## The Fractal (Foldables 2) You can find a movie showing how this folds together in two ways here. To understand how and why this works, let’s first look at a simple saddle: This is a polyhedron with a non-planar 8-gon as boundary. Its faces are precisely the four types of faces that are allowed in our polyhedra: All others have to be parallel to these four. The four edges that meet at the center of this saddle constitute the star I talked about the last time. Again, all edges that can occur must be parallel to one of these four. One can fold the saddle by moving the upwards pointing star edges further up (or down), and the downwards pointing edges further down (or up), thereby keeping the faces congruent. This works locally everywhere and therefore allows a global folding of anything built that way. For instance, the hollow rhombic dodecahedron above can be bi-folded. Now note that this piece is also a polyhedron with boundary. In fact, its boundary is exactly the same octagon as the boundary of the saddle.

Observe also that at the center of this piece we have a vertex in saddle form. This suggests to subdivide all rhombi into four smaller rhombi, remove the saddle an the middle vertex of the doubled hollow dodecahedron, and replace it by a copy of the standard hollow dodecahedron. This gives you Jiangmei’s fractal. Repeating this is now easy. Below is the generation 2 fractal (animation): And, just for fun, the generation 10 fractal: You can see it being bifolded here. So far, the two completely folded states of our polyhedra looked very much the same. We will see next week that this doesn’t need to be the case.

## Simple Beginnings (Spheres V)

The simplest way to arrange spheres in space is to use the cubical lattice. This is the obvious generalization of the checkerboard, and it lends itself naturally to a coloring with two colors such that neighboring spheres are differently colored. While this is not the densest sphere packing, it will be pretty dark inside. Leaving out the spheres of one color, painting the rest with most of RGB color space creates the following arrangement of spheres, and makes enough room for light to get through. Now imagine yourself inside of it, and all spheres being reflective in addition to being colored. The formerly simplistic object becomes a dazzling fractal-like maze. The original bicolored sphere packing is related to a packing of space by octahedra (one for each orange sphere).
Two octahedra share then at most an edge, and the gaps can be filled with regular tetrahedra of the same edge length. Minkowski discovered that octahedra can be packed much more densely. The gaps can still be filled with regular tetrahedra, but their edge length is only one third of the edge length of the octahedra. 