per Dennis Pence:
I have one minor question concerning how you selected the ?4 principal eye knots? in your diagram in Reply 87 of the Lehman8 Bend.
Dennis, I didn't declare any of the derived 'eye knots' to be
principal!
I ceased using the terms 'principal' and 'primary' some time ago - finding a better way to conceptualise relationships to the
parent bend via
homogeneity.
Key concept here:1. There is a parent 'bend'
2. The derived 'eye knots' are geometrically related to their parent bend.
3. There are four available
linkages between the S.Parts and Tail ends.
4. How these 'linkages' are exploited can be contentious, when only 4 choices are initially made (as I had done).
I had argued this salient point with Dan - stating that some choices are more logical than others.
Indeed, some
transpositions result in instability - where the derived 'eye knot' is not viable.
The ordinary dictionary meaning of
homogenous more closely aligns with my intent.
And so with respect to the
Lehman8 bend:
1) There are two
homogenous corresponding 'eye knots'
2) The original 'Lehman8' 'eye knot' was discovered by Dan Lehman many years ago - but his discovery was not via the systematic method I am employing here. His principal design goals were; jam resistance, and security. The Lehman8 is not 'TWATE' (Tiable Without Access To an End) - so Dan started with a simple F8 knot and then figured out a way of integrating a simple Overhand knot with the F8 - in order to achieve his design goals.
3) The derived 'eye knots' (A, B, and C from the Lehman8 bend) are
logical choices - and "D" being merely one choice out of 2 possibilities.
I had defined the concept of homogenous (and homogeneity) in a previous post.
Suffice to quickly summarise: A
homogenous derived 'eye knot' more closely aligns to its parent bend in terms of load segments.
The S.Part is congruent to an S.Part of the parent bend.
And one leg of the 'eye' is congruent with the opposite S.Part of the parent bend.
When I have done this in the past, I have picked one ?color? and made it the standing part in all of the first set (or principal) eye knots. For example, you chose the white rope for the standing part in A, C, and D, but not in B.
Again - I never used the term
principal - preferring instead the term 'homogenous'.
With respect to derived 'eye knot' "B" - the choice of the
blue rope results in a closer alignment with respect to the parent bend.
The
blue rope corresponds to the S.Part of the parent bend - and so, this choice results in homogeneity.
If you look closely at the parent bend (ie the Lehman8 bend), the lower
blue rope is an S.Part.
And the upper
white rope is the other opposite S.Part.
And so the 'choice' I made is logical if one is seeking
homogeneity with respect to the parent bend.
I included a side text box in my image - which explains that all of the derived 'eye knots' can undergo a transposition to reverse their polarity.
I assume that you noticed and read the content of that text box - yes?
I would also point out that preparing all of these knots for photography is a long and tedious (and thankless) exercise.
This is likely why Dan Lehman contributes exactly zero photos - because it is time consuming and fiddly, and a largely thankless exercise.
I will eventually show all of the transpositions - but it takes more of my personal time.
As my time is limited - I can only show what I believed was more 'obvious' and logical.
From my point of view, the derived 'eye knot' "D" was a toss up - I could have made the blue end the S.Part (but I chose to make the white end the S.Part).
Again - I did point out in the side text box that each of the shown derived 'eye knots' can undergo a transposition.
Of course, when you take transpositions, you will get all of the rest. I just wondered how you had selected the knots in you ?principal? list.
Again: I never used the term principal!
I was careful in my choice of language - because Dan Lehman had already gone to great lengths to assert his opinion with respect to how 'choices' of orientation are made.
Again - all derived 'eye knots' can undergo a transposition to reverse their polarity. I didn't have time to photograph all 8 derivations!
So I limited myself to what I believed were the most logical choices.
And here I use the term 'logical' in the sense that the choices were based on achieving
homogeneity with respect to the parent bend.
And the
parent bend is the key concept - the derived 'eye knots' relate to their
parent bend.
When deriving corresponding 'eye knots' - one needs a
reference frame (a source).
The source
reference frame is a particular 'parent bend'.
In this instance, the source reference frame is the
Lehman8 bend.
Using the Lehman8 bend as the 'source' - I then worked to derive the
corresponding 'eye knots'.
The
correspondence can only have meaning relative to something - in this case - the Lehman8 bend.
Does that all make sense to you Dennis?
...
Now, on to Mr Dan Lehman:
But this begs the question of Why are THESE knots "principal"?!
--and others come by some "transposition" process!?
I never used the term principal - Dennis Pence used that term (not me)!!
REQUEST: Can we please cease and desist in using the term principal or primary?
A few months ago, I was in search of the right language to describe the unfolding theory.
The term homogeneity had not yet been used.
I originally used terms such as primary and principal as placeholders to explain some concepts.
I have since dropped those terms in favour of
homogeneity.
As opposed to just generating the entire 8 (of a 2-tangle)
and calling them on their interelations as desired.
Firstly, for me personally, it is a huge exercise in time and effort to prepare and photograph all 8 derivations.
And it is a
thankless exercise.
Secondly, I do believe that the choices of orientation of corresponding 'eye knots' corresponding to the 4 linkages
can be made on a 'logical' basis.
The original 'Lehman8' eye knot is a logical choice in my view.
It is homogenous with respect to the parent bend - and it is secure and stable in loading.
Yes, the 'Lehman8' eye knot can undergo a
transposition - which reverses its polarity.
I will showcase a transposed Lehman8 when I find time.
But, I initially chose not to show it - because for me - it is not a logical choice.
And here I am sure we will get into arguments about 'logic' and the choices to be made.
One can advance an argument that the original Lehman8 is not a logical choice - and
that its transposed version is preferred. When attempting to present derived 'eye knots'
to the general public and world - I find it easier to just show 4 initial corresponding 'eye knots' that logically align to the parent bend, and also with respect to the available linkages. It can get confusing when you attempt
to show all 8 derivations...
