Bends
Of course I have examined carefully all the knots presented by yChang seconds after he presented them at the Forum ! However, I didn’t have his e-mail address, to congratulate him, and tell him my opinion on them, in particular, and on the issue of bends, in general ( on which I do NOT consider myself a master, as Roger E. Miles is - I only have tied some good tight hitches and one secure TIB bowline… )
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First, we have to tie every possible simple bend there is.
( I use the word ?is?/ =?exists?, but I could had used the world ?can be? / ?can be tied? as well ? simple knots are like basic mathematical numbers, shapes or theorems. We can say that they ?exist,? in a quasi-Platonic means, even if they have not been discovered yet by us, humans. We cannot say they are free, pure creations of the human mind, like art ? sooner or later, every sufficiently intelligent being, wherever and whenever in the Universe, would discover them ).
?Practical? bends, for many reasons, can not be but simple bends ? but ?simple? is not as simple a thing as it sounds! J Moreover, in knots ?simplicity?, even if it can be defined unambiguously, it can be estimated by different measures ? as I had tried to explain, in vein, in :
https://igkt.net/sm/index.php?topic=3740.0
In this thread I claim that multiple re-tuckings, and multiple turns, and multiply ( helically ) coiled segments, reduce the simplicity of a knot, because they increase the complexity of the path the rope follows in 3-D - regardless how easily can the knot be set-up and dressed during tying, and how conceptually simple and easy to remember is its shape and/or tying method.
Therefore, we have to place some limits on the complexity of the ?practical? bends we use. Otherwise, the ?practical? bend, which is a tool a piece of engineering, becomes too complex, un-practical and over-engineered.
https://en.wikipedia.org/wiki/Overengineeringractical
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Second, we have to test all those knots, and see which are secure and un-jammable.
( Ideally, ?maximally? secure, and ?maximally? un-jammable ). By ?maximally?, I mean under as heavy load as the knots themselves can withstand ? they should not slip, and they should not jam, even on the brink of their rupture.). And we have to test them on the ?common? rope materials AND on the very slippery materials like UHMWPE ? two distinct lines of tests ).
Needless to say, regarding the second task, nothing ( = 0 ) has been done? J
Regarding the first task.
Rioger E. Miles had thought that, by starting from the planar representations of bends, he would discover all the interesting knots that can serve as practical bends. However, he did nt consider two things :
First, that, oftentimes, the loaded knot acquires a very different geometrical shape than the initially set-up and dressed knot. Second, that identical planar representations can lead to different, geometrically, knots ( bistable knots ), and it is the geometry of the loaded knot, and not the topology of its initial tying diagram, which determines how the knot behaves regarding security and jamming. As, sometimes, knots with different, topologically, initial tying diagrams behave almost in the same way ( and we even call them by the same name, for example, the ?left? and the ?right-hand? common bowlines ), sometimes knots with the same, topologically, initial tying diagrams behave differently. Topology reigns only in mathematical knots ? in practical knots, it almost does nt matter.
It is no surprise that Miles did nt generate a complete catalogue if all the simple bends, and he just published tying diagrams and pictures of the dressed but yet unloaded of a few of them.
It seems to me that this over-estimation of the capability of the initial tying diagrams to generate all the simple knots is a common mistake ? possibly because we are accustomed to remember knots and ( one of their many ) tying methods together, we store them in the same corner of our brain, and we tend to confuse them? Noope, unfortunately, by starting from simple initial tying diagrams, and simple initial pre-bends ( the Reef, the Butterfly, etc. ), we can NOT discover all the simple bends there can be, and we can not produce a complete, exhaustive cataloguing and enumeration of the simple bends? And THAT is the task, all the knots that can be, not only some ?new? knots ( which may well have been tied many times in the past, but, just because they have not been members of a well-studied complete set of knots, and because they have not been tested, they have been forgotten?
I have tried to ?bypass? the complete enumerating and the testing phases by ?guessing? bends that would be easy to untie ? like the ?Easy to untie? bend, for example. How wrong I was? By actually testing those bends, under heavy or even moderate loading, all of my guesses have been proved so wrong. Many ?guessed? forms, during tensioning, change completely, even lose their initial symmetry, capsize, slip, become a jammed lump of material, you name it?
It is true that I have tied, and tried, most of the bends presented by yChang ( but not all ). And it is also true that, starting from simple, well known and easy to form configurations, and re-tucking them, one can discover ?new? knots. However, many ?re-tuckings? are tricky, and that is why they have not been explored systematically, and they have escaped the attention of knot tyers. The Ampersand bowline is just such a simple, single re-tucking of the common bowline, but it has not been found for many years. Also, I came to believe that this way of generating ?new? knots is like wandering in our desert of ignorance, hoping to un-cover something, but it is not the road we have to follow from now on. We are well beyond that point. We need our Mendeleev, a systematic, ?complete periodic? table of bends, based on some new concept, property, generating method, whatever, something that would produce all the simple bends there are. And then, we have to test them !
Under ?Xarax?, as well as under ? X1?, I have published some ?new? bends in the Forum ? with pictures, of course. yChang may search there for something identical, or similar to the knots he presents. He SHOULD also read Miles ! Also, he may find something interesting at :
https://igkt.net/sm/index.php?topic=3086.0
As a present, for his beautiful work, tell him to read one attempt, similar to his, to discover a class of bends - in which he will find some of the knots he published.
https://igkt.net/sm/index.php?topic=4445.0
In particular, I like the & bend shown at
https://igkt.net/sm/index.php?topic=4445.msg29585#msg29585
But that ( 'like" ) means nothing ! This bend jams even by looking at it !
