Origami: Pinwheel Dodecahedron

Pinwheel dodecahedron

Pinwheel Dodecahedron, a model by Meenakshi Mukerji

So here’s a really old model, apparently I made it in 2013?  It rounded out my set of platonic solids.  Yep, that’s the good old dodecahedron, with 12 faces, 30 edges, and 20 vertices.  I use one piece of paper per edge, so that’s 30 pieces of paper.

Looking back, I have some disagreements with how I made this model.  I chose to use patterned washi paper, but I should have used solid-colored paper instead.  The model already has patterns in it–the pinwheels, which are created by showing part of the backs of the paper.  The second problem is that I used paper with a bunch of different patterns, rather than sticking to just one or two.  The end result is a bit chaotic.  These days I try to make models with a more focused aesthetic.

Origami: When Rebecca Met Shuzo

Photo of When Rebecca Met Shuzo 1

When Rebecca Met Shuzo 1, by Robert J. Lang

This design is part of a set of 3, and you can find photos and diagrams on Robert J. Lang’s website.  Although his models are 3 inches wide, and mine is only 3 cm wide.

This model caught my eye not just because it looks nice, but because it looked relatively simple to replicate (which is not true of most of Lang’s models).  The biggest difficulty was that the very first step was to fold the paper in 5ths in one direction, and 13ths in the other direction.

To fold paper into 13ths, you could probably just use a ruler, or a printer for that matter.  I am not above such tools.  But if we can do it by origami instead, well wouldn’t that be nice?  So how do we use pure origami to fold paper into 13ths?  Instructions below the fold.

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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…]