppings" of any other polygon, such that the shape is always a polygon (after any amount of extensions of croppings). For example, I can have a square, and then "continuously" "push" a small piece out from the top such that the final polygon is like a small square on top of a larger square.

 

[mfb]

Something along the line of the Von Neumann paradox, but without the paradox part?