Two routes to complete foldability

Two routes to a completely foldable knotology basket. A knotology basket is completely foldable if it is directed by a bipartite map. There are basically two ways to turn an arbitrary map, m, into a bipartite map: subdivide edges, Su(m), or take the radial, Ra(m). The resulting baskets have the same crack+hinge graph, but rather different weaving.

Di Francesco has shown that a triangulation of congruent triangles can be completely phantom folded into a single triangle iff it is tricolorable (the vertices can be colored in three colors such that no edge joins vertices of the same color.) The radial map operation that turns an arbitrary map, M, into directions for weaving a knotology basket, Ra(M), creates a new class of vertices and returns edges that exclusively connect new vertices to old vertices. If the underlying map is itself bipartite, then Ra(M) will be tricolorable, and it will remain so even after the edges of M are added back as hinge lines turning the quad-faced map into a triangulation.

The image above illustrates two ways to obtain a bipartite map from an arbitrary map. One way is to use Ra(). This results in the same √2 refinement and rotation discussed in the previous post. Another is to subdivide edges with new vertices, which is the subdivide map operation Su(). The radial method results in extended weaving, the subdivide method results in chain mail.

A ring of knotology chain mail is a closed belt of eight, diagonally folded squares.