Dear Agent Smith et al,
In both these points of apparent contention:
Friction external to wood etc.
And
Cosine vs. Sine value pair
The effort is to, once again brainstorm into this open forum, and define most comprehensively individual components working inside of any Bowline. I try to show this by what they are individually first and what they are not to the total sum of the whole across all ranges.
The drawing with wood was highlighting HH component singularly outside of Bowline, to understand it more comprehensively inside a Bowline. Also, in that discussion, to show/highlight by contrasting how hard frictions work in nipping loop of HH in Bowline conserved only to plural rope parts; vs. how frictions serve on singular rope part + externally (here to wood) by defining contrast as lesser. The contrasts more fully illuminating the facets of the target nipping loop, and respects for it as a marvel too when host is own rope parts; not a 3rd party.
And, then too, the measure of Geometric Dimensions (plural) contrasts between knot forces in the same UNIQUE Dimension as Load, vs expressing NOT in that UNIQUE same Dimension as Load. Like how much need to keep major Bowline parts working in same UNIQUE Dimension of Load to lock properly vs. spreading eye apart outside said Dimension, lending to unlock. Such as when choked up tightly to large log or other spreading force vs. the lock works spaced far enough from log to allow locking function to predominate. The log can be different sizes too necessitate Bowline lock works be spread different distances from wood to command competently. The truest measure of how much and why is by how hard the specific material in the Bowline geometry is pulled open or close is the angles of pull into the locking function. This could be enumerated to decipher to aha moments with cos:sin value pair to then eventually know more intuitively without the numbers. But to truly appreciate and comprehensively comprehend, seems must put in the work and see the contrasts as well; just as anything else in this life. Cos:sin value pair here is from an engineering toolset, to define the engineering marvel of a Bowline by it’s Geometric forces; as the scientific method of many disciplines.
i’m not saying i’d lace the Bowline to the log; but have seen done or left behind like that many times.
Also, with our ropes and knots, cos:sin examination shows why radial(round or roundish) face host (and ropes) are mucho better than linear face host(square etc.) and (flat, flattish) ropes(webbing). That is what this is about from : Model/Prism of only 3 repeated root elements for organic/cornerless rope works
i truly believe this is the hand of the magician to watch to catch the secrets in this ‘magic show’ of these doings.