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.

Completely foldable surfaces

What kinds of closed, triangulated surfaces can be completely folded up into a single triangle?

The classic interest of origami is folding up a portion of the plane (usually a square sheet of paper) into a more appealing or useful shape, but physicists interested in 2-D quantum gravity have been looking at folding from a different direction: what kinds of closed, triangulated surfaces can be folded up into a small triangular portion of the plane?

Di Francesco and Guitter have shown that any vertex-tricolorable mesh of equilateral triangles can be phantom-folded to a single triangle. By phantom-folding we mean that the surface is allowed to pass through and coincide with itself in its folded state; by vertex-tricolorable (a.k.a., properly 3-colorable, or tripartite) we mean that we can assign one of three colors to each vertex of the mesh such that no two adjacent vertices receive the same color. When completely folded into a single triangle—or even just partially folded onto the plane—we find that the coincident vertices share the same color. Di Francesco and Guitter's result holds for any genus of surface, orientable or not.

Vertex-tricoloring a triangulation is rigid: coloring the three vertices of a single triangle forces all the rest. The coloring goes easily or not at all. The underlying graph must be tripartite for the coloring to succeed. Whenever an equilateral triangulation folds onto a portion of the plane, it has three vertex classes, and their spatial arrangement will conform exactly to a vertex tricoloring of the plane equivalent to this one. A subsidiary tricoloring of edges (not a proper edge coloring) results by simply mixing the vertex colors of each edge's endpoints. This edge coloring does direct a proper edge coloring of the dual, trivalent graph that describes the triangles' connectivity.

A surface mesh composed of equilateral triangles necessarily gives a rather crinkled approximation to a smooth surface (see image below.) Adding a requirement of vertex-tricolorability exacerbates this. A necessary (but not sufficient) condition for vertex-tricolorability is that there be an even number of triangles around each vertex. (Place three—or any odd number—of equilateral triangles around a vertex and try folding the ensemble flat!) That means surface curvature can only be coded by clusters of 6, 4, 8, etc., equilateral triangles around a vertex—we forfeit the option to approximate surface curvature with clusters of 3, 5, 7, etc., triangles. Completely foldable surfaces will in general look quite crumpled even in their fully extended state. C'est la vie.

A surface composed of equilateral triangles is crinkly-looking even before we require the triangulation to be Eulerian (i.e., to have an even number of triangles around each vertex, such a triangulation is locally tripartite.) Image quoted from Isenburg, Gumhold and Gotsman, "Connectivity Shapes."

By the way, to physicists, the curvature we are talking about is the quantized gravitational curvature of a 2-D spacetime, a curvature induced by the presence of matter. Constraints that we may find vitally useful (i.e., foldability) sometimes just make for more interesting behavior in their models.

Phase-correcting splices in puck weaving

A phase-correcting splice in puck weaving.

If an error is made, or the basket weaving is improvisational, the problem arises of joining two out-of-phase sections of composite weaver. These situations always arise in pairs—if you switch to the wrong phase in one place, you are going to have to switch back again before you come around the loop.

A way to do this is shown above. It does not compromise the strength, but adds one ply of thickness in two non-adjacent squares. Since corrections come in pars, each error causes one extra puck to be added to the work.

Puck weaving the dodecahedron

The skeleton graph of the dodecahedron is not bipartite, so, despite its high symmetry, making a dodecahedral basket is a project that introduces the real housekeeping chores of puck weaving.

We have a goal to finish at an all-outboard vertex in order to make the closing moves easy and familiar. More burdensome, we must keep track of all the Petrie paths in the basket to insure that we phase them consistently (i.e., should we place the next "unprecedented" puck centrally or outboard?). As far as the absolute phase goes, we only care about the three targeted Petrie polygons that intersect at our terminal vertex. For all the other Petrie polygons, we only care that we phase them consistently. Should we fail to do so we'll come to a non-sequitur in the weaving.

A vertex on the dodecahedron happens to have an antipodal vertex where the same three Petrie polygons also intersect. Since the half circumference of a dodecahedron (taking its Petrie polygons to be geodesics) is 5 edges, an odd number, an all-outboard vertex on the dodecahedron will always have an all-central vertex at its antipodes. Unfortunately, since we must keep track of the phases of all of the Petrie polygons anyway, this property does not really make the puck weaving of a dodecahedron any easier. 

The best working method seems to be to mark up an undip word for the dodecahedron with the correct phasing of the "unprecedented" weaver to be added at each open letter. Since central phasing seems a natural default, I mark open letters with a prime if the phasing of the unprecedented weaver is outboard. The whole weaving must be calculated in advance to get this right. Below is an example worked out by hand for the dodecahedron.

The three Petrie polygons that intersect at a vertex on the dodecahedron also intersect at its antipodes.

The phasing of all of the Petrie polygons must be calculated before weaving can begin. The pink line marks the Hamilton path traced by the undip word. The Petrie paths follow the crossovers (half-twists) at the midpoint of each edge. Each Petrie polygon has been labelled with a consistent alternation of 'c' and 'o' phasing (central and outboard.)

The undip word with its housekeeping markup is unnndn'unu'uu'pppdpdpdd .

Aluminum puck-woven cube

This cube was not made using an undip sequence. The working was crowded due to the need to pre-bend the springy aluminum sheet metal to approximately the final angle of its diagonal folds. It proved more practical to complete an all-central vertex at the "south pole," secure it on the outside with duct tape, add all possible extensions to that vertex (6 of them), and then weave in an equatorial belt of 3 puck weavers. That accounted for all 12 pucks. All that remained was an all-outboard vertex to be closed at the "north pole."

Increasing the radius at the rounded corners to about 0.5 cm helped in closing the outer flaps at the last vertex. 

Puck weaving from undip words

When puck weaving from an undip word, we build out a triangulated surface one triangle at a time, one triangle per undip letter. Half of those letters are "open", the other half are "close" letters. 

Each finished triangle is 4 plies thick. Each 4-square puck weaver effectively covers 4 * 2/3 triangles with a single-ply thickness. So, on average, we need to add 1.5 unit weavers per letter. (That checks since there is one undip letter per vertex in the primal map, and we already knew that we need 1.5 unit weavers per vertex.)

A possible weaving strategy is to add 2 puck weavers at each "open" letter, and 1 at each "close" letter. As an exception, add 3 puck weavers at the first letter (which is invariably "open") and 0 at the last (which is invariably "close."). This strategy always yields  1.5 puck weavers per letter.

4-square puck weavers extend pretty far. One placed centrally extends half-way through it's neighbor's neighbor; one placed "outboard" extends (on its longer side) half-way through its neighbor's neighbor's neighbor. These long extensions can make puck weaving visually complex compared to simple unit weaving, so it is desirable to have definite placement rules to follow. 

Bipartite maps are the simplest to weave. If we always shingle composite weavers the same way (the standard way is with the leading strokes of the w's in front,) then every all-central vertex will be identical, and every all-outboard vertex will be identical. If the primal map is bipartite, then it is possible to have just these two types of vertex in the basket. Such a solution is found simply by making these two types alternate around any cycle, for example the Hamilton cycle encoded by an undip word. This alternation in types will occur automatically along the length of any composite weaver, we only need to be careful when placing crossing pucks to avoid creating a hybrid vertices.

If we would like an undip word for a bipartite basket to terminate at an all-outboard vertex, we simply need to start it at an all-central vertex. 

Each woven peak can be characterized as "upstairs" or "downstairs" by circling around the peak in a counterclockwise direction and noticing the "steps" where each weaver underlies or overlies the next. This determination can be made on either side of the fabric and gives the same result. When composite weavers are shingled the standard way (with the leading stroke of the w in front) only "downstairs" peaks are correctly woven. Weaving correctly automatically assures that the flaps of the pucks are hidden on both faces. 

Oriented weavers: teleological weaving

Kagome weaving. In polyphase unit-weaving each of these weavers would have a specified direction (orientation.)

Polyphase weavers have a structure similar to fallen-over dominoes. This structure gives the composite weaver a definite directionality or orientation.

Weaver paths can wander all over a basket, sometimes crossing over themselves. In some cases a single weaver constitutes the whole basket.

The weaving elements used in traditional basket making have an orientation—the direction that the plant material, say rattan or bamboo, grew—but it is not usually a concern in weaving. A composite puck weaver has an orientation too.  The structure of a composite weaver is like a row of dominoes that has fallen over. Just as we can tell in which direction the dominoes fell, the structure of a composite weaver has a definite orientation. Either orientation will work fine in the weaving, so orientation would not be a concern if we were to make the whole weaver in one step. The problem is that we will be building small portions of weaver at a time, and generally not knowing which weaver we are working on. The only solution is to have complete foreknowledge of the basket being woven so that the orientation of each unit weaver being placed can be specified.

When weaving by undip code, we add one triangle at a time to the completed work. The triangle depicts three segments of weaver. The orientation of two segments are already fixed because  they also appeared in the preceding triangle. If the letter for the triangle is a close letter, then the orientation of the third segment is fixed by its appearance in the older triangle that is being closed to. Thus the orientation of the weavers in puck weaving requires the addition of one bit of information to each open letter in an undip code.