JPL - a math look.

[Karpeth]

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This is does not care about layering changes, and is a summary of what we discovered at MAIL.

Definitions:

JPL is only possible in "true" form at "odd JPL" numbers.
"JPL number" is the number of possible positions for the rings.
"JPL number" is (4x/2)+1, where x is the iteration.
The number of JPL variations possible are as such "JPL number"-1 for every JPL Grouping.
Given that mirroring produces identical chains, except for chirality, you get ("JPL number"/2)-1 different configurations, where the chain possibilities are numbered by "step x".
JPL is not fully rotate until 2 whole rotations.

The formula for where to put your next ring in relation to the first ring in degrees is as follows:
(720/"JPL Number")*(step+1)

As such, for JPL3, you get the following positions:
0
240 (apparent: 120)
480 (apparent: 240)

for JPL5, its:
0
144 (apparent: 144)
288 (apparent: 288)
432 (apparent: 72)
576 (apparent: 216)

For JPL5 step 1 its:
0
288 (apparent: 288, apparent 72)
576 (apparent: 216, apparent 144)
864 (apparent: 144, apparent 216)
1152 (apparent: 72, apparent 288)

The math continues in the same pattern forever.
JPL 7 has every new ring in 720/7 offset from the last ring.
JPL 7s1 has every new ring in (720/7)*2 offset from the last ring.
JPL 7s2has every new ring in (720/7)*3 offset from the last ring.

 

[Karpeth]

These offsets are what’s needed to produce the traditional orientation.

In older instructions and analyses, there’s talk of ”forward and back, alternating”, and ”flip the weave”. This is a practical solution to the theoretical issue of the 720 degree standard rotation.

If you always add the ring ”on top” with the Above offsets, the standard weave is produced. Constructions such as single oscillating JPL alternate this.

 

[Karpeth]

JPL 9, follows the same pattern, but as 9 divides by 3, you get a HP weave, I forget which.

0
240
480
720 (0!)
960 (240!)
1200 (480!)
1440 (0!)
1680 (240!)
1920 (480!)

As they have to share the same lean, you not only have the JPL lean, but the HP lean to account for.
This problem only occurs when the JPL is not prime.
JPL15, 21, 25, 27, 33 and so on have these HP crossover weaves.

 

[Karpeth]

This is much from memory - but the math is what falls out.
(there might be an error, and it would be that the doubling was of 180, not 360. As such, the long standing error on MAIL was perhaps viewing the 180 rotation as complete.)
EDIT: I am now leaning towards that my math was doubled, unintentionally, with the logic as follows. While math will help you get there, I loose the math when the weave starts to make itself with my hands.

What appears in JPL as adding rings at 60 degrees and flipping - is actually adding at minus 120 degrees.
As such, the following changes are needed above: JPL is not fully rotate until fully rotated.

720 and the degrees above should be halved.

As follows, the positions should be.
0
120 (-60)
240 (-120)

0
72
144
216
288

0
144
288
432; 72
576: 216

and so on.