# Cosine, my friend

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)

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

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💡

There is only one Kenny

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

Said like that ...

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..​

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Pics? Did I inadvertently block them?

um yes the 80/20 rule

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.

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!

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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.

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.

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I got the part about a child's swing.

The rest I'd need my hand held for.

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

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Below shows some of the places that the math decoder 'ring' has taken me, or sometimes just verified for me.
The preceding post shows in arcs how cos and sine are used for seating rope to host as below does too;
>>linears by contrast can only use sine xTension for same host seating, not cos(that is dedicated against load in this simpler form)
>>arc/radials can use (2x cosine + sine) xTension in this compound form by contrast!!!
From engines i find 'displacement' as very raw and exacting term in all this.
The Ancients gave 2 quantities to decode with :
cos% as amount of potential purely aligned to benchmark axis
vs.
sine% as amount of potential purely crossing benchmark axis
or force when that simple free travel is opposed by another displacer; "displacing original displacer" instance.
As a UNIVERSAL law/pattern, encompassing Earth and then extending far outside all known.
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Many times the numbers told me/re-enforced what already witnessed, also affirming larger pattern(as nips below)
>>other times just following the numbers, purely lit the way from darkness of not knowing, or thinking i might know...
>>either way , smacked forehead many times, many aha's as things fall into place as there all the time .
After ending up with same pattern as Ancient's so many times, take shortcut/just simply start there became the norm.
Nip pattern in single axis(like vertical gravity axis) is simplest study (w/single arc) of linears, arcs, radial positions etc.
Simplest model of arc0/linear ropePart(s) and single arc180 ropePart so simply stay on the input power axis , in this minimum model.
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Radial nips, grips, frictions to control a load expect as 1+ xTension fed; so greater than linear(usually MUCH greater for linears of<1 xTension host seating force)
>>also see that position in radial arc per input direction determines how the xTension of the rope is used, as radial force is by degrees/not distance :
nip force values translated for radial and linear
arc180 nip force = (2cos + 1sine) xTension (radials)
arc000 nip force = (0cos + 1sine) xTension (linears)

Seating force of rope to host powers nip force for a single position, same force also powers inside a range of frictions and/or grips, nip force is the single start position of rope seating usage.
A practical example of linear force into arcs and how changes nip is ABoK Lesson#0277 pg.049: Groundline also Groundline ABoK Lessons 1243 pg.224
>> Groundline used as Hitch showing how diameter of rope to host ratio can affect nip force in the arc
>> per the dismal-best nip regions given above

Sibling Bag Knot is similar but gets better start of Fig8 upgrade to Half-Hitch ABoK Lesson#1666 or Lesson#1668 fig8 upgrade to Timber Hitch both on page.290
>>in all 3(Groundline, Bag, Timber) get extra frictions, pressure, and spaced more towards arc apex with the strategy
>>Bag/Groundline ABoK Lessons #1242,1243 pg.224 can also use helpful correct slipped form to push nip towards TDC nip/best region
Especially like Double Slipped Bag as spacer when small rope/cord to host diameters.
>>similarly i view the final twirl in Timber/Killick as most positive nip trying to use the preceding twirls (each also a reducer) as spacers for the final twirl/nip to be in the best nip zone. i still give at least 3 tucks tho , not dropping a stitch cuz say can for #1668; and really plot for nip in the best nip zone as go, and generally can use the space of the 3rd tuck for that. i consider keeping the 3rd tuck minimum as a safety factor of using slippier ropes now, than the Naturals rope written for.

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Part of my linear imagery/logic proof is one that seems to break traditional thinking of tension degrading thru the friction path of a knot. It starts at one end at full bull and recedes to Zer0 in sequential pattern.
Also in building my own understandings, an arc90 changed the force flow from present reign of loaded axis across w/'Samson Angle' at cos0 /perpendicular angle , then continues to a new, now restated loaded axis for rest of following ropeParts.
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Not so i say with the continuous/both ends loaded form of Half-Hitch(HH) as put on nose of spar to pull to convert Timber to Killick etc., also in Bowline and Sheepshank as possible fail point of HH rolling out. The force instead seems to stay inline more from input of 1st arc90 passing to output of 2nd arc90. What seems to break the rules, instead screams the same ones back louder, the tension is least midway here, only peripherally loading grip around spar, as most tension favors the simpler, lesser as if 'electrical' short path. This is lost if the 2 hooks of arc90s un-hook; then we see the more expected gradual progression of greatest, mid and then least tensions working end2end like most knots. This then changes from a 1x pass of linear force bypass the arc180 mostly, to then fully loading 2 now arc180s, w/ 2 xTension and bypassing to 'crunch' thru a rope type host.

i think the force flow pattern changes here to more sequential/normal; only after unhook/rollout of the HH where loses the more direct path end2end before unhooks.
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i have many electrical path/logic theories for rope etc., under theory of force is force/electrical or not and it is all force flow conducted thru a device, even if not a wire device.

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If can see everything in the cos/sine aligned/crossing perpendicular pairs sense of total influences, the load would pull down the inner length of a rope equal to or less than xTension and 90degree deflection to outer side as then less than xTension. Linear can only use this lesser xTension, more nominal sideForce byproduct of deforming from pure line as nip etc. and then only in 1 direction at individual point(s), so nip must be (generally much)less than xTension. Just as when sweat/swig a line, the line tension is the greater force, not the (baby by comparison)cross force you impose. Radial nip by contrast can also use the greater load force with the more nominal side force for nip, generally to a sum greater than xTension itself, and also over a range in multiple directions. Literally the radial nip is as like the linear nip strategy on steroids. Same for linear vs radial frictions and grips. Radial is the machine in ropework of power to change, linear is more transfer at perhaps some nominal cost if deformed. In linears cost byproduct/tax over and above normal loading can then be capitalized on as nips, grips, frictions w/host seating sideForce byproduct, but not the primary load force orchestrated into host seating factors until corner or arc deformities i think. Corner at a point to linear faces for greater nip at apex of corner deformity only or arc over a range.
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i always thought the ship peg lessons of tinyurl.com/abok-chap20 'Belaying & Making Fast' showed this best/even as a conniving fight for it. As to even turn 2 perpendicular linear faces(face and 90degree shoulder/shelf to/on) to then extrude radial nip, and the fails if don't size properly etc. showed this best, cleanest/simplest(fewest ropeParts and functions); and even with arc90's of lesser than arc180 nipPotential. These simplest rigid host (as opposed to soft/rope host) single, simple Hitches, are real base examples i think. Ones that calls Slippery Hitch only after makes Bitter End a bight as to slip; to remotely dance/shake/flirt an unloaded rope free. Working End (WE) does not bypass SPart but serves immediately under it's greatest primary arc, w/only an arc90. But orchestrates secure nip in minimalist setting of fewest players; as study starting point for me. In next chapter defines right angle grab around host as 2 types: bypass SPart comes back around it to secure/nip around SPart as 'loose' Hitch or nip off around host instead as 'snug' Hitch, of which these Single Hitches are neither. So is another class in this chapter_20 set apart just before the right angle chapter_21 that leads to the lengthwise chapter_22 in logical flow.
i believe these to be some of the most simple, minimal examples to openly see the most basic hitching requirements of hook, tension rigidity reduction and nip.

Basic 3, all lacing wise match, but sizing and positioning/alignment of the ropeParts geometries is still critical.
A feel for cos/sine help to just read that from the descriptions etc. as another language besides that which is printed.

You are brilliant !!

Your're jest trying to make me blush Miss Virginia, ty;
But hold as by some creator command,
in the balance each are their own savant(means i can be big screwup other places perhaps).
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but sine is simply same pattern in reverse numbers and lateral deflection cross ways to cos.
from sine/deflection view perspective>>
Problem seems the cos/sine pattern of change hides in such plain sight*,
>>eye passes over easily what is so blandly buried in everything.
But virtual reality shows/sifts out back to us that if get the numbers of change wrong
>>eye (and even ear) discern vid/sound as fakey
>>and so does not draw you in so 'immersively' like part of action
It seems the deeper animal survival brain, that can't even count, can tell is fakey/off the numbers tho
>>not matching any known real experience, so not survival threat, so not immersive/emergency engagement,
>>observer feels more like detached observer; not tricked in thru this magic act thru eye perception
>>magician's for millenniums have been similarly fooling the eye pre-processor to brain!
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To me, when even driving (especially motorcycle)can see/feel these numbers.
>>cos is at right angle in/out of truck seat & bumper of front/back directions as inline axis, sine is sideForce on curve across seat & bumper
>>1minute clock deflection from straight is 10% side force movement feel.
>>2min:20%, 3:30% up to 6mins. deflection:60% side force movement feel(fair rounding accuracy to 100ths).
Sometimes tune into the numbers feel so well find they even feel so very hauntingly correct as drive around arc and think aha(as also tuning senses and brain as drive to rigging job) !
Relate driving feel to numbers, easier to more innately see in other things from this feel i think
(trying to process by feel what eye can get wrong).
Rope arc is same to me, 1degree slice of rope core's direction axis can model as cos, and forces 90degrees to side jacket/mantle against host as sine. Just like now rope driving around arc (1 degree slice at a time)instead of truck.
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Babylonians that gave us circle, degrees etc. gave us clock/calendar to show circle/cycles on; all in their base 60 math.
>>not sure of Babylonian's intention(find no pruf over decades of search), but numbers on clock like this for sine(found in highschool to get by multiple choice tests) simply work to decode on the fly down and dirty. No calculator, tables, sliderule, protractor, suit/tie, years of school, diploma, clean fingernails etc. needed to command; so can be anyone's.

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sideForce direction in tension vs. sideForce direction in compression
When pull a trailer, it follows as obedient puppy, that can taunt into bear pushing it backwards tho.
As pull trailer(tension) it wants to pull inline with any side force given when not inline
>>as the tension direction is all ways and always self-correcting until no correction needed at inline
But backing a trailer is opposite of compression direction, so any side force when out of alignment
>>works then by oppositely to push then even more out of line, more self de-stabilizing to jack-knife lockup.
5minutes on clock(@30 degrees) deflection, the sideForce is 50%
>>so greater deflection is greater than 50% sideForce,
>>Larger than 50%(30degrees+, past 5 mins.) sideForce backing tends to overpower lesser than 50% rest of trailer
>> and carry lesser than 50% part with with the more major direction to the side too...to jack-knife lockup
The numbers simply stack against effort until overwhelming, the bear will win if pushed too far
>>numbers can shift real sharply when trailer axle short across or closer to hitch type geometries.
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Most support fails are from the fast growth and impact of sine increase 10,20,30%... and so suddenly, not the loss of support column as cos drops(that is much less a loss/drop than sine growth especially initially). Cos loss only becomes greater than sine increase at midway 45 degrees/7.5mins(much usually happens before in support models). For this sine is the magician's other hand to watch for fail flags, as like trailer backing, plumb-bob etc.
Other/reverse strategy usages: same numbers become cosines on other end of scale w/this 'hack'.
>>like just can't slant 1,2minutes for real smoother ride over RR (or speed bump) as is only 10,20% softer crossing
>>now greater than 5mins/50%+ range is your friend for much softer crossing(vs. enemy when backing trailer) for now have 50% or more reduction in RR/speedbump crossing disruption! (still best to have tires squared to car/not pitched; when actually take RR/speed bump etc.)

*and counter-intuitively uneven force changes per increment, center 45degrees is not 50% and 50%cos is not found at same point as 50%sine but still equidistant from median 45 degrees etc.

This stuff is in EVERYTHING, as far as questions, solutions and constant examples; just as simple plumb-bob screams out. Crack this one code, and can decipher messages and examples all around; and then play along as if to command(as in sig). Things don't happen 'accidentally', they play out as commanded by their separate dimensions of each alignment vs crossing etc.; they have no choice or ambiguity in the matter(s). They are as if running out a computer program; faithfully following it's steps, fully staged in own 'environment'; logics of such understandings lead to computers. Some would say where the balls roll on pool table is pure accident, others say not...this same stuff is involved, and can decode..
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To read and to know many things, i use the clock in memory as my 'secret decoder ring'.
Sumerians really were showing the idea of 'cycles' as defined, time, calendar/seasons, degrees all on this 'clock' they gave us!
Their 'clock' of days and nights also shows 4x seasons, just as 4x 90degree quadrants of repeating cos:sin pairs, each season of 3'months'; i generally stay to using the 1st 'month' (as 0-30 degrees from noon gravity line of plum-bob) for reading with click cheats, and then know center median 45°, kinda got half the range, reverse the cos:sin values and have the 2nd half of the 90° range(past 45°) owned too !
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This pic shows sideForce = tan xLoad; and how to find it simply in most used region, from 0°-45° of the 90° range for defining 2 separate dimensions of concern.
>>also how this works in fixed, focused linear path or as extended to organic, dispersed radial path(only 2 types).

Usually to find that sideForce from the xLoad, divide by cos, then multiply by sine
>>to essentially swap cos out and replace with sine then xTension.
Tan(gent) is sin/cos, so filters out cos and replaces with sin in 1 move, that is also without division!
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Note how pretty much easy to find tan in 1st 5 minute range as ~11% X Minute guideline/has on click xLoad.
>>sine is actually about as easy, is only 10% per minute in this 5minute 0°-30° range.
>>whereby 24° = 4minutes (@6°per). So sin is 40% and tan is 44% .
>>cos trick on clock for this is more clouded in sequential subtraction from 100%.
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Can guess clock, and then cross verify easily are within the 30° , 5 minute range a la da Vinci architecture:

>>sat in many a tree double checking rig like that, and on ground for tree lean, pulling systems too, kinda 'handy' confirmation ! 30° is where sin(e) for side drift or sideForce has hit a full 50% of potential xLength or xTension already, in only 1/3rd of it's movement range of change !!
It takes the other 2/3 travel to complete the other 50% of xPotential, these are the things can see/read with this decoding on the fly! If tweak rigging angle of rope 1 more minute>>can 'see' change invoking.
(also almost shows 90°, which is as far as can lay down thumb from forefinger (more useful hand signs >> link) )
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rule the world, by following rules of universe as the encompassing domain..

The micro decimals between 0-1 that define cos:sin value pairs may deceptively seem insignificant.
1000# @100mph etc. in same equation more flamboyantly grab audience/'reader' attention than baby decimal.
i would caution against watching this wrong 'louder' hand of the magician purposefully distracting from
PIVOTAL moves of the less noticed hand; that is actually the total enabler/throttle of any xPotential in more obvious hand; whatever that other hand xPotential form or quantity is.
There is a whole nutha' BIGGER show playing in the smaller things of cos:sin , the less watched, but pivotal hand in the same magical equation of change!

(credit : Wikipedia Commons )
.
cos:sin value pair are filtering, limiters of the % the xPotential is expressed
>>to the solo aligned Dimension(singular)
>>vs: any 90° crossing Dimensions(may be plural).
Just as we saw with single aligned dimension of influence decoded by cos;
>>any crossing dimension(s) of influence are decoded by sin(inverse scale of cos fade from 1 to 0)
>>just as with aligned axis endpoints are of equal force, so are any crossing axis, all must balance.
>>but the crossing axises balance to themselves, TOTALLY independently of each other.
>>just as with aligned axis shows length displaced w/cos, so can show side drift w/sin

But this is all extra work co\$t to use tools at any deflection angle;
>>against same Load to hold 100#etc. still must be born on load axis + this 'side work' tax !!
cos:sin are as the filtering limiters of the actual, total distance, force etc.
>>expressed to the singular aligned axis(cos) or if not to any crossing axises(sine).
.

This stuff usually shown for distance, that can be speed over time as well;
but works for force too, as force is simply the inverse of distance ;
as is the loss of distance from potential that could be expressed on an axis.
Can have mix of 1000'possible;
>>500'held back as force as if satiated by equal/opposite mate
>>of equal distance force in opposing direction('married')
>>but rest of 500' of expression potential keeps 'wandering'
>>looking for equal/opposite 'mate' to exhaustion.
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So find distance and force to be such reciprocals;
so as to be able to swap in formulas.
Just as in perhaps more familiar terms:
let (d)=distances (f)=forces
mechanical leverage multiplier =
inputed(d)/outputs(d) =
outputs(f)/ inputed(f) .
>>force and distance reciprocally swappable to same sum
>>so can use in same types of formulas.

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All physical things and their changes, even as waveforms;
can be defined by cos:sin value pair throttling the expressions
of the full xPotential (xLength, xForce, xSpeed xEtc.)
throttled to the singular aligned vs. crossing axises/dimensions.

note how it may be very common to know the load more than the rope tension
>>and to not want to guess too much at unknown sideForce, plenty of rope/cable strength
cos xTension = aligned into load support tension axis
sin xTension = crossed from load support tension axis
must divide to find xTension to remove cos and replace w/sin then X same xTension
tan=sin/cos so automatically divides out the cos and replaces w/sin to show side force
>>in 1 move, w/o decimal division !
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cos:sin scale is intuitive on how flips,
>>but not the increments of individual change between
>>nor the impacts of changes then on these separate axises.
That sudden throw to side of sin, as just comes out of alignment cos;
can be a real upsetter, even killer.
.
The unique aligned and any cross axises are in different geometric dimensions.
Goldwing motorcycle sports a horizontal cylinder;
>>salesman trick was to put nickel on edge on cylinder while running
>>to show so smooth nickel stood on end/did not fall
>>to me because, yes it was smooth, but the action is not on that axis
>>to me would have tipped nickel over if same cylinder was vertical; and told'em so.

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Yin/yang is study of 2 dimensions: a full dimension state
>>and it's non(e) as another, totally separate dimensional state(such as vert/horz dimensional states).
The universal pattern of change between dimensional states, the Ancients found in the stars
>>is always the SAME between full and non(e) dimensions
>>is not an easy, static, incremental straight line of change
>>is rather a most organic bell curve between the 2 extremes of full and non(e)
>>Ancients had to go outside to stars, to see back into how things worked in/on Earth; but got it phenomenally right with what we would say virtually no tools
>>but then nor our industry of distractions as felt had to do something and no cars, lights, tv etc.
.
Cosine gives snapshots of the bell curve positions per the aligned dimension
>>cos1 of full 100% aligned peak potential expression of force (or support)
>>is at same point of sin0 of 0%, and vice/versa, as each other's non(e) dimension
>>both dimensional aspects giving the full reveal
>>math is a language of logical functionality, cos:sin the decoder as trace out where did this input go
>>as all must be accounted for to see all, no real loss collectively, only balance
>>if equal /opposites out of balance, then moves to balance to satiate/make up the difference
Distance allows us to visually see this pattern more tangibly
>>then trust it as expand to same pattern applied to invisible forces
>>this is mathematically correct as force is a distance reciprocal traded , as unrequited/unachieved distance
>>we see this reciprocal whether in zRig funnel 3x force into 1/3 distance or when opposed by equal and opposite of resistance>>still all can be accounted for to balance to the initial input, it simply must to the law/pattern of equality
That is not the end, but rather the beginning to see and trust this pattern thru
>>as it applies to gradients of light of opposing extreme dimensions of full light dimension and non(e) of full dark dimension
>>the organic pattern of change is of some of each dimension to sum total, revealed by value pair cos:sin
>>points between purebred extremes are hybrids inheriting per the the cos:sin value pair
>>this is so innate, that grunt brain that does not understand communication, math, knots etc. knows is wrong if not properly mixed in virtual 3D or look fakey/not fool brain into real feel
again, IF DON'T FOLLOW COSINE PATTERN IN VIRTUAL 3D WE INNATELY, DEEPLY KNOW IT IS WRONG,
>>and do not organically give the same response;
>>no matter what smarter brain says that hear more of it's own radio noise,
>>drowning out some softer tones other not so inflicted animals hear all the time, deep mind still catches some
.
Same for sound, electric, vibration, speed etc. etc.
>>once can define extremes of full potential and non(e)
>>'simply' can stretch to fit the universal bell curve between the parental extremes
>>to reveal the percentages of inheritance from the pure bred/totally void of each other parents
>>to this light gradient, sound fade, speed, tree fall, growth extension, force incurred etc.
.
Amazon shows this has much more acclaim now than when so discarded when i read decades ago,
and allowed me to Squint like Clint thru some stuff hear.
"The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism " -Fritjof Capra
.
Don't think even with that would have gotten this far w/o touching the rawness of these things thru tree work,
just as Doc Shigo invited to touch trees.
>>so bring the invite back again, please excuse.
.
truly, truly hear:
"Education is what remains after one has forgotten what one has learned in school"-Big Al (Einstein)
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As in 'my' Sir Francis Bacon sig quote, we don't make tree not hit house;
Just align to Nature pattern to allow Nature to carry tree away, by aligning to Nature's pattern correctly.
>>humbly, not cuz we good; but cuz Nature is, and we just humbly usher alignments to Nature to play out.
Peace

@theTreeSpyder Kenny, that was heavy chit as usual.

Thx for that book link, it looks very interesting and led down a rabbit hole of other good titles.

I recently looked for MyTreeLessons.com but struck out, is it no more? I learned much from it over the years

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