Chapter 7 delves into a handful of combinatorial problems in flat origami theory that are more general than the single-vertex problems considered in Chapter 5. First, we count the number of locally-valid mountain-valley assignments of certain origami tessellations, like the square twist and Miura-ori tessellations. Then the stamp-folding problem is discussed, where the crease pattern is a grid of squares and we want to fold them into a one-stamp pile in as many ways as possible.Then the tethered membrane model of polymer folding is considered from soft-matter physics, which translates into origami as counting the number of flat-foldable crease patterns that can be made as a subset of edges from the regular triangle lattice.Many of these problems establish connections between flat foldings and graph colorings and statistical mechanics.

In Chapter 9 we explore the work of Kawasaki and Yoshida from1988, where group theory is used to relate the symmetries of a flat origami crease pattern to the symmetries of its folded image.This is then applied to origami tessellations to show that if the tessellation’s symmetries form a crystallographic group of the plane, then the symmetry group of the folded paper must be isomorphic to the symmetry group of the crease pattern.