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.

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.

First attempt at 3x2 Z's in 12 gauge steel

These were formed from dull galvanized 12 gauge steel wire pretwisted to about a 60 degree angle (too much it proved). Some surface fracturing in the final shape, not elastic enough, and very hard to work by hand.

Possible non A-Trail vertices for Twongs and Z's

In the simplest, most versatile versatile vertex for Twongs and Z's, each element twines with its immediate clockwise neighbor, as in this three-talent vertex:

But there are other possibilities that may be useful:

Twongs flashback (bending)

Continuing the forming details for the 9-gauge 2-ply 'Twongs' presented at 2010 New York Maker Faire:

Schematic for bending

Bending jig in use

Completed bend

Completed Twongs

Twongs ready for the Faire