A bend is a single or monoaxial structure - the force line has a single axis.
And one could also state that a 'bend' is an end-to-end joining knot where both SParts are in axial alignment and 180 degrees opposed.
If an Alpine Butterfly loop is tied such as to excuse a portion of the cord, then it is effectively a bend and therefore also a monoaxial structure.
Although in reality, the eye of a #1053 Butterfly still remains (it exists) - when the Butterfly is subjected to a BTL loading profile (bi-axially through loaded) - the eye is isolated from load.
Irrespective of some closed "eye",
unloaded it is, and, well, it's not a loading factor;
so how are you making this, nevertheless,
BIaxial --when, in "through loading", there
is but this "through" axis loaded?
In order to re-classify as a 'bend' - there must (by definition) be a unification of 2 ends.
Since #1053 Butterfly does not have a unification of 2 ends, it remains an eye knot (not a 'bend').
Not by my definitions, beginning with some "tangle".
There, and eye knot might lack an eye --because I'm
looking only at the "loading profile" that puts (canonical)
End-1 vs. End-2 + End-A (which I don't look out beyond
their *endness* to assess connection. --a view that in
practical conditions runs into trouble vis-a-vis actual
loading (something that can obtain even WITH and eye,
but typically to a minor degree, brought about w/friction
such that eye-leg loading is imbalanced).
--which is to say that the above is a formal way of seeing
things, to look just at a "nub" and its "ends" only insofar
as they depart the nub (and might be loaded).
(not sure to what benefit this formality reaches, yet!).
As for "bend", since that word has rich enough historical
nautical meanings of making fast and e.g of "bending sails...",
I do not any longer follow Ashley's push to a new definition,
and will use "ends joint" and maybe just "joint" --to rid another
possible "E" initial in our terms (End / Eye / Either ...)!
However, there is a correspondence between a 'bend' and an eye knot.
All bends have 4 corresponding eye knots (within a chiral orientation).
What four do you see for butterfly knot put in my
notation of End-1 vs. End-A w/eye legs End-2 & End-B?
The traditional one is End-1 vs. End-2 + A, B unloaded.
If asymmetric, really we need to add End-A vs. End-B + 1,
2 unloaded. (If symmetric, one gets a duplicate here.)
Now, the butterfly already departs from this trad.
notion, by having (in reference to joint 1-v-A),
End-1 vs. Ends 2 + B (poor End-A going from full
loading to zero!). So, you then replicate this for
End-A (and this is an asymmetric knot), and ... four?
And not taking the joint's tails (2 & B) into having a
turn at eye-knotting? (which would beget a 2nd foursome).
This could/should be put out for the full exploration of
the *tangle*, perhaps; or at least the possibility noted
in some way, as some relation --it exists, after all
(not necessarily in a practical way, though).
I suppose one can see your four-counting making
better sense to be derived from a "
bend joint"
--as you keep one or other of the initial SParts--;
mine (getting 2 x four) for a *tangle*, which is
available to all loading profiles.
As for eye-to-joint correspondence, there are other
possible relations, as I've shown :: where one ties
one joint SPart to a SPart that is the pair of legs of
a bight (which becomes the eye), and figures out
which bight end to fuse with the SPart's piece's
other end (the End-2).
It astonishes me that he didn't focus on ... ,
it might have led him to the discovery of ...
Guilty! But "it might have led ..." can keep one from
eating dinner, getting to bed, going to the knot one
PLANNED to illustrate --so as to liberate a tied-up play rope--
but instead of that small forward step of documentation one
now has taken two steps backwards w/further *new* knots.
(Recently, on seeing a likeness in one knot-form/tangle to
another, which I'd found amenable to "fore<->aft'ing", an
operation on eye knots where one fuses end + tail and cuts
the eye to be their replacements,
I moaned "oh, no, not all that to do for this one, too!"
((
Hitchhiker's Guide to the Galaxy:
"But Marvin, there's a whole new life opening up in front of you," said Zaphod.
"Oh no, not another one!," came the reply.
))
It is possible to subject #1053 Butterfly to a tri-axial loading profile.
Some math wizards who addressed this term came to
the conclusion that it is, rather
BI-axial; I'm happier
with some note of *three*, and maybe thus
"3-way".
Be that as it may, the abused term carries the intended
sense to the not-too-educated masses!
--dl*
====