Adams "World in a Square II" on the tetrakis hexahedron

The Adams "World in a Square II" map projection tiles with itself to form a seamlessly periodic Earth. We use identically printed and cut pieces that are triangular after folding, but place rubber bands on only half of the triangles. Each triangle has a complete copy of the Earth's surface, the Western Hemisphere is seen on the face with interwoven flaps, the Eastern Hemisphere is seen on the smooth face.

(The Prime Meridian of the Adams tile has been shrunk to 89% of its true length, resulting in folded triangles that are isoceles and slightly acute to approximate the faces of a tetrakis hexahedron. This is, of course, a fudge to avoid calculating a proper conformal projection, what might be called "World in a (slightly rhombic) Square.")

Since there is never a need to reopen (unweave) the variety of triangles that wear the rubber bands, these can show their smooth side to the weaver, and thus show their Eastern Hemisphere to the weaver.

A quadrangulation can be reduced to an incrementally smaller quadrangulation by a face contraction: shrinking either diagonal dimension of one face to zero. The inverse operation, face expansion, incrementally grows the quadrangulation by opening a path of length 2 into a quadrilateral. Since there is only one sort of face on the cube, and the two diagonals of each face are essentially the same, there is only one way to decrement a cube by face contraction, so there is a particular shape that we can call Cube-1. Likewise, there is only one sort of path of length 2 on the edges of a cube, so there is only one way to increment a cube by face expansion, so there is a particular shape that we can call Cube+1. (The fact that the "squares" in this model are actually composed of four hinged triangles, lets us play this freely with solid geometry.)

In the strip of photos above, we start with Cube+1, then contract 2 faces to reach Cube-1. So what happened to Cube? To reach Cube from Cube-1, we must go back to Cube+1 and contract a single face. This indirection is necessitated by the limits of simulating topological transformations via physical folding. If our ambition was only to reach Cube from Cube-1, then we stuck with one quad's worth of unnecessary surface riding piggy-back on Cube.

Reducing Cube-1 down to a one-faced quadrangulation (photo strip above) proceeds directly because in each step we we are simply folding down a "flap" formed by a vertex of degree-2.