Making a completely foldable object

An equilateral surface is a closed surface composed of equilateral triangles connected edge to edge. Physicists have looked into the requirements for an equilateral surface to be able to fold down to a single triangle. The folding considered is both phantom (the material can pass through itself) and instantaneous (no worries about the intermediate states of the folding or the geometry of those intermediate states.) It turns out there is just one simple requirement for complete foldability: the triangulation must be node tricolorable, that is, we must be able to assign three colors to the nodes of the triangulation such that no edge has the same color at both ends.

From the viewpoint of constructing such a completely foldable surface, if we can assemble the surface from corner tricolored triangles (matching colors where triangles join) the completed surface will be completely foldable.

I have assembled some completely foldable models using cardboard and rubber band hinges.

The printed pattern is a node tricolored triangle grid. Print on 65-lb cardstock.

Press onto a thoroughly gluestick coated poster board. Allow an hour to dry.

Trim along triangle edges.

Separate triangle strips.

Separate triangles.

Punch indicated hole positions with an approximately 4mm diameter hole punch. This can be done in decks of two.

Trim along colored curves. This can be done in decks of two.

Using small scissors, make three cuts meeting in the center.

Use scunci miniature hair elastics. Unstretched, these are about 16 mm in diameter.

With the aid of a crochet hook, attach a single rubber band to make a hinge with an 'X' crossing on each side.

A completely foldable octahedron