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

Twongs flashback (cont.)

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

Gauging half-wavelength during the twining process

Completed 2x1 twines of 9 gauge galvanized steel wire

Untwining jig

Powered untwining

Cutting

Smoothing cut ends with a grinding wheel

Smoothed end

Twongs Flashback

I presented a 2x1 helical wire construction set called Twongs at the 2010 New York Maker Faire.

Four twongs formed from 9 gauge galvanized steel wire

Untwining the 2-ply in the forming step

Partially completed twong vertex

Completed twong vertex

As purchased wire

Wire being partially straightened winding "the other way" against a  18.5" radius template

Winding end of manual twiner

Sledge end of twiner

Man at work