Example of a square 2-ply Z with congruent wires

A square 2-ply Z with congruent wires

Wires arranged anti-parallel as they would be upon unwinding.

Wires arranged parallel to show congruency.

This Z is 1.75 mm diameter ABS with a wavelength of 15 mm, and an outside, peak-to-peak height of 5.2 mm in the coiled ribbon orientation. In terms of diameter (d): wavelength is 8.6 d; p-to-p is 3.0 d. Measured flat, outside p-to-p is 6 mm = 3.4 d.

Coiled-ribbon appearance of square 2-ply

Square 2-ply viewed against a lightbox.

Properly-made square 2-ply can be rotated to an angle where its silhouette resembles coiled ribbon. This is the correct rotation for bending Z's.

Schematic of the "coiled-ribbon" appearance of a square 2-ply. 

A common manufacturing fault that prevents such a resemblance is twist.

Z's from square 2-plies with congruent wires

End and side views of a square 2-ply. Lines mark balance points. Bue dot marks the midpoint of a Z that could be formed by making bends around the red pegs.

It is possible to form a square 2-ply Z from congruent wires. In the figure, the blue dot marks the Z's midpoint, and the red circles mark the bending locations for the shortest possible Z. The wires are still congruent after bending.

Longer Z's can be realized by incrementing both bend locations farther out, two balance points at a time (i.e., symmetrically adding a full wavelength to the interbend distance.)

Finding balance points

When a square compound 2-ply lies on a horizontal surface, a vertical view will reveal some of the balance points (half of them to be precise) as the places where silhouettes of the two wires cross. These are also points of minimum apparent width.

Cross-section of a square 2-ply pressed against a horizontal surface.

Balance points marked on a square 2-ply

Balance points and congruent wires in square compound 2-ply

For a cut length of square, compound, 2-ply to be composed of two congruent wires, the midpoint of the length must lie at what I call a balance point. Any length will do, so long as the midpoint is at a balance point.

At a balance point in a square 2-ply compound helix, the line joining the centers of the two plies is perpendicular to a radius of the major helix. For example, this 2-ply has been cut exactly at a balance point

For comparison, this 2-ply has been cut just short of a balance point.

When a length of 2-ply is centered on a balance point, diagonally opposite half segments are congruent, and therefore also, the two wires are congruent over their entire length.

As these end-views show, at a balance point the two cut faces of the wires lie parallel to a side of the 'square'. There are four balance points per wavelength.

Wire paths in a 3x2 square wire rope

Wire paths in a 3x2 square wire rope. (Actual cross-sections do not remain circular.)

In this animation contacts at the same level can be seen (intra-strand and inter-strand.) In a packed configuration there are additional contacts between strands crossing above/below.

A 2-ply compound helix (where twist and writhe have the same counter-rotating wavelength) always has a crossed-ellipse appearance in cross-section. These 2-plies can "stack" to accommodate any number of plies, e.g., 2x2, 3x2, 4x2, etc.

Using a straight wire for the scaffold strand

Wrapping a 2-ply helix around a straight center strand of the same wire diameter results in a low pitch angle: about 20 degrees vs 40 degrees for a true 3-ply helix.

It is tempting to use a straight scaffold strand in synthetic weaving since this would allow use of a current generation CNC wire bender like the D.I.Wire+. Unfortunately a low pitch angle results, which makes the helical staple strands prone to stretch at the crossings.

Synthetic weaving in a nutshell

Every mesh has a weave pattern. To find the pattern, connect the midpoints of the edges around each face (this is known as the medial construction.)

A single wire can trace the weave pattern without crossing itself. To find such a circuit, first find a spanning tree in the dual mesh. The circuit tries to avoid crossing edges in the primal mesh, while never crossing the spanning tree.

Helical 'staples' wind on to make the crossings, and overlap & interlock to form a 3-ply wire basket.

Why synthetic weaving?

A crossing in synthetic weaving. The continuous scaffold strand (orange) visits the entire fabric but makes none of the crossings. Short staples (purple and white) specialize in making the crossings.

The same synthetic weave crossing as seen from the 'wrong' side of the fabric showing the tag ends of the white staples.

Weaving is perhaps our most useful technique for constructing a fabric surface, but it can be unwieldy at large scale. How would you weave a building? If you've ever woven a basket, imagine manipulating a thicket of free ends some meters in length.

Unit weaving, Da Vinci style.

Unit weaving, IQ's style.

Unit weaving,Twogs style.

One solution is unit weaving (nexorades etc.) This is an idea that dates back to Leonardo Da Vinci. Unit weaving solves the problem of scale by weaving very short elements-- basically splices are placed everywhere. The structural weakness of the splices limits the practical application of this approach.

Pre-formed wire crochet.

Pre-formed wire crochet builds a surface from a continuous unspliced wire. The wire's path is loopy which makes it difficult to produce and maintain a precise surface geometry. One special advantage of pre-formed wire crochet is that information the crocheter needs in order to build the surface can be carried in the pre-bending of the wire.

Synthetic weaving.

Synthetic weaving is hybrid unit-weaving and pre-formed wire crochet that derives from mathematical insights into basket weaving, experience with unit weaving, and the scaffold-strand technique being used at Karolinska Institutet in Sweden to self-assemble nanoscale baskets from molecules of DNA.

I use synthetic in the sense of "put together." Even though a single long wire, the scaffold strand, constructs the entire surface (as in crochet) it fails to make any of the necessary weave crossings! That work is left to specialists: short helical unit weavers called staples. In some cases these staples can all be identical, interchangeable, and reuseable. The pre-bent scaffold strand carries complete geometric and working-order information for the weaving. While the scaffold strand does not participate in any of the crossings, it does contribute to the strength of the bond between the two helical strands that wrap around it.

Synthetic weaving is like unit weaving but with very strong splices where all the build information is in a pre-bent wire, and like pre-formed wire crochet but with the straight-line force transfer of weaving.