The relationship between 'bends' and eye knots

[Dan_Lehman]

@agent_smith post:20

Dan,

To my thinking via the construction of an EK given above, Ashley's #1043/5 & #1016 are all corresponding knots to the Sheet Bend --in one way more so than the BWL :: in that these knots do NOT have Xarax's *pure* "nipping loop" : they have a loop loaded principally on the S.Part end, but don't run then out into a half-force-loaded Outgoing Eye Leg!

I've attached an image of the #1043 [i]F8-Loop bend[/i] and corresponding eye knots. I don't see how #1043 knot core is the same as #1431 Sheet bend core - unless I've misunderstood you?

?? I was perfectly perspicuous. Your insistence on “identical”
for “corrsponding” blinds you to this insight, apparently. No,
the correspondence isn’t the identical structure you posit as
solely available for the claim, but the loading of the knot
is more nearly so --as I noted re the would-be-but-isn’t
Outgoing Eye Leg.

The notional concept of 'correspondence' is only valid from a strictly geometric point-of-view; where the cores of the eye knots are geometrically identical to the core of the parent bend. If you change the geometry, you change the correspondence to something else.

That's just your current opinion, not necessary to analysis here. One might just as well insist on like loading vs. the rather likely imbalance one gets via your method (one that isn't so imbalanced say with Fig.8/OH EKs (just shifting from exterior<=>interior loading). #1016 shows this relation conspicuously in its center image :: that ersatz "eye" on the left connects to what [u]would be[/u] the Outgoing Eye Leg (as per #1074 --and, as I've previously suggested, likely WAS so and the image maker botched this, authors dumbly copying its asserted purpose, but with no given rationale unlike what Ashley has offered and put for #1074). BUT, again, AS SHOWN, the center image nicely illustrates my offered another-way-of-corresponding (to #1034.5), and shows a Sheet Bend formed with a single "hitching" line to a U-folded U-fold providing the eye. And, again, the "nipping loop" is that in form but not in the usually intended loaded-on-both-ends sense (which for Xaras is part of its definition).

Here I'll again interject that correspondence can --even should?!-- have different meanings.

Not for me - sorry. I am of the view that terminology and concepts need to be defined --so as to remove as much ambiguity as reasonably possible. Words have meaning, otherwise it would be like the tower of Babylon and people would not be able to communicate.

Your strict view of "correspondence" hampers; but it's nice to see you coming around to liking good diction, after the previous defence of some rather misleading working!

I pointed to the process of building an EK from a e2e Joint by using 1 single end joining to the two ("twin") ends of a U-fold, and then fusing the single end into one of the others so to make a 2-tangle (unfused :: 3-tangle) and EK.

You like thought experiments that employ 'edge cases'. I like thought experiments - but they have to be grounded in reality. In my view, an end-to-end join will have 4 segments projecting from the knot core (and an 'eye knot' has 3 segments).

?! What happened to the 4th, which is what 2-Tangles have --for you, 2 of yellow, 2 of blue.

There is no 'fusing' in a real-world practical case where a person joins 2 ropes together by hand alone.

Knot designers/explorers might do this --some have, as I've shown. The work can be at times most easily done graphically --paper & markings don't move!

this means 8 segments exiting from the core (an even number again).

?! Do you have some single-ended cordage?!!

If a Sheet bend is tied with 2 bights, this has 6 segments exiting the core (an even number).

?! You'll have EIGHT ends, in a 4-tangle.

Imagine an OWK ("EDK", "Offset Water Knot") joining two ropes that are hanging from a narrow strip of steel:: and perhaps someone abseils or ascends these ropes. What type of knot is that? (It doesn't have force from opposite directions but the same, on either side of the steel strip.

I don't follow this... it seems to be another 'edge case'? I'm trying to visualise 2 ropes that are hanging from a narrow strip of steel (likely not load rated?). [?!! Hardly :: likely quite load-rated per human-body loading!] And these 2 ropes are then united with #1010 Offset overhand bend? And then a person abseils/ascends on these joined ropes? [i]No comprende[/i]...sorry.

One might be high up in a steel-lattice structure, and the rap-rope straddles some thin (horizontally) steel span. The OWK is loaded vertically by BOTH S.Parts.

–dl*

[agent_smith]

Dan,

Lets try to keep the language confined to a civil discourse - lets not use accusatory language.
I’ll reiterate some key points (again).

This entire body of work is new - nobody has attempted to do what I am doing now.
Harry Asher started on this subject matter in the 1980’s - but he did not advance any solid theoretical analysis.
There is effectively no peer reviewed body of work that I can refer to or cite.

I also think that language is complex - and it is inevitable that interpretations of typed words can be misconstrued or understood in a way that varies from the originators intent.
Lets take the #1431 Sheet bend and #1043 Eye knot as examples how things can be difficult to nail down and eliminate all ambiguity.
(I would also state for the record - that a high quality image is better than a thousand typed words attempting to describe that image).
Please examine the attached images below…

I define a #1431 Sheet bend as the union of 2 ropes/cords (ie an end-to-end join) - per image ‘A’ below.
It consists of a U fold (bight) + a loop (some may argue the ‘loop’ is a half hitch - but no degrees are declared - eg 540 degrees, 180 degrees, etc).
Per image ‘A’ - I see 4 segments exiting the core.
Per image ‘B’ - I see 6 segments exiting the core.
Per image ‘C’ - I see 8 segments exiting the core
Per image ‘D’ - I see 6 segments exiting the core (this is likely the geometric structure you were referring to?).
Image ‘D’ - corresponds to #1074 Bowline with a bight.
EDIT NOTE:
Image has been uploaded to show correspondence between #1074 Sheet bend against #1074 Bowline with a bight.

The Sheet bend variations (‘B’; ‘C’; and ‘D’) are not strictly #1431… they employ the same mechanism but the geometry is different.
I think of them as all belonging to the same ‘genus’.

I would also state that image ‘D’ parent bend corresponds to #1016 (per Ashley).
I note your comments re the differences between #1016 and #1074 - the ‘parent bend’ is per my image ‘D’ for both.
Once the ‘eye’ of the eye knot is connected/formed, then you can debate the orientation of the tail end ‘bight’ (ie which side you wish it to be oriented with respect to the ‘eye’).

Now, with regard to #1043 eye knot (as per Ashley #1043) - I don’t see any direct correspondence to #1431.
Now, you may have an edge case - and see correspondence, but I don’t.
This does not mean you should use accusing language such as being “blind” and/or use belligerent comments.
I think this really comes down to definitions and interpretation of language.
And so with a viewpoint where direct geometric correspondence is abandoned, one could unpick the F8 knot structure of the #1043 ‘bend’ - and yes, you end up with a Sheet bend.
But in doing this, you are changing its geometry - because you have changed from an F8 knot to a U fold - by unpicking the tail.
And again - from a pure loading point-of-view (rather than pure geometry) I can see how you make the case that #1043 bend is essentially based on a Sheet bend.
I would suggest that the #1043 parent bend is not a 1:1 perfect replica of the loading profile of #1431 Sheet bend. The differences are subtle but distinct enough to be regarded as belonging to a different family/genera of bends (I will assume that you will disagree!).

I am defining the word ‘corresponding/correspondence’ probably in a different way to you.
In my view, the term ‘correspondence’ is a geometric character (you appear to disagree - and that’s fine).
Per my attached image below - I can see the geometric correspondence between the #1043 ‘bend’ and the #1043 eye knot.
Now - loading is another matter - which (in my view) adds complications.
For now, I am proceeding on the basis of a direct geometric correspondence.
I think you are declaring something else - and I think it again comes down to ones definitions and interpretation of language.
I would say that you are positing a different geometric arrangement for the F8+Loop bend - perhaps with the inclusion of a U fold of sorts?
All I can state is that I am confining myself to a direct geometric correspondence - because to do otherwise complicates matters and opens up an array of possible derivatives.
You might argue that all possible geometries must be accounted for - and add loading profiles to the mix - and try to define it all.
Outliers do need to be examined and accounted for - I agree - but I prefer to start with simple/basic forms, and tease out the theory from that starting point.

With regard to the generic form of an eye knot tied end-of-line, with a distinct non-loaded tail end.
Take #1047 F8 eye knot as an example.
This ‘eye knot’ has 4 segments exiting from the core.
In terms of pure loading profile, there are only 3 loaded segments (1 S.Part in opposition to 2 eye legs).
Again - you may wish to disagree - but I think if there is disagreement - it comes down to definitions and interpretations.
I am confining myself to the geometric form of #1047 F8 eye knot, tied end-of-line (with a non loaded tail).

With regard to an end-to-end joining knot formed from 2 singular rope ends:
I see this arrangement as generically consisting of 2 opposing ‘S.Parts’ and 2 non loaded tail ends.
I included the word ‘generically’ because I accept that there will be bends that don’t conform to this rule.
It gets complicated when ‘U folds’ (bights) are present (eg #1074 / #1016 derived bends)
I don’t claim to have all the answers (and I never made such a claim).
#1074 Bowline with a bight is an example of a structure that has a U fold/bight.

Anyhow, for now, I am confining myself to a strict geometric correlation (or correspondence) between a parent bend and its ‘offspring’ eye knots.
Lets try not to snip out pull quotes to try to advance an argument to denigrate or slander ones perceived opponent.

I really think you need to get your camera out and try to use high quality images to support your arguments - rather than using a myriad of typed words to describe a geometric arrangement.

Also, can we try to keep the ship steered with a good tack - with a view to finding safe passage through the rough seas of language complexity

---

[Dan_Lehman]

@agent_smith post:22

All I can state is that I am confining myself to a direct geometric correspondence
–because to do otherwise complicates matters and opens up an array of possible derivatives.
//
Anyhow, for now, I am confining myself to a strict geometric correlation (or correspondence) between a parent bend and its ‘offspring’ eye knots.

We differ on the use of “correspondence” : you want it to apply
with the precision & limits of my tangle --for which I’ve yet
to try any particular names/IDs for the various loadings–;
I want to leave it a general connotation (which can take
qualifiers), accommodating of practical concerns and reflecting
loading. (Another e.g. :: if one ties the U-fold legs qua a single
“end” to a truly single end in the manner of the Thrun’s Bend
(aka “zeppelin”), arguably one has been well more faithful
to preserving e2e joint knottedness than if one loads the
S.Part’s Tail as Outgoing Eye Leg and has then only a 50%
loading in the path of the completion of that knot --which
might make a fine EK (seems to) and all, but is arguably
more of a departure from the e2e joint.)

Also, you are focused on a particular direction :: e2e → EK.
Going the opposite direction one has immediately an single
S.Part to convert, and “primary” l00ks appealing as its
descriptor (though we might resist)!

–dl*

[agent_smith]

I’ve uploaded the ‘U Fold’ bend and corresponding eye knots.

I am defining correspondence to be of a geometric character.
That is, the geometry of the parent bend core is congruent with the the cores of all the corresponding eye knots.
I am also defining some of the corresponding eye knots to be ‘primary’.
The primary eye knots are closest to the parent bend in terms of loading profile.
A good example of a ‘primary eye knot’ is the Zeppelin eye knot found on this webpage: https://knots.neocities.org/zeppelinloop
The content creator of this webpage has presented what I refer to as a primary corresponding eye knot.
(there are in fact 2 ‘primary’ corresponding Zeppelin eye knots).

In my view, it is more logical to use a ‘bend’ as a starting base.
From this parent bend - you can then derive corresponding eye knots.
I am also stating that there are 4 possible ‘linkages’ with all bends.
These ‘linkages’ enable corresponding eye knots to be derived.
If loading profile is taken into consideration, then there are a total of 8 possible corresponding eye knots that can be derived from a ‘bend’.
Not all of these corresponding eye knots will be stable (or viable).
For example; the Reef/Square bend produces no stable/viable corresponding eye knots.

I am of the view that to do it in reverse - ie to start with an eye knot, and then try to derive the bend is also possible but is illogical.
I’m not saying you cant do it that way!
I’m simply stating that it makes more sense (to me) to start with a ‘parent bend’ - and then try to derive the corresponding eye knots.

…

The ‘U Fold’ bend was sent to me by Xarax in June 2022 (correction; June 2020).
He might have discovered it prior to that date - he simply sent it to on that particular date.
At the time it was sent to me, Xarax did not assign a name to it. EDIT: I have since discovered that Xarax did assign ‘Bingo’ bend.
‘U Fold’ bend is a name that I have assigned (for now) - it is a placeholder name.
I like it (interesting structure)… note however that I have not investigated jam resistance.

EDIT NOTE:
Xarax might provide further details about this interesting bend (I hope).
I have not tested any properties of the corresponding eye knots.
Agaion, I have assigned a placeholder name for now - because Xarax did not reveal a name back in June 2020.

---

[siriuso]

Hi Mark,

What you say the U-fold Bend is named Heart Bend by Xarax.

yChan

---

[agent_smith]

What you say the U-fold Bend is named Heart Bend by Xarax. yChan

When Xarax sent me the ‘bend’ in June 2020 - he didn’t assign a name to it.
In other words, when he sent it to me (via email in 2020) - he didn’t reveal a name to me personally (at that time).
EDIT: In fact, Xarax did assign a name - it was identified as a ‘Bingo’ bend!

It seems that you have information about the ‘bend’ that I didn’t have at the time.
Thank you for updating the information about the bend!
(The image you uploaded is a dressing state - it is not energy stable.
The dressing geometry depicted in my images are energy stable).
[presumably Xarax wanted to present the knot in a heart shape - but he knows as soon as load is applied, it wont remain in that configuration].

[alana]

@agent_smith post:22

… a high quality image is better than a thousand typed words attempting to describe that image).
Please examine the attached images below…

these fig’s and pics are great

[DerekSmith]

“Re: The relationship between ‘bends’ and eye knots”

The relationship is that all knots are bends and that some groups such as Loops and hitches, tend to feature a single cord bent to itself.

My professional fields - Chemistry, Math and Programming, have well established lexicons, while my preferred hobby subject is, shall we be generous and say, fluid, and as such leads to a great propensity to steer us away from the challenge of conceptualising and understanding the inner functionality of KNOTS…

The IGKT is probably the nearest thing the world of knotting has to an International Reference Body, so perhaps it is up to the members here to try (once again) to distil some fundamentals of a lexicon for our field?

Derek

[Dan_Lehman]

@DerekSmith post:28

The IGKT is probably the nearest thing the world of knotting has to an International Reference Body,

Boy, is this ever a sad joke! Many people who might have
been constituents to such a group have left.
(Consider that Book Review at km160:15 --basically,
“There’s a Budworth book with spiral binding and modern
images; I recommend it.” --a now 10yr-old title
on a then a 10yr-old prior title (which was rebuked in a
2002? KM for having Ten Errors! (actually, more)).
And this got into KM!)

Knots are a deceptively hard thing to get a good grip on!
–even with entangled “simple” OH knots. Look at yChan’s
tranformation of the #1425a SmitHunter’s Bend into a
more-like-#1425 shaping (and quick jamming), quite
stable these two distinct dressings & settings of one
entanglement!! Egadz!

–dl*

[agent_smith]

First of all - Happy new year to all
Hopefully 2024 will be a year of many advances in scientific discovery.

…

With regard to the ‘correspondence between bends and eye knots’:

Clearly - I am biased because I do believe there is a relationship between a ‘parent bend’ and its offspring ‘eye knots’.
I had advanced that there are 4 possible linkages between the Tail(s) and S.Part(s) of a ‘bend’.
NOTE: If 2 different colour ropes are used to form a ‘bend’ - linkages can only be made between segments of opposite colour (refer to attached image).
And here I am defining a ‘bend’ to be a union between 2 ends of [a] rope.
It may be 2 independent ropes… or; it may be 2 ends of a singular rope to form a round sling.
Essentially, I define a ‘bend’ to be an end-to-end join.

With reference to the attached Butterfly image below:
Extraordinary claims require extraordinary evidence (Carl Sagan).
I am making a claim that #1053 Butterfly eye knot corresponds to the parent Butterfly bend.
Is this claim ‘extraordinary’?
Claim 1: Both knots are through loadable.
Claim 2: Both knots respond identically to a through-loading profile.
Claim 3: The tail segments in the ‘bend’ experience no load
Claim 4: Both legs of the ‘eye’ of the #1053 Butterfly experience no load
Claim 5: The core of both knots are geometrically identical.

Lexicon / Epistemology
I am using the term ‘eye knot’ - because the ‘eye’ of an eye knot has no chirality.
One must first have a robust definition of what a loop is.
A loop is formed by the overlap of one rope segment over the other to circularise with a distinct chirality (either S or Z) - no host object is required.
In my view, the ‘eye’ in an eye knot has no particular chirality.
Ashley did not conceptualise this clearly - although there are hints that he understood the concept of chirality - but in his classic publication, he chose to describe eye knots as ‘loop knots’.
Ashley was not particularly clear on the distinction between a turn and a loop…
I define a turn as a wrap around a host object - for example: A round turn and 2 half hitches; the ‘turn’ is performed around a host object.
And a turn can be further defined in terms of degrees - eg a 180 U turn, a 360 degree turn, a 540 degree turn, etc (540 degrees is in fact the ‘turn’ in a round turn & 2 half hitches).
I am 100% confident that some IGKT members will vehemently disagree on the use of the term eye knot in lieu of loop knot. For me, it is a question of defining what we actually mean by the word ‘loop’.
For some, I am of the view that tradition and culture strongly influences how a person defines something - and Ashley is regarded as a primary source - and it may be counter-culture to challenge the long held views of such sources. I think the word ‘change’ sounds alarm bells in the minds of traditionalists. Change can be frightening for some (and fear can induce anger and/or outrage).

To Dan:
#1425 has a very interesting derived Offset joining knot - only partially offset but morphs a little under load into… see below.
I’ll photograph and upload it…
[HINT]: #1425 is topologically equivalent to the ‘False Zeppelin bend’ (with crossed tail segments).

Also, just to be clear…
I find it logical to begin with a ‘bend’ - and then try to derive the corresponding eye knots.
Obviously, you can work in the opposite direction - that is, start with an ‘eye knot’ and then try to derive the ‘bend’.
In my view, this is not a question of ‘right’ Vs ‘wrong’… rather, it is matter of the logical ease of deriving and classifying possible eye knots.
It seems to me that to begin with an eye knot - and then attempt to derive the ‘bend’ - is more complicated particularly since you need to define which segments will become the ‘S.Parts’ (which opens up a branching chain of possible bends based on choice of tails and S.Parts).

I also am of the view that this whole subject matter is best dealt with in terms of geometric correspondence - that is; the core structure of the corresponding eye knots must match the core structure of the ‘parent bend’.
Dan - you may choose to disagree with these basic premises - and because of free speech and the freedom to express ones ideas in a democracy, you are perfectly entitled to disagree.
The ability to make scientific advancements is best achieved in an environment where there is freedom to express ones ideas/concepts without fear of cancellation, reputation destruction or retribution. If you have alternative competing ideas - please publish (preferably with quality images to convey your concepts).

---

[Dan_Lehman]

@agent_smith post:30

With regard to the ‘correspondence between bends and eye knots’:

Clearly - I am biased because I do believe there is a relationship between a ‘parent bend’ and its offspring ‘eye knots’.

I will simply again point out that w/o throwing anything
out of consideration (with a focused view) one has, from
whatever knot is up for consideration, a tangle;
and a tangle has all these possibilities re its knots
via the loading profiles.

But formalizing this is really tough. That yChan
dressing of SmitHunter’s Bend quite impressed me
–the “same” knot?! Not by my reckoning, but then
it shows how differences can sneak around formal
barriers.

[HINT]: #1425 is topologically equivalent to the 'False Zeppelin bend' (with crossed tail segments).

And I'd like to see its performance for strength. We've seen Thrun's Joint/PoorMan'sPride/zep. go out of form, in the HowNotTo guy's video (amazing!).

–dl*

[agent_smith]

from Dan:

I will simply again point out that w/o throwing anything out of consideration (with a focused view) one has, from whatever *knot* is up for consideration, a *tangle*; and a *tangle* has all these possibilities re its *knots* via the loading profiles.

Am unclear as to what the ultimate purpose of this comment is? I am not being offensive - I am simply being factual - in that I am left wondering what your point is? Ok.. the 'knot' that is [i]up for consideration[/i] is presumably a 'bend' (an end-to-end join). I'll refer to this "knot" (a [i]tangle [/i]per your words) as the [u]'parent bend'[/u]. [u]Note[/u]: Definition of 'tangle' = a confused mass of something twisted together I am of the view that any knot that is tied with an intended specific geometry is not a confused mass.

per Dan:

But formalizing this is really tough.

I am making the following claim:
A deliberately tied parent bend which has a specific intentional geometry has a defined number of [derived] corresponding eye knots.
These corresponding eye knots have a [core] geometry that is congruent with the [core] geometry of the parent bend.
Note: I am using the word ‘core’ to denote the part of the knot that is central to its existence or character.
I find this definition better than ‘nub’.

I am using the term ‘eye knot’ in lieu of loop knot.
An eye knot is analogous to an eye bolt - the eye being round/oval and permits connectivity (eg a carabiner).
The ‘eye’ of an ‘eye knot’ has no particular chirality (handedness).

I am (by definition) making the claim that the correspondence (or relationship) between a parent bend and its derived eye knots is geometric in character.
This claim is valid where load (or a loading profile) is not considered.
That is, the correspondence is purely geometric in character.
All knots respond to load in different ways - and the particular loading profile plays a significant role in this ‘response’.
Typical responses include (list is not exhaustive):
compression
distortion
extrusion of rope segments out of the core
instability
insecurity
collapse

Dan Lehman may wish to make an alternative claim (or perhaps make no claim whatsoever).
He is entitled to do so.
Dan may wish to reject a geometric relationship and consider loading profile as the dominant factor to consider.
If this is his preferred approach - this may explain his reference to “possibilities”.
There may indeed be a number of possible ‘eye knot’ derivatives - perhaps being difficult to quantify?
In contrast, if a geometric approach is taken, it ought to be easier to quantify the number of possible eye knot derivatives.
And again - my preferred approach is to begin with [a] ‘bend’ and then try to derive the corresponding eye knots (rather than the other way around).
In my view, the logical approach is to use a ‘parent bend’ as the basis for deriving corresponding ‘eye knots’.

That yChan dressing of SmitHunter's Bend quite impressed me --the "same" knot?! Not by my reckoning, but then it shows how differences can sneak around formal barriers.

In my view, this is potentially straying off-topic - and is best examined in a separate topic thread.

Quote

"[HINT]: #1425 is topologically equivalent to the 'False Zeppelin bend' (with crossed tail segments)."

And I'd like to see its performance for strength. We've seen Thrun's Joint/PoorMan'sPride/zep. go out of form, in the HowNotTo guy's video (amazing!).

Possibly best dealt with in a new topic thread. But, 'strength' is irrelevant in my view. Possibly you meant some kind of 'response to load'? Response to load (or from some particular loading profile) allows one to assess things like; stability, security, jam resistance, etc. [u]For example[/u], strength is not a relevant factor in the use of #1410 Offset overhand bend in abseiling/rappelling. Focussing on 'strength' leads one to wrong conclusions (because stability, security, knot footprint, and jam resistance are more important factors).

[Dan_Lehman]

@agent_smith post:32

from Dan:

I will simply again point out that w/o throwing anything out of consideration (with a focused view) one has, from whatever *knot* is up for consideration, a *tangle*; and a *tangle* has all these possibilities re its *knots* via the loading profiles.

Am unclear as to what the ultimate purpose of this comment is?

I mean to push towards reaching the Tangle level
and then going from there with whatever particular
loading profiles (eye knots, e2e joints, knot hitches…)
are of interest. Whereas, in this …

I find it logical to begin with a 'bend' - and then try to derive the corresponding eye knots.

... I felt a bit constrained. In a sense, one isn't so much [i]deriving[/i] ... but [i]showing[/i] --they're right there, in the Tangle, to be ID'd per the Tangle pieces (1-2 & A-B, for a 2-Tangle).

[u]Note[/u]: Definition of 'tangle' = a confused mass of something twisted together I am of the view that any knot that is tied with an intended specific geometry is not a confused mass.

... and that of course we're not using "Tangle"to mean this, but simply an entanglement of cordage devoid of loading. (Deliberateness might be overrated; in any case, one has THIS or that Tangle, however wrought, for the consideration of its various Knots via loading.

[u]Note[/u]: I am using the word 'core' to denote the part of the knot that is [b]central to its existence or character[/b]. I find this definition better than 'nub'.

What happens if "core/nub" is omitted --esp. re e2e joints!?

That yChan dressing of SmitHunter's Bend quite impressed me --the "same" knot?! Not by my reckoning, but then it shows how differences can sneak around formal barriers.

In my view, this is potentially straying off-topic - and is best examined in a separate topic thread.

Well, it points out a significant change of your vaunted geometry in a knot that is put together able to have such difference. (Consider also how there are different dressings of the Fig.9 EK.)

–dl*

[agent_smith]

This post is directed to Dan:

We are getting technical here…but it is useful to sort out definitions… because I think this is fundamental to any progress.
Also, this is a forum for technical discussion about knots - with deep dives into underlying concepts.
I don’t know of any other forum on planet Earth where these types of peer-level discussions are possible?
In this regard, I’m happy to explore this subject matter with a view to finding some common understanding and agreeance.
At the very heart of this subject matter is the English words we choose to convey complex ideas…Words have meaning.

In reply to your points…

I mean to push towards reaching the Tangle level and then going from there with whatever particular loading profiles

[u]We need to agree on definitions.[/u] I hold the view that the word 'tangle' has a particular meaning as follows: "a confused mass of something twisted together". A random knot that accidentally formed in a rope might fit this definition (eg due to wind, strong hydraulic water flow in a stream, shaking/jostling, etc). However, I am strongly of the opinion that when a person ties a knot he has a particular outcome and purpose in mind. The knot tier directs his mind to the task - applying cognition - to achieve a particular and/or specific geometric outcome. For example, I intend to tie a Butterfly eye knot (Ashley #1053). The result I end up with can be verified against a known standard (I can look up #1053 in a book and verify it). The end result (ie outcome) is either correct or incorrect. I am of the opinion that you can't say that a knot is [i]half [/i]correct (or 25% correct).

There must be some basis for establishing and agreeing upon certain standard geometric knot forms - for example, we all agree that a simple Bowline is depicted at #1010 in Ashley’s book.
In the same way, we have the ‘SI’ metric units of measurement - and we all agree on what a metre length is. And we all agree on what 1 kilogram mass is.
Without a standard model to compare against, there would be no way for an assessor to assess a trainee. A trainee could tie anything into a confused mass of twisted rope and demand to pass an assessment.
For example: An assessor asks a trainee to tie #1053 Butterfly. The trainee presents a confused twisted mass of rope to his assessor.
The trainee demands to be assessed as competent.
An assessor needs measurable criteria to enable decisions/judgements to be made about a student competence.
However, for this to be true - there must be an agreed standard for a knot that is assigned the name ‘Butterfly’.
Without an agreed standard - there can be no agreement - only confusion.
EDIT NOTE:
Can we assume Ashley and CL Day to be primary reference sources?
If I look up a knot that is assigned the name ‘Bowline’ - I anticipate finding the exact likeness as depicted at illustration #1010 in ABoK.
I would not expect to find the knot illustrated at #1047 (F8).
Humans assign names to things - eg I know what a tree looks like, and a chair, and a dog, etc.
If you asked me to tie a Zeppelin bend, I would assume that you had a specific geometry in mind - it wouldn’t be in the likeness of #1415 Double Fishermans?
If I tied and presented #1415 (instead of a Zeppelin bend) - I would surmise that you would look at me in astonishment?

You yourself have complained about which geometry is the ‘ASCii Bowline’.
You sent me a number of emails with ASCii code in an attempt to depict a certain knot geometry.
That is, you had a particular geometry in mind.

... I felt a bit constrained. In a sense, one isn't so much deriving ... but showing --they're right there, in the Tangle, to be ID'd per the Tangle pieces(1-2 & A-B, for a 2-Tangle)

In order to show or demonstrate that something is being related to something else - one must derive it from a 'source'. For me, the 'source' is the 'parent bend'. I begin with a parent bend - and I derive the corresponding eye knots by linking the Tail(s) and S.Part(s) in various combinations. I had posited that there are only 4 possible linkages that can be made.

Per your words… “showing they’re right there” suggests that you were able to identify something visually.
You observed some kind of congruence or correspondence from one thing compared to the other.
Could this “showing” be geometric in character? How else would you be able to "show’ something?
I had posited that it is the core structures that can be compared - the correspondence being that the cores have the same geometry.
Here again I use the word ‘core’ as my preferred way of identifying the ‘nucleus’ of a knot structure.
I define a knot core to denote the part of the knot that is central to its existence or character.

...per the Tangle pieces (1-2 & A-B, for a 2-Tangle)

I think you mean the segments that protrude/project from the [i]knot core[/i]? The 1-2 and A-B don't have a clear-cut definition - hard for a layperson to understand (you would need to establish definitions). Presumably you refer to the 4 segments (protuberances) exiting the knot core - and you are assigning an alpha-numeric coding to ID these segments? In 3D space, which orientation gives rise to assigning a particular segment as a "1" in contrast to a "2", or "A/B"? In other words, how do I determine which protruding segment is "1"? [u]EDIT:[/u] Given that a 'bend' has [u]2 Tails[/u] and [u]2 S.Parts[/u]... Perhaps assigning the following alpha-numeric values might make it easier to understand: S1, S2 and T1, T2 S1 = "S.Part 1" (one of the standing parts) S2 = "S.Part 2" (the opposite standing part) T1 = "Tail 1" (one of the tails) T2 = "Tail 2" (the opposite tail)

S1 can link to S2 (S1-S2)
T1 can link to T2 (T1-T2)
S1 can link to T2 (S1-T2)
S2 can link to T1 (S2-T1)
*Note that S1 cannot link to T1 (both originate from the same rope)
*And S2 cannot link to T2 (both originate from the same rope).

I find this alpha-numeric system and my descriptions to be more ‘logical’ than your 1-2 / A-B annotation.
You may choose to disagree (and that’s fine).

What happens if "core/nub" is omitted --esp. re e2e joints!?

In a 'bend', there will be 2 S.Parts and 2 Tails? I think you might also have contemplated another type of 'bend' which is an edge case - eg the linking of 2 eye knots? For example; I could link the [i]eye [/i]of a #1010 simple Bowline to the [i]eye [/i]of another #1010 simple Bowline. Would this be your alternative definition of a 'bend'? For me, I see this as [u]a composite union of 2 eye knots[/u] - with each eye knot possessing its own core. Perhaps a stricter definition of a 'bend' is that the union only has 1 [i]core [/i](not 2 separate cores). In the case of 2 cores (linking 2 eye knots) - this is a composite structure - and each knot core will respond to load accordingly. [it could be the linkage of #1010 simple Bowline to #1047 F8...an eye-to-eye link... Each eye knot responds differently to load.]

Well, it points out a significant change of your vaunted geometry in a knot that is put together able to have such difference. (Consider also how there are different dressings of the Fig.9 EK.)

I note the use of "[i]your vaunted geometry[/i]" phrase. It isn't 'vaunted' per se - its simply a logical choice. If I tie a Zeppelin bend - it will have a known geometry. (although again, there must first be an agreed definition/geometry for what constitutes a Zeppelin bend). If I tie #1411 F8 bend - it has a known geometry. Here I assume the energy stable dressings - the simplest most symmetric dressing that is demonstrably stable under load. With an 'F9', its simply a matter of declaring a particular dressing - again - the most energy stable dressing is logical. For all 'bends' and 'eye knots' one must settle on a dressing - nominally the most energy stable dressing. For example: An F8 bend (#1411) can be tied with a flat parallel dressing state - but this is unstable. One can also apply the same general principle to an offset joining knot - eg #1410 - where some dressings will be more unstable.

Again - we need to have agreed standard to reference against.
An F8 and an F9 eye knot can have different dressing states.
However, we know that there are geometries that are more stable in response to load.
Example:
A trainee is asked to tie an F8 eye knot.
The trainee presents the F8 to his assessor.
What is the criteria the assessor is making judgements about?
What evidence does an assessor require to form a judgment about the trainees competence?
What evidence is required for an assessor to declare a trainee ‘competent’?
Note: Assessment should also capture consistency of performance - to rule out random chance success, the trainee should accurately tie the F8 at least 3 times.
Accuracy and consistency ought to be part of the assessment criteria.

EDIT NOTE:
I’ve added an image to illustrate the concept of a standard reference.
An assessor could look up what an ‘F8 bend’ looks like in a primary source (eg Ashley).
An F8 bend is found at illustration number One thousand four hundred and eleven (#1411).
However, Ashley does not define what an energy stable dressing is… one can only assume that the depicted dressing is ‘optimal’ for loading.
I define an energy stable dressing state as:
“Being optimal for loading - to the extent that the knot remains stable and exhibits the least degree of distortion in response to load”.
With respect to the attached image below: Image ‘B’ is energy stable.

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[Dennis_Pence]

I know this is an old topic, but I just wanted to add one more example of what I think agent_smith is trying to show. Here is the basic structure that can be found in the Hanson patent document for what I like to call the Hanson Knot Family. {I have deleted the labels which were the 12 points of the BSA Scout Law for simplicity, but the Figure numbers come from the patent.}

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[Dennis_Pence]

Here are the 8 different loop knots that can be derived from the original Hanson Structure. Alden Hanson had 3 of them in his patent application, and I think that these are the best three from the eight.

The “sideways” loops in the last four become almost unrecognizable when you tighten them, and I am not sure they are very practical.

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[agent_smith]

Hello and thanks Dennis.

With regard to your mentioning of the number “8”, I can comment as follows:

A parent end-to-end joining knot (ie ‘bend’) has 4 corresponding principal eye knots that are derived by the linkages between available tail ends and S.Parts.
This is true for all end-to-end joints where there are 2 S.Parts and 2 tail ends.

With each of these 4 principal eye knots, the S.Parts can be transposed with the tails (a transposition).
This transposition results in a further 4 eye knots.

If we consider the 4 principal corresponding eye knots (created from the 4 linkages), + the 4 transposed eye knots, this means a total of 8 possible corresponding eye knots derived from a singular ‘parent bend’.

NOTES:

1. I prefer to start with an end-to-end joint (the ‘parent bend’), and then derive its 4 principal corresponding eye knots.
2. It is also possible to begin with an eye knot, and then derive [a] corresponding ‘bend’ and its other related eye knots.
   I personally find this approach less intuitive with multiple possible pathways (although Dan Lehman appears to favour this approach?).
3. The correspondence is geometric in nature (the core of the ‘bend’ is congruent with the cores of its derived eye knots).
4. I am using the term ‘eye knot’ in lieu of ‘loop knot’ (a ‘loop’ has chirality - S/Z - an ‘eye’ has no chirality).
5. There is no peer reviewed primary reference source for the relationship between ‘bends’ and ‘eye knots’ (the concept of correspondence has been known but not documented in detail).
6. Harry Asher may have been first to try to publish a basic theory of correspondence between ‘bends’ and ‘eye knots’ - Ashley apparently did not?

[Dennis_Pence]

For a more definite reference, I have a copy of Harry Asher’s The Alternative Knot Book (1989), where there is one section on page 81 titled “Loops from Bends.” His primary example is the Corrick Bend (which he claims is unnamed in Ashley, but I could not quickly find it). He gives only one “loop”, but given the extreme symmetry, there would not be 8 different eye knots that could be derived (but certainly more than one). On the next page, he mentions several other bends that may or may not lend themselves to this process. Also on page 82, Asher has a section titled “Bends from Loops.” Here he writes “The reverse procedure is vastly simpler.” His only example is the Angler’s Loop, and he suggests two different cuts to derive two different bends from this loop.

I have also looked at the different bends that you can derive from the Hanson Knot Structure. If you start with the basic Handson Bend (which Heinz Prohaska called a Ram Head Knot), you can switch the standing part and the free end on one rope (but not the other) to get one offset bend. Then you can switch the second rope (but not the first) to get a second offset bend. Finally, you can switch the standing part and the free end of both ropes to get the fourth different bend. Hanson did not have all of these clearly drawn in his patent document, but he seemed to imply them all.

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[Dan_Lehman]

@agent_smith post:37

With regard to your mentioning of the number “8”, I can comment as follows:

A parent end-to-end joining knot (i.e. ‘bend’) has
4 corresponding principal eye knots that are derived by
the linkages between available tail ends and S.Parts.
This is true for all end-to-end joints where there are 2 S.Parts and 2 tail ends.

But it’s not true in terms of distinct knots if the Tangle
is symmetric --for you’ll just get duplicates (which I will
count as such even were they to have different handedness).

With each of these 4 principal eye knots, the S.Parts can be [i]transposed [/i]with the tails (a transposition). This transposition results in a further 4 eye knots.

Which for me isn't transposition but a change in Loading Profile applied to the 1-2 & A-B (the two pieces) 2-Tangle.

[u]NOTES:[/u] 1. I prefer to start with an end-to-end joint (the 'parent bend'), and then derive its 4 principal corresponding eye knots. 2. It is also possible to begin with an eye knot, and then derive [a] corresponding 'bend' and its other related eye knots. I personally find this approach less intuitive with multiple possible pathways (although Dan Lehman appears to favour this approach?).

Wellll, one nice thing is to have a single S.Part --a key part! But things are really not so simple, doable. E.g., for our good ol' friend the (basic) [i]BWL #1010[/i], with S.Part "1", Outgoing Eye Leg "2", Returning Eye Leg "A" & Tail "B", how does one apply the Loading Profile of B-vs-2+A (i.e., this [<<corrected "B-vs-2+A] is a [i]"Tail-Loaded BWL"[/i]) ?!! Taking the Tail one way will yield an amphichiral knot, and geometries of the B-A part can differ!

–dl*

[agent_smith]

Hello Dan,

I had posited that 4 principal eye knots can be derived from a ‘Bend’ (they correspond to the parent bend).
This is due to the available ‘linkages’ that can be made between S.Parts and Tails.

With regard to your comment:

But it's not true in terms of distinct *knots* if the Tangle is symmetric --for you'll just get duplicates (which I will count as such even were they to have different handedness).

I disagree. Lets examine Ashley's #1411 F8 bend and its corresponding 4 eye knots. I would describe this bend as being 'symmetrical'. The 4 corresponding eye knots are 'distinct'. And here we need to precisely define what you mean by [u]distinct[/u]. You did comment that you would "count as such even were they to have different handedness". Which appears to conflict with your first statement. English language is complex - it is possible to arrive at 2 different conclusions with what you wrote?

I had already posted details of the eye knots that correspond to #1411 F8 bend.
It is true that 2 of the eye knots have opposite chirality (handedness).
Question: If 2 F8 eye knots have opposite chirality, are they ‘distinct’ from each other?
(that is, can the F8 eye knots be regarded as geometrically different?).
Or - are they the same?
That is, can an argument be made that they are identical?
I posit that no, they are not identical.
They are mirrors of each other - one is the opposite reflection of the other.
They are both of the same species (F8 eye knots) but they have differences in orientation.
I see this with humans - we are all from the human race.
But, all humans have differing appearance (unless they are identical twins).

The same applies to the Zeppelin bend and its 4 corresponding eye knots.
Two (2) of these eye knots have opposite chirality.

#1425A Riggers bend is a curiosity.
It of course has 4 corresponding eye knots.
BUT, two (2) of these eye knots are identical.
They have the same geometry - and same chirality.
Perfect identical twins.

The question for me is Why?
Why does the Riggers bend have this property?
Why are the Zeppelin bend’s corresponding eye knots all ‘different’ (ie no identical twins)?

#1415 Double Fishermans bend is another curiosity.
It has four (4) corresponding eye knots.
BUT, it has 2 sets of twins!
That is, 2 of the corresponding eye knots are identical twins.
AND, the other 2 corresponding eye knots are also identical twins.
From my research so far, #1415 Double Fishermans bend is the only knot that produces
a double set of identical twins.

I have no theory to explain this.
eg Why does the Riggers bend have a pair of identical twins, while the
Zeppelin bend does not?
Why does the Double Fishermans bend have 2 sets of identical twins?

…

I am still trying to figure all of this out…
#1415 Double Fishermans bend is symmetric (but its geometric shape differs from one side relative to the other).
The F8 bend is also symmetric, but one side is not identical to the other.
The chirality is unchanged, but the positions of the ‘collars’ relative the S.Parts has changed.

Hmmm.

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