Layer Sequencing and Flat-Foldability of Uniformly Gridded Polygons

Lunnon (1969) proposed a method to solve the Stamp Folding Problem which asks how many flat-foldable stack patterns there are for a given M by N rectangle grid (map). This project generalizes the original problem to include arbitrarily shaped polygons, including M by N maps. The purpose of this project is to develop a mathematical theorems that would determine whether or not a uniformly gridded polygon with a given stack pattern is flat-foldable. Furthermore, the theorems will determine the two mountain-valley assignments by which a gridded polygon with the given stack pattern can be folded flat. The results of the number of flat-foldable stack patterns for a given M by N map computed by this method is coincided by the results derived by Lunnon’s method. However, our method extends the scope of the original problem to more arbitrarily shaped polygons. This method will aide in the designing of foldable metal or paper structures.