Tessellation symmetry

This is (the last) part of my series about symmetry in origami.

A tessellation is a set of tiles that fill up a 2D plane. And I do mean the entire 2D plane, infinite in extent. When we talk about origami tessellations, these are models that could hypothetically fill a 2D plane, if we had an infinite amount of paper. In practice, an origami tessellation is finite, but for the purposes of discussing symmetry, we will imagine them to be infinite.

example origami tessellation

An example of an origami tessellation, the Rectangular Woven Design by David Huffman

Previously, I only discussed two kinds of symmetry transformations: rotation, and reflection. However, many tessellations have repeating patterns, and this in itself is another form of symmetry. Are there other kinds of symmetries that we forgot? Let’s take an inventory of all the possible kinds of symmetry transformations.

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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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One: the universe’s favorite digit

This is a repost of an article I wrote way back in 2011.  I’m still proud of figuring this one out.

Out of all the digits, from zero to nine, one is the most common.  This has to do with the log scale.

The log scale captures an important fact that is true of many quantities in life.  Take money for instance.  If you have one dollar, then earning another dollar is great because you’ve doubled your money!  If you have a million dollars, earning another dollar does not make much of a difference.  Small changes matter less the more you already have.

This is true on a log scale too.  On a log scale, 1 is the same distance from 2 as 100 is from 200.  The higher you go up, the more the numbers all get smooshed together.  What does that mean for the digits from zero to nine?

A picture of a log scale, highlighting the regions that have 1 as their first digit (eg 1-2 and 10-20)

In the above picture, I show a log scale.  And on that scale, I highlighted in blue all the regions where 1 is the first digit of the number.  You should see that the blue regions cover more than one tenth of the log scale.  In fact, they cover about 30%.  And so, if we pick numbers randomly on the log scale, about 30% of those numbers will have 1 as their first digit.

Just for fun, let’s apply this concept on the fundamental constants of nature.  I will compare two hypotheses: [Read more…]

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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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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