Author Topic: The relationship between 'bends' and eye knots  (Read 2141 times)

agent_smith

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The relationship between 'bends' and eye knots
« on: November 20, 2023, 04:30:33 PM »
All 'Bends' have corresponding eye knots ('loop knots').
There are 4 possible linkages that can be made between tail(s) and S.Part(s) - from which 'eye knots' are derived (aka loop knots).
A further 4 eye knots can be derived based on loading profile where the tail and S.Part of an eye knot are swapped/exchanged.
Thus, a total of 8 possible eye knots can be derived from a 'bend'.
Not all of the derived eye knots will be necessarily stable/viable.
[Note: Linkages can only be formed between segments of opposite colour - refer to images below which employ 2 different coloured ropes to form a bend. With reference to the images, it is not possible to create an eye knot by linking 2 segments of the same colour].

All of the corresponding eye knots will share the same common core structure - thus being of the same 'genus' type.
The relationship being geometric in nature.

It appears that Harry Asher may have been the first to publish a rudimentary theory about this subject area - in his book: "The Alternative Knot Book".
However, Asher did not expand on the theory or offer any detailed insight.
Clifford Ashley and Cyrus Day don't appear to have published anything about the relationship between bends and eye knots.
I haven't found anything from Budworth.

It would be interesting to catalogue a range of end-to-end joining knots (ie 'bends) - and then derive the corresponding eye knots.
Depending on the particular bend and resulting loading profiles, some new offset bends may also be discovered with this approach.

I do seem to recall that the great knot master Xarax may have stated: "If you have tied a bend, then by definition you have also tied all of its corresponding eye knots".
Or something similar to that effect?
I am not entirely convinced of this - since some interesting eye knots emerge - with similarly interesting response to various loading profiles.

I've offered up 4 different 'bends' and their corresponding eye knots.
It would be a huge undertaking to catalog all of the known bends and their corresponding eye knots - which might be too time consuming or too difficult a task...

EDIT NOTE:
None of this information is exhaustive or the final word on the theory behind bends and their corresponding eye knots.
Its an evolving field of study - one that I think is worthy of attention.
« Last Edit: December 29, 2023, 08:27:22 AM by agent_smith »

Dan_Lehman

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Re: The relationship between 'bends' and eye knots
« Reply #1 on: November 21, 2023, 01:57:24 AM »
All 'Bends' have corresponding eye knots ('loop knots').
There are 4 principal corresponding eye knots[/b] to any given 'parent' bend.
There are an additional 4 secondary eye knots[/b] - due to choice of 'S.Part' and tail end.
WHERE ARE YOUR DEFINITIONS OF THESE PROPOSED TERMS?!
(You also need to choose between "principal" & "primary", I think;
IMO, the latter better corresponds to "secondary".)

Let us posit that common/simple end-2-end joints & eye knots
are "2-Tangles" --i.e., as viewed so that all parts ("ends") that
exit the entanglement (the Tails, and for EKs the Eye Legs)--
then the end-2-end joint will be seen as that knot of the 2-Tangle
loading ends 1 & A in opposition to each other, Tails being 2 & B.
Eye knots are :

 on notation ::
     "1-v-2+A" means "1 is loaded in opposition to 2 & A(the eye legs )"

And I leave blank line for "ditto" repetition,
giving only as last character that of the differing
character for that line --e.g., in line 2 for "B" vice "A".


----------S.Part-loaded
1-v-2+---1st
        +B  ---2nd
A-v-B+1  ---3rd
        +2 ---4th

---------TAILoaded
2-v-1+A ---5th
        +B ---6th
B-v-A+1 ---7th
        +2 ---8th

My surmise is that the first set might be called "primary"
because each has an e2e S.Part qua EK S.Part;
the lower four load an e2e joint's Tail qua EK S.Part.
But another thought is that only the 1st &3rd cases
should be "primary", as these preserve the e2e joint's
S.Parts and ADD one Tail for making an EK!?


Quote
I've offered up 4 different 'bends' and their corresponding eye knots.
It would be a huge undertaking to catalog all of the known bends and their corresponding eye knots - which might be too time consuming or too difficult a task...

Your sample sets break any rule; you're apparently
operating by some sort of *feel* of propriety rather
than a logical definition.

Let's take them in presented order.
First up is the capsized Carrick Bend, for which your
"A" & "B" EKs are the traditional choices, my 1st & 3rd cases
defined above, where e2e joint S.Parts are loaded.
Then comes your "C" in which a chosen joint's S.Part
connects to the other piece's Tail; but you only give
one of these --of the TWO defined--, I guess so to keep
to your pronouncement of "four" --you got both ends
loaded for A&B but don't give a yellow-rope's play
at being S.Part (my 2nd or 4th cases).
Instead, you make your fourth be a Tail-loaded case,
but again with only one color's end qua S.Part, the
other-rope complement not counted.
Note that this case is of a symmetric knot.

Next comes the more elucidating asymmetric and
natural case (and simple!) of the net-knot 2-Tangle,
home of familiar sheet bends, BWLs, becket hitches.
You have the most commonly cited 1st case, but miss
the complementary (other rope playing S.Part)
3rd case --that, yes, awkward thing loading a U-fold
end and necessarily sending the other into a trying
backflip!  But that's logic/definition, not whim.
(It is your "C" with Tail & S.Part switched --blue Tail.)

Your third set (Butterfly) is puzzling re what the four
knots are --as the first/top case with hashed eye
connection has TWO EKs, per the blue, white options
qua S.Part (asymmetric knot, too) --and not some
new thing put in devoid introduction of "mid-line" bent!
Similarly, the bottom case also offers --as you note
by "EEL"-- two EKs, not one schizophrenic one!

Finally, to the one begetting this interesting deliberation,
with the "Lehman8", brought out by Alan's Tail-swapped
variation.  Your "B"&"C" are my 1st & 3rd, traditional
cases of one piece forming its part and then arcing
out to make the eye and return qua opposed joint S.Part
to close out; so, a yellow S.Part'd EK, and blue one too.
But then your "A" has Alan's variation --a returning yellow
eye leg into joint's blue Tail--, but not the corresponding
one for the blue eye leg fusing to a yellow Tail.  Rather,
you make your fourth case a joint's Tail-loading for EK
(for which there is the complementary yellow-Tail'd case
to do, if you fancy that.  Well, again, maybe by mere
*fancy*, but not by logic of Tangles and knot definitions.

Finally, to the assertion that tying the e2e joint means
you've tied all EKS, well, why be joint-centric in this ::
tying ANY *knot* means you've tied all of that knot's
Tangle's knots ... !?  But IMO having tried to work
with this notion of Tangle and Knots-are-loadings-of
-Tangles notion, things are tough; dressing can play
a role.  I found the Tail-Locked BWL not by plying
the fun, SIMPLE loadings for the common net-knot
(BWL, sheet bend...), but seeing the TLBwl In The Wild;
whereas in going around the logical loadings of that
Tangle, I got only something well inferior (IMO) known
as sort of Crabber's EK !!

And there is the issue of "primary" :: if your Tangle *seed*
is an EK, you have just the one S.Part, in case you have
some sort of "primary" feelings for "1" vs. "A"; with the
e2e joint, you have two.   And then each such knot can
demand full treatment for itself, where IT is the definer
of derivatives.  Consider Alan's twist on the Lehman8
qua e2e joint, e.g.!


--dl*
====


agent_smith

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Re: The relationship between 'bends' and eye knots
« Reply #2 on: November 21, 2023, 03:55:37 AM »
Thanks for your reply Dan.

Some assumptions that I think you have made - need to be addressed.

In the first instance - my opening post (to open this new topic post) - is not exhaustive or intended to be the final word on this subject.
It seems that in your reply, you make the assumption that the theory is already well established and settled (when it is not).
In this regard, I added an edit note to my original post pointing out that this subject is not exhaustive and is very much an evolving work.

The second point I'd make is with regard to definitions (in the English language).
I have been playing with the term 'principal' and 'primary'.
I haven't settled on a best candidate descriptor...
Although given my use of 'secondary corresponding eye knots' - this does suggest 'primary corresponding eye knots' as a better candidate.

For the 'primary' derived eye knots - they adhere to the following general rules:
1. They are based on the chiral orientation of the parent bend (and here I am using the term 'parent' to refer to one specific geometric form).
2. The choice of S.Part is derived from the parent bends opposing S.Parts - preserving the loading profile of that bend (where possible to do so).
3. The primary eye knots can only be formed by linking segments of different colour (where the parent bend is tied from 2 different colour ropes).
4. There can only be 4 primary corresponding eye knots.
5. In some cases, not all of the primary corresponding eye knots will be stable/viable (it is false to declare that 100% of all corresponding eye knots will always be stable in every case).

The Xarax quote was from old memory - I do recall him making such a statement many years ago - but I have no concrete written proof.
Be that as it may, I understand the point he was trying to make.
I did make the comment that I was not entirely in agreement with him - for various reasons.

...

I am happy for you to advance your counter theories - as I think this is a subject area of significant importance.

For example, if you wish to declare a competing view that there are in fact more than 4 'primary' corresponding eye knots, I think you should take the time to back your theories with clear and unambiguous photographs (so it is clear to the viewer/reader).

I think it is open to some debate as to how 'loading profile' plays a role in the classification of corresponding eye knots.
For example, we could dispense with load altogether - and only examine corresponding eye knots without load.
This might alleviate your seeming problem with choice of S.Part and tail end.

I am in favour of preserving the general chiral orientation of the 'parent' bend.
'Primary' corresponding eye knots should (wherever possible) - follow the general loading profile of that bend, and/or be the most stable configuration.

...

As for the rest of your criticisms of each of my initial offering of 4 bends and their corresponding eye knots...
A lot of this is incoherent and appears to a source of irritation on my depicted choice of 'S.Part' and 'tail end'.
Again, I have tried to preserve the chiral orientation of the parent bend - and to use the bends existing 'S.Parts' (where possible).

Your seeming irritation of choice of 'S.Part' and 'tail end' could be somewhat alleviated by dispensing with loading profile and only looking at the geometry of eye knot.
This is something that needs further thought...

In any case, I'll keep posting while I am motivated to do so.

My next offerings will be the infamous 'Riggers bend', the Zeppelin bend, and the Figure 8 bend (#1411)..
The derived corresponding eye knots from these bends are interesting.

Amen.
« Last Edit: December 29, 2023, 08:30:39 AM by agent_smith »

Dan_Lehman

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Re: The relationship between 'bends' and eye knots
« Reply #3 on: November 21, 2023, 05:48:35 AM »
It seems that in your reply, you make the assumption that
 the theory is already well established and settled (when it is not).
Not at all; I pointed to the glaring absence of statement
of your theory --i.p., how do you define your terms.
To which purpose I laid out the logical possibilities for
a 2-Tangle's 8 eye knots ::

----------S.Part-loaded
1-v-2+A  ---1st
        +B  ---2nd
A-v-B+1  ---3rd
        +2 ---4th

---------TAILoaded
2-v-1+A ---5th
        +B ---6th
B-v-A+1 ---7th
        +2 ---8th

Now, it is for YOU to define your "primary/secondary" definitions
using these articulated cases.  I showed how your offered sets
of supposed primary knots followed no rule, but some whim.

Now, you give this --which fails to satisfy--:
Quote
For the 'primary' derived eye knots - they adhere to the following general rules:
1. They are based on the chiral orientation of the parent bend
This tells us nothing; and "chiral orientation" is unwanted.

Quote
2. The choice of S.Part is derived from the parent bend's opposing S.Parts
 - preserving the loading profile of that bend (where possible to do so).
Again, "is derived from" says nothing; it begs the question HOW?
(One might state, e.g., "One of the joint's S.Parts will be the EK's S.Part.)

Quote
3. The primary eye knots can only be formed by linking segments of different colour ...
This gratuitous and unwantedly incorporating a presentation aspect
(nicely contrasting colored ropes) : an EK is just that, and so clearly
not a link.
Quote
4. There can only be 4 primary corresponding eye knots.
!!??? Whosever ordained this limit?
Rather, one might guess/hope that this statement is merely
a plain consequence of the yet-to-be-given definitions of
primary/secondary EKs of a joint.  (And beware the case(s)
where in fact there are not so many distinct knots --albeit
yes the filling out of my EK possibilities which number 8 :
there might be duplicates.


Quote
For example, if you wish to declare a competing view that there
are in fact more than 4 'primary' corresponding eye knots,
I think you should take the time to back your theories with
clear and unambiguous photographs (so it is clear to the viewer/reader).
Photographs --of logical entities?
Their representation is perspicuously given now twice;
ANYone can attach the "end" labels and see the results
of treating some knot in Tangle analysis.

Quote
I think it is open to some debate as to how 'loading profile' plays
a role in the classification of corresponding eye knots.
For example, we could dispense with load altogether
--and only examine corresponding eye knots without load.
This might alleviate your seeming problem with choice of S.Part and tail end.
?????  This is nonsense :: the EKs are defined by loading profile
of a given Tangle --as e.g. in the BWL 1010 & 1034.5 from the
loading of that common 2-tangle.  The loading profile on the
Tangle is how EKs are defined!

Quote
I am in favour of preserving the general chiral orientation of the 'parent' bend.
'Primary' corresponding eye knots should (wherever possible) follow the general
loading profile of that bend, and/or be the most stable configuration.
Well, this shows you have only a *feel* and whim about
what you will choose to accord the gold star of "primary";
that said,  I don't understand how you come to think there're
only four --your sets, as I've shown, have in them knots
that have a complementary (colors reversed) equally
qualifying knot that you've left out.

Quote
As for the rest of your criticisms of each of my initial offering of 4 bends and their corresponding eye knots...
A lot of this is incoherent and appears to a source of irritation on my depicted choice of 'S.Part' and 'tail end'.
Your seeming irritation of choice of 'S.Part' and 'tail end' could be somewhat alleviated
by dispensing with loading profile and only looking at the geometry of eye knot.
  It is not at all incoherent but carefully stated,
and your choice of S.Part/Tail is exactly what defines a knot.

Quote
My next offerings will be the infamous 'Rigger's bend', the Zeppelin bend,
and the Figure 8 bend (#1411)..
The derived corresponding eye knots from these bends are interesting.

No, please not.  You stand in want of a definition that can be applied
by people to understand things; they are not waiting to see how YOU
understand things one whim to the next.  (Or even if that, how it is
you can assert that four is the limit.)


--dl*
====

agent_smith

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Re: The relationship between 'bends' and eye knots
« Reply #4 on: November 21, 2023, 07:12:11 AM »
In reply to Dan,

I read your post(s)... And after sorting through the negativity and generally incoherence;
its hard to avoid reaching a conclusion that there is something 'off' here.
Your closing remark for instance:
Quote
No, please not.  You stand in want of a definition that can be applied
by people to understand things; they are not waiting to see how YOU
understand things one whim to the next.  (Or even if that, how it is
you can assert that four is the limit.)
I am not sure if this is in the spirit of this forum?
Okay, I could just simply yield and stop posting any further bends and derived corresponding eye knots.
But is that in the best interests of exploring this subject?
Should I just yield and leave it to you to further this on your own?
I'll point out (again) that this entire subject area is new and explorative - it is certainly not settled.
There are no existing research papers or peer reviewed works that I can draw from.
I am forging my own path - where none exists (so to speak).

Quote
yes the filling out of my EK possibilities which number 8
Hmmm, I see you have offered up a number!
8, it seems :)
(I also had previously pointed this number out - which was met with debridement - but now is okay if you arrive at the same number?).

Now I could serve up the same sort of narrative back to you.
To para-phrase you: 
Quote
!!??? Whosever ordained this limit?

Can you prove that there is such a limit?  ;)

I wont go on here because it would serve no useful purpose.
Other than to say that I still posit the following general principles:
1. For any given 'bend' - there will be corresponding eye knots. EDIT: There are 4 possible linkages - from which a corresponding eye knot is derived.
2. There will be a maximum of 2 'primary' corresponding eye knots and up to 6 'secondary' corresponding eye knots (a total of 8).
3. This 'correspondence' is based on the fixed chirality and geometry of the parent bend (ie the 'chiral form') - a different bend means another set of different eye knots.
4. The 'primary' eye knots are closest to the parent bend in terms of loading profile.
5. A further 4 eye knots can be derived based on choice of S.Part (the tail and S.Part are swapped) - therefore a total of 8 possible eye knots can be derived from a parent bend (but not all of which will necessarily be stable/viable).

These statements are not settled - that is, it is not an established/agreed theory.
This is a relatively new approach to understanding the link between a given 'bend' and its 'corresponding' eye knots.

The source of irritation with Dan appears to be linked to:
1. What is the definition of a 'primary' corresponding eye knot?; and
2. What is the method be which an 'S.Part' and 'tail end' is chosen? (to distinguish it as being 'primary').
3. The total number of possible corresponding eye knots - 8 - may be in dispute?
4. The gratuitous use of two different coloured ropes to depict a parent bend.

Its a pity that the source of irritation and shouting cant be managed in a more civilised manner.
When you think about it, there are no power struggles here, or attempts to dominate, or to vanquish an opponent.
Its just an attempt to try to open some new lines of thinking - which may have been started way back with Harry Asher's book (1989?).
« Last Edit: December 29, 2023, 08:35:31 AM by agent_smith »

Dan_Lehman

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Re: The relationship between 'bends' and eye knots
« Reply #5 on: November 22, 2023, 01:50:55 AM »
... the negativity and generally incoherence;
Let's see exactly what you find of generally incoherence?
My words are there in hard print for all to see, to refer to
exactly, not in general.  Perhaps I might clarify something for you?


Your closing remark for instance:
Quote
You stand in want of a definition that can be applied
by people
to understand things; ...
Pretend that you're too busy to treat those threatened
further cases of joints begetting eye knots, and so you
solicit helpers :: now, how do you instruct those helpers
to examine the joints and show what you call "primary"
(& secondary) corresponding eye knots for the joint?
THIS instruction, the definitions that must guide any
to follow in your discriminations (at least, if they are
to be generally followed), is what you continue to lack.


Quote
Quote
yes the filling out of my EK possibilities which number 8
Hmmm, I see you have offered up a number!
8, it seems :)
(I also had previously pointed this number out - which was met with debridement -
but now is okay if you arrive at the same number?).
"debridement"??
The number is quite evident in the definition given
of a 2-Tangle, but that only.  I have also noted that
there are other means of making what might be
called a "corresponding eye knot (to a joint)".
But, of a 2-Tangle, there are 8 EK loading profiles, by definition.

Quote
Now I could serve up the same sort of narrative back to you.
To para-phrase you: 
Quote
!!??? Whosever ordained this limit?
Can you prove that there is such a limit?  ;)
But that is no limit, but mere fact of the 2-Tangle --four ends
to be an EK S.Part, and two loadings for each in this.  (The
count of *knot*s as unique is another matter :: e.g., in the
Carrick Bend's case, you showed yellow & blue S.Parts each
making the same knot --the obvious, traditional one.)

But YOU have not given any hint of a reasoning for your asserted
limit of four --nothing; and your sets show varied inclusions/exclusions,
some of which are clearly illogical (a yellow S.Part in, but the matching
blue S.Part case not, e.g.).  Now, you seem to want to *feel* around
for behavioral niceness in order for you to award some EK "primary"
status; well, accepting that that's one way to look at knots, it begs
the question of Why (only/as many as) four?!  To which you've had
no response --coherent or other wise.

Quote
I won't go on here because it would serve no useful purpose.
Other than to say that I still posit the following general principles:
1. For any given 'bend' - there will be corresponding eye knots.
2. There will be 4 'primary' corresponding eye knots
and 4 'secondary' corresponding eye knots (a total of 8).
HERE YOU GO AGAIN :: WHY, OH WHY, ARE THERE >>4<<
"Primary" EKs??
1) What makes an EK primary (for SOME e2e joint)??

Quote
3. This 'correspondence' is based on the fixed chirality
and geometry of the parent bend (ie the 'chiral form') - a different bend means another set of different eye knots.
What is THE chirality of an e2e joint in which one piece's
handedness is Z but the other's S?!  (E.g., the shared-eye
joint known sometimes as Two BWLs, which give a sort
of "mirrored BWL" look courtesy of the two hands!?

In any case, I don't see the relevance of this #3 criterion,
in that one has A specified joint --whatever it is, however
you name it-- and that is that from which corresponding
EKs are sought.

Quote
4. The 'primary' eye knots preserve the general loading profile and orientation of the parent bend,
and are the most stable forms - where possible to do so,
Here I see you as mixing guidelines/rules :: your initial "preserve ..."
suggests that one need examine each of the 8 logically possible
2-Tangle EKs (this particular treatment, not including other ways
of seeing "correspondence") for behavioral nicety.  Ok, that might
be a way some want to go, especially from a pragmatic standpoint;
but then I don't see how the number four is reasoned --which smacks
of the sort of 2-Tangle figuring that I laid out, and not the process
of examination for behavior (or "whim", as I've remarked)!?
Perhaps I could understand four as a limit, but not as an
assured There WILL Be These set, come hell or high water:
one gets such assurances by definitional implications, not whim.

Quote
5. The 'secondary' eye knots arise from the fact that for each corresponding eye knot,
there is a choice of 2 'S.Parts'.
NO :: for a (simple, 2-Tangle) joint there are four ends;
same for an EK.  For each of the joint's ends when taken
to be an EK S.Part the Outgoing Eye Leg is fixed (it is the
other end of that piece giving the S.Part) can return as
the joint's yet-to-be-assigned loading-- S.Part or Tail.
For the common case of the Sheet Bend, the turNip
has the S.Part as one end and the other is the Outgoing
Eye Leg, which can take either the Sheet Bend's
other S.Part (=>#1010) or the Tail (=>#1034.5).
And so on.
But we aren't choosing such options from an EK.

Quote
The source of irritation with Dan appears to be linked to:
1. What is the definition of a 'primary' corresponding eye knot?; and
2. What is the method be which an 'S.Part' and 'tail end' is chosen? (to distinguish it as being 'primary').
BINGO!  And alllll of these many words later we remain wanting these!

Quote
3. The total number of possible corresponding eye knots - 8 - may be in dispute?
There is no dispute that if you take the joint as a 2-Tangle
then there are 8 per definition EKs (possibly with duplicates).

But there is another way to see "correspondence" of an EK
to a joint.  I remarked about Ashley's #1043/5 --and I should
add #1016 (or some version of...)-- being seen as corresponding
to the Sheet Bend; in a way, they do so in the spirit I take of your
looking for preserving loading profile and so on --which they do
by altering the reference joint from a 2-Tangle to a 3-Tangle
in which one piece forms the eye and presents thus its two
ends qua single end to fill the path of the joint.  Having done
this, then look to how to *fuse* the single-S.Part'd piece's
Tail into one of the twinned pieces' Tails to restore a 2-Tangle. 

Quote
4. The gratuitous use of two different coloured ropes to depict a parent bend.
Entirely missing the complaint :: the use of colored ropes is GREAT,
but that aspect should NOT be part of your definitions.  (One might
print out such things in B&W and still there should be clarity and
the rules of correspondence need no altering.)

Quote
It's a pity that the source of irritation and shouting can't be managed in a more civilised manner.
When you think about it, there are no power struggles here, or attempts to dominate, or to vanquish an opponent.
Its just an attempt to try to open some new lines of thinking - which may have been started way back with Harry Asher's book (1989?).
I'm sorry that you feel that way about strongly expressed arguments,
but my expression reflects my own struggles at coming to ways (even
one!) to better understand knotting,
and your continued failure to answer clear questions aggravates
that frustration.
As I said above, we want from you your rationales for the otherwise
seemingly arbitrary, even capricious assertions of how things are.


--dl*
====

agent_smith

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Re: The relationship between 'bends' and eye knots
« Reply #6 on: November 22, 2023, 01:03:58 PM »
I've evolved the underlying theory for this subject matter.

I decided to use excessive gratuitous use of coloured dots to identify the various segments of a corresponding eye knot relative to its parent bend.
This enabled me to devise a stricter definition of what constitutes a 'primary' corresponding eye knot.

The primary corresponding eye knots have a congruent S.Part + one congruent eye leg.
That is:
[ ] The S.Part of the primary corresponding eye knot is congruent to an S.Part of the parent bend
[ ] One eye leg of the primary corresponding eye knot is congruent to the opposite S.Part of the parent bend.
(for now that's as clear-cut as I can be until I examine this subject further).
The primary corresponding eye knots are therefore closest to the parent bend in terms of loading profile.
As such, I now surmise that loading profile is the best way to examine the corresponding eye knots - and to determine congruence with the parent bend.

Using this approach, it appears that there is a maximum of 2 'primary' corresponding eye knots.
The remaining corresponding eye knots can be 6 or 7 in number.
Interestingly, the Sheet bend only has 1 primary corresponding eye knot.

Feel free to shout and launch a counter-offensive theory :)
« Last Edit: November 22, 2023, 04:45:31 PM by agent_smith »

Dan_Lehman

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Re: The relationship between 'bends' and eye knots
« Reply #7 on: November 22, 2023, 11:07:56 PM »
I decided to use excessive gratuitous use of coloured dot
Good!  (-;
Quote
Using this approach, it appears that there is a maximum of 2 'primary' corresponding eye knots.
What might see the limit not reached (i.e., there being but 1 PCEK)?

Quote
The remaining corresponding eye knots can be 6 or 7 in number.
Interestingly, the Sheet bend only has 1 primary corresponding eye knot.
This implies that the answer to my question above is one of
judging the resulting knot's "stability" or the like --the EK not
rising to the level of decent behavior.  (The same-side Sheet bend
yields a sort of Gnat-H Noose which CAN behave qua EK, in some
materials, at some forces --a YMMV situation.)

Hmmm, one can wonder if there is some knot that will behave
erratically in either "primary" loading!?


--dl*
====

agent_smith

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Re: The relationship between 'bends' and eye knots
« Reply #8 on: November 23, 2023, 07:34:06 AM »
I have uploaded the corresponding eye knots for the following 'bends':
[ ] Riggers bend (#1425A)
[ ] Zeppelin bend

The Zeppelin bend has 2 'primary' corresponding eye knots.
Interestingly, the Riggers bend only has one primary corresponding eye knot (due to its symmetry).
The geometry of the primary corresponding eye knots derived from the Riggers bend are in fact identical.

...

I looked at the Reef knot (bend) #1402 , however, there are no stable corresponding eye knots.
This is due to its structure, which is the integration of 2 opposing 'bights' - which are essentially 2 U folds.
Any attempt to create a corresponding eye knot results in instability.

Interestingly, a 'Granny knot' (#1405) does enable somewhat stable corresponding eye knots to be created.

The Ring bend (water knot / tape knot) #1412 does have stable corresponding eye knots.

EDIT NOTE:

Tugboat 'B' bend and corresponding knots added.
This bend had no apparent name - other than the eye knot published by Dave Root (although I have no knowledge as to actual discovery and creator info).

There is also a Tugboat 'A' bend - although it does not appear to be as stable as the 'B' version.
The difference being a mirror reversal of the Crossing hitch.
« Last Edit: November 26, 2023, 11:51:56 AM by agent_smith »

agent_smith

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Re: The relationship between 'bends' and eye knots
« Reply #9 on: November 27, 2023, 09:30:22 AM »
#1412 Ring bend (Water knot/bend, Tape knot, Overhand bend) and corresponding eye knots.
This 'bend' consists of 2 integrated Overhand knots of the same chirality.

I am only showing the 4 principal ways of linking tail to S.Part based on S/S chirality.
Obviously, the 'Ring bend can also have 4 linkages based on Z/Z chirality.
Linkages can only be formed between segments of opposite colour (blue to white).

These 'linkages' all form corresponding 'eye knots' (aka loop knots).
Of these, only one (1) is a 'primary' corresponding eye knot (2 linkages are in fact identical).
The rest are all 'secondary' corresponding eye knots.

Note:
This is all a new area of rapidly evolving concepts.
The terms 'linkages', 'primary corresponding eye knots', 'secondary corresponding eye knots' are my own.
Harry Asher likely originated the correspondence between a bend and eye knots in his book "The Alternative Knot Book" (1989).
However, he did not expand on his work or offer detailed insight to the underlying theory.

Dan_Lehman

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Re: The relationship between 'bends' and eye knots
« Reply #10 on: November 28, 2023, 12:12:37 AM »
#1412 Ring bend (Water knot/bend, Tape knot, Overhand bend) and corresponding eye knots.
This 'bend' consists of 2 integrated Overhand knots of the same chirality.

I am only showing the 4 principal ways of linking tail to S.Part based on S/S chirality.
Obviously, the 'Ring bend can also have 4 linkages based on Z/Z chirality.

And in general consideration, mirror images are the SAME knot
--you don't get New Knot Awards for inventing such a knot, e.g.!
Quote
Of these, only one (1) is a 'primary' corresponding eye knot (2 linkages are in fact identical)
And so too are there 2 linkages (of the 8 logically coming
for a 2-Tangle) in fact identical for each of the possible
other loadings.

You'd do well to observer the Tangle ID scheme of labelling
each end of each entangled piece respectively "1" & "2",
"A" & "B" (and for a 3-Tangle, "X" & "Y" --which letters I
think will be better accepted/perceived-as-different-pair
than say using Roman or Greek numerals).
As shown above, if one is specifically looking to find EKs
out of a end joint's 2-Tangle, you run through each of the
four ends (1,2,A,B) playing EK S.Part and see what comes.

----------S.Part-loaded   [here, parens added of ends joined in loading "v" other(s)]

1-v-(2+A)  ---1st
      (  +B)  ---2nd
A-v-(B+1)  ---3rd
      (  +2) ---4th

---------TAILoaded

2-v-(1+A) ---5th
      (  +B) ---6th
B-v-(A+1) ---7th
      (  +2) ---8th

In your graphic, you wrongly assert that in addition to
the two-which-are-identical primary EKs (resp. 1 & A qua S.Part)
there are "7 secondary eye knots" --wrongly merging
the two equal-result loadings (different *material*/color
if you tie it, but same result/knot) as "1" and subtract
from logical 8 ... which misses the point that your 1
=2 loadings and so too will those other loadings have
"identical" (but for colors) knots!  Half of 8 = 4 -1 =3.

You've missed the case of loading the EXTERIOR twin
--Tails-loading, here.  (E.g., your image A but with
white Tail (upper end) qua S.Part.)

Quote
Note:
...  The terms 'linkages', 'primary corresponding eye knots',
 'secondary corresponding eye knots' are my own.
And I can see reasons to abandon them, in time ::
they have *quality*/*value* connotations, and it
will probably be better to be more objective --e.g.,
as I have come by abandoning "strong/weak form"
(given w/o good basis, even) of twin-parts knots
such as this & Fig.8 for "exterior/interior", resp.
of the loading on twinned parts.

The Butterfly's established/defined EK loading is not
that given for e2e joints per our thinking here; it is
either 1-v-(2+B) or A-v-(B+2) --the case with both
joint S.Parts loaded (one the Outgoing Eye Leg) is
abnormal tying (cutting away the B.'s OELeg will
yield a novel joint --or two, as it's an asymmetric
tangle!).


--dl*
====

agent_smith

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Re: The relationship between 'bends' and eye knots
« Reply #11 on: November 28, 2023, 02:51:07 AM »
Dan,

You're making some assertions here with : you typed, "wrongly assert"...
I think we need to set the facts straight and sort out definitions.
I've attached a graphical representation of an 'end-to-end joining knot ('bend').

My first point:
With respect to any bend, there can only be 4 principal 'linkages' between tails and S.Parts segments (refer to the graphical representation of the a bend below).
Syntax tails...S.Parts... protruding segments... whatever!
Note the use of the term 'linkages'.
Note also the use of the term 'principal'.
A linkage can only occur between segments of opposite colour.


Now you either agree with that premise - or you don't.
If you disagree, I'd like to see your evidence to the contrary.

My second point:
Each of these 4 principal linkages will form corresponding eye knots.
That is, each eye knot corresponds to its 'parent' bend.
Each of these corresponding eye knots has 2 possible choice of S.Parts - which makes a total of 8 possible combinations (if we think in terms of loading profile).

My third point:
Of the corresponding eye knots, some will be 'primary' and some will be 'secondary'.
Definition of 'primary corresponding eye knot':
The primary corresponding eye knots are closest to the parent bend in terms of loading profile.
The S.Part of a primary corresponding eye knot is congruent with the S.Part of the parent bend.
One leg of the eye is congruent with the opposite S.Part of the parent bend.

...

With respect to #1412 Ring bend and its corresponding eye knots...
There are 2 linkages which create 'primary' corresponding eye knots.
It turns out that both of these eye knots are in fact identical.
As such, I count that as one (1) - not 2.

Now, if you swap/exchange the S.Parts of these particular primary corresponding eye knots, they are no longer 'primary'.
(refer to my definition of 'primary').

And once again - in case its not clear...
I am forging a new trail here - where none exists.
I have had to evolve my own definitions and underlying theoretical concepts.
You may disagree with these concepts - and that is how science is done.
Someone publishes a paper and others will either refute it or find that their test/analysis results support it.

If you have an opposing theory - I'd like to try to understand it and your supporting definitions.

EDIT NOTE:
With respect to your original hypothesis of 'strong' versus 'weak' forms of #1047 F8:
Strength is irrelevant.
There is no load that a single climber/abseiler/canyoner/caver can generate that will reach the MBS yield point of a knot.
So you are 'correct' to abandon this origiunal theory.

In reality, what needs to be brought to the forefront of thinking is jam resistance, and stability.
A sloppily tied #1047 F8 will rapidly deform (refer definition of 'stability').
A well dressed F8 tends to maintain its geometry and does not deform to any significant degree.
The jamming threshold will be higher in one particular dressing state than the other (which corresponds to your original 'strong/weak' assertion).
Not much testing has been done - there needs to be some peer reviewed studies/tests (with a 'control' group) to settle the mater.
« Last Edit: November 29, 2023, 04:01:10 AM by agent_smith »

Dan_Lehman

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Re: The relationship between 'bends' and eye knots
« Reply #12 on: November 29, 2023, 01:25:33 AM »
You're making some assertions here with : you typed, "wrongly assert"...
And one is right on target.

Quote
I think we need to set the facts straight and sort out definitions.
I've attached a graphical representation of an 'end-to-end joining knot ('bend').

My first point:
With respect to any bend, there can only be 4 principal 'linkages'
between a tail and an S.Part segment
(refer to the graphical representation of the a bend below).
Wrong per your image :: you have a linkage between Tails;
between "S.Parts & Tail" we might reject the S.P-2-S.P linkage, too
--4-2 = 2.

Quote
Note also the use of the term 'principal'.
A linkage can only occur between segments of opposite colour.[/i]
Oh, indeed I noticed "principle" appearing on the stage!

Quote
If you disagree, I'd like to see your evidence to the contrary.
--just cited.

Quote
My second point:
Each of these 4 principal linkages will form corresponding eye knots.
That is, each eye knot corresponds to its 'parent' bend.
Each of these corresponding eye knots has 2 possible choice of S.Parts
--which makes a total of 8 possible combinations (if we think in terms of loading profile).
--agreed.

Quote
Definition of 'primary corresponding eye knot':
The primary corresponding eye knots are closest to the parent bend in terms of loading profile.
The S.Part of a primary corresponding eye knot is congruent with the S.Part of the parent bend.
One leg of the eye is congruent with the opposite S.Part of the parent bend.
This definition should cut to the chase,
and just give the loading profile --as written
it seems to suggest some decision process.
Well, no, you can point out --what you've NOT here
written, nb-- that you want to preserve joint geomentry.
Witness the exclusion by you of one sheet bend EK
with U-fold's S.Part qua EK S.Part : unstable.


...

Quote
With respect to #1412 Ring bend and its corresponding eye knots...
There are 2 linkages which create 'primary' corresponding eye knots.
It turns out that both of these eye knots are in fact identical.
As such, I count that as one (1) - not 2.
Which you then mis-use in subtracting 1 from 8 to assert
"7 secondary EKs" when in fact it will be that similar
"identical" EKs arise in them to reduce to a trio --half & half.


--dl*
====

agent_smith

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Re: The relationship between 'bends' and eye knots
« Reply #13 on: November 29, 2023, 04:39:15 AM »
Dan,
You seem to focus on syntax/grammar errors - rather than the underlying concepts.
This appears to prevent you from seeing the trees because the forest obscures your view :)
It appears that you take the extreme literal interpretation of each and every typed word.
The English language is complex - I think you should not jump to a (wrong) conclusion because of a syntax/grammar construction that you perceive to be erroneous.
This makes it very tedious to communicate with you - because I would need to closely examine every possible literal interpretation of typed words - and account for every possible outlier.
In a face-to-face conversion, it would have a retarding effect - because you would need to carefully think about every word you wanted to speak out loud - before uttering those words.
Example: You ask a question... there would be an awkward silence for 15 minutes while I calculate in my mind every conceivable way you could interpret my words, and then I reply.
Then you ask another question, and again you wait in silence for 15 minutes while I calculate another reply... and so on!

For example: (read very carefully)
Below is an example of typing words in the English language to convey a concept.
Q. Can you find different ways to interpret A, B, C, D - and arrive at completely different conclusions?

A. There are 4 segments projecting from any bend.
B. There are 4 projections exiting from any bend.
C. All bends have 2 S.Parts and 2 tail ends.
D. There are 4 segments protruding from a bend - if you 'cookie cutter' a bend - you will see the 4 segments.

Another example:

A. There are 4 possible linkages between all segments that protrude from a bend.
B. There are 4 possible linkages that can be made from segments of opposite colour.
C. There are 4 possible linkages that can be made between all of the tails and S.Parts that protrude from a bend.
D. There are 4 possible linkages that can be made between any tail end, and any S.Part.
E. There are 4 possible linkages that can be made between any projection exiting from the core of a bend.
F. There are 4 possible linkages that can be made between any segment of opposite colour that exits from a bend.
and so on...

In my view, there are syntax/grammar differences due to complexities of the English language.
I can see that you appear to adopt an extreme literal interpretation of each and every typed word and sentence.

In terms of A - F above, I can see the underlying concept that is present in each statement.
I imagine that you see conflict in the meaning of each sentence - which is fine - but can you glean any meaning/concepts?
Example:
per Dan Lehman:
Quote
you have a linkage between Tails
You have noticed a dotted line in my posted image linking a tail-to-a-tail.
There is also a linkage between an S.Part and an S.Part!
Dan - this is intentional.
I intended to link a tail segment to a tail segment!
I also intended to link an S.Part to an S.Part!
There is no error with this depiction :)
It was all intentional :)

With reference to the attached image below:
It is a graphical representation of a 'bend'.
It shows 4 possible linkages between all of the segments of opposite colour exiting from the core of a 'bend'.

Possible interpretation risks:
The word 'linkages'.
The word 'segments'.
The phrase 'exiting from the core'.
The phrase 'opposite colour'.

In one sentence, I have tried to capture the essential concepts - but I appreciate due to the complexities of the English language - different interpretations can be arrived at.
I could add an additional sentence such as; 'Only segments of opposite colour can be linked' - to further tighten the definition.

« Last Edit: November 30, 2023, 12:06:37 AM by agent_smith »

Dan_Lehman

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Re: The relationship between 'bends' and eye knots
« Reply #14 on: November 30, 2023, 12:14:24 AM »

Quote
per Dan Lehman:
Quote
you have a linkage between Tails
You have noticed a dotted line in my posted image linking a tail-to-a-tail.
There is also a linkage between an S.Part and an S.Part!  [as I noted}
Dan - this is intentional.
I intended to link a tail segment to a tail segment!
I also intended to link an S.Part to an S.Part!
There is no error with this depiction :)
It was all intentional :)
The question wasn't to your intent, but to your
description of it --to wit (which you conveniently
omitted in quoting me above) ::
Quote
Quote
My first point:
With respect to any bend, there can only be 4 principal 'linkages'
between a tail and an S.Part segment
(refer to the graphical representation of the bend below).
Wrong per your image :: you have a linkage between Tails;
between "S.Parts & Tail" we might reject the S.P-2-S.P linkage, too
--4-2 = 2.
The words "between a tail and an S.Part segment" do NOT
include the cases of linkages between Tails & between S.Parts.
If you intend to describe the full set of linkages, you should
amend this conflicting description to (say) "There are 4 linkages
between the Tail & S.Part of one piece to those of the other (not
of a piece's own ends to each other)".  You might do well to
speak here of "ends", noting that ends include Tails & S.Parts.
"The are four possible linkages with an end of one piece and
that of the other."  (This not easy speaking --lotta pitfalls!)

Your C & D are also wrong (D for the same reason as above,
and also denoting like-color'd linkages, as does C).
Quote
C. There are 4 possible linkages that can be made between all of the tails and S.Parts that protrude from a bend.
D. There are 4 possible linkages that can be made between any tail end, and any S.Part.

Technical writing such as giving descriptions requires adherence
to rules of the language, and imposing clarity where the language
might be ambiguous.  And this clarity can be tricky to achieve
(esp. if one acts as one's own editor).
It bears repeating :: "This not easy speaking --lotta pitfalls!"


--dl*
====

 

anything