Multiphase weaving strips for locked crossings

If all weave crossings will be locked at the same angle, the same notch pattern can be repeated with high frequency along the length of the weavers. The wavelength can be an integer divisor of the width of one crossing, for example, in the photo and the diagram below, the repetition wavelength is 1/2 the width of one locked, 90° crossing.

This allows a variety of shapes to be made with the same set of notched weaving strips since the distance between successive crossings can be adjusted to any integer number of wavelengths. On the other hand, if the strips are custom notched, any angle and spacing of crossing can be accomodated, but the strips may only be useful for the intended construction.

Correction for engagement windows

The geometry of an oblique locked crossing is not fundamentally altered if the engagement windows are too large to be neglected. The radius, r, used in the calculation, remains the same, but the mechanical radius of the hole cut into the weaver, rm, must be made larger. If the 'thickness' of the engagement window (seen as the cross-section of a double convex lens) is t, then rm = r + t/2.

The best value for t needs to be determined by experiment, as it will depend not only on the thickness of the weaver, but also its elasticity, the desired stiffness of the joint, and how acceptable permanent plastic deformation of the weaver might be.

Geometry of oblique locked crossings

When two weavers of equal width cross at an angle, say 2θ, their area of overlap is a rhombus. If the weavers are thin enough that the engagement windows of their oblique locked crossing can be approximated as points, those four points also form a rhombus (dotted lines in diagram above) that shares the same diagonal lines as the former, but with angles slightly different—assuming the radius, r, at the bottom of the notches is not zero.

In the diagram, weavers are only indicated out to the width, 2h, where the centers of the notch radii are located. The point where the centerlines of the two weavers cross makes a natural origin for the diagram; in particular, we choose the perpendicular to the centerline at that point on our chosen weaver (the gray one) to define the line of x = 0 for our coordinates. Taking h, r, and θ as given, the geometric problem is to find the x-coordinates of the centers of the four holes (the geometry of the other weaver will be simply the mirror image of this one.)

The solution requires repeated use of trigonometric identities for geometrically similar right triangles, all having acute angles of θ and 90°-θ. Such an analysis yields these expressions for the marked dimensions indicated in the diagram:

xL = r/cosθ

xR = r/sinθ

sL = htanθ

sR = h/tanθ

The x-coordinates of the circle centers on the inner edge (upper edge in the diagram) can then be calculated from:

xL = cL - sL

xR = sR - cR

For the circle centers on the outer edge, just multiply by -1.

Oblique locked crossings

When weavers cross at a non-perpendicular angle, they can still be locked using four notches in each weaver.

And they engage in the same manner as for a perpendicular crossing.

The centers of the engagement windows no longer form a square (as they do at a perpendicular crossing) but a rhombus. When the notches terminate in a circular radius, the circle centers do not form a rhombus, rather a parallelogram.

The acute angle of parallelogram (and thus the acute angle of the weavers crossing) ends up being a little wider than the acute angle of the rhombus. In the diagram above, 33 degrees vs. 30 degrees in terms of half-angles. The bottom line: we can program oblique locked crossings with same shape notches used for perpendicular locked crossings, we just need to make some adjustments in their positioning along the length of the weaver.

Sinusoidal notches for locked crossings

The best profile for the notches seems to be roughly sinusoidal, as seen in this template which I have been using with a 3/16" hole punch and strips of aluminum flashing that have been averaging 1.47" wide (I had aimed for 1.5"). The black discs are for visually centering the punch when the cardboard template is punched. Even at this relatively coarse scale of weaving the notching must be done at an accuracy that is pretty demanding for handwork (all of this could be avoided with a steel rule die to punch the whole 4-notch crossing in one blow.)

The image above shows via backlighting the four little lens-shaped openings or engagement windows that are the inevitable consequence of the material's non-zero thickness. A precise design needs to accomodate this geometry.

The fourth notch is engaged by bending the sides of both weavers upward, as in the photo below:

The square pencil packing in Diamond Weft

The weaving elements in the densified version of Diamond Weft (stereogram above) wrap helically around square tubes that are packed in this arangement.

I haven't found the name for this structure, but Sigbjørn Björklund has made a pencil stacking video that makes its structure clear.

The NbO net and 3D weaving

In "Three-periodic nets and tilings: regular and quasiregular nets," Friedrichs, O'Keefe, and Yaghi dub the net of straight lines found in the D-surface 'nbo' after the chemical bond structure in niobium monoxide, NbO (shown above.) The same net is discussed in Peter Pearce's "Structure in Nature is a Strategy for Design,"

and in A. H. Schoen's "Infinite periodic minimal surfaces without intersections",

but without either obtaining a name for it.

From its similarity to the square grid on the plane-- both consist of straight lines meeting at 90-degree angles-- I expect that the nbo net and its underlying Schwarz D-surface will be important in the future of 3D weaving. Of course, there is another famous 3D grid that is composed of straight lines meeting at 90 degrees: what is special about nbo is that the lines meet only two at a time as in normal weaving.

Denser than Diamond Weft

Diamond Weft is a rather low density sketch of the Schwarz D-surface, and its material elements don't actually lie in the D-surface (it's the missing hypotenuse of their right-triangular cross-sections that is actually in the surface.) What happens when we make the sketch denser by shortening the distance between crossings? The stereopair above shows a single saddle hexagon after the segments between crossings have been shortened to squares. Each folded weaving element wraps a tunnel of square cross-section. While this is just an assemblage of six folded strips of paper, it gains a certain rigidity when the cycle is closed.

Weaving with locked crossings

Recently I've been interested in weaving with quad-faced weave openings because of the ability of this traditional form of weaving to change shape by deforming in shear, and then I paradoxically switched to wanting to lock the crossings at 90 degrees so that very open weaves can stand up on their own.

Thin, flat weavers can be locked together at a fixed position and angle of crossing with four side notches in each weaver. In an x-ray view of the completed locked crossing, the respective notches just barely overlap. For a 90-degree crossing the overlap areas form a square (the notches on each weaver may not appear to be arranged in a square because the locus of overlap is eccentrically located on each notch.)

These photos are of Diamond Weft. I used an earlier version of locked crossings with just two notches per weaver in They Urned It (a data sculpture based on the expansion of the Fed balance sheet), but using just two notches relies on a certain interplay between the surface curvature and the notch location to keep the crossing locked.

While it may seem it would be difficult to engage all four pairs of notches at a crossing, if the material is thin and flexible enough, engaging the fourth pair of notches is a move similar to getting the last corner down in the familiar weave method of closing the flaps on a cardboard box.

Here are some accurately cut paper weavers with a 1/8" punch used to shape the bottom of the notch, along with an "X-ray" view of the interlocked crossing.