Coloring connections with counting mountain-valley assignments

TL;DR

This paper surveys methods for counting mountain-valley crease assignments in origami patterns, providing solutions for specific tessellations and exploring complex cases like Miura-ori through graph colorings.

Contribution

It introduces new enumeration techniques for crease patterns, extends existing methods to broader families, and establishes a bijection with grid graph colorings for complex origami structures.

Findings

Solved counting problem for square twist tessellations

Generalized enumeration method to broader crease pattern families

Established a bijection between Miura-ori foldings and grid graph colorings

Abstract

We survey more recent attempts at enumerating the number of mountain-valley assignments that allow a given crease pattern to locally fold flat. In particular, we solve this problem for square twist tessellations and generalize the method used to a broader family of crease patterns. We also describe the more difficult case of the Miura-ori and a recently-discovered bijection with 3-vertex colorings of grid graphs.