In this chapter, we consider two of the simplest types of crease patterns, with the goal of characterizing when they arise as the crease patterns of origami, particularly flat origami (see Figure 12.1).

In the first type of crease pattern, all creases are parallel to each other. In this case, the shape of the paper is not important, so we can imagine a long thin strip of paper with all creases perpendicular to the strip. In fact, we can imagine the paper as one-dimensional, a line segment with points marking creases.

In the second type of crease pattern, all creases are incident to a single common vertex. Again, in this case, the shape of the paper is irrelevant, so we view it to be a unit disk centered at the sole vertex of the crease pattern. At a high level, the two types of crease patterns are related: the first type is a limiting case of the second type in which the radius of the disk is infinite.

We explore two natural problems about flat foldings of either type of crease pattern:

1. Characterize which of the crease patterns can be folded flat, that is, for which there is a flat folded state using precisely those creases.

2. For each such flat-foldable crease pattern, characterize which mountain–valley assignments correspond to flat foldings.