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.

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.

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.

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.

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.

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.