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.

Opening and closing at an all-outboard vertex

At an all-outboard (mmm) vertex, the six unit weavers appear as two sets of three "flaps."

The three flaps on the outer face open like a cardboard box tops, as though a box had three tops, all arraged with rotational symmetry around a corner.

Opening an outer flap at an all-outboard corner.

Opening the third outer flap.

Opening the three outer flaps leaves a flower-like arrangement with three inner flaps still closed in the center in a reciprocal, nexorade-like pattern. Simply lift these up.

Opening the three outer flaps reveals a flower-like pattern with the three inner flaps still closed in the center.

Making puck weavers from aluminum flashing

A puck weaver made from aluminum flashing.

Cut strips that are about 0.05" narrower than the intended distance, s, between the "square" folds (in the pictured weavers s = 2.5". The strips should be at least 4s long.

Make the square folds, trimming off any excess beyond the fourth square. When folding begin in the middle and work outward—never reaching the edge. Never press closer than about 1/4" from the edge to prevent bending the edge in too sharp a radius.

After cutting and making the square folds.

Make one set of diagonal folds, ideally, due to the narrowed width, these should follow a 45-degree line and fall just short of intersecting the square folds. (Intersecting folds over stress the metal.) If the piece is flipped over so that the square folds show as mountain folds, the diagonal folds show as valley folds. Note that you must skip a square when making these two diagonal folds.

After the first set of diagonal folds.

Now make the remaining two diagonal folds which are aligned 90 degrees to the first.

After folding.

With scissors or tin snips round off the four corners.

After rounding the corners.

Theoretically you should adjust the square folds to a 90 degree dihedral angle and the diagonal folds to a 70.5 degree dihedral angle, but you may want to compromise with your intended surface. The final state of the square folds will always be 90 degrees of dihedral angle, but such tight bends may be difficult to work with in a very small basket. The final angle of the diagonal folds will vary with surface and whether you are working points in or points out.

Frequency-1 puck weaving of the tetrahedron

A tetrahedron in frequency-1 puck weaving. The forward corner is an all-outboard vertex. (The tetrahedron is self-dual. Putting cube-corner pyramids on the triangular faces of the dual tetrahedron makes a cube.)

In frequency-1 weaving of the tetrahedron, the diagonal corrugation folds open out to 180 degrees, effectively disappearing. We are left with a set of three composite weavers essentially having only the square folds. These form a cube-like surface (actually a tetrahedron decorated with cube corner pyramids) from three orthogonal, belt-like paths. Two unit weavers overlap to form each belt. The total number of unit weavers used is 4*1.5 = 6.

Within each composite belt, body diagonals of the "cube" connect points of the same type. Therefore, if we start at an all-outboard point we can finish at the body-diagonally opposite corner of the "cube" at another all-outboard point.

All-central = www. All-outboard = mmm.

A 4-square puck weaver in standard position with points labelled.

A more self-explanatory terminology for vertices that are mmm or www, is all-outboard and all-central.

We wish to end our weaving at an all-outboard vertex (formerly mmm) so we can tuck it closed like the flaps of a box.

A completed all-outboard (formerly mmm) vertex in puck weaving.

A completed all-central (formerly www) vertex in puck weaving.

When finished, an all-outboard vertex naturally looks tidy because no unit weavers come to an end at its folds. An all-central vertex will look messy if the ends of unit weavers are visible, as in the photo above.

1.5 unit weavers per vertex

Only unit weavers measuring an even number, N, squares in length are permissible if we wish to hide the splices on both of the faces. If the phase shift used in building up the composite weaver is 2 squares, the thickness of each composite weaver is N/2 plies. But, since the weavers cross two-by-two in what is locally a tabby weave, the basket itself will be N plies thick.

Each vertex in the trivalent primal map "owns" one triangle in the triangle-faced dual map. In turn, this triangle "owns" three half-squares of tabby weave. Since these three half-squares are N plies thick, each triangle (and in turn each primal vertex) "owns" 3*N/2 squares of unit weaver. Since the unit weavers themselves are N squares long, the consumption of unit weavers is 1.5 per primal vertex, regardless of length, provided the phase shift is 2.

Ending up at an mmm vertex

The points-out side of a frequency-1 korgome weaving. The "points" correspond to the vertices of the primal, trivalent map.

We'll take the trivalent map as the primal description of a korgome basket because, for example, we probably have better intuition about the cube (which is trivalent) than about its dual the octahedron, and about the dodecahedron (which is trivalent) than about its dual the icosahedron.

Taking that viewpoint, the "points" in korgome weaving correspond to vertices of the primal trivalent map, and the triangles (so evident from the points-in side of frequency-1 korgome shown below) are faces of the dual triangle-faced map.

The points-in side of a frequency-1 korgome weaving. The evident triangles correspond to faces of the triangle-faced, dual map.

The path of a korgome weaver corresponds to a zig-zag (a.k.a. a left-right path or Petrie polygon) in the primal, a triangle strip in the dual, and a central circuit in the medial of either primal or dual since Me(M) = Me(M*).

The complete network of folds in frequency-0 korgome is the radial of either the primal or the dual since Ra(M) = Ra(M*). However, frequency-0 korgome is only possible if the primal map is bipartite.

(An abstract graph is bipartite if and only if its vertices can be colored "black" or "white" such that no two vertices with the same color are adjacent. If such a vertex 2-coloring of the graph exists, it is unique up to color rotation and is easily discovered by simply starting coloring.)

The complete network of folds in frequency-1 korgome is ko of the trivalent primal, Ko(M), and kis of the triangle-faced dual, Ki(M*). Frequency-1 korgome is possible for any trivalent primal map. 

Every polyphase weaver which is composed of unit weavers of even length has a w-side and an m-side (assuming, as always, we are observing the points-out face of the fabric.) Backtracking from the vertex we wish to finish weaving at, we can make that vertex mmm simply by correctly orienting its three polyphase weavers. In general, this leads to complicated constraints on the construction sequence since we must correctly orient three polyphase weavers wherever each is first encountered in the weaving.

The situation is much simpler if the trivalent primal is bipartite. Then we can start with a www vertex and sequentially add unit weavers subject to the rule that every vertex must be either www or mmm. If the primal is indeed bipartite, all will go well. At the end we will either find ourselves at an mmm vertex or right next door to one.

The cube is a simple example of a bipartite map. If we dangle a cube from one corner, we can color the bottom corner white and follow the bipartite coloring rules to color the other corners until we finish with a black corner at the top. Accordingly, in korgome weaving the basket described by this trivalent map, we should start at the bottom with a www point and make only www or mmm points until we end up at an mmm point at the top.

The cube has a bipartite map.

Being bipartite, this map admits a frequency-0 korgome weaving. In order to finish at a point-out, mmm vertex at the top corner of the cube, we can start at a point-in, www vertex at the bottom corner.

Puck weaving started with the aid of spring clamps. A completed www vertex is below, a completed mmm vertex is above. The plane can be woven with vertices of just these two types.

Another bipartite trivalent map is the (infinite) map of hexagons tessellating the plane.

The tessellation of the plane by hexagons is a bipartite, trivalent (infinite) map.

Weaving and Petrie duality

The hemicube, Petrie dual of the tetrahedron. Sculpture by Carlo Sequin. Image quoted from  www.cs.berkeley.edu.

The universality of weaving can be expressed by stating that a basket is an embodiment of the Petrie dual of a map. The familiar sort of map duality (Poincaré duality) interchanges faces and vertices. It is a rather friendly duality because it does not take us to a different surface. But every map also has a Petrie dual. The Petrie dual interchanges faces and zig-zag paths, and, generally speaking, the Petrie dual describes a different surface. For example, the tetrahedron, which is a map of the sphere, has as its Petrie dual the hemicube, which is a map of the projective plane.

The sculpture of the hemicube by Carlo Sequin shown above exemplifies a way to visualize the Petrie dual of a spherical map. A spherical map can be realized as a wire frame with soap films stretched across its faces. To model the Petrie dual, stretch the soap films, not across faces, but a across the zig-zag circuits, that is, the sets of edges formed by alternately taking left and right turns at each vertex.

Of course, weaving elements do not follow zig-zag circuits, rather they follow central (straight ahead) circuits in the medial of the map. Deza and Dutor in "Zig-zags and central circuits for 3- or 4-valent plane graphs," show that the zigzags of a plane graph G are in one to one correspondence with central circuits of Med(G). A visual argument extends this result to non-planar trivalent maps by considering a truchet tiling of the map's triangle-faced dual, as shown below.

Both the zig-zag and the central circuit correspond to a loop of triangles in the dual—which may or may not be a Mobius strip. 

Nederland-Skoviera-Zlatos, 2001: The Petrie dual of an orientable map M is orientable if and only if M is bipartite.

Closing up PUCK weaving at an mmm point

The advantage of an mmm point as a point to close up a surface is that only outboard "flaps" of six weavers are involved. It may be possible to close these flaps up the way the flaps of box can be tucked under each other.

Under the standard rules, the three weavers that expose their trailing stroke belong underneath the three that expose their leading stroke. In two steps the closure could be made by tucking the trailing-stroke weavers together flaps-up, and then tucking the leading-stroke weavers together flaps-down. 

Closing moves in PUCK weaving

The final weaving moves that are needed to close up a woven surface do not arise in most basket making for the simple reason that most baskets are not closed surfaces. Nonetheless, an ability to patch up a hole is indispensable in any fabric technique used to make topological spheres and all higher genus surfaces.

At the local level, PUCK weaving is simply tabby weave: weavers cross one another at 90 degree angles, over-one-and-under-one. Each crossing is a square of double-thickness fabric. In order for both ends of a PUCK weaver to lie buried inside the weaving, the weaver must measure an even number of squares in length. Since the zig-zagging, diagonal folds follow alternate diagonals of successive squares, a weaving unit that is an even number of squares in length will have mirror symmetry (an odd length would yield 180-degree rotational symmetry.) For example, the diagonal folds of a 4-square PUCK weaver trace out a mirror-symmetrical "W" or "M" depending on how you look at it.

For simplicity, we'll standardize the assumption that we are weaving a points-out basket from its outside. That orientation makes the diagonal folds appear as "valley folds" to use origami terminology. Note that if the four valley folds in 4-square PUCK weaver look like an "M" to you, a 180 degree rotation in the plane will make them look like "W". Again for simplicity, we'll standardize on the "W" point of view. 

When 4-square PUCK weavers are shingled together (inevitably with a 2-square phase shift) to compose a longer, multi-ply weaver—depending how they are shingled—we will see shingle edges at either the leading or trailing stroke of the "W". (Recall that we are always viewing weavers from the points-out side.) Expecting that only complexity canl be gained by permitting different composite weavers to be shingled in different ways, we further standardize on leading-stroke shingling.

Once we have formed a composite weaver by shingling multiple W's together, notice that we now have two kinds of "points-out" points: points that appear to be at the bottom in the W-orientation, call these w-points, and points that appear to be at the bottom in the M-orientation, call these m-points. The two kinds of points alternate along the length of a composite weaver—and they are physically different. At a w-point we find the middle of a unit-weaver coinciding with two cut ends. At an m-point we find we find the overlap of outboard portions of two unit-weavers, and no cut ends. 

In terms of individual unit weavers, each 4-square unit weaver has one w-point and two m-points. 

These two physically different types of points complicate the situation in closing the last stitches of a woven surface. We may ultimately find ourselves finishing at any of four kinds of point. Ignoring the possible chiral variations, we may find ourselves  finishing at a point that is mmm, mmw, mww, or www.

I am guessing that the mmm closing is easiest. In a korgome basket based on a chess-colorable triangulation, the composite weavers can be phased in a coherent way that makes every alternate point www or mmm, giving a 50/50 chance that any given point is mmm. But chess-colorable triangulations are shape limiting. In the general, non-chess-colorable case, only one eighth of the points will be mmm. However, if we know which point we will be finishing at, we can teleologically orient the three composite weavers that will cross at the finishing point to make that point mmm. 

Starting a korgome icosahedron

The finished side of an icosahedron woven in sheet metal using the corrugated kagome (korgome) technique.

These aluminum weavers were cut on a paper trimmer from 10"-wide flashing that is about 0.2 mm thick.

The width of the strips is 1.21".

The weaving units are 8 squares in length. I tried for 1.25" spacing between perpendicular folds—the measured value is 1.23-1.28". I tried to make the diagonal folds at 45 degrees.

In the icosahedron, the little pentagonal engagement windows end up measuring about 0.12" from base to peak.

In flat korgome the nominal value for the dihedral angle in the valleys is 70.5 degrees. I found that in making such a strongly curved, points-out surface, it helps to open the valleys somewhat before starting the weaving.

The same korgome icosahedron, here viewed to the inside. From this side the cube corner pyramids act like retroreflectors. 

Cube corner pyramids

A manufactured surface decorated with cube corner pyramids.

Define a cube corner pyramid as a pyramid specified by taking one corner of a cube as the pyramid's apex, and, as the pyramid's base, the equilateral triangle formed by face diagonals of the three faces of the cube adjacent to the chosen corner.

This diagram (far right) clarifies the geometry:

A cube corner pyramid (far right.) Image quoted from Wolfram MathWorld.

Plugging in the formula for the volume of a pyramid, V = A* h/3, we can calculate from the numbers given in the Wolfram chart for a unit cube:

1/6 = h * sqrt(3) / (2 * 3),

so the height, h = 1/sqrt(3).

Since the sides of the equilateral triangle that is the base of the pyramid measure sqrt(2), the ratio of the height of the pyramid to one of the sides of its base is 1/sqrt(6) = 0.40825...

If a surface mesh is composed of equilateral triangles, then decorating the mesh with cube corner pyramids gives us a surface that we can weave with straight, constant-width weavers that always cross each other at right angles, over-one/under-one. This sort of weaving can be seen a tabby weave or kagome weave depending on how local your perspective is. I think it is easier to see it as corrugated kagome.

Whether to point a given pyramid inward or outward can be decided independently for each equilateral triangle in the mesh. Here are some simulations of corrugated kagome surfaces, mostly with points out. These surfaces are the same as those that can be assembled from Mitsunobu Sonobe's unit origami module.

An icosahedron decorated with cube corner pyramids.

A tetrahelix decorated with cube corner pyramids.

In the following two images the mesh triangles are not equilateral, so the weavers will not be straight or constant-width. One shows a mesh with points out, the other the same mesh with points in.

A non-equilateral decoration decorated with cube corner pyramids. (The resulting weavers are not straight or constant width.) Model courtesy INRIA.

The same model decorated with points-in cube corner pyramids.

A origami project using sonobe units by lonely-soldier.

Sphere packing, circle packing, and kagome weaving

A packing of spheres centered on a surface generates an attractive triangulation. Image quoted from J. Wallner and H. Pottmann, "Geometric Computing for Freeform Architecture."

Any triangulation of a surface directs a kagome weaving. Some of the prettiest triangulations result from sphere packings or circle packings (which are nearly the same thing.)

A circle packing is a configuration of circles having specified patterns of tangencies. Adding the condition that circle interiors may not overlap, such a circle packing can be called a packing of discs. A disc packing on the plane is associated with a plane graph, called its carrier, that has a vertex at each circle center and an edge connecting the vertices of any two circles that are tangent to each other. A packing of discs is called compact if its carrier is a triangulation.

The incircles in the carrier graph of a plane compact circle packing constitute a circle packing having a carrier graph dual to the first. The two sets of circles are orthogonal. (Only the red circle belongs to the original packing.) Image quoted from M. Hoebinger, " Packing of circles and spheres on surfaces."

Mathias Hoebinger has shown that a compact disc packing in the plane has as its carrier a triangulation in which circles inscribed in the triangles (incircles) are mutually tangent. In effect, a second (non-compact) circle packing lurks inside the carrier of the first. The carrier graphs of the two packings are graph duals of each other. The points of tangency of the two packings coincide, and the two sets of circles are orthogonal at those points. See diagram above.

The graphenes of theoretical carbon chemistry  are closely related to circle packings, and both topics are related to deltahedral surfaces, which are the surfaces we can kagome-weave with straight weavers. For example the haeckelites, which are graphenes having 5-, 6-, and 7-carbon rings, can have the same arrangements as disc packings with two sizes of disks. These 3-regular patterns can be converted into triangulations by dualization. Triangulations direct kagome weaves.

Haeckelite nanotubes. Image quoted from Terrones et al. "New Metallic Allotropes of Planar and Tubular Carbon."

A compact packing of discs of two sizes (5- and 7-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (5- and 7-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (5- and 8-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (4- and 8-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (5-, 6- and  9-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (5-, 12-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (4-, 10-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (4-, 12-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (4-, 18-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."