Hitches : Are they "knots", like bends ? If they are, where are their nubs ?

Knotting Concepts & Explorations
1

Personally, I tend to think of “knots” as “local” rope-made machines - not as “expanded”, spatially extended ones, as the Versatackle or the Trucker s hitch, for example. So, when I think of an eyeknot (loop), I imagine it as the three-loaded-ends “knot”, the very nub of the eyeknot, which does not include the whole eye..
What makes me think like this ? Two facts :

  1. There can be knots even in the absence of any friction, which rely on topology, and topology only, to be and to remain knotted. A simple overhand knot stopper, for example, loaded by both ends. When we want to clarify what a “knot” is, we should better start from the simplest cases, where the knots can be revealed in their naked, most essential form.
  2. When all the ends of one of those knots are being pulled ( and they have to be pulled, all of them at the same time, in order an already knotted knot remains knotted, otherwise the non-loaded end / tail will be swollen into the knot and then slip out of it ), what will this knot become ? A local maximally tightened tangle, a nub, that will reach a state of minimum rope length. Therefore, one has to conclude that the “ideal knots” that have lost all their “physical” properties except their mathematical ones. when they will settle in a final stage of maximum shrinking, they will become nubs - local maximally dense tangles of segments of minimal ropelength .
    That is why, when I say " knot", I mean the knot s nub, the dense part of the rope mechanism which may include many other elements, but which is concentrated at a “local” area of the space. In this view, the “knots” of the Versatackle and the Truckee s hitch are their eyeknots - meaning the very nubs of their loops - all the other are “external” knot elements, which, together with the “knots”, form a compound knot, or a spatially extended rope mechanism.
    So far, so good. One would not find very difficult to agree that there are compound knots, indeed, i.e., spatially extended arrangements of knots connected through tensioned lines - arrangements which constitute rope mechanisms, not “knots”. Then, one would be ready to agree that a loaded knot will settle to a tangle of minimal ropelength, where all the excessive portion of the free ends would have been consumed, and the only thing that would be left would be nothing more than the “knot” itself = the nub…until -
  • until hitches pop out ! :slight_smile: What are those rope mechanisms ? Are they “knots”, or compound knots, spatially extended, non-local rope mechanisms arranged around a pole or a main line ? If they are but “extended” rope mechanisms - rope tangles plus tensioned segments of ropes in between them -, where are the “knots” of those mechanisms ? One would show me a hitch and tell me : Show me your “knot” s nub ! What am I going to respond, in the case of a Clove hitch, or a Cow hitch, for example ? Is the main line or the pole an important, functioning part of the knot, because it does not allow it to degenerate into the unknot and become a straight line, or it is only a neutral, non-functioning element, a part of the rope mechanism ?
    A related question can be this : Is the main line or the pole that penetrates the compound knot, a part of it ? There are many cases where, if we remove this main line or pole, the hitch will collapse to a nub, and other cases when it will simply disappear - the Constrictor, for example, and all the other TIB hitches.
    This question has been addressed in the past (1), for the hitches as well as the binders, but was not answered . Now, with this amusing game of the 5-slot minimal collection of “knots”, where the available “knot” places are very limited, ( so it pays if are not oblized to include the Versatackle or the Trucker s hitch and cover valuable slots with other, more useful knots )(2), the question came back. The hitches, be them TIB or not, are they “knots”, or only compound knots, non-local rope mechanisms ?
  1. http://igkt.net/sm/index.php?topic=3610.msg20701#msg20701
  2. http://igkt.net/sm/index.php?topic=4418.msg28122#msg28122
2

Elementary, my dear xarax… :slight_smile:
After a night s sleep, I awake with a completely different mood - with no more questions, but with some (tentative) answers:
I thought of the exactly opposite situation : Let us have a proper “knot”, a knot s nub, and penetrate it from the one to the other side with a main line or a pole, loaded through both its ends. Will this “knot” become less of a “knot” ? No, it will be simply an ordinary “knot”, just wrapped around a “neutral element”, regarding its own function. We can now imagine that the cross section of this main line or pole expands, and widens, forcing the "knot’ to widen along it ( because it would have to remain tightened around itself, AND around the main line or pole ). So, what will this "knot : become ? A hitch !
I had already thought of this “thought experiment” :slight_smile: , albeit in a slight.ly different context - but, to my surprize, I had not made the small step needed to actually define what a hitch is, starting from the already conquered ground, the image of the tight, dense, minimizing ropelength tangle we use to call the knot s “nub”.
The interested reader is kindly required to read the following lines, replacing the “neutral element” with “main line” or “pole”

One is not obliged to concentrate about the question of the TIB or not nature of the "surrounding knot", as I had done in this paragraph, which is only of a secondary importance ( a TIB or not TIB hitch, is a hitch nevertheless ). The “surrounding knot” is a not but a knot s nub, a proper “knot”, in the case the diameter of the main line or pole shrinks to zero. We are not allowed , of course, to alter the topology of the hitch, that is, no segment of the hitch can cross this axis of the zero diameter cross section of the main line or the pole. Therefore the hitch is bound to be knotted, literally, around this axis, and it is nothing different than a proper “knot”, what most people would call the nub of the knot, the local, tight, dense part of it.

3

( Second writing ) I decided to keep them both, so there is greater a chance of my awful language could being understood by some brave reader !

Elementary, my dear xarax !
Just after a night s sleep ( and even if I was awaken at one moment by a nightmare involving a certain roodent making grunting noises and trying to bite me :slight_smile: ), I saw the light of the day ! My mood has changed : No more questions, just some (tentative) answers.
I just made a “thought experiment” of the exact opposite. Instead of starting from a hitch, and trying to see if and how we can define it as a knot, we better start from a “knot”, and see if and how we can define it as a hitch. In a split second, everything became much more enlightened !
Imagine that we have our local, tight, dense tangle, the “nub”, which I have defined as the “knot” proper. How we can make a hitch out of it, literally ? Simply by penetrating it, from side to side, by a tensioned main line ( in the case of a hitch around a rope ) or by a rigid pole ( in the case of the common rope-to-object hitches ). What will happen ? The ex-knot will become a hitch, wrapped around the main line ot the pole, and bound to be knotted, literally again, by the mere presence of the main line or the pole inside its core. In short, a “knot” transformed into a hitch, which can not alter its existence, so it was, it is and it will be a knot, ever.
Now, imagine the diameter of the cross section of this main line or pole becomes zero - and, also, the straight, rigid line of the axis of the main line or the pole becomes curvilinear and flexible. What would have happen ? The ex-hitch would become a knot, again, wrapped around a curvilinear line of zero area cross section. In short, the hitch would become what it never ceased to be : a “knot”.
Why this knot-hitch-knot is a “knot” condemned to remain knotted ? Because of its topology, the fact that no segment of it is allowed to cross this imaginary mail line or pole of zero diameter, so, if it was knotted when it was wrapped around a real main line or pole, it would remain knotted, in theory, and so it should be considered as knotted, when it will be wrapped around this invisible, imaginary path.
I had discovered that I had already though of something like this, albeit in a slightly different context, which was dealing more with the (secondary) issue of a TIB or not-TIB hitch. However, to my surprize, I had not made the small step needed to be able to define what a hitch is, before examining if there is a difference between TIB and not-TIB hitches. The interested reader is kindly requested to read the following paragraph, thinking of main lines or poles instead of “neutral elements” - and realizing, as I did, today (!) , that what I am actually describing is the relation between the set of knots and its subset, the hitches.

4

[/quote]
The “Neutral” parts are , imo, part of the knot. I don’t feel that these so called neutral parts are in fact neutral. They play an important role in those particular knots. Most knots, again my opinion, are compound structures. Yes, we can make a knot that does not do any work, decorative ones for example (though they can do work or a job). A drawn knot or a knot-board’s entries does nothing, but they are still knots.

We can see all manner of things within a simple or complex tangle we call a knot. We can even see this in un-knots, such as the Blackwall hitch, that I have used for years as a method of tying off mason’s twine to layout foundations of houses.

Hitch: To connect or attach.

So, within many/most knots there are hitches (not all!). A compound structure for sure.

SS

5

( Third attempt, with an almost one-liner answer ( in comparison to the dull sheets of the previous posts :slight_smile: )

Imagine that the diameter of the the main line or the pole around which the hitch is wrapped, shrinks to almost zero - but it does not disappear altogether. The wraps of the ex-hitch become rings (toruses) of almost zero diameter, but not balls (spheres). The geometry changes, but the topology does not - because we are not allowed to alter the most fundamental characteristic of any knot ( be it a mathematical, an ideal or a “real”, physical knot), its topology.
The ex-hitches now becomes the knot s nub, i.e., a properly defined “knot”. This “knot” remains knotted, even is the hitch was TIB - because no segment of the “knot” is allowed to cross the axis of the main line or the pole. Those elements may not be “functional” parts of the “knot”, as the rope segments on which it is tied are, but they still dictate its topology.

6 
Thank you SS369. A brave reader !  :)

Sure they do ! However, I had to distinguish them from the segments of the “knot”, through which the tensile forces run.

7

I do not think that we can eliminate the compressive forces and leave just the tensile.

Even though the topology of the segments are the same, with or without the “neutral parts” they both are knots of potential, till they are employed. Or enjoyed.

SS

8

We can think of “ideal knots”, where the circular cross section of the segment is assured - so the compression forces can not and do not alter it. However, I understand what you mean - because you “feel” the compression forces, and the friction, heat, etc, they generate - and there are the compression forces that will lead a “real” knot to its destruction.

9

With “ideal” knots we can “feel” tensile force from one end to the other, but with the addition of compressed point along the route, I don’t think there is equal tensile load along the line. Some is absorbed into the body of the tight nub (knot) and transferred.

So for me it is hard to translate the perfect into the real. Like predicting weather or fluid motion.

There are hitches that have nubs as you call it and there are a few that have nothing.

SS

10

If a hitch has one wrap - and all hitches have at least one wrap, of course -, it will have one nub, too - because we are not allowed to alter its topology. Therefore, however small the diameter of the main line or the pole would become, the small ex-wrap = present “ring” would always remain curled, unstraightened, and, together with something else, whatever else would be next to this little curl ( a half hitch, for example ), it will form a “nub”. Even a TIB hitch can not disappear, because there would always be one or more little “rings” packed together in a small area, a dense lump of material, a “nub”.
The trick is to imagine the same, topologically, hitch, tied around a diminished, different, geometrically, diameter of a main line or a pole. When you do this, you “see” the nub ! :slight_smile:
It is the geometry, the large scale of the wraps of the hitch, that does not allow us to “see” the “knot” in it. Change the geometry, shrink the wraps, without altering the topology, and the nub will manifest itself at once. A “knot” can be and can remain knotted due to its topology, and to its topology only. If we do not change its topology, a “knot”, any knot, can not change, by definition. So, if a "hitch , when its wraps are shrunk, is a “knot”, it should have always been a knot, and should always remain a “knot”, whatever the dimensions / the scale of its wraps were or would be.

[i][b]Knot tyer, shrink the wraps[/b] ![/i]    :)
11

So does this description/definition(?) include such as the Blackwall hitch in your mind? I just see a crossing, no interlacing.
I am fairly sure that I don’t consider laying a rope across another constitutes a knot. :-))

Honey, I shrunk the knots… ;-)))

S

12

You got it ! :slight_smile: :slight_smile: :slight_smile:

( You should nt try as hard, and go as far as Blackwall hitch, to discover a counter-example ! :slight_smile: ABoK#35, 36, 37 and 49 would be enough ! )
Yes, indeed, I think that the single ring the Blackwall hitch would be srhunk into, can not pass through the eye of a needle, so it constitutes a nub !
Any definition reaches its limits, at the limits ! Zero is an even number, but many people would find hard to believe it. Zero to the power of zero is 1 ( 00 = 1 ) - or is it knot ? :slight_smile:
Also, with the following assumptions:

http://upload.wikimedia.org/math/5/9/f/59fbcec15fbbc8744c0a4309c126a8a8.png

The following must be true:

http://upload.wikimedia.org/math/d/2/e/d2e283e91dee2cad966314a84da9f1d5.png

Dividing by zero gives :

http://upload.wikimedia.org/math/4/6/9/469c83d5e0959e4caaceea20df153c53.png

Simplified, yields :

http://upload.wikimedia.org/math/c/4/c/c4c9b852c938da096b69fc257a7a8d82.png

- Honey, I duplicated everything - including the knots ! :slight_smile:
- Could you, please, honey, duplicate the number of the knot tyers in IGKT Forum, too ? :slight_smile:

13

Wasn’t trying too hard to come up with the Blackwall hitch. Just one that I remembered the name of. Can’t do much about remembering the ones with numbers as names only. Point was made and I understand the eye of the needle reference. Still it is hard to accept in real usage.

As far as your math goes, I don’t buy into 1=2 unless it is perspective and then anything goes. ;-)))

And I don’t buy into that you have duplicated everything. Long way to go!

Duplicating the number of knot tyers in the Forum is an everybody endeavor. Bring someone in… Another thread please.

SS

14

ABoK#35, 36 and 37 are not “hitches” around lines in the ordinary sense, they are the most simple tangles that are topologically equivalent to the unknot, but “knotted” nevertheless, in the most simple way - although their “knottiness” is depending in knothing but geometry, not topology, it is made possible only by the particular location / arrangement of the four ends. However, they, too, pose the same question : Are they “knots” ? And, if they are , where the f… is their nubs ? That is the limit where the “Honey, I shrunk the wraps” picture has been magnified too much, and we can only see some black and white dots, but no shape at all…We can say that it should be expected, and that “any definition reaches its limits, at the limits” - but we can also say that the limits are the ultimate test beds for the validity of all definitions.

15

The geometry, the geometrical scale of parts of a knot s nub can not change the fact that a knot is a knot is a knot - provided we do not change the topology, of course ! ( If we were allowed to do this, we would have been able to transform every knotted rope into an unknotted one, and vice versa…)
Change the geometry, shrink the round turns, and you will see the nub that was concealed within them.

P.S. The main line or the pole may not be visible - but that does not mean it does not exist ! :slight_smile: On the contrary, when we see a “knot” that remains “knotted”, although we can not “see” how, we must suppose that it is entangled with some parts of a very small diameter.

16

I don’t buy the “machine” definition --something advanced
by the (USA) Cordage Institute, and which is contrived and
a mis-reading of the dictionary definition of “machine”.

“knot” has various uses/meanings, and I’ve often thus typed
at times ‘knot’ to signal something special (whether I’ve
been consistent re what that something is is another issue!).
Indeed, I’ve wanted to define “knot” as a particular loading
of a “tangle”, where a tangle is the interlaced material; but
I came to realize how much of geometry must be assumed
in order to have mere, specified loading achieve the intended
result --and that put a wrench into my (mistaken) thought
that tangle could serve such a useful, geometry-neutral
purpose.

And I’ve followed the thinking about an eyeknot being a “nub”
loaded in a certain way by 3 of 4 ends --i.p., of 1 end opposing
2 others, 1 of which is the end of its part (so, not a bight
hitch)–; then it follows that one might question whether
there is in fact any such implied “eye”, or might the opposing
2 ends NOT be connected --be separate ropes, e.g.!?

OTOH, Dick Chisoholm’s articulation/definition of “nub” goes
some way to show that “knot” might be bigger than what
some think of it as, and indeed one really does include the
eye, and --it follows?-- the mis-span parts of the sheepshank;
that still, IMO, leaves structures as the versatackle needing
to be seen as “compound knotted structures” and not “knots”.

As for hitches, my definition of them (in these forums) is a
“knot that entails an object”, and the object is part of the
nub. No, it isn’t (necessarily) the same cordage or cordage
at all; but it is necessary to the structure of the knot,
and I just see that as a basic fact and so define it.
Note that in the venerable --and perhaps most-cited
“hitch”-- two half-hitches I see that as a compound
structure, whose knotted part is a clove hitch of the
tail around the SPart of the structure. The pile, say,
is the structure’s object tied to; the rope is the knot’s
object, hitched to.

–dl*

17 
     [url=http://etymonline.com/index.php?term=machine&allowed_in_frame=0]machine (n.)[/url] 1540s, "structure of any kind," from Middle French machine "device, contrivance," from Latin machina "machine, engine, military machine; device, trick; instrument" (cf. Spanish maquina, Italian macchina), from Greek makhana [ μηχανη ], Doric variant of mekhane "device, means," related to mekhos "means, expedient, contrivance," from PIE *maghana- "that which enables," from root *magh- (1) "to be able, have power" (cf. Old Church Slavonic mogo "be able," Old English m?g "I can;" see [url=http://etymonline.com/index.php?term=may&allowed_in_frame=0]may[/url] (v.)). 

     Main modern sense of "device made of moving parts for applying mechanical power" (1670s) probably grew out of mid-17c. senses of "apparatus, appliance" and "military siege-tower."         
     The word [i][url=http://en.wiktionary.org/wiki/machine]machine[/url][/i] derives from the [url=http://en.wikipedia.org/wiki/Latin]Latin[/url] word [i]machina[/i], which in turn derives from the [url=http://en.wikipedia.org/wiki/Ancient_Greek]Greek[/url] ([url=http://en.wikipedia.org/wiki/Doric_Greek]Doric[/url] μαχανά [i]makhana[/i], [url=http://en.wikipedia.org/wiki/Ionic_Greek]Ionic[/url] μηχανή [i]mekhane[/i] "contrivance, machine, engine", a derivation from μῆχος [i]mekhos[/i] "means, expedient, remedy").
     A wider meaning of "fabric, structure" is found in classical Latin, but not in Greek usage. This meaning is found in late medieval French, and is adopted from the French into English in the mid-16th century. .. The modern meaning develops out of specialized application of the term to [url=http://en.wikipedia.org/wiki/Stagecraft]stage engines[/url] used in [url=http://en.wikipedia.org/wiki/Elizabethan_theater]theater[/url] and to military [url=http://en.wikipedia.org/wiki/Siege_engine]siege engines[/url], both in the late 16th and early 17th centuries. The [url=http://en.wikipedia.org/wiki/OED]OED[/url] traces the formal, modern meaning to [url=http://en.wikipedia.org/wiki/John_Harris_(writer)]John Harris[/url]' [i][url=http://en.wikipedia.org/wiki/Lexicon_Technicum]Lexicon Technicum[/url][/i] (1704), which has:[i]Machine, or Engine, in Mechanicks, is whatsoever hath Force sufficient either to raise or stop the Motion of a Body... Simple Machines are commonly reckoned to be Six in Number, viz. the Ballance, Leaver, Pulley, Wheel, Wedge, and Screw... Compound Machines, or Engines, are innumerable.[/i][i]         [/i]

The German scientist Reuleaux provides the definition “a machine is a combination of resistant bodies so arranged that by their means the mechanical forces of nature can be compelled to do work accompanied by certain determinate motion.” In this context, his use of machine is generally interpreted to mean mechanism.

It is interesting to notice that a great percentage of the early machines were machines-made-with-ropes much more than machines-made-with-solid-objects. The situation was reversed much later, when the accuracy of the man-made solid objects used as tools was improved. Weaving should be the first use of light elements able to withstand tensile forces, connected to each other by “knots”.
The word appears in Homer a number of times, with exactly the same meaning it has today ! Perhaps because of this I believe I can understand what a “mechanism” ( μηχανισμος ) and/or “machine” ( μηχανη ) is, and that I do not have to read the dictionary ! :slight_smile:
Knots are static machines, no question about that. People are often misled by the fact that, most of the time, these machines are in a state of equilibrium, so they do not "convert one form of energy into an other " any more. However, we always have to be careful not to identify “machines” with “engines”. Also, we can see that the slightest slippage of a segment of a knot due to an increased load, generates friction, friction generates heat, heat re-arranges the molecules of the materials within the segments until they reach a new state of equilibrium, and so on.

The existence of the connection or not between the 2 ends is irrelevant, I believe : a 3-limb knot is absolutely equivalent, regarding the “knot-ness” of the tangle, to a two-rope bend, where the one rope is loaded by both sides.

I believe I said exactly the same thing. I used the honey-I-shrunk-the-round-turns thought experiment, only to “show the nub” of the hitch. We should be free to change the geometry of a knot, to see if it remains knotted or knot. What we are not allowed to do, is to change the topology. If topology is sufficient to keep a tangle knotted, it is and it will remain a 'knot", and it will necessarily have a nub - although we should play with the geometry a little bit, in order to reveal it. If we change the geometrical scale of the round turns of a hitch, and shrink them, we can see its nub, and convince ourselves that hitches are knots, indeed, just like the stoppers and the bends.