General > Knotting Concepts & Explorations

Model/Prism of only 3 repeated root elements for organic/cornerless rope works

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KC:
Looking at a corner as an immediate, forced, un-natural, inorganic, flow interruption;
vs. the more natural, gradual, organic flow of arc for best architecture when loaded.


Other than in-organic corner, the 3 elements are : linear(0 degrees) and 90degrees, 180degrees arcs.
Actually i look for them in reverse order, starting with 'most evolved' arc180?.
>>as much a marvel in this usage as for bridge or Roman Colosseum, for same reasons.
Then 90s, to find arc0s/linears remaining

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Rope is a material, as like other materials, of attributes that can be enumerated.
As universally with other materials, geometric dimension and direction of input;
determine how fully and where finite input can be expressed or not.

agent_smith:
Thanks KC.
I put myself in the shoes of a layperson / casual reader, and from that
perspective, its hard to make sense of what this theoretical information
is intended to apply to?

A layperson would (maybe) make a wild guess and assume that you
are thinking about hitches?
But that might depend on whether a layperson actually has a working
definition of what a hitch is?

Its difficult to see how this information might apply to a knot:
(a knot being a self-supporting structure that has an intended
distinctive geometric form, which enables it to be recognised).

I surmise that perhaps a "Round Turn and Two Half Hitches"
might fit the intent of your post?
The 'Round Turn' being performed around a 'host' (eg a pipe),
and the 2 'half hitches' being performed around its own
Standing Part (S.Part).

We know that hitches don't work so well around square profile
- round profile being optimal. That is, the shape of the 'host'
matters.

...

I see that you persist in using Sine and Cosine as part of your
narrative. Your use of these trigonometric functions makes
no sense.

In trigonometry:
[ ] cos 90 degrees = 1
[ ] sin 90 degrees = 0
These values are based on the 'unit circle'.

A more realistic mathematical analysis of what's going on
within a 'hitch' might involve the capstan equation.
Although, the capstan equation does not include the effect
of crushing/squeezing (eg one rope segment squeezing/crushing
another encircled segment).

I would also comment that you don't appear to examine 'knots' - you
appear to have a bias toward 'hitches'.
For example, I'd like to see your theoretical analysis applied to
a Zeppelin bend - and perhaps explain why it is resistant to jamming.

KC:
Dr. Smith, et al...
i show this for all knots, ropework; but simplest to start with Hitches and Bends that port linear force thru a SPart/Standing Part into the receiving 'knot' microcosm, whether it be termination(Hitch) or continuation (Bend) at the given node swell (from otherwise clean running/flowing rope line). 
.
Pure alignment vs. pure 90degree crossing from that benchmark alignment of input force direction are the separate dimensions needed to enumerate by cos:sine patterned %'s ratio.  100%pure cosine is totally devoid of any sine, vice/versa.  Anything between these dimensions of pure alignment or 90degree crossing to alignment has some of alignment dimension and some crossing dimension at play, decoded to cos:sine percentages of the available potential to the crossing dimensions of consideration. 
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This proposed model/prism of view for 3 organic rope elements uses cosine/sine in real world more dynamically tho than fixed un-dynamically to a horizontal axis statically; as introduced in school sterilely on graph paper.
This cosine usage looks at where an aligned bullet should go as sighted(w/o drift) and the recoil together as cosine benchmark of most loaded axis formed by these opposing directions(bullet and recoil paths).  Any value from drift of projectile down, up, or to any side is revealed by sine as drift from the original benchmark cosine of most loaded axis.
So dynamically can set cosine benchmark to the projected pure linear (force)path; anyplace/angle that firearm is set to. Firearm does not have to be horizontal to show cosine, as it's force defines the cosine benchmark if bullet path is what we are tracking etc.
Same could be said of this vertical stone drop to calm waters, to give the resultant sine waves outward from the benchmark cosine linear axis(vertical w/cosine hear) :

The vertical stone drop is the unique benchmark cosine axis, sine/waves as drifts from unique benchmark cosine axis.  Full considerations of alignment and deflection(s), as in all things.
.

--- Quote from: agent_smith on October 07, 2024, 01:47:35 AM ---We know that hitches don't work so well around square profile
- round profile being optimal. That is, the shape of the 'host' matters.

--- End quote ---
TY, this is my why to that, and how it fits into this model; as projects forward for workings of triangular or even octagonal stop sign shaped geometries as hosts, and where friction is/isn't in these models.
Also, should always view things as focused linear/s(where corner is not travel but rude/sudden/inorganic change) or dispersed/unfocused radial of organic gradual change.  Flat rope has thus a focused linear cross-axis of consideration in contrast to round rope's evenly dispersed non focus; as far as same considerations for same reasons of rope material around linear of radial faced host encompassed.
.

--- Quote from: agent_smith on October 07, 2024, 01:47:35 AM ---A more realistic mathematical analysis of what's going on
within a 'hitch' might involve the capstan equation.
Although, the capstan equation does not include the effect
of crushing/squeezing (eg one rope segment squeezing/crushing
another encircled segment).

--- End quote ---
The capstan equation is part of the root of this model for the frictions, as also capstan theory highlights the most evolved arc180 element of building an architecture in rope; just as in stone bridge.  Also, it is my contention that the capstan equation w/o the friction value logically leaves the seating value of the rope to round host,  This seating value gives hug/grip when operating on both sides of host in opposing directions across; arc180 again superior for the reasons i tried to highlight(only need arc180 on both sides for full effect).  It is all the geometry, as like in arc bridge and glass bottle sides etc..
.
For these forces as shown here in rope, are really in all things; and when enumerated to graph show same in gradient fades of light, vibration, wind, electric etc. All are the same patterned graduated rate fade from dimension of full potential to their dimension of non(e) of that same enumerated attribute potential, universally.  Rope is just 1 facet to view central works of same gem of how all this stuff works.  The receding cosine makes up it's own factor of economic "Law of Diminishing Marginal Returns" to this predictable path. 
.
Things change by rigidly displacing against their non.  Occupied space can displace against it's non of freespace as distance; or against a non of equal/opposite displacer as force.  As cosine rises, sine decreases etc.
Rigids may resist in 3 dimensions, and in both tension and compression directions in those dimensions.
Rope, as a flexible device, offers it's lessons in some ways simpler, in that it only natively resists force along it's length dimension and then only in the tension direction of that dimension.  Thus , fewer considerations to rope, except perhaps counter-intuitive idea that a flexible(rope) can rigidly displace against anything!
.
These rules shown are UNIVERSAL, that miniscule Earth must then inherit even to our humble rope, even all atoms and their electrons.  These things are so innate to us, that if the calculations of change for motion, speed, sound, light, shadow etc. do not follow these patterns, they fail as Virtual Reality.  The deep, inner, lowest level of brain can tell are out of pattern, and said gatekeeper is not drawn into the 'immersive' level that VR can lend.  We are so surrounded and saturated with these things as normal base, we cannot just simply see them separately anymore than a fresh mold spore on petri dish can taste the gelatin.  The gelatin is just that which is to the mold spore, there is nothing without it to it; it is just that which is.  The cos:sin patterned ratio is like that to us i believe; as has no non.
.
Peace.

KC:
Agent Smith !
-et all;
i think these are key factors not to miss, please. 
Aligned dimension% vs. opposite of perpendicular/crossing dimension% reveals in value pair ratio cos:sine.
Applicable as reach and support column% : byproduct of host hug (for frictions, nips and grips in rope)!
Been trying to get more time drawing etc. figured would just post what have.
.




--- Quote from: agent_smith on October 07, 2024, 01:47:35 AM ---I see that you persist in using Sine and Cosine as part of your
narrative. Your use of these trigonometric functions makes no sense.

--- End quote ---
.
Sorry, cannot leave this behind; i totally, completely disagree; and really find that value pair cos:sine are the key definitive here, and everywhere.
This is an Universal pattern shown in these numbers; viewable now from front row seat of raw rope work, that could be taken as fairly simpler model of cos:sine ratio of expression in everything.
.
i model 100% efficient virtual column as correctly aligned to maintain support column constitution longest against load, breaking only at highest/peak capacity tensile strength possible, cos=1; so 100% as benchmark potential. Physical column may slant/deform to lower % virtual column constitution maintained against load as force vs. perpendicular dimension of shear across to destroy that same support column by contrast; so breaks at lowest possible load; as extreme yin/yang contrasts of influences to the sum of the total whole.  So as to reveal both sides of coin at once, even during the flip.
.
The cross axis load of support (sine)is as byproduct co$t of achieving needed support on aligned axis to load
>>so loss of cosine as efficiency costs more input to same target; AND carries greater cross force byproduct cost to be carried at same time both challenging against efforts
>>pure aligned cosine 100% has no byproduct; sine=0% ; so no host hug side force for frictions, nips nor grips.
>>cos:sine  in ratio to support column:hug as giving force to frictions, nips and grips. 
Trade-offs of how much of total sum tension is expressed to the 2 parent dimensions of aligned vs. not aligned(crossing),







Several standing common examples of tensile strength lost or maintained per support column deformity of SPart input to similar knot architectures:
A HH  shears across the single column of support harshly, but Tensionless Hitch does not.
A Cow/Girth shear across the SPart(s) too;
>>but less so when sporting 'bull nose' on either
>>that i think grips around SPart(s) to pull more correctly along the SPart(s) as a column, rather than just shear across as less evolved Cow/Girth as spreads out contact.
So we can see that a simple single Turn/arc180 is a 1D pull shearing across SPart column or other host, 2xarc180 gives 1D grip around, and a 3rd arc180 for RT 'component' architecture of more 2D grip and higher tension reductions from frictions*.  Here it's utility of grip serve to comb/pull/groom more along the SPart support column 'properly'(cos), as also reduces forces across(sine) same support column(s).**
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Cat's Paw and Rat Tails take it a big step further; almost lacing a re-usable splice the way they pull along, not across the support column(s) to maintain cos, to your cause(support). 
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A simple Turn around host and Friction Hitch(FH) back to  SPart column can help keep 'clean' column; the pointier the 'teepee', the more strength maintained.  But less hug as a trade-off of higher cos:less sine.
Round Turn(RT) on host before FH can help resolve, but then less power in FH grip from the lowered tension.
These are trade-offs incurred in the manufacture of the 'device' made of the rope architecture at rigid/tension loaded.
.
This is not just about the strength/tensile preserved tho, but rather the pattern used to express in aligned vs. crossing dimensions, that also find in rest of things, as the Ancients tried to gift down/forward.
Organic/Natural fade from full potential (as a dimension) to the opposing non(e) of potential is always same pattern.  Even with organic fades of sound, light, electric etc. fades to non(e) on graph.
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Distance and therefore reciprocal of force(potential distance unachieved) are just most visible parts of this deep iceburg of pattern.




*A key component in may things, even bringing Cow/Girth to Prusic's open or closed, by RT not single Turn to each side etc. 

**Kellig is interesting 'twist' on pulling more along columns, so less shearing across;
>>but also can tweak/crank rope, especially if stiff/rigid organically and/or rigid at high load, so like kink in chain can count against efforts to maintain strength.  Does not flow so cleanly as unkinked finds subject to more/easier/earlier 'corruption' of column constitution.  Some varying trade-offs to be suspected here.

KC:
My contention offered is that these things shown below happen to all things, UNIVERSALLY;
as we can show them outside a Hitch, the universal pattern ruling even the stars;
then doesn't stop at knot node border as disappears inside the Hitch's microcosm.
>>and rope's property of tension powers the knot node w/these forces; to these numbers (less any frictions) as travels around.
.

.
i use potentials of length and force here that match; so can see they change in co-ordination as angle changes.   Then set to 100units of whatever to show cos as root math most clearly.  Pattern continues to follow cos scale even when length and tension don't match, and are not such an even clear number to play with as follow them thru the visual model of Geometry.
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Geometry does give a visual, enumerated example of fades from full expression of potential to none that are found also in physical, electrical, light, vibration, wind, water etc. force changes.  That then give the range of all possibilities of expression of each principle's individual expression, and where in the range of change they happen.  Just from knowing the extreme endpoint positions of full and non, hear is gifted the rest of the 'tweens'.  (tweening:  making transitioning images from a 'keyframe' to next in animation as to show change/action/movement, the time slices betwixt).
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The organic gradient fade of a property from expression of it's full potential to non(e) always follows the cos scale.
It just so happens that this also applies to going from fully aligned to fully crossing in geometry, but goes much further beyond those most visibly enumerated points.
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"The world is full of magical things patiently waiting for our wits to grow sharper." ~Bertrand Russell

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