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

3x2 strand construction for wire Z's

3x2 strand construction. All three strands have one white and one orange fiber.

The strands have 3 times the twist frequency of the super helix.

3x2 is a coiled coil construction for 3-plies that might work with DNA since the sub-strands are double helices. It also has the advantage of using fewer wire sub-strands (six vs nine) than the 3x3 construction.

Below is a 3x2 construction realized in 24 gauge galvanized steel wire by twisting with a reversible drill. Tension was about 20lbs (dragging an anvil on a rug along a tile floor).

3x2 coiled coil construction in 24 gauge galvanized steel wire

Unwinding the 24 gauge steel coiled coil shows that the frequency ratio is 3:1

Except for a phase discontinuity that formed in the middle of the work, this structure was regular and easy to make. Small irregularities are more evident upon unwinding.

Unwinding the compound helix shows some irregularities.

A sloppy tetrahedron from 3x2 compound helices of 24 gauge galvanized wire.

Metal Z's from compound helical arrangements

A 3-ply of 3-plies. In each turn of the superhelix, each component 3-ply completes two turns. Ordinary rope lay is used: the superhelix has twist opposite to that of the three component plies.

The winding-on/winding-off action of Z's requires a wire that can bend around a radius 25x its diameter without permanent deformation. Polymers like ABS can just manage this, but the best spring wire would need to be improved by factor of about two to accomplish this feat.

An alternative for metal Z's is to use wire rope in place of the solid strands. The configuration above naturally locks together when the superhelix is twisted oppositely to the three 3-ply components.

Another possibility is a 3-ply of 2-plies, also ordinary lay.

A 3-ply of 2-plies. Components complete 2 turns in one turn of superhelix.

Z-bent Springs poster at SMI 2017

I presented a poster last week at Shape Modeling International 2017 in Berkeley, CA summarizing the technique of assembling wireframe meshes from triple helices. The presentation included a hands-on exhibit  of Z's made from 1.75 mm ABS filament.

DNA inspired baskets: Weaving Z- bent springs

Nanotechnologists are hard at work figuring out ways to self-assemble mechanical things from segments of DNA. The wire sculpture below gives an idea how they are able join several double-helix DNA molecules by strand-exchange. Such a structure can be self-assembled from a soup of single-stranded DNA if the single strands have uniquely complementary sequences, so that strands pair up as one intends.

The sculpture also reveals a mechanical weakness in using DNA in this scheme: strand exchange using double helices leaves a single-stranded section in the center of each strut where the splices are made.

Since, unlike DNA, at macro scale we are not limited to double helices, we can move up to triple helices. Triple helices can be joined with have a triple-stranded middle sections (see the octahedron in the top image) and splices near the ends of the strut where they cause less trouble mechanically.

Z-shaped segments can be used to make any mesh since every

edge has a unique clockwise neighbor at each end--automatically

giving a triple cover of each edge.