Symmetric bends - definitions

[agent_smith]

Hello IGKT members,

Am gathering momentum to write a new paper about symmetric bends.
I have read Roger Miles book (1995) but I find that it is unclear in some aspects and the images are not the best. I feel that he missed a few opportunities to explore the subject further.

So here I am some 22 years later - hoping to add some clarity to the subject.

I find the Zeppelin bend and #1411 (F8 bend) have interesting 'mirror-like' geometries (although not true reflectional symmetry).

In contrast, #1415 (Double Fishermans bend) has differing front/back geometry (as does #1425A Riggers bend / hunters bend).
#1415 is shown on page 116 as photo D1 in Roger Miles book.

I am still studying Roger Miles definitions of 'symmetric' versus 'asymmetric' bends as well as his + and - aspects.
I find it interesting that he only shows one image of the Zeppelin (photo B8 at page 109) - and not its reverse side.
Same goes for #1411 which is shown on page 116 as photo Cc3.

It appears to me that Roger Miles has made certain assumptions, and explored the math behind it - but, I think there is more work to be done.
With better photographic technology available today - images of knots can be better presented.

Some geometric definitions that could be further investigated and how they apply to 'bends'...

[ ] reflectional symmetry
[ ] rotational symmetry
[ ] translational symmetry

EDIT NOTE: Some of the images attached with this post are unstable when load is applied. The F8 bend is in a flat parallel dressing state that will quickly deteriorate when load is applied. The same goes for the Carrick bend..it will 'transform' into a different geometry under load. These knots are shown purely for academic study/curiosity!

Mark

[agent_smith]

Some better close-up images with side-by-side comparisons...

Just trying to get the best possible image quality so details can be gleaned.

I know some IGKT members are expert in image analysis...

Mark G

[agent_smith]

Still gathering momentum on this project.

I am hoping that someone who is expert in image symmetry / geometry can assist.
Access to a copy of Roger E Miles 1995 book on Symmetric bends would also be helpful.

Had my first attempt at trying to represent the Zeppelin in a 2D line drawing... not easy to do.

Mark G

[agent_smith]

Just adding a few more knots...

#1402 (Reef knot) is an interesting base from which to create secure bends. Last played with tucked reef knots in 2011 - but thought it worthwhile to revisit.

Unfortunately, the tucked reef knot jams...pity because it has a nice symmetrical shape. I bounced on it with my own body weight (100kg) - and it took quite some effort to untie... In all other respects, it seems rather stable and secure.

Mark G

[Dan_Lehman]

Just adding a few more knots...

#1402 (Reef knot) is an interesting base from which to create secure bends.

You've labelled it "symmetric"; this can be, but also
it can be(come) asymmetric :: in the particular setting'
given ..., one tail sometimes is drawn out of the roughly
plane of other parts while the other remains co-planar;
with flat material (tape/webbing), formed particularly
as a sling-2-sling joint, one bight will be such that one
side of it is exterior always/all-'round, but the other
bight inverts interior=>exterior as it flows into the knot
--and this side seems to be what has been reported to
break, in sling testing!?

Unfortunately, the tucked reef knot jams...

There are various other extensions via tucking,
i.p. one wherein the tails serve as a sort of toggle
in the middle of the knot (which I think Xarax has
presented (images of)), and ... .
AND, given allowance for such security-enhancing
extension, one might begin with also symmetric
"thief" and others!

 

[agent_smith]

I actually had photos of #1207 (Thief knot) but hadn't got around to compressing them to fit within the 100KB file size restrictions imposed by this website.

So here are the Tucked Thief knot images...

Interesting in that it reminds me of #1425A Riggers bend (per Phil Smith) and the Zeppelin.

I haven't conducted load testing on this structure...so I cant comment on whether it is vulnerable to jamming as with the tucked Reef knot. I presume it is but, I wont know for sure until tested. Hmmmm.

Mark Gommers

[SS369]

Same side tucked reef worth considering?

SS

[Dan_Lehman]

Same side tucked reef worth considering?

SS

Thanks for this --one thing I was thinking of!

 

[agent_smith]

Same side tucked reef worth considering?

SS

Scott, this is quite an amazing creation...well done! It seems at first glance to be stable and secure. It has an unusual flattened/matt form that doesn't appear to capsize (in the way that a carrick does). No idea about vulnerability to jamming though...

I just now tied a little variation by reversing the direction of one of the tucked tails to create a zeppelin / cruciform geometry.

I'll photograph both these structures and post back here.

Thanks for your work

Mark G

[KC]

A fave Lehman Lesson that may apply;
A Double Fisherman's with a Thief in the middle.
.
i came to see this as Square/Reef not as good a man-in-the-middle;
because pulled on each other's sPart as Equal & Opposite;
so does not 'power' each 'Fisherman' properly.
But the way Thief pulls on it's mirror's tail (rather than sPart) allows more of a force pass to Fisherman to work properly(?theory).
.
It seems that the man-in-the-middle Thief as a force passer rather than inhibitor (like Turn passes more force than RT)  can function to:
>>more 'squarely' serve forces to each Fisherman(?)
>>with softer bend, more noticeable in stiffer line devices(?)
>>buffer impacts to each FisherMan
.
Effect not inherited by Grief Knot it seems;
as the tails can't float around and align to position of pulling out, like the free tails do.
>>the Fisherman's are easier to uni like this it seems, so seems again less force passed to them
>>to me Grief floats pretty close to Evil Imposter, that turns almost like gear wheels churning each other's/mirror  tail's out to freedom
effect more pronounced starting straight into Evil Imposter; and easier to see closeness to Grief, by pulling EI's tails and 'squaring' to Grief i think.

[agent_smith]

I am still gathering momentum to write a new paper about 'bends' - in particular, to closely examine symmetric versus asymmetric bends (per Roger E Miles).

Xarax has been in contact (which I appreciate) and sent me this really interesting link from the IGKT forum...

Link: https://igkt.net/sm/index.php?topic=2088.0
It starts off well enough then devolves into some interesting discourse about symmetry.
And finishes with some wise comments from Scott (very interesting comment about slipped zeppelin).

I particularly like the posts from DDK.
Is he still around? His informed commentary about symmetry is really well done - and I would like to bounce ideas with him

Mark G

[agent_smith]

I would like to connect with DDK if that is possible?

I have been informed via private message that DDK has left/retired from the IGKT forum...which saddens me deeply.

DDK - if you are by chance reading this post, please contact me - I would like to collaborate with you in a new paper dealing with Symmetric bends.

If you are still in contact with any of the moderators on this IGKT forum, maybe you could send a message through them?
Alternatively, I can be contacted via personal email (email address can be found on this site...: Link  http://www.paci.com.au/knots.php (email address on the first pdf file in table)

If any moderators are reading this...can you assist (or not)?

Mark G

[agent_smith]

I've extracted some comments from our friend DDK...which I think are important.

From 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.

Interestingly, the geometric symmetry of the "X" bend under discussion is neither the point inversion symmetry of the Zeppelin Bend nor the axial inversion symmetry of the Smith/Hunter's and Ashley Bend.  It could be considered a marriage of both symmetries, i.e. a marriage of the Smith/Hunter's with the Zeppelin.

DDK summary of comments - compiled for each knot:

[ ] Zeppelin bend : has point inversion symmetry

[ ] Zeppelin X bend: Marriage of point inversion symmetry + axial inversion symmetry
DDK further commentary: 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. Those that say that the 'Zeppelin X bend' is not symmetric are exactly right in my opinion.

[ ] #1425A Riggers Bend (aka Hunters bend): has axial inversion symmetry

and commentary from 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).

Mark G

[sgrandpre]

You will find that the double fisherman's knot is in fact symmetrical if you simply twist one of the halves 180 degrees about the standing part of the other half. 

It will be either point symmetrical or rotationally symmetrical, depending on whether you tie both halves with the same handedness. 

[agent_smith]

You will find that the double fisherman's knot is in fact symmetrical if you simply twist one of the halves 180 degrees about the standing part of the other half. 

But this disturbs the inherent stability of the knot... and, this dressing state is essentially fiddling with the structure to force it into something else (...If I have understood you correctly). It is not a solution in my view because it is forcing a sqaure peg into a round hole...

It will be either point symmetrical or rotationally symmetrical, depending on whether you tie both halves with the same handedness.

In the correct dressing state that is #1415 (Double Fishermans) - each half - ie, each double overhand knot strangles the SPart in an opposite direction relative to the other. For example, one side strangles an SPart in a clockwise direction while the opposite side strangles an SPart in an anti-clockwise direction.

If both sides strangle the SPart in the same direction - it wouldn't be #1415 anymore. It would be a 'derivative' of #1415 deliberately tied in such a way as to achieve a  particular symmetrical outcome. So again, this is simply forcing a change in the basic structure to achieve an outcome.