From flicker to flame then perhaps>>
Things of single change with a known Zer0 and known potential extreme (of reach and/or force);
>>cos/sin pattern decodes the rest of positions and forces in between the 2 known, extreme, bordering, benchmarks.
>>cos of directness, sin of deflections of displacing against distance and or force.
>>if engaging at all, must do so to the categories of direct and not(deflect) to encompass all.
Can be only 1single organic, premium,pure cosine of 1D , and numerous deflections from
>>commonly from linear force is shown as 2D(like pendulum/swing) with 2 outer sine extremes to the single, sole cos as pattern repeats over and over. i like child's swing for everyone can feel the breathless Zer0 point before each return, harder to sift the Hero/full bull point but that is logically directly down with gravity.
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Another application is simple rope arc180, and decoding it's magic as a supreme machine at each occurrence,
and then even in groups (like specifically Round Turn/RT). We find aligned cos1 and cos0 at 90 to each other
>>but arc creates an even greater external force at apex than internal tension feeding that force (2x point in pulley)
>>work then as 2xTension on load axis for cos, and then 1xTension on cross axis for sine i think
>>to reveal how arc frictions, nips and grips are sooooo much greater than linears(arc0) as a conversion machine.
In this pic of arc we then know Zer0 to Hero(raw/full bull) points and values as fed/electrified with 100# input tension
>>pic is for nip values of single point on arc choices, but all points work in friction together and grip as well; not just the single nip point. It does this while friction reduces the tensions from input to output, as also reduces the rigidity of the output.
i think this is the most powerful 'single stroke'(arc stroke back completes to a cycle like in engine) rope element for rope systems (knots, machines etc.). It gives the most usable conversions and outputs when include apex especially.
ABoK Lessons# 1662-1668/pg.290: Start of chapter_21 :HITCHES TO SPAR AND RAIL (RIGHT-ANGLE PULL) very first Half Hitches and Timbers
shows specific positioning notes in one of the best chapters for us of real work with rope.
and by these maths , this specific position of nip is not by chance:
ABoK Lesson# 0465/pg.076: "Sailor's Hitch"/un-named
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These are just my words and logics to what i think ABoK faithfully shows more comprehensively in between the lines thru his tales and turns.
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Also, note a complete 'gear shift' in way knot carries as work an 'early'/pre-Bitter End rope part to nip at apex opposing source /Standing Part direction of pull in Sailor's Hitch, or even close sibling Pile (harder to manipulate to 'top nip' on rail/spar but lends very nicely on correct size rope to hook ratio), or even the humble 'Crossed Cow' shown recently, contrived to tap into the arc science/avail efforts to it. These knots carry one way, with so much pressure against Standing Part, but shift gears/attitude, but bring from a 'sloppier' side nip purposefully to nip early at opposing apex and knot seems to 'soften', relax as if it knows is in best position, there is less pressure against the Standing Part to soften that weakening deformation/ but then less grip in trade as straighten cosine, lose sine across of hug to host.
high nip cow post