We all know that cardboard cubes are rigid, which is why we get our packages in boxes. We also all know that if we remove two opposite faces from a cube, we can fold it together. This started to interest me when I noticed that the polyhedral approximation of the Schwarz P surface is surprisingly flexible. This summer, I showed this to our local Origami and Paper Folding expert, Jiangmei Wu from our School of Art and Design, and she became interested. A few days later she came with a paper model that looked like this:

She called it a *simple variation* of the polyhedral P-surface. Hmm. This is a triply periodic polyhedral surfaces tiled with rhombi. To understand it, we build it out of smaller units (which we called *butterflies*):

The really cool thing about it is that it can be folded together in two different ways, like so:

You can find an animation showing the continuous deformation here. We stared at this for a (long) while, until we realized that this has to do with rhombic dodecahedra. The structures up above are composed of the rhomboids from last week that tile a rhombic dodecahedron. The latter has, as the name hints, 12 faces, which occur in opposite pairs. Like the cube, it is rigid per se, but becomes foldable if we remove two pairs of parallel faces, leaving us with four faces to use, which are distinguished by color up above.

Above you can see the four hollow parallelepipeds (which we called *hollowpeds*). The almost trivial but nevertheless mind bending realization is that everything you build out of these hollowpeds becomes a structure foldable in two different ways. Next week I’ll show Jiangmei’s second model, a foldable fractal… If you can’t wait, check out this.