Improved completely-foldable octahedron

Completely foldable octahedron with Bassetti hinges and box closures on the outside.

View of the rubber band hinges and the unfolded shape of the triangle units.

Triplets of Dyck words code the physical configurations completely folded surfaces

In phantom configurations of a completely folded surface, the triangles and hinges can intersect and coincide. In semi-physical configurations of a completely folded surface, triangles do not coincide (they have a physical, stacked order,) but hinges may cross through each other. In physical configurations of a completely folded surface, triangles do not coincide and hinges do not cross. Since a surface in a physical configuration has no self-intersection, such can never be a non-orientable surface without boundary.

A completely folded surface in either a physical or semi-physical configuration has a stack of triangles—and consequently three stacks of edges—hinged together in a particular way. We assume in the following that the surface has no boundary: therefore every triangle edge is connected to one other triangle edge in the same edge stack by a hinge.

If hinges are permitted to cross each other (the semi-physical condition), then the hinge connections in a given edge stack are a perfect matching of those edges.

If hinges are not permitted to cross (the physical condition), then our boundary-less surface must be orientable, and the hinge connections in a given edge stack are a non-crossing partition of those edges. A natural way to code a non-crossing partition is as a parenthesis word, or Dyck word. When written out as a Dyck word, it is easy to imagine the non-crossing arrangement of the hinges in the edge stack by simply connecting the paired parentheses with arcs above the Dyck word, e.g.:

( ) ( ) ( ( ) ).

Since there are three edge stacks, a triplet of Dyck words of the same length can encode a completely folded physical configuration.

The number of Dyck words of length 2n are counted by the nth Catalan number:

1, 1, 2, 5, 14, 42, 132, 429, 1430, ....

and triplets of Dyck words are counted by the cube of the nth Catalan number:

1, 1, 8, 125, 2744, 74088, 2299968, 78953589, 2924207000, ....

For triangle-faced maps that are, as usual, without boundary, but possibly composed of multiple disjoint components, the same daunting integer sequence counts their completely folded configurations when 2n is the number of triangles in the stack. In using the cube of Catalan numbers we are counting as different every possible difference in the completely folded configuration: e.g., simply rotating the stack, flipping it over, enantiomorph-ing the surface or the folded configuration, etc. More significantly, since the map is permitted to have multiple disjoint components, we also count every possible way the multiple components can be arranged in the stack, including the different ways they can enclose each other.

Though it is appealing to think that we can now dial up the phone number of any completely folded configuration that will ever be designed, the combinatorial explosion frustrates "cold-calling" as a technique to discover new designs. Nonetheless, dialing from even the shortest phonebook puts us in touch with shapes that we humans have largely avoided thinking about. A phonebook with n<=2 has 1 + 1 + 8 = 10 entries in it. The configurations that pick up the phone are the empty set; the triangular pillow; a stack of two triangular pillows; a triangular pillow enclosing another triangular pillow; two triangular "ears" that have seamed along a double edge and folded along the seam—in three rotations; and the same "double ear" with one ear tucked inside the other—also in three rotations. 

Origami vs. Complete Folding in 3D printing

ORIGAMI:
A large machine makes a creased sheet that folds to a small object.

COMPLETE FOLDING:
A small machine makes a folded stack that unfolds to a large object.

Regular {3, 2n} maps

Weddslist has a listing of regular maps.

A necessary condition for a regular map to be a CFTM is that its Schläfli formula be {3, 2n} for some positive integer n.

Regular maps meeting this condition include:

Orientable genus 0
di-triangle {3, 2} (it seems the only configuration is folded)
octahedron {3,4}

Orientable genus 1
{3, 6} (1, 1)  (only one vertex)
{3, 6} (0, 2)  6 triangles ()(()); ((())); (())() (it seems the only configuration is folded)
{3, 6} (2, 2)  8 triangles
{3, 6} (1, 3) 14 triangles
{3, 6} (3, 3) 18 triangles
{3, 6} (0, 4) 24 triangles
{3, 6} (2, 4) 26 triangles
{3, 6} (4, 4) 32 triangles
etc.

Orientable genus 2
S2:{3, 8} 16 triangles

Non-orientable genus 1
hemioctahedron {3, 4} 4 triangles

Non-orientable genus 6
C6: {3, 10} 10 20 triangles
C6: {3, 10} 5 20 triangles

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.

Completely foldable octahedron with Bassetti-style hinges

A completely foldable octahedron with Bassetti-style rubber band hinges and extra parts.

The Bassetti "Poly-O" style rubber band hinges work better for completely foldable models because they maintain accurate alignment after re-folding. Here the flaps are extended to form a triangular box closure in order to keep them folded flat. The triangle sides are 5 cm, the notches are 0.25" diameter holes centered on the triangle corners.

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