Earlier I shared a video of me folding a cube made from Fuse’s wireframe modules. While I was working at the University of Minnesota’s Math Centre for Educational Programs, I learned a different way of folding a wireframe cube. I don’t know the original designer, because the video I used to reference is gone. Here is a video of me slowly folding the module (in case you want to learn yourself) and then a time-lapse of me assembling it into a cube:
https://echo360.ca/media/f2f16367-8010-4097-b635-2ab4debbd49d/public
The cube is one of the five platonic solids (named for the ancient Greek philosopher Plato). A platonic solid is a solid whose faces are all the same, whose faces are all regular polygons, and whose vertices all have the same number of faces meeting together. Faces of the cube are regular 4-gons (squares), and at each vertex three faces meet. Now, if you put a vertex in the middle of each face and then connect two vertices together if and only if their corresponding faces share a mutual edge you get another solid. We call that the dual of the first one. It turns out that the dual to a platonic solid is always also a platonic solid. Grab a cube and see if you can work out what its dual is!
Go ahead; I’ll wait.
You could always fold one yourself; you just need 12 pieces of paper!
The shape you have (perhaps) just drawn or visualized is called an octahedron because it has eight faces. Each face is an equilateral triangle. In this next video I’ll show you how to fold a model of an octahedron, which I also learned at the Math Centre for Educational Programs. It’s made out of six “water bomb” modules. While at UMN, I regularly helped teach local teachers how to fold these and help their grades 3 and 4 students assemble the model, as part of the Girls Excel in Math (GEM) program there. That means that this model is not too hard to put together; it is certainly easier than the wireframe cubes.
Watch to the end to see a demonstration of their duality!
https://echo360.ca/media/e7b2d811-dbd2-4bf6-be4d-9f3082d9a705/public