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.
Cite
@article{arxiv.1601.02727,
title = {Coloring connections with counting mountain-valley assignments},
author = {Thomas C. Hull},
journal= {arXiv preprint arXiv:1601.02727},
year = {2016}
}