Two variations of the Zeppeln bend

[Dan_Lehman]

Contrary to what is said in #22, the curvature will not necessarily become less sharp when there is more fill in the Zeppelin, but curvature may increase, because the curvature that is relevant to strength is not necessarily the one within the knot, where the nub sits, but might be the one at the entrance of the knot, where the collar hugs the standing part.

Good point.  I think that a *doubled* collar will help, here,
in ameliorating the effects at this deflection point.

When a knot breaks, it seems always to be just outside but never inside its structure.

I disagree : I have found and have had tested-to-rupture some knots
in laid rope and insofar as I can trace the breaking point it appears to
be at a point at the initial deflection and on the concave/"inside" side
of the bend --contrary the outside-fibres-take-the-load theory.  And in
some cases where a break is seen outside, it comes in material that was
formerly inside the knot and which got abraded and degraded in being
pulled out.

This is a tricky thing to test, but note that in some slow-pull testing
it will be the case for laid rope that ONE (or two) strand will break
and the other two will remain intact (and there will be a temporary
drop in tension by this yield of material, and some devices I surmise
automatically stop (this is how my test specimens have come back,
incompletely broken, sometimes).  Which then enable one to try
to determine where the broken strand lay at its parting point.

--dl*
====

[[Inkanyezi] gone]

The pit is aft.

Your picture shows a winch almost amidships...

Oh, it is a small boat with a large cockpit, but also, it is artistic freedom. It is a sketch, showing a principle, not a picrure of the actual boat. The principle can be used for any boat. It was my frustration with the mooring fingers not having a convenient belaying point that made me think up a better solution that suited my boat, and I adapted the same principle as when I tie with a buoy, but reversed the attachments. At first I tried double springs attached to the center of the Y-beam, but then, when I reversed the springs, attaching them to somewhere more central at the boat and the ends of the beam, it became a lot simpler to handle. The eye that I use for those is a fig8 tied in the bight. First I didn't use any eye, but it was more convenient, because it puts the boat in exactly the same position each time in a very simple way. 

[knot4u]

Interestingly, the geometric symmetry of the "X" bend under discussion is neither

I continue to fail to see symmetry at all,
when --as has been stated now several times-- the tails
have different relations to the knot (and of course vice versa)
--one exits out through the collar of the opposite rope,
the other does NOT.  THAT, to me, shows Asymmetry (not "a symmetry")!?
Please explain the contrary position.

--dl*
====

I didn't even notice that at first.  A quintessential feature of the Zeppelin Bend is that it's symmetrical.  I don't think Zeppelin Bend X should have the word "Zeppelin" in it because Zeppelin Bend X is NOT symmetrical.

[knot4u]

Even if the Zeppelin Bend X looks symmetrical initially, the bottom line is that it is not symmetrical.  In a symmetrical bend, each rope takes a "mirrored" path through the knot.  In other words, there is at least one axis around which you can flip the knot and the knot will look the same on the opposite side.  Accordingly, the Zeppelin Bend and the Carrick Bend are symmetrical.  I think the Ashley bend is symmetrical too.  However, the Zeppelin Bend X definitely does NOT fall within the meaning of symmetrical.  Neither does the Butterfly Bend.

It's OK to be wrong.  Just don't be loud and wrong.

[knot4u]

[Dan_Lehman]

   Why DDK understood it better and you failed to do the same ?
   Try again...

&

I have already spent much of the limited energy resources of our planet, [ad nauseam]

I quite agree, and am hoping for DDK to answer the question, not you,
as you seem unwilling to consider the case.

Are two Overhand components of "X" geometrically identical --no, they
are (1) opposite-handed (which only *colors* possible symmetry)
and (2) differing in how their tails exit : one tail after crossing the SPart
arcs around without much variance from its position to cross it on the
opposite side (if first deemed "under", this one "over) as it completes
the Overhand ;
but the other tail cannot follow such a flight plan, as it must dip farther
away from its SPart to allow the first tail's clearance.

Now, DDK, have you some reasonable explanation for your position,
and how the issues I et al. have remarked can fit into "symmetry"?

Let's give XaraX a rest.

--dl*
====

[Dan_Lehman]

Oftentimes it is better to leave things to settle down by themselves with time, rather than with insistent arguments.

What was insistent was an assertion; what was missing was argument.

I have argued that the Zeppelin X is somehow symmetric, ... , while most other members of this forum have not seen much or any symmetry at all, so they argued that it is asymmetric.

We saw a departure from symmetry and pointed this out;
we judge a knot not isolated parts of it --"not with
themselves" thus points to the issue:  "with themselves" is
a partial picture, of parts, not of the knot.  We have pointed
to how the asymmetry manifests itself in practical effects on
the knot's SParts.

I have also pointed out that the two interlocked parts of the bend, if viewed separately, are  mirror symmetric, ...

No, you have asserted this, but frankly it is clearly NOT the case.
For the R.Z. bend, the mirror would be put *alongside* the Overhand
component, so though mirrored SParts might both be turning down
--say, of horizontal flow *into* the knot--,
they would then show their opposite-handedness by moving say both
away from the mirror plane, and so on;
but when it comes time to flow into the tail exit,
they necessarily diverge from mirror-matching as they respectively
must flow around the opposite tail in different ways.  And why you
don't see this (or, how you can explain this as yet keeping symmetry)
is beyond the rest of us.

And so I'm awaiting for DDK to answer this challenge to an assertion
that he, too, has made.  He certainly has many words explaining the
challenge to work with.

--dl*
====

[DDK]

Re: Geometric symmetry
I have used the phrase "geometric symmetry" in the technical sense which may be unfamiliar to those who have not worked with symmetry groups (an interesting and not too difficult mathematical discipline).  In the technical sense, every configuration has geometric symmetry, that is, it has symmetry operations (a "group" of them) which leave the structure invariant (the new structure will lay exactly on top of the original structure).  Sometimes the only symmetry operation member of the group is the trivial identity operation (i.e. you do nothing to the original structure)!  We have been using the term asymmetric for such structures while the phrase "lacks symmetry" is likely more appropriate.  By this we are saying it lacks symmetry elements other than the identity operation.   

Re: The "marriage" of two symmetries
Let's be clear, the "X" bend lacks symmetry.   Having said that, I should explain what I meant by the "marriage" of two symmetries.  I was not implying that we have combined two symmetry groups into one.  For example, a combination of a twofold rotation element and a mirror plane can produce the central inversion symmetry group found in the Zeppelin Bend.  This is a legitimate non-trivial symmetry group.

In the "X" bend, if one follows the standing parts into the knot, you find that they are related by an "APPROXIMATE" central inversion symmetry, e.g. Zeppelin-like, and at some point rather smoothly transitions into an "APPROXIMATE" axial inversion symmetry, e.g. Smith/Hunter's-like.  The transition has to occur within a particular plane in the bend, and thus, the structure is fascinating from that perspective.

Re: What is a "good" symmetry?
In my opinion, there is no such thing.  For example, both the Zeppelin Bend and the Thief Knot have exactly the same symmetry.  The turns, curves, crosses, etc. (mechanical elements) of the bend determine its performance.  Symmetry may at times allow us to understand how these mechanical elements work in harmony or at cross-purposes within a particular bend.  Symmetry does not allow us to compare one bend to another.  If one introduces new mechanical elements, which often happens with even minor changes to the structure or symmetry of a knot, obviously all bets are off.  It is my understanding that as a species we find symmetry "attractive".  I believe we should recognize this instinctive infatuation for what it is.

Re: The naming of "X"
Given my own in infatuation with symmetry, I would admit that the use of "Zeppelin" to describe the "X" bend rubs me against the grain.  Certainly, it does no justice to the interesting "marriage" aspect of the structure.  That is my issue.

DDK

[Dan_Lehman]

It should be obvious (and, moreover, expected) that "everyone" of
normal knotting interest does not come with special mathematical senses
& sensibilities, and that the common understanding of "symmetric" is
different from what I understand from DDK.  In common senses, a bend
is *symmetric* if one half can be replicated to form the other --either in
pure copy or by mirror-image copy-- and joined exactly so (regard the first
half formed of rigid material (steel, e.g.), and posit that the symmetric
complement is materialized into place to complete the knot.  THIS cannot
be done with bend "X"; the halves have different shapes.

When I use the term "geometry", I mean the physical shape of the
structure (or component).

--dl*
====

[DDK]

Xarax, I feel you are taking too lightly just how approximate is the symmetry of the individual components of the X bend.   This will assuredly lead to a differing behavior for the component and its, in some cases not so similar, "twin", even in a very localized sense.  The relation of the individual components to each other is important and is one of the reasons that one places a great deal of importance on the symmetry of the knot as a whole.  Those that say that the X bend is not symmetric are exactly right in my opinion.

To possibly explain my perspective on Symmetry to Dan Lehman, I might use this analogy (perhaps not a very good one, but the best I have for the moment).  If one discusses the Quantity of an item one has (let's say in inventory), he might say he has 1, 4 or maybe 0.

DDK

[SS369]

"I see it as one symmetry in space that is transforming into another in time, if with "time" we mean the intrinsic time of a point travelling along a line/curve/tangled rope in 3D space. "

I think we have entered the Twilight Zone now.
Perhaps this should go to the Knot Theory & Computing section.

I personally am having a small difficulty following where this thread has traveled and would like to see some real data about these ZB variations. How do they indeed compare to the "parent" ZB?

Perception, and that is what I think is being argued with and for here, will always be from the side of the speaker.

So how is the "X" version improved or nor improved if you introduce even more crossings in the center?

My own personal opinion is that the the ZB variation where the working ends are circled around and re-tucked back into the nippers is the most secure.
Even if making it a slipped ZB (tripled bundle inside).

Scott