F8 + Overhand bend

Knotting Concepts & Explorations
1

The image below showcases five (5) different methods of integrating
an F8 knot with a simple Overhand knot.
The shown methods are not exhaustive (there are other methods).

I have confined myself to geometries where an opposing S.Part
enters the knot core on an initial parallel pathway.
Reason: This allows for exploring various offset joints (F8+OH).

Note that the ‘Lehman8’ bend is image ‘C’.

Given the 5 different integrations I have shown, I am pondering if
Mr Lehman also went down this path - and settled on “C” as being the
optimal geometric integration for producing his infamous ‘Lehman8 eye knot’?
Or, did he stumble upon “C” at first instance and tied his Lehman8 eye knot
without considering the other options?

Note that for each illustrated end-to-end joint (ie ‘Bend’), there are 4 available
linkages - from which corresponding ‘eye knots’ can be derived.

Dan Lehman will be pleased that I avoided stating 4 corresponding ‘eye knots’,
thus avoiding responsibility for declaring how one chooses a particular ‘eye knot’ orientation.
And I still hold the view that most choices will be logical - being stable - because they
more closely align with the parent bend.

I’ll be showing all of the corresponding ‘eye knots’ in future posts.

F8-OH bends_WEB.jpg

2

Four, actually --as the rightmost two are mirror images,
with this aspect slightly disguised by rotating one on a
horizontal hinge (top<=>bottom rotation, i.e.). This
is an interesting tangle, and I’m sure that you’d pass
muster for having correctly tied a gym-required “Fig.8” EK
were you to do so using this form, esp. with the end having
the twin parts entering & arc-ing together before they go
OH- & F8-specific ways!

I am pondering if Mr Lehman also went down this path --and settled on "C" as being the [i]optimal [/i]geometric integration for producing his infamous 'Lehman8 eye knot'? Or, did he stumble upon "C" at first instance and tied his Lehman8 eye knot without considering the other options?
He (=me) was working from nothing with the design goal to make an EK with F8 strength & BWL ease of loosening-to-untie. All I could figure re the former was that the S.Part must gradually give off force via being twinned and bearing into its twin before making a rather hard (1dia) turn around the eye leg(s) --on loading, the initially contacted eye leg is drawn away from resistance and the hard turn comes around the other leg. ((I now think that F8 strength must come more from the two Eye Legs' wrap & bearing upon the S.Part as S.Part comes into the knot, and less about what happens soon thereafter.))

So, I inserted the RELeg and brought it back towards
the eye laying in the believed-2-B-helpful born-into part
and then … looked to finishing the knot to leave it
non-jamming. (With some looking at the final tuck
going out this or the other side of things --more or less
disturbance (“more” being the form shown here).

(Only later did I come to realize that this knot could be seen
as a securing of what I call the “Quick 8” --just the first two
tucks of the RELeg, no bringing back to cushion S.Part’s turn.
(And to secure the Q8 one might tuck the Tail back between
eye legs --a knot that held in urethane-coated Dyneema to
break at a non-inspiring 38%? or so?)

–dl*

3

per Dan Lehman:
In relation to images “D” and “E”:

Four, actually --as the rightmost two are mirror images,
No, I count 5 (not 4). The right most images ("D" and "E") are not mirrors of each other. If you look closely at the F8 knot, its [i]chirality [/i](handedness) has not changed. The F8 is left-handed [i]chirality [/i]in all images.

Therefore, in my opinion, each ‘bend’ has a distinct geometric integration.
I don’t see any ‘tangles’ either.
A tangle isn’t a repeatable geometric outcome - it is something random.
All of these geometric integrations are deliberately and intentionally tied.

…

He (=me) was working from nothing with the design goal to make an EK with F8 strength & BWL ease of loosening-to-untie. All I could figure re the former was that the S.Part must gradually give off force via being twinned and bearing into its twin before making a rather hard (1dia) turn around the eye leg(s) --on loading, the initially contacted eye leg is drawn away from resistance and the hard turn comes around the other leg.

And this is one of my underlying reasons for tying this specific set of geometric integrations.
I wanted a side-by-side comparison - using an F8 knot with the same chirality throughout.
It enables me to visualise potential corresponding ‘eye knots’.
I pondered: If Dan had gone down this path, would he have settled on the same 'Lehman8 ‘eye knot’?

((I now think that F8 strength must come more from the two Eye Legs' wrap & bearing upon the S.Part as S.Part comes into the knot, and less about what happens soon thereafter.))
And I would point out that MBS yield (strength) is largely irrelevant. [u]What matters is:[/u] 1. Stability 2. Security 3. Resistance to jamming (which I think was one of your original design goals) 4. Amount of rope consumed to tie the knot 5. Relative tying complexity (eg Scotts locked Bowline scores well here - you get a lot of 'bang for buck' for something so simple).
4

Well, an F8 is amphichiral, so … . All I’ll say now is
“I look forwards to your two sets of derived EKs for resp. D/E.”

I don't see any 'tangles' either. A tangle isn't a repeatable geometric outcome - it is something random. All of these geometric integrations are deliberately and intentionally tied.
Spare us this diversion --we've suffered enuff already. "Tangle" per its use in knots discussions has specific, technical senses --at least as by the late (COVID, alas) John H. Conway & differently by me. "Entanglement" & "intersplicing/-weaving" and other terms could do but are longer than the simple two syllables. In short, it means the knotted-together, "nub/core" part of what is commonly called a "knot", bereft of any particular loading. (And, alas, this can lead to some awkwardness & challenges --in trying to have a loading-irrelevant state but yet ... enable a variety of loadings for that state.)

And so you show Tangles insofar as we ignore their knot-defining
Loading Profiles. You’ll be looking to put on those profiles for
defining EKs, essentially, by giving linkages. (For me, I’m happy
to ignore physical “linkage” and define per loading :: e.g., I’ll
not care that “eye legs” are joined, simply that the two
“ends” are loaded jointly in opposition to the S.Part.)

Here is what Patil et al. wrote regarding “tangle” ::

We were interested in tying two lines together so that they form a stable longer rope, a task known as ?tying a bend? among sailors (2). Mathematically, this configuration describes [b]an oriented 2-[i]tangle[/i][/b], ... .
And they certainly have repeatability & order in mind. (And, yeah, "tying a bend" isn't something one can find in knots books, i.p. their cited ABoK, tsk tsk!)
I pondered: If Dan had gone [u]down this path[/u], would he have settled on the same 'Lehman8 'eye knot'?
Well, again, I started pretty much from nothing --and not from a given knot and exploring derivatives, variations of loading. This F8 + OH is fascinating, with prospects.
And I would point out that MBS yield (strength) is largely irrelevant. [u]What matters is:[/u] 1. Stability 2. Security 3. Resistance to jamming (which I think was one of your original design goals) 4. Amount of rope consumed to tie the knot 5. Relative tying complexity (eg Scotts locked Bowline scores well here - you get a lot of 'bang for buck' for something so simple).
You might also point out that "strength" isn't all so well defined & tested --e.g., knock-about weakening of a knobby knot in usage; cyclic or dynamic loading (repeating).

–dl*

5

Ah, point to A_S :: my rotation yes gives the mirroring
of the two, BUT the loading differs : D loads the blue OH
where it turns with a twin white part into the knot;
E loads the OH where it & white part immediately
go separate ways.

–dl*