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.

Periodic, isotropic weaves from overlays of three quasi-cartesian grids

Three copies of the cartesian grid cannot be overlaid into a pattern that is both periodic and isotropic. This is a familiar problem in designing halftone patterns for printing. The traditional arrangement of three square-grid halftones avoids distracting moire patterns with a 120° rotation dispersal that results in a pattern of dot 'rosettes' that is isotropic but not periodic. Wang and Loce show the way towards periodicity. If the orthogonal grids are not square but 1:0.866 rectangles (i.e, based on the base-to-altitude ratio of equilateral triangles), and still arranged 120° to each other,  isotropic and periodic patterns are possible.

Three copies of a rectangular grid with aspect ratio 0.866 can be rotated 120° to each other and still form periodic patterns.

We may prefer, instead of sacrificing the equilateral property of the cartesian grid, to sacrifice its orthogonality. Three copies of a slightly sheared (skew) cartesian plane can be overlaid with 120° rotation dispersion to form periodic patterns. (The sheared grid is formed by rotating two copies of a parallel ruling ± 43.898 degrees.)

Three copies of a slightly sheared cartesian grid can be rotated 120° to each other and form periodic patterns.

A cartesian grid has two line directions, after three copies are dispersed 120° from each other, there are six line directions (see figure below.) It is somehow easier to perceive these six directions as three narrow-angle 'flashlight beams' with (half-angle) beam-spread of 15°.

Three right-angles rotated 120° to each other tend to be perceived as three narrow beams with half-angle 15°.

In Panda, Maulik, Chakraborty, and Khastgir there a several periodic solutions for billiards played on an equilateral triangular table. Any of these solutions that preserve the full symmetry of the triangle could decorate the equilateral triangles of a deltahedron resulting in a surface wrapped by geodesic lines.

Periodic solutions to billiards on an equilateral triangle table (from Panda, Maulik, Chakraborty, and Khastgir.)

The solution labelled (m=2, n=5) caught my eye, me thinking that the lines were orthogonal (in fact, they are just slightly oblique.)

Flat weaving in this pattern would look something like this:

Flat weaving the (m=2, n=5) periodic solution to billiards-on-a-triangle. Heavy lines emphasize the kagome-like organization.

Extended in this way, it becomes easier to see that the lines are not quite orthogonal. This geometric construction shows that an angle that needs to be 15° for orthogonality is actually atan(sqrt(3)/7) = 13.8979°.

For lines in the periodic pattern to cross at 90°, the apex angle of the green triangle would need to be 15°. This geometric construction shows that it is instead arctangent of the ratio of one equilateral altitude to 3.5 equilateral sides.