Counting Miura-ori Foldings

We consider the problem of enumerating the different ways in which the classic Miura map fold crease pattern can be folded flat. Specifically, we aim to count the number M(n,m) of ways to assign mountains and valleys to the creases so that each vertex in a m by n Miura map fold will be able to fold flat. Recurrence relations and closed formulas are found for small n and arbitrary m. We also prove that the array of numbers generated by M(n,m) is equivalent to the number of ways to properly 3-vertex-color a m× n grid graph with one vertex pre-colored. Research supported by the National Science Foundation grant EFRI-1240441 “Mechanical MetaMaterials from Self-Folding Polymer Sheets”.