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Physics of the random overhand
Your inquiry has to do with the mathematics of topology. First off, here''s a website for you:
I think it''s helpful to use an analogy to understand what happens to this "neatly coiled rope" as we feed it out. Imagine a chinese yo-yo (a strip of paper coiled about itself and fastened to a stick). It''s analogous to your neatly coiled rope. Imagine you grab the end of the paper strip and pull it as far away from the stick as you can, if the paper doesn''t kink what you will end up with is a twisted strip of paper, looking like an archemedial screw or helical auger bit. Thus the coiled rope has twists incorporated in it when it is coiled in the conventional way.
This is why your handy garden hose neatly coiled at the slip develops kinks when you deploy it to wash your boat or flush your engine. Unless you carefully untwist it when you deploy it you will retain the twist it has in its coiled state.
Flaking is, of course, a way to stow line without twisting it. Another thing you can do is coil a hose or a rope without twist by reversing the twist on every other coil (sort of like adding coils alternately to the back and front side of the stack of coils). You can demonstrate this with a long rubber band or a long strip of paper. Take the long rubber band (or the strip of paper with the ends taped together endwise) and coil it into three smaller coils. If you do it the right way you will have three coils without twist. You will be able to trace your finger along the outside of the coils without encountering a twist. If you pick up these three coils and look at them, you will see that there is a point at which the strip will cross from one side of the coils to the other, crossing over the middle coil. Additional pairs of oppositely twisted coils can be added to form a coiled rope or strip of material with numerous coils, with no net twist in it. Unfortunately, because of the crossing that the coils must do to form this zero "net twist" state, it''s probably not a good way to stow line for quick deployment, as in paying out the line it might get knotted or knuckled. What do you experienced cruisers with rope anchor rodes think about flaking vs. untwisted coiling?