This is the fourth part of a series about symmetry in origami.

**Given an origami model, what are the possible symmetric colorings?**

This is a question I posed in an earlier post, and I said I didn’t know the solution. I thought about it a lot, and I found the solution. I will write up a mathematical proof in a later post. Here I will just explain the result.

A brief review: Each origami model has an associated **shape** symmetry group, which is the set of all transformations (rotations and reflections) which leave the shape unchanged. There is also an associated **color** symmetry group, which leaves the shape and colors unchanged. Then there is the **pattern** symmetry group, which may swap the identities of some of the colors, but leaves the color patterns unchanged. In this post, C is the color symmetry group, and P is the pattern symmetry group.

I defined a “symmetric coloring” to be one where P is “large” and C is “small”. I didn’t say how large P needed to be, or how small C needed to be, but it doesn’t matter. I found a method that can construct any and all symmetric colorings.

### Fundamental domains

Suppose we want to find a symmetrical coloring of a cube. First question: Are we assigning a color to each face, each vertex, or each edge? We want to find a general solution that will work in all three cases, as well as any more complicated case. So the first thing we need to do is divide the cube into **fundamental domains**.