Generating completely foldable triangulations

Tripartite subdivision of an icosahedron.

When the triangles of the above mesh are made equilateral, the resulting polyhedron (a stellation of the rhombic triacontahedron) is completely foldable. The paper model itself is rigid, but the surface has an alternate conformation as a stack of triangles.

As shown by Di Francesco and Guitter, the necessary and sufficient condition for a closed surface composed of equilateral triangles to be completely foldable is that its graph is tripartite. Given a surface mesh (or, topologically speaking, a map,) they teach a simple way to generate a tripartite triangulation (triangle-faced map) of the same surface.

Construction: Given a map with black vertices, bisect every edge with a white vertex; place a pink vertex in the center of each face and connect it to the white and black vertices incident to that face.

The construction is equivalent to the map operation Meta (a.k.a. barycentric subdivision, full bisection, 2-D subdivision, dual triangle quadrisection,) and as well the construction that locates the preimages of 0, 1, and ∞ in a dessin d'enfants.