If you build a wire frame into the shape of four consecutive edges of a regular tetrahedron, dip it into soap water, and carefully pull it out again, you get a piece of the Diamond surface. If you cheat and just span wires between corresponding points of opposite edges, you get a doubly ruled surface, the hyperbolic paraboloid. Here is one such surface, together with a mirror image. The eight corners coincide now with the eight corners of a cube.
As a digression, we can fit a total of six such paraboloids into the same cube, creating a curved version of Kepler’s Stella Octangula.
But let’s return to paper making. The home recipes include the usage of a mold, which is a wireframe that is used to get the right amount of paper pulp into shape and, most importantly, dry. For flat paper one can just use a flat wire frame, like a window mesh screen, which is purchasable. The hope is that, using modest force, such a screen can be stretched into tetrahedral shape. We’ll work on that later.
For the moment let’s delight in previewing how the paper would bend.
Up above you can see three sheets. The darker bottom one is the actual hyperbolic paraboloid, while the two lighter and greener ones are bent versions that are still attached to each other along the middle straight line that is pointing towards us. This will be our spine. Here is a top view:
The Inner Frame 