Distortion

Distortion is a soccer ball (also known as a truncated icosahedron), where each vertex is an origami dodecahedron. It is made out of 1350 Post-it notes folded into units of 4 different types (C, W, S, and M).

The name comes from the fact that it is mathematically impossible to arrange dodecahedra in this way rigidly. If you try to glue rigid dodecahedra into a ring, you will need about 5 and a half of them. As you can see below, 5 are too few and 6 are too many.

Fortunately, paper is flexible, so both of these rings can actually be made with origami, using a bit of distortion, which sacrifices some stability.

After I realized this, I had the idea of building a structure that had both pentagons and hexagons -- one pushing and the other pulling -- balancing each other's forces to keep the structure stable. And the obvious choice was the soccer ball, which consists of 12 pentagons and 20 hexagons interspersed around a sphere.

The challenge was the sheer size required to build such a thing out of paper. With each of the 60 nodes being a dodecahedron made up of 30 Post-it notes (15 of which are shared with neighboring nodes), the total Post-it note count came out to 60 * (15 + 15/2) = 1350! I had a feeling that such a massive structure would likely collapse under its own weight.

I ended up building two of them -- one out of regular Post-it notes, and one out of the Super Sticky kind, which has thicker, sturdier paper . The former is rather flimsy, but the latter is what you see in the video, and it turned out pretty well.

Note that I'm not actually using any of the sticky properties of the Post-it notes. As you can see during the first minute of the video, the sticky strip is hidden away during the initial triple fold that makes the square piece of paper into a 4-by-1 strip. After that, each strip is folded into a unit of one of 4 types -- C, W, S, or M -- named to loosely resemble their shapes.

It is actually possible to build a giant ball, called a rhombic triacontahedron, out of dodecahedra without distortion. But that thing would need to be even bigger and heavier than this one, and I'm already pushing the limits of paper rigidity here. Here is a video from Matt Parker in which people use golden ratio rhombus shapes to build the massive ball that does not rely on distortion, but the materials they use are quite a bit sturdier than mine.

Update [2020-03-12]: after a few months of sitting in the closet, the giant ball started losing pieces of paper here and there. Surprisingly, it continued to hold its shape very well. I needed space in that closet, and I didn’t want to keep fixing it, so my son and I decided to give it an honorable final send-off. It was a peaceful sight… at first.

And here are some more shots of my finished sculpture.