i've always tried to find patterns in things, as mnemonic of the items, order, functions etc.
>>in time respected more, to find that mnemonic pattern in the functional elements ;
>>made more sense as it also gave lessons , and more defined targets of those lessons.
chasing cos/sin/tan was no different.
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Defining a circle in terms of a clock is one of these patterns to me.
As a relationship; both 360 circle and 12hr clock were defined 4000yrs ago
>>in Ancient Babylon with their 'strange' base_60 math
>>obvious remnants still persist, even as much else has faded..
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The leg of the feed from inside the knot to outside world, that draws input force into the knot is called the Standing Part.
This should be a straight iBeam of support against the confronting force for greatest 'strength' / efficiency.
Zer0 deflection is cosine 1.00 (i think of as 100%) multiplier of length and force usages against target (gravity etc.) direction
ANY deflection from this line drops cosine as efficiency rating to squarely meet against direction/force
Slanted or otherwise deformed from trued, straight iBeam of SPart, drops the cosine efficiency
>>so needs more length and force to oppose against the same vertical distance and force
>>as is not using it's length nor force contained as efficiently as when properly aligned
height (or force) achieved against = total length (or force) used X cosine or
height (or force) achieved against divided by cosine = total length (or force) needed
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The clock 'hashes' (or seconds/clicks/minutes) give the angle from center, deflected from 12noon as TDC
>>while the associated numbers 1-7 give the approximate cumulative loss in cosine efficiency
>>Babylonians calc'd PI as 3 1/8 (3.125) so where .016 off, so some err is expected/inherent( in the hundreths tho).
(http://mytreelessons.com/images/clock-wise-thrumb-rule_cos_2.png)
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verify: thumb
min. deg. rule cosine
0 0 1.00 1.0000000000
1 6 0.99 0.9945218954
2 12 0.98 0.9781476007 start compounding
3 18 0.95 0.9510565163
4 24 0.91 0.9135454576
5 30 0.86 0.8660254038
6 36 0.81 0.8090169944 add 1
7 42 0.74 0.7431448255
7.5 45 0.71 0.7071067811 memorize .707
( ea. clock min. = 6 degrees )
The reduction in co(lumn)sine as support column efficiency(rising against target to your cos/cause as mnemonics);
leverages higher tension in rope, for same task/load(just as requires more length to same height as task)
>>working rope and supports, along w/other linked events more than required by nominal loading itself
>>can reduce cycles to failure, cause higher recoil if fail etc. as now more over-loaded weapon.
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A wire, displaces space physically so calc'd like this too
>>but it's internal force payload displaces electrical force, so the displacements are calc'd differently
A rope, displaces space physically so calc'd with cos/sin
>>but it's internal force payload displaces physical force, so the 2 PHYSICAL (space and force)displacements are calc'd w/same maths
>>thus describing physical position of rope clues forces carried, not so true with wire etc.
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Thinning of the rope channel can affect strength some, also w/it's impacts of change.
But mostly i think the drop is due to geometry, just as in rigid devices.
>>deflection from load /extension axis PHYSICALLY IMPEDES length and support of column given
>>the device whether rigid or flexible or even waveform must abide by these same rules, as given by the Ancients.
So, when say, knot reduces 'strength'/efficiency by 30%, decode as ~45degrees column deformity leaving ~70% to your cos(cause)
Sine, by contrast is the non-inline to the target force/direction, the side forces.
As the co(lumn)sine exists in the simplest, inline, single dimension of column rise/extension or force support ea. on given axis.
>>the sine as it's non, covers all else, the other 2 dimensions not of the simple, inline single dimension
thus, as cosine alone is single dimension, the mere occurrence of sine indicates multiple dimensions of forces/lengths
The amount of distance difference made is reciprocally linked antagonistic against total force in same equation/instance.
>>in that one of the reciprocals is consumed, to increase the other (works either way)
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Sine is even easier, to find by clock reference, in fact what i started with.
>>Mostly dealt with angles just off of TDC alignment against force and or distance between refs 12/noon and 5mins./1 o'clock
(where clock strategy/metaphor is also most accurate w/o adjustment)
>>lean of tree, angle of rigging line ( then extending to SPart deformity considerations later)
This 'early range' (12-1) off of most proper TDC/Noon against gravity direction where as seen above,
>>the changes in cosine are most minimal/muted 1% change per clock minute to start
BUT, this is the most volatile rate of change for the sine,
>>for as the cosine of efficient support/travel only decreases 1%, the sine is jumping to the side 10%
>>for 10x the impacting change in same moment
Thus many system fails are not from reduction of the required cosine of support
>>but rather from the much greater (in this range) destabilizing, impact of change of the sin(e)
>>destabilizing structure thru buckling under compression; exists in tension in too.
>>in compression the side forces push out from center as the linear ends push in
>>but in our ropes, tension forces pull to center as ends pull apart
i picture a type of elastic response, whereby can't be long/wide on both force profiles, and must be long/wide on 1
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(http://mytreelessons.com/images/clock-wise-thumb-rule_sine_1.png)
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min. deg. rule actual sine
0 0 0 0.0000000000
1 6 10 0.1045284633
2 12 20 0.2079116908
3 18 30 0.3090169944
4 24 40 0.4067366431
5 30 50 0.5000000000
6 36 59 0.5877852523 -1
7 42 67 0.6691306064 -2 more
7.5 45 71 0.7071067811 memorize .707
( ea. clock min. = 6 degrees )
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Cosine/sine is in all physical displacements of space and force, and any other kind of line or curve.
Even a light beam, shadow, water fall etc. are defined by these numbers.
Even in vibration, electricity, light, heat, water etc. waveforms on a graph, have sine waves, preceded by cosine waves!
Thus their amplitude changes over time impacting in that world, just as same as in physical devices
Without 'Ancient Astronuts' , the Ancients keyed into all this Naturally somehow,
>>w/o our distractions and also w/o tools as we know them!
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In forces, i think of cosine as conjured first as a requirement of the loading
>>then immediately following the sine echoes in from sides to also be carried as extra load in the work
>>my imagery follows cosine waves pulse first even within physical devices of rope etc., with sine following 'half a stroke' (from 2/4 cycle motor type definition) of 90degrees
Pulse wave imagery in waveform, makes it easier i think to visualize impacts of change, especially sudden
>>as also such pulse wave impacts in rigids and flexibles, to more truly field and confront them(i think).
Cosine and Sine absolutely rule all straight and arc forms of spatial and opposing force purposeful/displace-meants;
and even permeate deeper into our world as also find same spatial and force displacements in waveforms of light, heat, electricity/magnetism, sound etc. all the same.
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The Ancients handed these down over the milleniums to describe the consistently same, minimal definition, pivotal, forces found in their known basic known devices so well, even holds together in the waveforms they could perhaps not define, beyond their nagging senses except for perhaps water waves. These were about the same lines of force in all, irregardless of the transferring devices, that turns out can even be air itself as mute testimony to how far these things go across the board.
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As in all other basic examples they found cosine/sine to rule in support work against loadings etc.
rock as a non-malleable example
metal as a malleable example
rope/fiber as a flexible example
wood as a stiffened fiber of at times both malleable and flexible in ranges type hybrid
>>that could also be carved and used for fuel unlike the malleable/non-malleables.
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Rope work is just one lesson of lessons in these things, as a leading example of the flexibles category/family/class.
Flexibles are uniquely different to rest of classes in that:
>>doesn't resist on the cross axis/ only the inline axis
>>and then only in the tension/not compression direction on that given inline axis
>>is in it's malleable/form-able context at room temperature w/o hammer, drill, nor saw
>>only takes it's forged rigid form in loading, and only to the limits of that loading
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Tangent is a very useful pre-configured shortcut; simply sine divided by cosine
>>so if multiply times a cosine value (load force), the cosine value drops out as x/x=1 for multiplier, leaving the sine value(side force) only, thus revealed.
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(http://mytreelessons.com/images/rope-mechanix-of-sine-cosine-and-tangent-shortcut_1.png)
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This saves from going from load force to calc the line tension by division of cosine, to then multiply by sine to get side force
>>So can be very handy, as saves a step, AND more likely to know the load, than angled line tension...
>>Tangent, also, can be plotted on the clock fairly accurately to the common 45degrees point (7.5clock mins)
>>most Naturally accurate again, in most used range first 5 minutes on clock; 12-1
(BUT does not have the flip to reveal other half of 45-90 degrees as sine and cosine oblige to so well)
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(http://mytreelessons.com/images/clock-wise-thumb-rule-tangent_1.png)
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Most system and device fails from destabilizing influences of side/sine cutting across co(lumn)sine of required support against loadings.
In compression , side force value direction is asserted outwards from center
>>buckling from this to far outwards propulsion compromising column.sine(cosine), is easier to see .
>>and most expressed in center, as most leveraged from each end.
But, also, can have tension buckling, harder to even visualize as side force in tension pulls towards more correctly inline.
>>perhaps at some point too much as 2 force lines can 'slide' past each other (?)
>>to separate but in perhaps curiously in a half of a water meniscus shape AND angles
(effect exemplified in popular rigids under tension with lateral slider joint in between)
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In both compression and tension directions on axis,
seem to have and 'elastic model' of response(how i keep them straight) for the side force directions:
>>in compression side forces swell outwards as ends compress in (like would expect compressing foam pillar)
>>in tension side forces pulls inwards as ends pull outward (like would expect stretching foam pillar)
But, in each case, failure seems to come from destabilizing lateral/cross forces more so, than outright longitudinal/inline causes.
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These numbers decode the cryptic patterns, sifting to the cosine and sine piles.
edit/forgot:
min.deg.rule tangent
0 0 00 0.0000000000
1 6 11 0.1051042353
2 12 22 0.2125565617
3 18 33 0.3249196962
4 24 44 0.4452286853
5 30 57 0.5773502692 plus 5 %
6 36 72 0.7265425280 plus 10 %
7 42 91 0.9004040443 plus 20 %
m 45 100 1.0000000000 memorize
All 3 scales showing the clock number pattern and related angles (w/o err mitigations for clarity to pattern)
(http://mytreelessons.com/images/clock-wise-thumb-rules-cos-sin-tan-without-errs-shown.png)
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First 5 mins as mostly used and least errs, that cos/sin duplicates as a mirror flip to last 5mins of the 90degree clock model
(http://mytreelessons.com/images/clock-wise-thumb-rules-cos-sin-tan-first-5-mins.png)
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Cos/sin/tan all 3 lined up, show the repeating patterns,
>>and curiously most accurately in the first 5min division from 12-1 on the clock.
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Base_60 math, to 12 divisions , while also defining same shape as 360 ,
>> are all curious choices, but related somehow
>> like them inventing the wheel, and also giving us calendar as round cycle of 12 lunar months*
I don't know if cos/sin numbers on clock as I show are purposeful(but always wanted to think so)
>> or even observed at the time; as they hit the mark in so many other fashions
(But can't believe I'd be the only to see, but can find no other reference over the years.)
I think tangent scale simply happens as sum of change between cos + sin
>> but fails when cos really starts dropping, as a divisor, simple sum can't keep up.
(find another chance pattern in tangents after 45 degrees, as10% of primes series)
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As i stay focused on the cosine being the single, linear dimension of support to my cos/cause
>>and sine anything else but to this target axis, the sin across that must now also be carried
>>should lose sight of neither hand (cos/sin)of the magician, as things change!
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All 3 scales again, to midpoint @7.5mins of 45 degrees w/err mitigations:
(http://mytreelessons.com/images/clock-wise-thumb-rules-clock-angles-numbers-decode%20-cos-sin-tan.png)
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12 also comes up in Geometry of the Egyptians, much of which inherited from Babylonians (cuneiform as well):
>>Divide rope into equals of 12parts
>>make 1 flat side of triangle w/3parts
>>another flat side w/next 4parts
>>close all parts by slant w/5parts
Now have 3-4-5 right triangle, right angle between the 2 minor parts
This then later lends into the Greek Pythagorean theorem,
the numbers allowed them to name values, that ruled their lives and all around.
Their perception of these as the minimal, pivotal points that then covered all ranges
>>was so true in what they witnessed, that it carried on into all other things,
>>even beyond what the could possibly know or perhaps even imagine a hint of.