Cosine, my friend

[Mellow]

 

[theTreeSpyder]

In some aspects of this repeating pattern, yes.
Roo's NotableKnotIndex pages talk some about friction.
Offers a picture showing 6x arc180 with rope to metal CoF of .3
input a HOLD(not trying to lift )of 7# effort LEVERAGES to holding 2000# monster force (by comparison)at bay with this 'simple' math.
Notable Knot Index: The Power of Friction
My version way back when was drawing of a baby sitting on the end of a rope going to tree warps, holding back a bus!
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Notable Knot Index: Pipe Wrench from Rope shows an interesting old-timey usage of this ever present force to harvest to such services.
and also lists this Google Calculator readout showing the formula again , and the keys used for E, PI, exponential etc. to the 1ton sum of hold from the 7# effort!
7x(e^(0.3 x 6 x PI)= ~2000# is all it takes to peel back the curtain and peek at the wizard workings hidden in microcosm of a knot.
>>so how to command this and other things around you; with these 'simple' mechanical commands, as like computer commands.
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As friction reduces the tension of the rope, it also reduces the rigidity, to a more 'nippable soft' (link)

 

[theTreeSpyder]

Previous post reference to Roo's Friction page seen in chart below.
line_14: 'triple round', 1080 degrees , 6x arc180 0.3 CoF gives 285.68 friction factor leverage in chart
>> 7# holding effort X 285.68 leverage = 1,999.76# force/load held
wish had such an easy 285x lifting lever, but 285x easy brake leverage is nice to know too !

By the engineering numbers, a nylon rope on an Aluminum vs. Steel round ropeBrake device;
gives exponentially wider and wider frictional differences with successive arc180 turns.
The only variant is CoF Nylon w/Aluminum @ .25 vs. w/steel @ .40
Stuff like this can be much more of a mystery until can pull the curtain back with the numbers;
and start to read as like on the right frequency, with less garbled dead spots etc. in what you do get.
Always tell self more of an open the door, rather than peek underneath the door; look at the world.


Radial vs. Linear Frictions for Rope Rescue and Nylon/Aluminum CoF: .25
rest of quoted CoF’s: EngineeringToolbox.com
fuller spreadcheat: tinyurl.com/radial-frictions-spreadcheat

 

[cory]

💡

There is only one Kenny

 

[theTreeSpyder]

cosine, my friend(s)
cosine....

 

[Marc-Antoine]

Said like that ...

 

[theTreeSpyder]

This stuff predicts, as if to command, star movements.
("Nature, to be commanded, must be obeyed" -Sir Francis Bacon)
This same 'organic' pattern that encompasses wider than the stars, also engulfs all below the stars to abide by same pattern.
All from the actual counter-intuitive cosine scale of change vs. our expected scale of change, judged from pivot and organic benchmark of most efficient.
The pictures, of ball on rope, slant from vertical ,
help see trig triangle in all. (that then is simply a 'closed' '0' size triangle at vertical hang of focused potential).
The non slanted faces are the measures of vertical and horizontal influences from the actual slanted usage.
Thus, accounts to us for aligned and drifted to cross-alignment axis maths against distance and force from the potentials.
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Picture below shows columns of any material as before to compare together side by side.
Note how if always 1000#tension supported load is just cosine x1000#; this is cos1 organic point, Nature chooses as most efficient and our benchmark single dimensional cosine.
>>just as if always 10' long , distance pure vertical downward from pivot drops from 10' to cosine x10' also.
The deformity in a rope column to load reduces efficiencies of reach and strength in a rope column to knot just as hang from pivot.

The pattern for change in the range from Zer0 to Full(0-100%) of these potential values is not to an intuitive easy, incremental scale of change but rather counter intuitive cosine scale of change.
This is for distance(therefore speed) and force (therefore waveform) etc. that can predict not only planetary movements accurately but also water waves and realistic shadowing etc. Just by realizing the scale of change from Zer0 to Full expression of a potential (length or force etc.) is cosine, not incremental !
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Pareto , i personally think of as "pare to" for my L-earning and attack purposes, was an Italian town that found 80% property owned/ruled by 20% of people, a pattern found typical in many things now known by such aliases as 80/20rule ,law of sparsity of pivotal principles, law of the vital few etc.. This cos/sine of alignment and deflection factors for any connected part in scenario are a true pareto/'pare to' to me.
The cosine scale mirror flipped into sine so that as one declines the other increases(alignment vs deflection to sum of all elements now available to work for and/or against you) . Trying not to focus on sine, but inevitably must to show full glory and comparison etc. to cosine; and show how are L-earning sine as do cosine.
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cosine my friend(s), cosine..​

 

[lxskllr]

Pics? Did I inadvertently block them?

 

[cory]

um yes the 80/20 rule

 

[theTreeSpyder]

Cosine/Sine accurately predict, as if to command, the movements(even distances, sizes and speeds), of the stars and planets.
Whatever engulfs them, locking all to this same pattern of changes, engulfs us too; and we are even less powerful against, rather only subject to!
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Just as pictures above in posts 30 and 32 show decoding to cosine as efficiency works for even Standing Part of rope as a column structure into any knot/Bowline; we can decode, even reverse engineer, other supports (even distances and therefore speeds) with this pattern. If know Zer0 value point and full potential point and value:
Cosine gives all the values and explanations between Full and Zero for supports*, extensions/displacements against space and per those extensions over time therefore speeds. Force is needed to reciprocally trade for distance, therefore the same formulas work even in these expressions of force displacement against other force. i find these 'few'-er things to be the root principles, to really read, forecast as if to command what setup hauntingly accurately at some times; as it all sinks in on how long ago Ancients tried to show us...
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This drawing shows same amount of rigid support material diameter for strength and length for reach as separate properties of separate values. Double length and strength needed when cosine =1/2 at 60 degrees. 4 units total needed, to get double length and strength are needed at 60 degrees deflection from load confrontation for what the single unit at 00 degrees deflection from load confrontation can do for 1unit co$t by this math. You may choose to steer vs. trial & error to the same math imposed. These are decisions as used in how we make temporary supports of rope etc. on the fly. Math is the language to show expressions of reach and strength to and against load from the known potentials revealed at 00 deflection of cos1.

* Even if must flip it backwards to show things that capitalize on sine (like leverage as opposed to these support column examples capitalizing on cosine value ) . Still trying to leave sine alone, but that is the over view that should be known.

 

[theTreeSpyder]

Another actual usage example on how this stuff defines/rules/decodes all what can see, touch and even much more: 90 degree bracing.
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Looking to cos as the efficiency of alignment, therefore support column efficiency, a 90degree angle gives cos0(0%) as worst mechanical angle of support/best angle of strike against.
>>but also a most useful, and so a most connected and braced angle(even if just at the joining seam of glue etc.) seen in many forms to same math.
Looking at bracing across as connection chain, whole system fails when the weakest part blows as fuse.
45degrees median will be the only point that both connecting ends of the brace/glue/weld would be equal, everything else being set equal.
The cosine of 45degrees is 70.7%, if narrow to 44degrees on one side cos goes up for more support column lent, but the opposite side then must be 46degrees with lower cos and thus earlier breaking point(as like purposeful fuse that limits loading to rest of system)in this chain of events to support.
So, just follow the numbers to 45degrees!

Note also how the more closed angle to 30degrees as always also effects the distance, and so leverage of support, that is then lost on the 60degree side.
>>So greatest connection and leveraged position of that connection(another factor) for both brace angles together is again 45 degrees.
>>compounding effect of both as same maths to the whole of support, as how pulls along column support and gives leveraged support across!
45degrees is the peak of this balance.
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Cosine is 'simply' the pattern that the support column efficiency etc. degrades at.
So if know the extreme positions and values, can define the whole range between.
cos as amount of alignment, sine (reverse cos scale)amount of deflection; if none of neither is not logically part of scenario anyway!

 

[theTreeSpyder]

Kind of a twist hear.
Your primal grunting mind knows cosine, but not as math; that only educated non-grunting mind fights with.
That is the secret to virtual reality: calculate all gradient changes of size, speed, shadow, sound etc. per the natural, organic cosine pattern.
>>then tricks primal mind into thinking is real/gets by the gatekeeper, and is described as being 'immersive' as draws you in to feel like in the scene, surrounded in real time.
Anything that does not pass 'cosine test' by primal, is passed over as not real; not a primal fight/flight/freeze threat. Watching news, not in it.
That is how deep cosine pattern is ingrained in self and all around so much that can't sift it out, struggle to read it when do; but innately know and depend on it so!
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When i started drawing shadowed gradients, they weren't push button yet, still the raw form of had to write small program to evenly calc shadows correctly; it was always based on competing antagonists cos/sine to look real. Just like as we see with the raw work of trees, i was into the raw work of shadowing to L-earn and cross compare cosine across both/all as a single topic. Even extending into waveforms of electric, water, wind, light etc. on graphs as all the same lesson(s). Which means you can L-earn one from the other, and each echoes the others depth of importance across all and to a connected/contiguous more tangible fabric of understanding instead of individual loose points trying to remember to in turn command.
Doc Shigo invited to touch/even embrace trees to know them, have done so with cosine too.

 

[theTreeSpyder]

This 'primal grunt brain' knowing cosine better than higher order civilized mind's math is not my only point of contention of psyching self/opening eyes to already know part of this; to then throw harder, more confidently into it.
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Lots of things i try to describe are not only things witnessed visually, that can show mathematically but then too that have felt in gymnastics, exercising/climbing etc. for a deeper familiarity. From across the sea i think that is what some of the martial and yoga arts binding to a knowledge is all about, watching and feeling these forces inside self to more innately know and flow with thembetter.
Cosine felt in a child's swing is won of those, so simple; and universal sitting in one of the simplest of dynamic/moving models: pendulum
>>lessons in cosine rite hear as if for a child; if we key into them:

The same pattern of change is in tree fall, eventual drop of horizontally fired bullet, force of rope pull sideways, shadows, waveforms of wind, water, electric, light, speed of planets etc. Just usually, we are shown the revealed pattern as a loss of column efficiency of strength, as not an end to itself, but a window into this key organic pattern that controls/reports so much.
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In theoretical simplest models look at as perpetual/no friction etc. degrading for pendulum horizontal, spring vertical etc.
>>so if know 1 side/sine extreme(cos0) and the organic natural primary cos1 can use cosine pattern to extrapolate the inbetween points
>>and in perpetual motion mirror back the same to the opposite extreme to complete the 'stroke'
>>return to start completes a cycle as engine terminology when piston has these values
Accurate enough to send peoples into space, around the moon and confidently back whole, from following these numbers on a plastic slide rule.
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Tree forces are huge, but so are the even larger whole planets that must follow this organic pattern, just as tree forces must.
As we dial volume of friction up from Zer0, the system dynamics fade to less within the stated potential, and over time grow quiet.
But only calm when settled/satiated at the pure vertical hang of cos1; until pry it from this natural 'home point' again, to start swing over.

 

[cory]

I got the part about a child's swing.


The rest I'd need my hand held for.

 

[theTreeSpyder]

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