Origami: Basket Weave

Basket Weave tessellation
Basket Weave, by unknown

This comes from Eric Gjerde’s book, Origami Tessellations, although Gjerde says it’s a “common design”.  It was one of the earliest tessellations I made, apparently dating to 2014.  They were quite challenging to make at first.  It might be interesting to fold another one just to see how much I’ve improved.

I remember people being very impressed by this one.  I’d tell people, no it’s just one sheet of paper.  The strips of paper that appear to be woven together are not actually continuous, that’s just an illusion.

Origami: Pair of dragons

two dragons, one smaller green one and larger purple one, looking at each other

Dragon by Jo Nakashima

You might have noticed that I have a very strong preference for non-representational origami.  Non-representational origami does not seek to represent any particular object, and only seeks to be itself.  (Arguably many of my models represent mathematical concepts, but I’m declaring that this doesn’t count.)

But I do dabble.  I was part of the origami group at my university, and they would usually fold things like this.  These models represent dragons.  Yes, for some reason I folded two of them.  You can learn how to make these from Jo Nakashima’s website.

I am not particularly skilled at making these, I mean, considering how much origami I do.  The models shown are messy, and I often have trouble interpreting the diagrams.  But I do like these dragons.

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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