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

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Main Question or Discussion Point

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.
 

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