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 origami.  Previously, 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”.

Umulius

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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Colorful origami subgroups

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.

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What is a symmetry group?

This is the first part of a series about symmetry in origami. Here I will explain what a symmetry group is through a series of examples.

An origami heart

This image is sourced from a video with folding instructions.

This heart illustrates one of the most basic forms of symmetry. A symmetry is a transformation that preserves the shape and orientation of the object. In this case, the transformation is a reflection. If you reflect the heart across a vertical line, you get back the same heart. But with further examples, we can see that this is not the only kind of symmetry.

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