Symmetrical coloring theorem

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.)

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Origami: Cube Tessellation

A cube tessellation folded from a science poster

Cube Tessellation, designed and folded by me.

This model is a sendoff of my Ph.D.  It’s folded from a poster about my research, created while I was writing my dissertation.  It’s really big, about 25 inches (63 cm) across.  It’s the third tessellation I’ve made from one of my science posters, thus the title of this piece is “A Deconstruction of My Research 3”.

For this third model, I wanted to do something special, so I tried designing my own tessellation completely from scratch.  I was inspired by the Rhombille tiling, which looks a bit like a wall of cubes.  So I tried making a wall of cubes.  And after a long design process, I was able to perfect it.

Below the fold, I have diagrams and prototypes to document the design process.

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Origami: Thunderbolt

Thunderbolt by Ekaterina Lukasheva

This is just a basic model I made, that I don’t have much to say about.  Icosahedral symmetry, 30 units.  Happy New Year!

Origami: Compound of Five Octahedra

Compound of Five Octahedra, by Meenakshi Mukerji.  From the book Exquisite Modular Origami.

At first when you look at this it just looks like a many-pointed star.  But look carefully: each color forms an octahedron.  Yep, it’s five octahedra arranged symmetrically.  Classic.  I’ve already talked your ear off about symmetric colorings and polypolyhedra, so I’ll just leave it at that.


If you’re wondering, yes those are the aro flag colors (or at least the most common version).  Black, gray, white, light green, dark green.  It’s a good color scheme.

How to make symmetric colorings

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.

A cube. Each face of the cube has been divided into eight triangles.

Each triangle on the cube’s surface represents a fundamental domain. In total, there are 48 fundamental domains.

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The impossible symmetry

This is the third part of a series about symmetry in origamiPreviously, I established the idea of a symmetry group, a set of transformations that leaves a model’s shape unchanged.  Next, I talked about how the colors of a model define a subgroup.  In this post, I will explain the concept of a normal subgroup.

First illustration: The Umulius

We begin with a case study of one of my favorite models, Thoki Yenn’s Umulius.  “Umulius” is a Danish insult meaning “impossible person”.


Ignoring the colors, the Umulius nearly has cubic symmetry.  Here I have a series of diagrams “cleaning up” the details to make the underlying cubic symmetry clear.

A series of diagrams showing how the Umulius can be fit into a cube

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Origami: Octahedron Skeleton

Octahedron skeleton

Octahedron Skeleton by Robert Neale. I hope nobody was expecting an actual skeleton. Halloween was yesterday.

I know somebody who runs an art gallery, so every year I run a workshop for kids where we do modular origami.  The hardest part of designing the workshop is picking the right models.  I’ve been quite surprised by which aspects the kids find difficult, and which aspects they perform with ease.

Anyway, this is one of the models I picked this year.  It’s on the easy side, and the kids thought so too.  And that’s great!  Art doesn’t need to be technically challenging to be good.

If you’d like to try this one out, I made some fancy diagrams to print out and pass to the kids.  Check them out below the fold.

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