Origami: Cube Plus Alpha

Cube plus alpha

Tomoko Fuse’s Simple Sonobe 12-unit Assembly Plus Alpha, from Unit Origami: Multidimensional Transformations

Fuse has a series of models that consist of basic polyhedra, with extra pieces of paper attached as embellishments.  This model is a cube (made of 12 pieces of paper) with a pyramid added to each face, and 3 spikes to each vertex.  All in all, that’s 42 pieces of paper.  This is a pretty neat idea.  Since it is made of three distinct types of units, it defies the usual convention of making modular origami from many identical units.

This one’s a fairly old model, apparently made in 2013.  I gave it away as a gift so I don’t know if it’s still living, or deceased.

Origami: Lilia

The Sparaxis, a spiky ball that fits in my hand

Lilia, by Ekaterina Lukasheva. I’m not sure I got the name right, but I’m absolutely sure about the author.

In the past year I’ve been dabbling a lot into other kinds of origami, such as traditional origami, tessellations, and minimalist designs.  But still, it’s good to make some modular origami models.  This is a kusudama model of very standard design.  30 units, 5 colors arranged symmetrically.  Although, I actually made 2 of the 5 colors identical, which is a cheeky way of making it slightly asymmetrical.  This was made as a gift for a relative.

Origami: Curvature experiments

Pizza wedge model sitting on top of a copy of "Folding Techniques for Designers: From Sheet to Form" by Paul Jackson

Pizza Wedge, a design/experiment by me

I acquired a copy of Folding Techniques for Designers: From Sheet to Form, by Paul Jackson.  It has a rather unusual, but refreshing perspective.  Basically, it tries to avoid the origami tradition entirely, and instead focuses on folding as an element of design.  Several chapters are occupied by simple ideas about pleating paper.  The reader is encouraged to experiment, and this is just one basic experiment.

The Pizza Wedge is not technically challenging to create, but its simple and abstract nature leads one to contemplate the little details.  One emergent property of the paper is the negative curvature (i.e. the saddle shape).  When you crease a paper back and forth, on the macro scale the paper compresses in one direction.  When I added the “crust” of the pizza, that suppressed the creases, which has the effect of stretching the paper on one side.  The stretching and compressing leads to negative curvature.

I include a second experiment below the fold.

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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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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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