Author Topic: The 20 "&" bends ( the 4 basic, and the 8 Xs and 8 Bs variartions of them )  (Read 5661 times)

X1

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   What is the difference, if there is any, between a knot tyer and a knot user
   Trying to satisfy both sides of the equation  :) , I came uo with the folowin definition :   
   The knot tyer is happy when he can tie as many, different, unknown knots as possible - while the knot user is happy to tie the same, few, memorized knots over and over again - and, on top of that, he is more happy if everybody else does the same, so he advertizes the few, same, memorized knots over and over again. The knot tyer is a humble victim of the vast size of the KnotLand, while the knot user is just another proud fashion victim.
   In this thread I would attempt to offer something to both of them :   
   To the knot tyer, so he ties many knots he probably would have not tied before, and he tries to think of yet other variations of those knots, or even extensions of the very "class of knots" those knots belong to.   
   To the knot user, so he can continue what he has been doing all the time : ignore any "new" knot presented in this Forum, and hope that, at the end, ( self-) defence will make him keep parooting, and SURVIVE  :)  !   
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    Even by a quick look at those knots, one can easily see their common characteristics . I do not see any reason for any definition here !  :)
    Just to explain my labelling convention a little bit : o denotes "over" itself, where "it" is the segment of the link with the Standing end at the right side of the picture, the yellow segment - or "over" the corresponding segment of the other, the blue link, when it comes to the relative position of its tail . l denotes that the working end of the one link ( the yellow link ) makes a left turn, and r that it make a right turn ( the first, Standing part s turn / curve is not described ). X = crossed denotes crossed tails - so, Xo and Xu denotes that the crossed tail of the one link ( the yellow one) goes "over" or "under" the tail of the other link ( the blue one), when it crosses it.  B = beyond, is related to the fact that two parallel and adjacent tails ( not crossed ) can be arranged in two symmetric ways : so, the B denotes that the tail of the one (the yellow link s tail) is located beyond the tail of the other ( the blue link s tail), in relation to its ( the yellow link s) Standing end.
 
  That s all, folks !   :)  Happy knotting !
« Last Edit: June 23, 2013, 12:16:18 PM by X1 »

X1

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2

X1

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  The X = crossed tails variations. Xo and Xu denotes that the tail of the yellow link , when it first meets the tail of the blue link, it goes "over" or "under" it, respectably.
« Last Edit: June 23, 2013, 01:29:25 PM by X1 »

X1

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X1

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X1

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   B variations : B = beyond, meaning that the one link s tail ( the tail of the yellow link ) is located beyond the other link s tail, in relation to its Standing end - as we are describing the path of the yellow s link working end, we also mean "beyond the yellow s link Standing end".

X1

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  The oru Bu "& bend" - a beautiful knot.
  However, the (structural) beauty of all those bends, is that their two links are topologically equivalent to the unknot ( the bends Roger R. Miles denotes by "A" ). Therefore, they can serve as bases for post-eye-tiable eyeknots (loops) - like the Tweedledee loop, for example, which is based on the Tweedledee bend ( M. A 24 ). Also, it is convenient for somebody to know that, if he unties the one link, by pulling the one, only, end out of the knot s nub, the bend will be untied at once, leaving no "relic" knot still tied on the other end - a not-functioning any more knot, that has to be untied at a second stage ( an overhand knot, or a fig. 8 knot, for example, the knots that are the usual ones by the interlocking of which most bends work ) .
« Last Edit: June 23, 2013, 03:17:22 PM by X1 »

TMCD

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The oru Bu "& bend" is pretty awesome looking! I tied the very first one in the series but decided I liked the Zeppelin Bend a little better.:)

X1

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   I liked the Zeppelin Bend a little better. :)
   
   I like the Zeppelin bend a LOT more  :) , but it is still a two interlinked overhand knots based bend.
   However, the Zeppelin bend is almost unique in that, in a sense, it does not use those two overhand knots  ! - meaning that it does not use the most fundamental characteristic of them, their topology ! It uses only their bights, as parallel rings, not hooked to each other, and their tails, as pivots that penetrate those rings, and keep the knot in one piece, as a rope-made hinge.
   See another Zeppelin-like bend, where the tails are not direct continuations of the overhand knots, at the attached pictures (1).
   For loops based on the mechanism ( not only the superficial "looks" ) of the Zeppelin bend, see (2).
 
   1. http://igkt.net/sm/index.php?topic=3716.msg21527#msg21527
   2. http://igkt.net/sm/index.php?topic=4095.0
« Last Edit: June 23, 2013, 03:58:27 PM by X1 »

xarax

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   Two pictures of the most compact of all those bends - which is also very pretty - in its loose form, before the final dressing
   Try to retrace, visually, the rope of each one of the two links, without paying attention to the other. Then, try to understand why the two ropes remain linked - how they are interweaved the one within the other.
« Last Edit: August 18, 2013, 08:48:34 AM by xarax »
This is not a knot.

xarax

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"Supersymmetric" bends - and a conjecture about them
« Reply #10 on: October 26, 2014, 02:01:27 PM »
   The notion of "Symmetric bends" is already debatable ( does it mean only bends where the one link is point- or line- ( = mirror-) symmetric to each other ? ), so imagine what the notion of "Supersymmetric bends" would be !  :)
   
   In short, I characterize as "Supersymmetric bends" those symmetric bends where :
   
   1. The Tail Ends leave the knot s nub towards opposite directions.
   2. They are "face-symmetric".
 
  Note : Just a few words about what a "face" of a knot is / what we mean by this : Some simple bends ( in fact, most of them, but that is irrelevant for our discussion here ), even when they are tightened, ( i.e., not in any initial loose, but in their final, most compact form ) are more "flattened" in relation to one plane going through their axis { and more "rounded" in relation to another - we can easily understand this if we realize that the areas of their cross-sections in relation to all such planes should have a maximum and a minimum : the maximum corresponds to their "flat" section, and the minimum to their "round" section ). We call "face" the aspect they present when they are projected perpendicularly to the plane on which the area of their cross section is minimum. We can distinguish the two "faces", the "front face", which we choose to be the face which shows more details of the knot ( that is, more details of how the most important segments of the knot are related to each other at their mutual crossings ( "over" or "under" )), and the "rear" face. When the "front" and the "rear" faces are indistinguishable regarding their geometric form, the bend is "face-symmetric". ( Which should we name as the "front" and which as the "rear" face of the bowline, was something that proved to me how stupid can clever knot tyers become, when they are "talking" to each other ( = talking past each other ).

   Now, it has been some time I had noticed something about all he "Supersymmetric bends" I know, which I present here in the form of a conjecture ( = precise but unproven statement ):
   
   All Supersymmetric bends are either Zeppelin-knot like, or Fisherman-knot like.

   ( Note : In the later case, that of the Fisherman-knot like bends, the two interpenetrating links can also be linked / hooked to each other, at their middle )
   
   Trying to get an indication about the truth value of this conjecture, I have recently been examining again the pictures of the bends I keep in my files ( I keep only the simpler and more beautiful ones, but I am afraid that one has to actually tie a knot, in order to become able to decide if it is really  "overcomplicated, which will never be taken up by mainstream users because of its complexity, when far simpler variants work perfectly well." (sic)  :) :) :) - and, in general ( so, by definition ) knot tyers are those who do not tie other people s knots !  :))
   So, what about the beautiful, most compact & bend of the 20 shown in this thread ? At first sight, it does not seem to be Fisherman-knot like - but that is true only at first sight, and sight alone does not help knot tyers, as it was proven over and over again !  :) One has to TIE the knot, a number of times, to see if it is "overcomplicated'(sic), or anything else he would had wished it to be...
   It turns out that, before its final dressing, this & bend is a Fisherman-knot like bend, indeed ( and, although it is not very simple, it is not "overcomplicated"(sic) either ! ). However, to start from this loose form ( which reveals its simple topology at a glance - one can also start from the slightly different loose form shown in the previous post, which leads directly to the final compact form ) and dress it in its most compact form, passing through others, less compact ones, is an interesting knotting exercise. See the attached pictures, of the loose and the most compact form of this & bend.
   Therefore, the conjecture about the Supersymmetric bends was not disproven - but that does not mean it was proven, of course ! ( The antonym of the "simplest" is not "overcomplicated" !  :))
   
   
« Last Edit: October 26, 2014, 06:14:48 PM by xarax »
This is not a knot.