This is the second part of a series about symmetry in origami. Here I talk about the role colors play in reducing symmetry.
Let’s return to the ninja star that I showed you last time. I said that it has a symmetry group of order 4, because there are four transformations preserve the shape of the ninja star: rotation by 0, 90, 180, or 270 degrees.
But suppose we want to preserve more than the ninja star’s shape. We also want to preserve its color. The only tranformations that preserve shape and color are rotations by 0 and 180 degrees. So the ninja star actually has two kinds of symmetry groups: the shape symmetry group of order 4, and the color symmetry group of order 2.
The color symmetry group is always a subset of the shape symmetry group. We have a special name for groups which are subsets of other groups, we call them subgroups.
Some facts about subgroups:
- Every group is a subgroup of itself. Suppose the ninja star were all one color, then the shape symmetry group is identical to the color symmetry group. And therefore, they are subgroups of each other.
- The trivial group is the group containing nothing but the identity transformation (i.e. rotation by 0 degrees). The trivial group is a subgroup of every group. So we could paint the ninja star a bunch of random colors, and then the color symmetry group would be the trivial group.
- If you take the order of a group, then it is always divisible by the order of its subgroups. This is known as Lagrange’s theorem. For example, the ninja star has a shape symmetry group of order 4. Therefore the color symmetry group must have order 1, 2, or 4.
Now let’s talk about a few other examples.
This Ray Cube has cubic symmetry. If we include only rotations, then the symmetry group has order 24. But if we also include reflections, then the symmetry group has order 48. Now what about the color symmetry group? There’s a set of 3 rotations, all around the same axis, which will lead to the preservation of the cube’s color. The rotation axis is shown below. You may rotate around this axis by 0, 120, or 240 degrees.
So the shape symmetry group has order 48, while the color symmetry group has order 3. Sanity check: is 48 divisible by 3? Yes it is! Math works!
Here’s another example, my Etna Kusudama. Note that if you rotate the model by 90 degrees, all the colors change, yellow to black, black to dark green, dark green to light green, and light green to yellow. But there’s something that doesn’t change, which is the pattern that the colors make. I declare this to be yet another symmetry group, the pattern symmetry group. The color symmetry group is a subgroup of the pattern symmetry group, which is a subgroup of the shape symmetry group. Or, in mathematical notation:
color symmetry pattern symmetry shape symmetry
It’s difficult to determine the symmetry groups of the Etna Kusudama because the photo only shows one side. So here’s a skeletal diagram of the coloring:
One thing that is considered aesthetically desirable in origami is a “symmetric” coloring. There isn’t a clear way to define a symmetric coloring, but I propose the following criterion: The pattern symmetry group must be large, and the color symmetry group must be small.
For the Etna Kusudama, the shape symmetry group is the full cubic symmetry group, order 48. The pattern symmetry group is the cubic symmetry group without reflections, order 24. The color symmetry group is the trivial group of order 1. This sounds like a symmetric coloring to me!
An interesting math problem would be to determine the possible symmetric colorings of regular polyhedra. And by “interesting” I mean that literally nobody knows the answer, as far as I know. However, I did once see a blogger consider the question using slightly different definitions, so the answer is probably within reach.