This is an appendix to my series about symmetry in origami. Here I will provide a proof that my construction of symmetrical colorings works.

While I try to make the series accessible to people who do not know much about math, I don’t think there’s much point in trying to make this proof broadly accessible. This is intended as a reference for people with some experience with group theory. (You need to know about cosets at the very least.)

### Definitions

Suppose we have an object. I will define the following things:

- The
**shape symmetry group**S is a set of transformations that preserve the object’s geometrical shape. - The
**color symmetry group**C is a set of transformations that preserve the object’s geometrical shape and colors. - The
**pattern symmetry group**P is a set of transformations that preserve the object’s geometrical shape, and permute the colors without changing their pattern. Note that . - If g is a transformation, and x is a piece of the object, then g.x is the piece of the object that we get when we apply the transformation g. This is called
**the action of g on x**. - x is a
**fundamental domain**of the object with respect to group G, if covers the whole object without double-covering any part of the object. The set is called a**fundamental domain partition**(my term). - Given a fundamental domain partition, a
**simple coloring**is an assignment of a single color to each fundamental domain.

### The Symmetrical Coloring Theorem

Theorem: Suppose we have an object with a fundamental domain partition with respect to a group G. Given a group , and a single fundamental domain x, we can construct a unique simple coloring that fulfills the following conditions:

- is the set of transformations in G that preserves the color of x.
- .

**The construction**

Begin by constructing the set of all left cosets of in G: . There are a finite number of distinct cosets, so let’s enumerate them as , , , and so on up to . Only is a subgroup of G, and the others are just subsets.

Now, let’s define subsets of the fundamental domains: . Each domain in is assigned the color k. And that’s it, that’s our simple coloring.

**Proof that the construction works**

First, note that each fundamental domain has exactly one color. This follows immediately from a well-known theorem in group theory, which says that left cosets are disjoint, and their union is the whole group.

Second, we show that is the set of transformations in G that preserve the color of x. When we apply a transformation , the final color of x is equal to the initial color of . This preserves the color of x if and only if . Thus, the set of transformations that preserve the color of x is which is just the same as the set .

Finally, we show that the pattern symmetry group P contains G. I begin with a formal definition of P: . In other words, P is the set of transformations such that each color i is transformed to another color j (and j may be either identical to i or distinct from i). Suppose , then we can show for some j. Here we’re using the fact that left multiplication of a left coset always produces another left coset.

**Proof that the construction is unique**

Obviously you can construct distinct colorings by switching your greens and blues, or something like that. When I say that the coloring is “unique”, I mean that if you have a coloring that fulfills the desired conditions, then you will always come up with the same sets .

Given that preserves the color of x, we know that has to be a single color, and that no other domains are the same color.

For every , we require that g is also part of the pattern symmetry group P. Thus, for some number j. We know , and thus each left coset must define one of the sets . From there the rest of the construction follows.

### Further associated theorems

**Exhaustiveness of construction**

Theorem: Given an object and a fundamental domain partition with respect to G, the Symmetrical Coloring Theorem can be used to construct *all* possible simple colorings that have .

Proof:

Suppose that we have a simple coloring, and that . Let be the set of all fundamental domains that have been assigned the color k. All we need to do is choose a fundamental domain , and show that , the set of transformations in G that preserve the color of x, is a group. From there, we can use the Symmetrical Coloring Theorem to construct a simple coloring, and by uniqueness of that coloring, we know that it is the same coloring that we started with. The hard part is showing that is a group.

To do this, we’re going to construct some additional sets: . Not all of these sets are groups, but it turns out that is always a group. I start with the observation that always contains the identity transformation. Next I note that if p and q are both elements of , then we have , and hence . Therefore . This is sufficient to show that is a group.

Finally, we show that , and is therefore a group. First, suppose , then p of course preserves the color of all fundamental domains in , and therefore . Next, suppose . We know that , since P contains G. According to the definition of P, that means there exists some j such that . Hence . If we consider the fundamental domain , we know that this is an element of , and we also know that it’s an element of since the transformation p preserves the color of the fundamental domain x. Since the sets are all non-overlapping, this implies that j = 1, and therefore .

**Number of colors**

Theorem: The number of colors is .

Proof: This follows directly from basic theorems about cosets.

**The color symmetry group**

Theorem: The color symmetry group C is the normal core of in G.

Proof:

First let’s formally define C: . In other words, C is the intersection of all as defined earlier.

Now what we want to show is that the groups form a complete set of conjugates. Suppose we have , we can show that , for some value of j. We can conclude that , meaning that is conjugate to . I’m skipping a few steps, but it’s easy to show that all are conjugates of each other, and that no conjugate sets are left out.

Since , we can see that C is the intersection of all conjugates of . This is one of the definitions of the normal core of in G.

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