Parity embedding: completely foldable triangulations from Tait colored maps

The tetrahedron has a Tait coloring, but its dual is not a CFTM. We fix this by half-twisting every edge that has the same color orientation at each end.

If a cubic map has a Tait coloring, then it is possible to find a new embedding for its edge-colored graph such that the dual—a triangular map—is node tricolorable, and thus a completely foldable triangular map (CFTM).

Represent the Tait-colored cubic map as a ribbon graph. Apply a half-twist to any edge that has the same color orientation around the nodes at each end. This edge-selective application of the skew operation yields the ribbon graph of a map having the same number of nodes and edges, but (probably) a different number of faces, and therefore a different Euler characteristic and a different surface topology.

In the new map we always see color orientation reversing at each successive node in a walk on the surface. Thus, if a walk closes after an odd number of steps, our sense of orientation must therefore have reversed. Similarly, if a walk closes after an even number of steps, our sense of orientation must therefore have been preserved. Such an embedding, where every even closed walk is orientation-preserving and every odd closed walk is orientation-reversing, is called a parity embedding. In this new, custom made embedding, the Tait-colored graph has a CFTM as its dual.

Since a walk around the boundary of a face can never be orientation reversing (a face is a topological disk,) every face in a parity embedding is even.

For example, the Tait coloring of the tetrahedron is unique (deleting edges of any one color leaves a 4-cycle.) The unique Tait coloring gives the same color orientation to all four nodes. We need therefore to apply the skew operation to all the edges. That gives us the Petrie dual of the tetrahedron: the hemicube, a map on the projective plane with three 4-sided faces. Its dual, the hemioctahedron, also in the projective plane is the completely foldable triangular map we are looking for.

Edges inherit their Tait colors throughout this process. In the last step, the Eulerian triangulation generated by dualization has just two edge colors incident to each node. Picking the third color to color each node gives a tricoloring.

A nearly physical folded configuration for the hemioctahedron (a pair of hinges must pass through each other) can be imagined as the folding onto a central triangle numbered 0, of three "ear" triangles, numbered in order of folding, 1, 2, 3.

This scheme gives us already one hinge on each side of the stack of triangles—thus we have no choice in hinging together the remaining two edges on each side.

The parenthesis words describing the hinge connections on each side of the folded stack of triangles:

On the side where triangle 1 folds down: ( ) ( )

On the side where triangle 2 folds down: ( { ) }

On the side where triangle 3 folds dow: ( ( ) )

Thus only one side of the stack has a non-physical fold.