Friday, August 11, 2017

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.

Wednesday, August 9, 2017

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.

Friday, August 4, 2017

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.

Monday, July 31, 2017

Non-helical scaffold strand in synthetic weaving

Synthetic weaving with a helical scaffold strand (orange.)
There are good reasons to want to avoid the use of a helical strand for the scaffold. Foremost is that a helical scaffold imposes differential lengths in the weaving due to the problem of like-turns vs alternate-turns.

Since staples go straight, there is nothing special about the radial vertex. Any integer number of vertices (half-wavelengths) is OK. But the scaffold strand must land on a vertex that naturally turns in the desired direction. A non-helical scaffold strand would not have this limitation.

Anothet advantage of a straight (non-helical) scaffold would be the use of standard NC wire forming machine to bend it.

Synthetic weaving of the tetrahelix map

Cyclical bends in the scaffold strand (green) in a synthetic weaving of the tetrahelix map. Work order is from right to left, spiraling upwards: start, 60 up, 120 down, 120 down, 120 down, 60 up, 120 down, 60 up, 120 down, 120 down.

Synthetic weaving technique

Synthetic weaving progressing from left to right, just prior to twining-in the second staple at the vertex in the center of the view.

Synthetic weaving works on any surface

The system of the walk in the medial avoiding the dual's spanning tree is general—it works on any surface, not just the sphere. Weaving of course is also general, so synthetic weaving works on any surface, not just genus zero or orientable, and their is always a free choice of working order.