Origami: Ace WXYZ

Four intersecting triangles, in the colors purple, white, gray, black
Skewed WXYZ, a model designed entirely by me.  It’s about 6 cm (2.5 in) diameter.

There are two things to talk about in this model.  First, the color scheme is taken directly from the ace flag, which feels timely because last week we were just marching in pride parades.  I really like the color scheme of the flag, and also like how easy it is to make these references even in a very abstract art-form like non-representational origami.

Second, there’s the model’s design.  This is an entirely original design, although it’s based on Tung Ken Lam’s WXYZ Triangles.  Folding diagrams, discussion, and mathematics below the cut.

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Origami: Tsuru Rose

Shown: a paper crane, but the back of the crane is a four-petaled rose

Tsuru Rose, or Rose Crane, by Satoshi Kamiya

Instructions for this model are freely available online, for instance see this video.

This is a neat model that combines ideas from traditional origami, and origami tessellations.  (I last talked about tessellations here.)  You start out by folding a “rose” right in the center of the paper.  The rose isn’t a tessellation, but I believe that in principle it could be turned into one if you repeated the rose infinitely.  But here we just have one rose.  Then we fold the rest of the crane around it.

By the way, there’s a trick to making those wings curve so smoothly.  I press the wing against a toothpick (the side, not the point).  Then I slide the toothpick along the wing several times until it curls.

Origami: 360-piece polyhedron

A polyhedra made out of 360 edges
Double Sided Concave Hexagonal Ring Solid by Tomoko Fuse

The local origami club wanted to make something big for an activities fair, so we worked together to fold 360 pieces and assemble them together. You can see the result above (along with a few other models they had for display).  Assembly was quite tricky, because at this size the weight of the model pulls itself apart.

I don’t think this polyhedron has a name.  Let’s see, there are 20 hexagons, 12 pentagons, 90 squares, and 60 triangles.  All in all, there are 182 faces, 360 edges, and 180 vertices.

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Origami: The failure files

Sometimes an origami model just doesn’t work out.  Here’s a collection of some of my failures.

failed flexible polyhedron
One thing I’m interested in doing is finding unusual polyhedra, and designing origami around them.  This here was meant to be Steffen’s polyhedron, which is a flexible and concave polyhedron.  “Flexible” means that it can be deformed even when each of the faces is rigid.  “Concave” means that some of its edges are bent inwards instead of outwards.  Cauchy’s Rigidity theorem states that convex polyhedra cannot be flexible, and Steffen’s polyhedron is an example of why it doesn’t also apply to concave polyhedra.

Anyway, this is tricky to design because I basically need to make a bunch of triangles of arbitrary sizes, and I need some way to attach them together.  At some point, I got the bright idea of making triangle edges rather than triangles.  And I didn’t even have to design my own edges, I just took the “Jade” units from Ekaterina Lukasheva, which are designed to be of arbitrary length.  I carefully cut the paper to size (which required a bunch of oddly dimensioned rectangles, like 10:17), and started putting pieces together.

But it turns out, the design was fundamentally flawed.  The geometry of the jade units doesn’t work out, and you just can’t put arbitrary triangles together with it.  Well, back to the drawing board.

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Origami: Arrow illusion

So, you know how things in mirrors always have left and right reversed?  This origami model is no exception.

An origami arrow in front of a mirror. The arrow points to the left, while the reflection points to the right.

Arrow Illusion, my design.

The arrow illusion was inspired by a much more impressive optical illusion, the Ambiguous Cylinder Illusion.  Video below the fold. [Read more…]

Origami: Six Intersecting Squares

Six Intersecting Squares

Six Intersecting Squares, a model of my own design

Description: 6 squares, each made of four sheets of identically patterned paper.  The squares are organized into three pairs of parallel planes.  Each square has a square hole cut out from the center so that you can see straight through it.

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Origami: A nontrivial knot

Origami in the form of a trefoil knot

Knotted Toroid, designed by me.  Based on Thoki Yenn’s Umulius.

Although this blog has a standing ban on nontrivial knots, this piece of origami defies the ban because it knows it can get away with it.

I have two comments on this model.  First, I’ll explain how the choice of paper presents a philosophical problem.  Second, I’ll talk a bit about the inspiration for the model.

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