Thursday, August 31, 2017

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.)

Wednesday, August 30, 2017

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.

Tuesday, August 29, 2017

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.

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.

A-trail (working order) in synthetic hexagonal weaving

An A-trail (green) through a hex-grid portion of the base map. Dashed lines are a portion of the chosen spanning tree in the dual map.

Synthetic weaving

A small sample of synthetic weaving on a square-grid map. The orange scaffold strand follows an A-trail in the medial map.The short staple strands each cover almost two medial edges, producing stiff, 3-ply strut-like sections between crossings.

Staple strands extend straight-across the crossing, producing an 'elastic hinge' joint for the four struts that meet there.

Two A-trails in the medial from two different spanning trees in the dual. No computation needed: the A-trail mirror reflects off of primal edges and spanning tree edges, respecting the latter when there is a conflict.

Sunday, July 30, 2017

Medial manufactures A-trails

OK, now I get it. The map operation Medial turns any map into a literal forest of A-trails—there's one for every spanning tree in the map or its dual!

Find a spanning tree in the dual of the map, then trace along the medial staying within a face of the  primal until blocked (reflected) by the spanning tree.

Baskets have A-trails!

Bridges Waterloo 2017 was great. Thanks to the organizers and Craig Kaplan especially.

I am posting a pdf of my presentation on google drive.

Baskets have A-trails!

Too late to include in my talk, I learned of Anton Kotzig's proof from the 1960's that every genus zero 4-regular map has an A-trail. Wow. Given any map on the sphere, the map operation Medial (which also gives the path of the weavers) converts it into a map with an A-trail. It follows that the weaver pattern (not the weaving) can be constructed by pinching an untwisted loop (like the kids were doing in todays Family Day activity.) By the way, great work guys.

Monday, July 17, 2017

Surface color patterns on 2-color 3x2 Z's with 4:1 turns ratio

A section across a single-phase portion of bicolored 3x2 compound helix with 4:1 turns ratio reveals a symmetrical arrangement of colors

Color pattern on surface of section shown above.
Color pattern on surface of a bi-phase portion of the same compound helix.

Saturday, July 15, 2017

Making some Z's

A two-ply Z made from white and orange Hatchbox ABS.
My current technique for making 3x2 Z's uses just a single anneal, which might be practical to achieve with a hot air gun. 2.44 m lengths (measured between clamps) of white and orange 1.75 mm ABS 3D printing filament from Hatchbox were stretched using as weights two drill press vises (7 lbs. each) dragged along a carpet.
A slow-speed drill was used to twist the pair of filaments counterclockwise. I believe the total number of turns were about 440 (there were some mishaps in the twisting,) or about 180 turns per m of original length.

Three such 2-ply strands were made. The twist was preserved in each strand with a heavy clamp at the drill's end.

Then the three strands were twisted together in the clockwise direction to a pitch of 0.47 turns per cm on the finished rope. My hope was attaining a 4:1 ratio of strand turns to rope turns to keep a neat phase relationship between the two colors. The actual ratio proved to be 3.8:1, which is not nearly accurate enough to maintain proper color phasing along the length of a Z. Oh well, this is just an appearance issue, they twine together fine. The rope was annealed while stretched in place using a hot air gun. This annealing was not as thorough as might be wished as indicated by the slightly pale orange in the pigmented filament.

The 3x2 annealed rope was then unlaid into its three separate strands in preparation for bending and cutting the Z's.

An all-but-finished tetrahedron woven from the Z's.

Wednesday, July 12, 2017

3 vs 2 twists per major wavelength

Three twists per major helical wavelength (left) is a tighter structure than two (right.)
Both the 3-twist (upper) and the 2-twist (lower) are attractive when assembled into a 3-ply.

Phase relationships in 2-color 3x2 Z's

There are two possible phase relationships in twining three 2-color, 2-plies into a 3-ply:

Color phase relationship in a 3x2 compound helix where the nearly axial rows are 1B1W.
Color phase relationship where the nearly axial rows are 3B3W.

When the first two 2-plies are twined, they can be screwed past each other into configurations that (in the direction parallel to the axis) pair either BB and WW, or BW and WB. The latter pairing leads to the 1B1W phase relationship if the third 2-ply is phased to continue the WBWB alternation. Any other color phasing of the 2-plies results in the 3B3W phase relationship.

Monday, July 10, 2017

Orderly 2-color 3x2 Z's

An orderly vertex for 2-color 3x2 Z's.

The center of an orderly 2-color Z must be a center of rotational symmetry for both colors.
In a Z consisting of a two-color 2-ply, the coloring needs to be identical at both bends, therefore the center of the middle section must be a center of rotational symmetry for both colors. That condition requires a 'stacked' configuration with one favored color lying directly atop the other when viewed from the 'z' side.

If there is a desired phase relationship between the minor and major helices at the vertex, the frequency ratio must produce this phase difference in the distance between the center and the vertex. In the current design, that distance is 1.0 wavelength (from top to top) + 0.5 wavelength (from top to bottom) + 0.25 wavelength (from bottom to half-way). So an integral number of half-twists must be completed in 1.75 major wavelengths. The pictured model completes two full twists in each major wavelength, so 1.75 x 2 x 2 = 7 half twists, satisfying the condition, but solutions with 6 or 8 might also be useable.

In the top image above, a relatively tight structure shows the two colors also lying 'stacked' at the center of the vertex bend.

Friday, June 30, 2017

Counting frequency ratio in a compound helix

The easiest way to count the frequency ratio in a compound helix is to count (starting from zero) the outward appearances of a single sub-strand in full major helix wavelength, and then subtract one for the counter-rotation effect of following the major helix.

This 2-ply, unwound from a 3x2 compound helix, exhibits a 3:1 frequency ratio.

Pitch angle = pitch angle

In ordinary-lay compound helices, a stable configuration tends to occur when the pitch angle of the major helix equals the pitch angle of the minor helices. For example, the case below of a 3x3 compound helix: the 70 degree pitch angle (measured from the equatorial plane) of the major helix matches that of the three 3-ply strands. The frequency ratio in the 3x3 compound helix  is 2:1. The equality of pitch angles together with the counter-rotation of the ordinary lay, make exposed strands look like they run straight along the rope.
3-ply strands with 70-degree pitch.

Compound 3x3 helix with 70-degree pitch.