# Origami: 360-piece polyhedron

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.

# Origami: The failure files

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

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.

# Origami: Arrow illusion

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

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, 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.

# Origami: A nontrivial 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.

# Origami: Square Star and other tesselations

Square Star, by Ekaterina Lukasheva.  The squares are on the side, not visible in this photo.  Ekaterina has her own fancier photos here.

The past month has been astonishingly productive, in terms of origami.  I discovered that there was a nearby origami convention so of course I had to go.  Most people were doing one-piece origami, so of course I ended up trying a lot of one-piece origami myself.

I was, however, pleased to see some modular origami representation, and in particular there was Ekaterina Lukasheva, of Kusudame.me.  She gave a presentation on the connection between modular origami and origami tessellations.  And afterwards, as a demonstration of principle, she showed people how to make the Square Star, shown above.

I think perhaps few people understood her talk, but as someone who is interested in the design of both modular origami and origami tessellations, I for one found it inspiring.  Further discussion and origami below the fold.

# Origami: Hydrangea cube

A cube made from six copies of Shuzo Fujimoto’s Hydrangea

The instructions for the Hydrangea are freely available online in diagram or video form.  It’s not too difficult, except for one step (8:32 in video, #23 in diagrams) that involves inverting a pyramid, which is the source of all the wrinkles in above photo.  In theory, you can recursively add smaller and smaller petals ad infinitum, but for some reason I chose not to.  These flowers only go 3 levels deep, which is quite shallow but people seem to be impressed by it anyways.

This model was inspired by Origami Inspirations, by Meenakshi Mukerji.  She included a single photo of a cube made of Hydrangeas, and it was fairly easy to reverse engineer.