4 BENDS : Zeppelin, Hunter , Ashley, Alpine Butterfly a COMPARATIVE

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4 BENDS : Zeppelin, Hunter , Ashley, Alpine Butterfly

A COMPARATIVE STUDY.

23 pages of text the rest, 36 pages, is illustration : photo, figures, tables.

SUMMARY page 2

OBJECTIVE O F THE STUDY page 2

DIFFERENT TYPES OF BENDS page 3

THE BENDS Page 4

EXISTING SITUATION page 5

INTRODUCTORY REMARKS page 6

MATERIAL AND METHODS page 11

DATA COLLECTING page 33 Zeppelin vs Hunter page 33 Ashley#1452 vs Alpine Butterfly page 35

CONCLUSION page 38

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SUMMARY:

Four bends ( Zeppelin, Hunter’s, Ashley #1452, Alpine Butterfly) are examined on their sole morphology and about an alleged “topological similarity”. There is no intent of evaluating the comparative practical merits or lack of therein. Beyond the simplistic similarity that can be stated as : they are bends, the joining of two ropes, in these cases using two knots to create a locking mechanism, NO NON- TRIVIAL SIMILARITY was found.

OBJECTIVE OF THE STUDY

There is absolutely no intent of evaluating “practical merits or lack of therein” but only the legend of similarity (topological or not).

The author holds the view that the use of the noun ‘similarity’ in relation with any pair in this “gang of four” is an abuse of guarded and educated language. (similar == identical but for a difference in size)

The author’s contention is that only a low quality observation can find any medium or high order similarity beyond the very trivial one :

All are comprised of two lengths of rope made into a functional single length using a joining structure ( a bend ) consisting of two interlocking knots made with the WEnd of the ropes.

Each WEnd makes an individual knot for its own AND participates in its interlocking with the knot of the other side.

One of the two original SParts keeps its SPart nature while the other mutate into the WEnd of the newly made single length of rope.

The interlocking knots are thumb knots ( what is called “ “simple knot” in French because it is the smallest knot that can exist ; no knot can exist under 3 crossings) that can take 4 forms: “S” OVER hand, “Z” OVER hand, “S” UNDER hand “Z” UNDER hand ( see next page about some of the types of bends )

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DIFFERENT TYPE OF BENDS

*** the two WEnd take their own part in the making of the joining knot- Opposite side tails (#1406)

*** each WEnd makes its own knot and also contributes to its interlocking with the opposite knot. Same side tails (#1408 #1425 #1453 are two thumb knots)

*** the two WEnd follow a unique cordage route to form one joining knot. Same side tails. (#1410

*** each WEnd makes its own knot and also contributes to its interlocking with the opposite knot. Opposite side tails (#1412 #1426 are two thumb knots)

*** the two WEnd form each a knot which is not interlocked with the opposite knots as the bend is formed by the two knots abutting to each other and so locking the bend. Opposite side tail (#1414)

The two WEnd form each a complete knot which is not interlocking with the opposite knot but the SPart of the opposite knot pass through it and so the two knots are abutting to each other and locking the bend (#1416)

*** the two WEnd each form a part of a common joining knot. Same side tails perpendicular) (#1418)

*** the two WEnd do not form a real knot ( 3 crossing minimum) but each makes a half-hitch to which the SPart of the opposite half-hitch give a “support” in cordage and the two half- hitch abut against each other so locking the joining. Opposite side tails ( #1420)

*** the two WEnd each form a half-hitch which interlocks forming a knot. Same side tails (#1422)

*** just a intertwining of the two WEnd. Opposite side tails (#1423)

*** the two WEnd each form a half-hitch. The half hitches are interlocking to form the joining knot. Opposite side tail ( #1428)

*** each WEnd form a loop and loops are interlocking. (#1455)

Those are some of the types.

It is obvious that one type cannot be similar to another type or they would no be really individualized ‘types’ but a single type.

The question that remains is the one about the similarity between bends belonging to the same type. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 4 sur 59 ©Charles HAMEL aka Nautile

THE BENDS

Photo 1 & 2 THE BENDS MADE WITH COMPARABILITY IN-BUILT AND MAINTAINED ( as explained latter)

The Roger Tory Peterson ‘s system (ornithology) is used to point the discriminators. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 5 sur 59 ©Charles HAMEL aka Nautile

EXISTING SITUATION

There exists quite a lot of parroting about “ topological similarity ”.

181 entries in Google with "It is topologically similar to the Zeppelin bend" in the search field on 2011 Oct 21th.

Wikipedia [open quote] The Hunter's bend (or Rigger's bend) is a knot used to join two lines. It consists of interlocking overhand knots, and can jam under moderate strain. It is topologically similar to the Zeppelin bend. [end quote ]

This is the single one I kept as “sample”, eight samplings I deleted : it is indeed just boring reading as it is only tedious and unimaginative as all plagiary is. ( a heartfelt thank you to Constant XARAX for pointing to me that non information redundancy )

There exists also another kind of plagiary who wisely refrains from using “topology” ; only seven samplings of those were deleted for the same reasons the eight previous ones were deleted.

[open quote]… zeppelin and Hunter's Bend interesting rigging knots that are very similar [end quote]

When it happened that those persons thought to precise the nature of the thumb knots they speak of “OVERhand”, never of UNDERhand, and never say a word about an important topological parameter : chirality. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 6 sur 59 ©Charles HAMEL aka Nautile

INTRODUCTORY REMARKS

The THUMB KNOT can take several forms, it can shows four different aspects of itself to the observer if that observer is forbidden to manipulate the knot from its “position” directly resulting from the making of the knot.

Ashley is showing #46 which is by structure a ‘Z’ UNDER HAND while saying it is an OVER HAND.

The thumb knot must be distinguished in Flat Land, (2D) diagrams, not only as OVER hand and UNDER hand but since each of the two can take two chiral forms we also get : “S” OVER hand, “Z” OVER hand, “S” UNDER hand “Z” UNDER hand. There is no possible similarity between a “Z” and a “ S” as they are plainly different at first sight.

OF COURSE , in the cordage as opposed to ‘in the diagram”, as you have no information on the “spatial position” in which the structure was made, there is a collapsing of details and there are only two ‘types” of orientation to take in account : “ S” and “ Z”. As shown in Fig 0 bis page 14 the CW OVER “ S” is equivalent (Reidemeister’s moves) to the CCW “ S” ( CW UNDER “ Z” to CCW OVER “ Z”). This is so because once the knot is made we lose, by manipulating it, very important information about its orientation in the making phase, information which is kept in the V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 7 sur 59 ©Charles HAMEL aka Nautile diagrams representations which show beyond any reasonable doubt that there are FOUR “forms” of formation processes of the Thumb knot.

Four ‘ forms’ that by not being cognizant or by deliberately ignoring or by destroying important information about the actual making can be pragmatically collapsed into the TWO types “ Z” and “ S”.

The use of left /right hand(edness) about any entity devoid of a brain is egregiously improper as handedness is a neuro-psychological preference. Better say ‘ left /right oriented’ so that one can avoid to commit the illogical thinking : "left -handed" (right -handed) because it was made left -handed ( right -handed).

The left /right orientation notion does not pertain to the " maker' nor to the ‘making’ but to the " made " and simple rules taken from topology allow to consistently determine the orientation of the knot, which has absolutely nothing to do with the orientation of the " maker " or of "the making ".

When with a ‘in the cordage’ “ S” thumb knot *made* CW OVER you show it as having the appearance of an “ S” thumb knot *made* CCW UNDER it is *only* at the cost of introducing a very important ‘existential change” in the knot : you need to give it some energy to make a flip (turning it like a book page). It is identical to the other member of the pair only by ignoring this important fact : a flip was introduced in one and not in the other which *prove* that the making was quite different and sustain the FOUR “forms” for TWO “types” “ Z” and “ S”.

Suppose that it is forbidden to flip a knot in the cordage from its ‘making’ position without putting a paint mark on it "to keep the existential memory" : then you will have indeed two quite distinguishable 'forms' having been made differently. From the author’s point of view this is simple logic. One must keep the comparability of what is shown to the observer, one must keep the ‘aspect’ of the original structure before any move is applied to it.

Adding a flip to get indistinguishable forms just *prove* that before that necessary flip that make them visually identical they were indeed visually "different" : proof = you have to make something special to one of the two to render it *apparently* identical. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 8 sur 59 ©Charles HAMEL aka Nautile

Suppose we live in Flatland then you cannot flip a knot so we stay with 2 forms for each ‘type’. Another proof : when a knot in the cordage is flipped that does not change its orientation, an “S” stay an “S” , a “ Z” stays a “ Z” but that is not so in Flat Land where the traced on paper diagrams dwell. Diagrams “keep all the information” of the making, no possibility to cheat on how the knot was drawn hence the FOUR forms that can be of TWO types only by loss of information. When a diagram is flipped on a sheet of paper that cause the diagram to disappear from view as the sheet is turned over, when it is flipped on a computer screen that creates a mirror image and so change a, “S” into a “ Z” see the illustration in the preceding page. (see annexe about geometrical manipulations ) IT IS HIGHLY IMPORTANT TO NEVER CONFUSE WHAT HAPPEN ON PAPER WITH WHAT HAPPEN ON SCREEN DURING GEOMETRICAL MANIPULATIONS

Remember, we are after comparability so we need to “keep all bits of information” in order to built a valid (exhaustive, no double, no missing) database which is shown in Table 2 page 34 .

If we discard the four diagrams forms then we are led to build our comparability with the egregiously improper Table-2-0.

Once again those table-2-0 / table-2-1 / table 2 are for the drawing of the diagrams, they describe the ‘aspect’, the visible face of the bends offered to analysis and only that, consider them as alike to the head and tail faces of given coins.

Obviously were you asked to draw those coins ‘as seen’ you would produce different drawings according to the face that you are shown : idem for those diagrams. Were you to draw several sorts of coins it would be best to keep comparability by drawing them all from the tail (or all from the head) point of view and certainly not in a mix of heads and tails.

Table-2-0

Table-2-1 is less precisely expressed than Table 2 page 35 but it still ‘acceptable’. (barely so).

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Table-2-1

As to make the comparisons easier we will be using diagrams we need, all along the way, to take in account the making of those diagrams so as to make certain of the comparability of those we use, hence the insistence on the four ‘forms’ instead of the two ‘types’ to be used.

Anyway even if one is unable to mentally conceive of the very simple logic that shows FOUR ‘forms” this point is not important for the argumentation (as long as one admit that indeed the comparisons are valid due to in-built comparability in the diagrams) because this point is important only for building the comparability . Not being able to “see in one mind eye” the four forms does not detract from the validity of the comparison made as each of the diagrams ( and photo) show a comparable ‘aspect’; the first knot in the bend is always in the same orientation.

When a thumb knot is seen in isolation, “free”, not knowing how it was made, you may not know if a “ Z” ( or a “ S”) as been made as an OVER hand or an UNDER hand but when “locked” in the “was made so” position by another knot you may not chose, while still keeping the same structure, to see it from a front view or from a back view without also changing the orientation of the other knot. (two red , two green are front view/back view or mirror) “S” CW OVER / “ S” CCW UNDER and “Z” CW UNDER/ “Z” CCW OVER Each pair is a pair of isotopic knots, they are mirror images (one show the front view, the other the back view of the same structure).

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That is why in the diagrams the “blue knot”, to insure comparability between the bends studied, has been arbitrarily chosen as being constantly “ S” CW OVER so this ‘blue knot’ gives the ‘fixed point” for the ‘red knot’ which is decided by the structure of the bend. It is obvious that, with comparability in view, four choices (see Table 2 , p 30) of ‘blue knot’ were possible for the standardisation of the diagrams or the pictures.

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MATERIAL AND METHODS (Subtitled : building common ground)

Fig 0 Look at Fig 0 : *** Triangles A, B, C, D are similar *** Triangles A and B are similar AND congruent ( just flip one –make the front the back and the back the front -)

*** Triangles A and B are mirror of each other and ONLY because they are 2D figures so in ‘plane geometry’ they are similar BUT 3D mirror images CANNOT BE similar : just look at the two knots in FIG 0 one is the mirror of the other and yet due to chirality they cannot be similar ( the yellow is “ S” and the green is “ Z”. The similar to the yellow CW “ S” OVER hand in Fig 0 is its isotope CCW “S” UNDER hand which is the yellow one as seen after a front to back flip (Fig 0 bis )

The knots in Fig 0 would be isotopes IFF the WEnd of one were to become its SPart and its SPart its WEnd. You can spent the next billion years trying once every nanosecond you will never get one from the other just using Reidemeister’s moves.

Reidemesiter’s theorem : two knots are topologically equivalent ( NOT similar but equivalent please ! ) IF AND ONLY IF their projections may be deformed into each other by a sequence of three moves shown in Fig 0 ter .

Fig 0 bis

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Fig 0 ter (the THREE Reidemeister’s moves : necessary AND sufficient)

Well just try with Reidemeister’s moves to make any of the four bends into any of the other bends, if you succeed (a snowball chance in hell of doing that) then you will have established the topological equivalence of the bends.

Those two knots in Fig 0 are not isotopic, they are quite separate. ( in Fig 6 it is through a series of isotopic stages that one passes from ABoK #525 to Fig-of-9. A Fig-of-Eight knot can be, using only Reidemeister moves, made into its mirror image == amphicheiral knot . V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 13 sur 59 ©Charles HAMEL aka Nautile

Chirality and sign of crossing Illust A

Illust B Illust C

In Illust B the top pair are OVERHAND knots while the lower pair are UNDERHAND so you see that leaving it, with the flock of parrots, at “two overhand” for all the 4 bends just show inept procedure of observation. You will, please, note that Ashley made drawings that are impossible to validly compare as he mixed in the greatest randomness both “altitude” (over-under) and CW /CCW hence a mixing of chirality. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 14 sur 59 ©Charles HAMEL aka Nautile

Topology What is topology ? Fig 1 Simple !...... it is that which makes that Fig 1 is still a Bugs Bunny representation.

Like it or not, believe it or not it is Bugs ! This is what is called "rubber sheet geometry"

Of course the geometry of angle and distance is destroyed.

Only the relations between elements/parts are respected.

Another character which is respected is relative direction, or in case of a knot SPart and WEnd if only because it is a “relation” and relations are preserved.

That is why when using a morphing software one can reverse the process step by step. Had the relations been destroyed it would no be possible to reverse, to back track.

When in an attempt to disguise your appearance you pad your cheeks with cotton wads you respect the topology but you so alter the geometry that you hinder fast human visual identification.

Topology is what makes that, essentially, the intact flat sheet of paper is 'not topologically altered but stays 'equivalent' , when you are crumpling it. ( only crumpling, no cutting, no tearing, no gluing please - just a tiny pin-point hole and it is not any more topologically equivalent as there would be one “hole” more and as we will see ‘holes” are very important ‘discriminators’)

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Fig 2 This alphabet regrouped in 3 homogeneous sets of topological equivalence –Fig 2 -should help you understand the “principle” (‘holes” are the discriminator).

D'Arcy Thomson (Sir D'Arcy Wentworth Thompson 1860 - 1948 ) wrote On Growth And Form published in 1917, a magnificent work, here is an illustration taken from it.

In each pair of fishes one is topologically equivalent to the other. Fig 3. You see how direction is kept by seeing that heads are still heads and tails are tails Fig 3

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Fig 4 Fig 4 shows counts of “holes” that make the two knots topologically different.

Fig 5

The bends, in each of their thumb knot component, have “non occupied” ‘hole’ or “non occupied compartment ” or ‘ cells’ . Not occupied by the interlocking with the opposite knot –Fig 5 - and it is also plain to see that there is no similarity in the repartition of those “free compartments”, plus Zeppelin and Hunter have two and the other two bends have only one free cell . Even discarding the ”non occupied compartments” just with the type of tails ( opposite or side by side ) you get two different groups. Fig 6

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In this case it is absurd to think about topological similarity or equivalence as one cannot transform ( using Reidemeister moves which are used to establish topological equivalency ) any one bend into any of the others as it is possible in some other cases as shown in Fig 6 . Note that Fig 6 does NOT prove “ topological similarity ” but more properly “ topological equivalence ” between ABOK #525 and Figure-of-9 because the transformation was done without any cutting, gluing, un-threading and re-threading of a part or parts of the knots but just rearranging the cordage using the three Reidemeisters’s moves.

Fig 6 bis An historical topological equivalence is the PERKO’s pair : one is the homeomorphism ( isomorphism it would be if the “isometric” was respected ;-) ) appearance of the other : EXACTLY THE SAME KNOT MODEL UNDER TWO NON IDENTICAL BUT EQUIVALENT APPEARANCE ! ( Fig 6 bis )

This does not exist for any of the four bends compared to any of the other ones ; one may hold the hope that it exists for one form of a given bend to be made into another form of it but then you need to demonstrate that formally or at least to show it pragmatically like I did for other knots in Fig 6.

Equivalence ( ), is not equation ( = ), is not identity ( ≡ ), is not logical similarity ( ~ ) , all those concepts are different one from the other.

One may have equivalence without having equality or identity and one may have equation without identity. You may wish to look up isomorphism and isotopic.

Dual illustration Fig 7 shows the difference between topo graphy (the graphical representation of topo logy ) and geometrical reality.

It is fortunate that direction (departure / arrival) and relation (relative position without taking in account angles or distances) are kept intact : That is why the right side diagram is topologicaly similar to the left side map and is as useful in term of data but is easier to use.

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People imagining topological similarity in those fourbends are just forgetting SPart / WEnd and/or the chirality of the component knots and/or the “holes”.

Chirality is kept in topology because the relations are kept even if angles and distances are distorted compared to the physical reality ( scaled or 1/1)

Fig 7

Let us see the notion of TANGLE

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Fig 8 Fig 9

Of course every one will have seen that the topological tool call TANGLE shows that the two cordages routes (the trail followed by the cordage but with no mention of the type O or type U of the crossings made ) are identical ONLY if they are AMPUTATED of their direction, orientation, that is to say are without indication of which end is SPart and which is WEnd while the crossings stay identical.

The tangles being similar and congruent are so identical but with the addition of SPart and WEnd data any identity or similarity vanishes.

In the real bends what is SPart of one is WEnd in the other and what is WEnd in one is SPart in the other. Fig 10

The upper part with the vertical black bars in Fig 10 shows the TANGLE ( in fact in topology tangles are drawn in a different manner using a circle instead of bar. If you make the tangles of each of the four bends you will not find any similarity like in the two cases just evoked.

------

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--- first point to make about methodology : Fig 11 If one wants to have the tiniest chance of making a valid comparison one has to use isometric grids so as to be constrained in a constant manner in all three directions AND follow some simple rules ! Schaake has shown that quite a number of time.

It is the only type of grid that does not hide some “relations” which are immediately apparent in isometric.

Isometric drawings properly made respect the topology and so respect “relations”, they also, by nature, respect the iso-metry in the three directions and for that reason are used quite often in industrial documents.

For all the Thomases the Unbeliever here is Fig 11 showing the strict topological equivalence of 2 representations of thumb knot cordage route .

Isometric is a trick toward rigor not a trick towards cheating or manipulation. (‘working’ area is between the two horizontal yellow lines )

In this paper ‘comparability’ has been in-built as far as possible in the way the bends are represented.

As long as topology, which is relations ( departure/arrival and relative position), is respected you may do what ever you want to distort the geometry which is angles and distances .

Topology is about qualitative while geometry is about quantitative …

As long as “the relations” (relative positions and so ‘direction’ : whatever the manipulations done you can always follow the ‘circuit from start to end) are fully respected and as long as you do not HIDE one or several crossings and do not ADD useless crossing(s) comparability will always be possible.

Economical but truthful !

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Fig 12 Note that it is a botched work that Ashley did with his crossings : See! there are hidden or piled up crossings hiding crossings and also added crossings ( a compounded no-no –caveat searcher do not make comparisons using ABoK drawings - ) as in #1426 just for a single example among many.

Fig 13 You can see in Fig 13 that if the inside tail is prolonged then you will make a non structural (non nipping, non functional) crossing so I pushed the tail extremity and part of the loop outside the “working area”. The other solution, much less satisfactory is to make the false crossing and put a warning circle or star on it.

It is a finding of topology that ANY knotted structure CAN be drawn. This is a given, what is not a given is the ease with which the tracing can be done!

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Fig 14 shows the addition of useless (non structural) crossings that I did for the sake of demonstration. NEVER DO THAT.

Another cause for anxiety for some readers (perceived as opposed to real) could be the “parti pris’ chosen for drawing. As long as the diagrams which are subjected to comparison are kept comparable by respecting the topology and by the use of an identical ‘parti pris’ all over the compared sets then even if it change the external appearance of the data as it will do that in a comparable manner so the analysis of the data will stay valid thanks to “comparability maintained” all over. The ‘error in ‘’measure’’ will be constant all over. Mandatory : only show the minimal number of crossings (but all of them) that is compatible with a correct structure . Fig 15

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Fig 16

Photo 3

This shows that isometric does not detract from ‘deep reality’ as with Reidemeister’s moves you may get back to the disposition existing in reality.

In Photo 3 several severe distortions have been applied to the photography of the knot in Fig 16 but all distortions were 100% respectful of the topology even if oblivious of the original geometry. As it is plain to see the relations in the knots and between them in the bend are unaltered ; just as when using an isometric grid which has the advantage of keeping constant the ratio between dimensions hence the name ISO-metric. It is the most performing grid of all to study knots in my experience. Fig 17

Fig 17 shows an isometric “”””””””“””similarity”””””””””””””, rather ‘equivalence’. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 24 sur 59 ©Charles HAMEL aka Nautile

As long as the relations between crossings is respected and no crossing is hidden/omitted or created (useless crossing) then there is perfect comparability between the cordage route in the actual knot in cordage and its representation in the isometric grid. ( cordage route : no type O/U of crossing is indicated, only their relative positioning – this is the equivalent of the topological ‘shadow’ )

Fig 18 shows what is NOT to be done : creation of a fictive topology with crossings that do NOT exist in the knot. Fig 18

Never do, under any account, what is shown in Fig 18

NEVER either ADD or HIDE or SUBSTRACT any crossing. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 25 sur 59 ©Charles HAMEL aka Nautile

Photo 4

Photo 4 shows the faults to be avoided: - adding useless, non structural crossings that can easily be made to disappear using one or several of the three Reidemeister’s moves ( 4 orange line pointers) - piling up crossing hiding some of them and modifying the “hole” count ( 2 red line pointers)

Fig 19

Fig 19 is there just to show three different manners of representing a knotted structure.

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Fig 20 So as not to destroy the isometric keep an identical “unit” in all directions. ( Fig 20 )

Just as one must go for the lowest possible number of crossings ALWAYS go for the “most compact possible” tracing . Do not put in useless spacing.

Avoid useless “turn or change of direction in the tracing ”.

Again : the “working area” is between the two yellow horizontal limits.

Fig 19 shows equivalence but one is not compact enough and so visually the onlooker can get a deformed impression or you must deform all the compared diagrams in a similar manner. Easier, surer and shorter it is to go “compact”.

Fig 21 Fig 21 shows a bad spacing (inconsistent use of “unit” leading to two (green arrows) “space” or “hole or “cell” that could easily have been made smaller giving a more “compact” tracing.

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Fig 22 BEWARE: Fig 19 to 24 CONTAIN MISTAKES THAT I INTENTIONALLY KEPT TO SHOW WHAT CAN HAPPEN Fig 23 I made using some diagrams found (Fig 22 ) on the Net. Their author used KnotMaker (a useful program if your intent is NOT comparison but merely showing “instructions for making” (of course then the drawing not be faulty!).

This Fig 22 is not the best way to represent knots diagrams when the intent is to make a valid comparison between what is being examined.

This Fig 22 has all the appearance of having been made to create and to reinforce any germ of similarity .

Some big mistakes : The diagram shows the RIGGER’s with side by side tails while this bend really have opposite tails. FAULTY .

The diagram shows #1452 with opposite tails while in fact it is side by side tail FAULTY . So ipso facto this manner of drawing falsify the reality and induce wrong conclusions.

Unfortunately making the knot with the diagrams leads to a wrong Hunter with side by side tails and leads to a #1452 with the correct side by side tails.

Fig 23

Fig 23 shows an imitation in isometric of those ( faulty ) diagrams in Fig 22. It shows that even with isometric if you do NOT COMPLY with ‘rules’ then you obtain biased results when following a mistaken model. Intentional manipulation is always a possibility if one want to “trick” so… V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 28 sur 59 ©Charles HAMEL aka Nautile

SO Fig 23 is as (deliberately so as a teaching about keeping your brain ‘sharp’) mistaken as Fig 22 is : the waste paper basket is their proper place.

Just to show the “reality”, of those bends unaltered by the system used for their representation here they are “in the cordage” in ‘exploded” view. STILL BE WARNED THAT THE FAULTY DIAGRAMS WERE USED TO MAKE THEM HENCE THE FAULTY RESULTS THEY YIELDED in Fig 24 to 27 .

Fig 24 Fig 25

Fig 26 Fig 27

Those knots above were made following the faulty diagrams and as you can see the resulting bends are not correct. Table 1 ZEPPELIN RIGGER’s or ASHLEY Alpine HUNTER’s #1452 Butterfly B. REALITY of tails OPPOSITE OPPOSITE SIDE by SIDE by SIDE SIDE Tails gotten with the OPPOSITE SIDE by SIDE OPPOSITE SIDE by “faulty diagrams ” SIDE REALITY 10 12 12 12 CROSSINGS CROSSINGS IN 12 12 12 12 FAULTY REALITY HOLES 9 11 11 11 HOLES in FAULTY 11 11 11 11

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---second point :about methodology and procedure. One needs to built COMPARABILITY from the very start and to maintain it or if not complying with that then you need to take in account the “garbage factor” that about junk any study not having ‘comparability’ all over.

Comparability in the method used (here isometric for all diagrams and a constant ‘parti pris’) and in the “material” being compared, bends.

It is impossible to validly compare ‘Z’ with ‘ S’.

‘Z’ or ‘ S’ are much more than the way the first crossing is made.

It is a concept that is fully amenable to a logical process which is reproducible! (don’t dare to say ‘handedness’ as handedness –a neuro psychological preference- cannot exists in a brainless inert object such as a knot).

It just happen in the case of the Thumb knot (3 crossing so strictly alternating O-U-O or U-O-U) that the sign of all its crossings is the same as the sign of the first crossing made which also give the sign or chirality of that particular knot. That is not so for all knots .

Most often the so-called “”handedness”” (chirality), attributed by ‘tradition’ is, in fact, false when the sign of *all* the crossings in a knot is taken in account.

Many persons will say that a thumb knot made CW is a ‘ right ” overhand and made CCW it is a “ left” overhand but that is not so.

A CW made overhand is “ S” ( that is left if you want hence just the opposite of what is said).

A CCW made overhand is a ‘ Z’ ( just as when making laid rope and the direction of the rotation of the crank)

Fig 28 The four bends may be considered similar *only* on two points:

-“joining two ropes”

- “using two thumb knots” (See annexe for more details)

Fig 28 shows the four ways a THUMB knot may be obtained.

This is the basis of the following table ( Table 2 )

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Table 2 Once you have made certain that the “base” from which data will be extracted has in-built comparability between the “individual items” being studied you still have to take the greatest care to collect that data maintaining all the while the comparability which must be preserved till the last final ‘.’of the conclusion.

After data collection, verification and validation then you may begin the analysis phase still maintaining comparability by applying an identical procedure to all ‘individual items’.

Choices made can be “arbitrary” because “arbitrary” is not opposed to “reasoned and logical”

So I chose the first two lines in Table 2 but I could have used any of the other blocks of two lines because comparability existing each and the final conclusions are not changed by that ”arbitrary” choice.

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Fig 29 Just for the sake of the demonstration I chose the ZEPPELIN BEND and made the FOUR forms of it shown in Table 2 .

Those forms are illustrated with photographs of front and back for each. ‘Option n’ refers to the “blocks” in Table 2

You could exchange yellow for green and vice versa ( you may also imagine instead of colours, slightly different diameters ) and you would have EIGHT different forms ; again on those eight imagine exchanging SPart for WEnd and vice versa and it is SIXTEEN different forms you get.

Photo 5 Photo 6

Photo 7 Photo 8 Even though closely related (would be better to say : analogous ) those FOUR forms are NOT SIMILAR (geometry conceptualisation) beyond the mere trivial similarity already stated , that even with the added notion that option 1 and option 3 are mirror of each other as option 2 and option 4 are mirror of each other. Do not say they are similar say they are “mirror”. With the modern braided cordages may be the CW / CCW way of making knots is not all that important but with the old laid/twisted cordage then the choice was not V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 32 sur 59 ©Charles HAMEL aka Nautile indifferent as it affects the lay of the cordage and the “ratchet’ effect of one part of cordage against another part of it.

Not being similar beyond the trivial level of similarity they all have distinctive “traits” that can be pointed to using Peterson’s system and that immediately and irremediably differentiate one from any of the others .

You can already see the blunder of first magnitude it is to speak of “ two overhand knots ’ in those bends as it is IMPOSSIBLE to have all four bends made with two overhand if comparability between the bends is to be maintained.

It is absolutely impossible to get all four bends made with two overhand knots because the Zeppelin can NOT have two knots of the same “altitude” (O-U) in the first crossing made : either two OVERhand or two UNDERhand. It always will be a mix lot : blue :OVER with red : UNDER or blue :UNDER with red : OVER. (The Alpine Butterfly will always be a mix of CW and CCW made component thumb knots that can be both with a common ‘altitude’ : OVER or UNDER)

Already the low quality of the observation “It consists of interlocking overhand knots” is established.

On that point alone you may junk the plagiary’s conclusions concerning ZEPPELIN and HUNTER’s bends ( both with same type of tails ) without any fear of loosing anything of value by doing so.

Only the ASHLEY#1452 and the Alpine Butterfly bends ( both with same type of tails) can both be at the same time two OVERhand or two UNDERhand but in the very first illustrations ( Photo 2 ) the Peterson’s system visually shows that they are NOT similar.

I suggest that you print the Printer friendly Table 3 in the very last page before you continue reading.

It will come handy when reading and verifying illustrations and text. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 33 sur 59 ©Charles HAMEL aka Nautile

DATA COLLECTING

ZEPPELIN versus RIGGER knot (so called HUNTER knot)

Fig 30

Fig 26 was suppressed The first made knot or the chosen ‘constant knot’ (blue ) is in both cases an OVERHAND knot made clockwise so ‘ S’. Comparability is perfect as the second knot will be chosen by the bend structure itself.

It happen in both cases that the second knot made (red ), is also made CW but for the Zeppelin it is then a ‘‘Z’’ UNDERhand while for the HUNTER it is ‘ S’ OVERhand. Comparability is still there but similarity has flown the coop.

Both have the tails on opposite sides. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 34 sur 59 ©Charles HAMEL aka Nautile

Again comparability is there; had one be with tails on opposite sides and the other tails on the same side then comparability would have been destroyed ( Just as any idea of similarity is destroyed by the presence of different “tails types.)

Zeppelin : 10 Xings (Xing==crossing) Hunter : 12 Xings Difference in count does not detract from comparability but surely destroy any hope of any alleged ‘ topological similarity ”

Half of the Xings in each bend is of one sign the other half being of the opposite sign.

When ones looks at the isometric representations with a clear attention and a sharp focus then entertaining the notion that those bends are ‘similar’ ( understand this noun in is ‘real’ meaning : “ nearly identical” except for size) seems to the author quite inappropriate and an abuse of an educated use of the word ‘similar’.

The author concede that they are similar in that they are both made with cordage and both have two thumb knots components but going beyond that then similarity as in “extremely close resemblance, a near sameness” is a bit far fetched for the author personal taste.

Lets us junk the non discriminative word “thumb” and distinguish two sorts of thumb knots : OVERhand and UNDERhand. (really we should go much farther and have “Z” OVERhand, “ S” OVERhand, “ Z” UNDERhand, “ S” UNDERhand.

So the author may agree that both are similar in that both are made of one OVERhand interlocking with another thumb knot ( you may this time not precise over or underhand) *but* the locking mechanisms are vastly dissimilar if only because the second thumb knot is “Z” UNDERhand in the Zeppelin and “S” OVERhand in the Rigger’s

Fig 31 shows the crossings distributed in logical sets according to “geography” : on the SPart, between the SPart, outside the SPart left and right.

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Fig 31

Note : the locking mechanism may be described from two perspectives :

- from the perspective of the ‘ coding pattern’ (the sequence of crossings) and that is what is done in the “set of crossings” just above Fig 31

- from the perspective of the cordage route ( or knot shadow : the tracing of the cordage but without any mention of the nature of the crossings made, just their position.) and using the “ component thumb knot cells “ number in a matrix like in Table X1 in page 40 . Table X1

RED locks with BLUE

1 3

2 2 3 1

Zeppelin and Hunter are both ‘a bend made of two thumb knots’ but as soon as one gives more personality to those “thumb knots” by calling them OVERHAND and UNDERHAND knots and giving them their chirality it makes it more tricky to maintain the fiction of overall similarity.

Hunter’s bend has 11 ‘holes” and Zeppelin’s bend has 9 , Hunter’s is interlocking of two ‘ S’ OVERhand knots while Zeppelin’s bend is one ‘ S’ OVERhand interlocked with one ‘ Z’ UNDERhand there flies away, with no return, any illusion of “ topological similarity ” ; it cannot really be sustained with success.

ASHLEY’s BEND vs Alpine Butterfly BEND

Neither ZEPPELIN nor HUNTER is similar to either the ASHLEY or the ALPINE BUTTERFLY if only because the former have tails on opposite sides while the latter have tails side by side on the same side. (a different disposition is ‘same side but not adjacent’ as they are adjacent here.)

The reference knot ( blue ) made clockwise is in each case an “S” OVERHAND knot.

The second made ( red ) is an anti-clockwise ‘ S’ OVERHAND knot for ASHLEY but an anti-clockwise “Z” OVERHAND for the Alpine Butterfly.

ASHLEY 12 Xings, all but 2 signed ‘minus’ so ‘ S’ ALPINE BUTTERFLY 12 Xings 7 of minus sign so ‘ S’

We may compare those two bends which have the same chirality.

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The only similarity is that it is an interlocking of ‘S’ OVERhand with another Thumb knot (either an ‘S’ OVERhand or ‘ Z OVERhand ) but with non-similar locking mechanisms. Fig 32

For the difference of locking mechanism just look at the following illustration that put the Xings in different SETS : “on the SPart”, between the SPart, outside the SPart. All four knots are treated with the same methodology so as to keep comparability.

There is no similarity whatsoever beyond the simplistic fact that it is the interlocking of two “thumb knots”, similarity disappears as soon as I chose to call them “S” OVERhand , “Z” OVERhand , and “Z” UNDERhand knots and put the crossings in ‘homogeneous” sets

Fig 33

As far as the author is concerned seeing similarity here has to rest upon some incredibly imaginative credulity about “external appearance”.

Only superficial observation can leads to a conclusion of similarity.

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CONCLUSION

COMPARING THE LOCKING MECHANISMS OF THOSE FOUR BENDS

Fig 17 bis

Fig X1 of the four forms of a same bend, the ZEPPELIN, just shows that the diagrams are not altered in comparability by chirality as far as the ‘ emprise ’ is concerned, hence reproducibility and comparability from start to finish. In this Fig X1 the ‘emprise’ is 2 ‘units’ Fig X1 The pale green areas show the part of the “red knot” which escapes the emprise of the “blue knot”.

This is a sort of ‘integration’ of the ‘compactness of the interlocking’.

Table X1

RED locks with BLUE

1 3

2 2 3 1 Table X1 show which “portion’ of one component knot locks with which ‘portion’ of the other component knot

So as to show the reproducibility and comparability of the method let us try it on the four forms of the zeppelin Fig X1 .

We may now proceed with the four bends themselves.

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You will immediately note that the HUNTER’s bend stands out by the absence of any green “liberty area”: the intermeshing of the two thumb knots components is 100% compactness. The three other bends have “green liberty areas” of equal size : 2 ‘units’, but this it on bends that do not have the same diagrammatic “height”: Zeppelin : 2 // 9 Hunter : 0 // 6 Ashley : 2 // 9 Alpine : 2 // 8 2 // 9 are not to be considered ‘similar” as one is for 2 ‘pairs of holes’ locking (Zeppelin) and the other for 3 ‘pairs of holes’ (Ashley).

.Fig X2

Table X2 zeppelin RED locks with BLUE 1 3 2 2 3 1

Table X2 Hunter RED locks with BLUE 1 2 1 1 2 1 3 3

Fig X3 Table X3 Ashley RED locks with BLUE 1 2 1 3 2 3 1

Table X3 Alpine Butterfly RED locks with BLUE 1 2 1 1 2 1 3 1 3 It seems obvious that the locking mechanisms are not similar at all either as expressed by the “sets of crossings” or as expressed by a component thumb knot cells matrix. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 39 sur 59 ©Charles HAMEL aka Nautile

Fig 34

The *only*, quite basic, similarity is that all four structures are different locking mechanisms to join (bend) two ropes using a pair related knots “ Z” OVERhand , “ S” OVERhand , “ Z” UNDERhand .

There exist the most trivial order of similarity : They are bends using two interlocking knots. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 40 sur 59 ©Charles HAMEL aka Nautile

Table 3 ZEPPELIN HUNTER ASHLEY ALPINE BUTTERFLY Blue knot orientation CW CW CW CW Blue knot chirality S S S S Red knot orientation CW CW CW CCW Red knot chirality Z S S Z Over Equality Equality S S All chirality minus/plus minus/plus Type of blue K OVERhand OVERhand OVERhand OVERhand

Type of red K UNDERhand OVERhand OVERhand OVERhand

PAIR OF TAILS OPPOSITE OPPOSITE SIDE BY SIDE BY SIDES SIDES SIDE SIDE Number of “holes” 9 11 11 11 Number of Xings 10 12 12 12 Nb of Xings in SPart 6 6 7 7 SET NB of Xings 0 6 0 0 Inbetween SET Nb of Xings above 2 0 4 5 SPart SET Nb of Xings under 2 0 1 0 SPart SET

As soon as the SETS of crossings are taken in account it is impossible to sustain the similarity hypothesis which is only an improbably imaginative hypothesis stemming from lack of proper observation and thinking process.

Even when discarding the SETS of crossing criterion : the “holes’ counts by themselves kill all possibility of “”””””topological similarity””””””””( very long pincers to touch that assertion ) between Zeppelin and any of the other three bends.

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Table 4 SHOWS THEY ARE SIMILAR ONLY IF ONE PUTS ASIDE ALL THAT CAN SHATTER THE ILLUSION OF SIMILARITY .

ZEPPELIN HUNTER ASHLEY ALPINE BUTTERFLY Blue knot orientation CW CW CW CW Blue knot chirality S S S S Red knot orientation Red knot chirality OverAll chirality Type of blue K OVERhand OVERhand OVERhand OVERhand

Type of red K PAIR OF TAILS Number of “holes” Number of Xings Nb of Xings in SPart SET NB of XingsInbetween SET Nb of Xings above SPart SET Nb of Xings under SPart SET

Table 5 NO SIMILARITY beyond the most trivial one. (read rows ) ZEPPELIN HUNTER ASHLEY ALPINE BUTTERFLY Over Equality Equality S S All chirality minus/plus minus/plus

Type of blue K S CW S CW S CW S CW OVERhand OVERhand OVERhand OVERhand

Type of red K Z CW S CW S CW Z CCW UNDERhand OVERhand OVERhand OVERhand

PAIR OF TAILS OPPOSITE OPPOSITE SIDE BY SIDE BY SIDES SIDES SIDE SIDE Number of “holes” 9 11 11 11

Number of Xings 10 12 12 12

Nb of Xings in SPart 6 6 7 7 SET NB of Xings 0 6 0 0 In between SET Nb of Xings above 2 0 4 5 SPart SET Nb of Xings under 2 0 1 0 SPart SET

Those 2 by 2 comparisons show the absence of any “similarity”.

Table 5 bis ZEPPELIN HUNTER ASHLEY ALPINE BUTTERFLY Over Equality Equality S S All chirality minus/plus minus/plus

Type of blue K S CW S CW S CW S CW OVERhand OVERhand OVERhand OVERhand

Type of red K Z CW S CW S CW Z CCW UNDERhand OVERhand OVERhand OVERhand

PAIR OF TAILS OPPOSITE OPPOSITE SIDE BY SIDE BY SIDES SIDES SIDE SIDE Number of 9 11 11 11 “holes”

Number of 10 12 12 12 Xings

Nb of Xings in 6 6 7 7 SPart SET NB of Xings 0 6 0 0 in between SET Nb of Xings 2 0 4 5 above SPart SET Nb of Xings 2 0 1 0 under SPart SET

Table 5 ter

ZEPPELIN ASHLEY ZEPPELIN ALPINE BUTTERFLY Over Equality S Equality S All chirality minus/plus minus/plus

Type of blue K S CW S CW S CW S CW OVERhand OVERhand OVERhand OVERhand

Type of red K Z CW S CW Z CW Z CCW UNDERhand OVERhand UNDERhand OVERhand

PAIR OF OPPOSITE SIDE BY OPPOSITE SIDE BY TAILS SIDES SIDE SIDES SIDE Number of 9 11 9 11 “holes”

Number of 10 12 10 12 Xings

Nb of Xings in 6 7 6 7 SPart SET NB of Xings 0 0 0 0 in between SET Nb of Xings 2 4 2 5 above SPart SET Nb of Xings 2 1 2 0 under SPart SET

Table 5 quater

HUNTER ASHLEY HUNTER ALPINE BUTTERFLY Over Equality S Equality S All chirality minus/plus minus/plus

Type of blue K S CW S CW S CW S CW OVERhand OVERhand OVERhand OVERhand

Type of red K S CW S CW S CW Z CCW OVERhand OVERhand OVERhand OVERhand

PAIR OF TAILS OPPOSITE SIDE BY OPPOSITE SIDE BY SIDES SIDE SIDES SIDE Number of 11 11 11 11 “holes”

Number of Xings 12 12 12 12

Nb of Xings in 6 7 6 7 SPart SET NB of Xings 6 0 6 0 In between SET Nb of Xings 0 4 0 5 above SPart SET Nb of Xings 0 1 0 0 under SPart SET

Table 5 quinte ITEM N°1 ITEM N°2 ITEM N°3 ITEM N°4 Character N°1 Character N°2 Character N°3 Character N°4 Character N°5 Character N°6 Character N°7 Character N°8 Character N°9 Character N°10

Item 1 and item 2 are SIMILAR

Item 3 and Item 4 are ANALOGUE V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 45 sur 59 ©Charles HAMEL aka Nautile

DISTANCE – SIMILARITY CELLS MATRIXES The locking cells matrixes show there is no similarity !

Virgin matrix locks RED with BLUE 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3

ZEPPELIN HUNTER 5 discordant out of 5 (100) locks RED with BLUE locks RED with BLUE 1 1 1 1 1 2 1 2 1 3 1 3 2 1 2 1 2 2 2 2 2 3 2 3 3 1 3 1 3 2 3 2 3 3 3 3

ASHLEY Alpine Butterfly 3 discordant 2 concordant out of 5 (60) locks locks RED with BLUE RED with BLUE 1 1 1 1 1 2 1 2 1 3 1 3 2 1 2 1 2 2 2 2 2 3 2 3 3 1 3 1 3 2 3 2 3 3 3 3

ZEPPELIN ASHLEY 1 discordant 2 concordant out of 3 (33.33) RED locks BLUE locks with RED with BLUE 1 1 1 1 1 2 1 2 1 3 1 3 2 1 2 1 2 2 2 2 2 3 2 3 3 1 3 1 3 2 3 2 3 3 3 3

ZEPPELIN Alpine Butterfly 4 discordant 1 concordant out of 5 (80) RED locks BLUE locks with RED with BLUE 1 1 1 1 1 2 1 2 1 3 1 3 2 1 2 1 2 2 2 2 2 3 2 3 3 1 3 1 3 2 3 2 3 3 3 3

ASHLEY HUNTER 4 discordant 1 concordant out of 5 (80) RED locks BLUE locks with RED with BLUE 1 1 1 1 1 2 1 2 1 3 1 3 2 1 2 1 2 2 2 2 2 3 2 3 3 1 3 1 3 2 3 2 3 3 3 3

Those matrixes clearly show the absence of any “similarity” which would mean that we have ZERO discordant and 100% concordant. Version date 2011 Oct 19th RELEASE CANDIDATE Page 47 sur 59 ©Charles HAMEL aka Nautile

Using the “ distance criterion ” the following statements can be made :

ASHLEY and HUNTER are not too far from each other but both are a greater distance from both ASHLEY and Alpine Butterfly who are separated from each other by a distance greater than the distance separating ZEPPELIN from HUNTER .

IF similarity is denoted

Z H with similarity implying logically ≡ZZ .. ≡ HH .. ≡AA .. ≡ABAB A AB ( ≡ is math sign for identity ) ZZ HH AA ABAB ⇔ the logical equivalence symbol could have replaced ≡

The author is blind to that similarity and cannot see anything else than Z ≠ H ≠ A ≠ AB ( H ≠ A ≠ AB≠ Z ≠ A ≠ AB≠ Z ≠ H AB≠ Z ≠ H≠ A ) ≠ symbol for « different »

Z ≠ ≠≠ ……. ≠…… ≠ H ….. ≠____≠…….. ≠…… ≠……. ≠…… ≠… ≠ ≠ ≠ AB.…. A≠ ≠ ≠ ≠ ….. .

All considered those four bends can be seen as being ANALOG or ANALOGUE or ANALOGOUS. Version date 2011 Oct 19th RELEASE CANDIDATE Page 48 sur 59 ©Charles HAMEL aka Nautile

ANALOG / ANALOGUE / ANALOGUOUS

in http://www.merriam-webster.com/dictionary/analogy

Origin of ANALOGOUS Latin analogus, from Greek analogos, literally, proportionate, from ana- + logos reason, ratio, from legein to gather, speak — First Known Use: 1646

a correspondence or partial similarity :

in http://oxforddictionaries.com/definition/analogous

analogous(analo|gous) - adjective *** comparable in certain respects, typically in a way which makes clearer the nature of the things compared : they saw the relationship between a ruler and his subjects as analogous to that of father and children

*** Biology(of organs) performing a similar function but having a different evolutionary origin , such as the wings of insects and birds. Often contrasted with homologous

In Oxford Advanced Learner's Dictionary Academic

(formal) analogous (to/with something) similar in some way to another thing or situation and therefore able to be compared with it

This illustration is in congruence with the strict meaning, the only one acceptable in a technical discussion, of ‘similar’, ‘similarity.

The four bends are about as similar between each other as those “wings” are similar from the point of view of the function “flight” that is another manner to say they ARE NOT AT ALL similar ( not all their ‘traits’ are common) but may be considered as being analogous . ( a very small number of common ‘traits’ seen in a weak form : one knot is of the same ‘sort’ ‘ S’ OVERhand OR ‘ Z’ OVERhand OR ‘ S’ UNDERhand OR ‘ Z’ UNDERhand while the second must be ‘weakly’ describe as ‘ a thumb knot )

Table 6 S OVERhand Z OVERhand S Z UNDERhand UNDERhand S OVERhand * HUNTER * ALPINE * ZEPPELIN * ASHLEY BUTTERFLY Z OVERhand * ALPINE * HUNTER * ZEPPELIN BUTTERFLY * ASHLEY S * ZEPPELIN * HUNTER * ALPINE UNDERhand * ASHLEY BUTTERFLY Z * ZEPPELIN * ALPINE * HUNTER UNDERhand BUTTERFLY * ASHLEY

If two bends are in different cells then they CAN NOT BE SIMILAR beyond the most trivial similarity : they are bends !

If two bends are in the same cell then if they have different number of “holes and/or of crossings and/or have different “tails out” ( same side vs opposite ) then they CAN NOT BE SIMILAR beyond the most trivial similarity !

The two bends sharing commons cells are distinct by the type of tails : opposite versus side by side ( also by the distribution of sets of crossings )

As you can see there seems to exist a potential for exploring. ( WHITE cells )

BUT… if we do not bias the selection by taking just those four bends but taking all the bends with interlocking thumb knots in ABok then we have this table 6 bis

Table 6 bis S OVERhand Z OVERhand S Z UNDERhand UNDERhand S OVERhand HUNTER ALPINE #1408 ZEPPELIN ASHLEY BUTTERFLY #1425 #1408-9 #1412 #1426 #1453 Z OVERhand ALPINE HUNTER ZEPPELIN #1408 BUTTERFLY ASHLEY #1425 #1408-9 #1412 #1426 #1453 S #1408 ZEPPELIN HUNTER ALPINE UNDERhand #1425 ASHLEY BUTTERFLY #1408-9 #1412 #1426 #1453 Z ZEPPELIN #1408 ALPINE HUNTER UNDERhand #1425 BUTTERFLY ASHLEY #1408-9 #1412 #1426 #1453

If we are searching for trivial similarity then it is not 4 but at least 10 “trivially similar” bends – 2 interlocking thumb knots that we could find and if we don’t bother with ‘holes’, Reidemeister, tails, chirality… we can be happy fools.

The #1408-9 #1408 #1412 #1425 #1426 #1453 I redrew without the unacceptable points shown in ABoK.

This article use “standardised” drawings so as to get and maintain comparability with the previously shown bends. DO NOT, UNDER ANY ACCOUNT, DIRECTLY USE ASHLEY’s DRAWINGS FOR COMPARISON PURPOSE ; THEY ARE TOO OFTEN UTTERLY HOPELESS FOR THAT .

Ashley botched many of his drawings (he made a really mammoth job for which I have respect but I have no worshipping blindness. His book is the reflection of average education and intelligence sustaining an obsessive leaning but without great methodology.).

He was not rigorous enough, you will need to verify what he gave and make your own drawings while taking the greatest care to comply with “comparability” and correct representation.

There are two disgusting mistakes about ‘appraisal’:

--- first one is unconditional acceptance for the sake of blind admiration with no critical objective intellectual evaluation done.

In fact this is just an insult as a work must be accept after deep and long analysis made with an open and critical stance and I hold the view that Ashley is very much a victim of that with so many persons uncritically swallowing all that is given!

--- second one, no less and no more stupid than the first, is immediate rejection without a deep and long analysis made with an open and critical stance.

The blind Ashley’s worshippers will be often led astray if they give their trust to his drawings for comparison purposes, may be even for “making in the cordage” purpose in not so rare cases.

Photo 9 yellow = CW ‘S’ OH green= CCW ‘S’ OH

Photo 10 yellow = CW ‘S’ OH green= CCW ‘S’ UH

Photo 11 yellow = CW ‘S’ OH green= CCW ‘S’ UH

Photo 12 yellow = CW ‘S’ OH green= CCW ‘S’ UH

Photo 13 yellow = CW ‘S’ OH green=CW ‘S’ OH

Photo 14 yellow = CW ‘S’ OH green= CCW ‘S’ UH

Fig 35 The cordage routes

Fig 35 to 37 do not offer the tiniest similarity beyond the obvious trivial one : they are bends using two thumbs knots The pale green areas show the part of the “red knot” which escape the emprise of the ‘blue knot.

Table #1408-9 locks RED with BLUE 1 1 1 2 2 1 3 1 3

Fig 36 The full diagrams Table #1408

locks RED with BLUE 1 1 1 2 2 1 3 1 3

Note #1408-9 and #1408 Table #1412 locks RED with BLUE 1 3 2 1 2 2 2 3 3 1

Fig 37 the sets of crossings locking formulas Table #1425 locks RED with BLUE 1 2 2 1 2 2 3 1 3 2 3

Table #1426 locks RED with BLUE 1 2 2 1 2 2 3

Table #1453 locks RED with BLUE 1 1 Fig 38 the cells formulas 1 2 2 2 3

Table summary of interlocking as cells formulas

Fig 39 Going back to #1408-9 and 1408 which is an interesting couple: - drawings in ABoK : no comparability, hidden crossings, added crossings, side by side tails shown as opposite tails.

They have locking mechanisms that have Identical formulas both in sets of crossings and in cells numbers matrix. The same cordage route or shadow can be used for both. They have 11 crossings that are common BUT They differ by ONE crossing which is quite sufficient to forbid “similarity”.

None of the ten bends is similar to any of the other, absolutely none.

Only analogy exists here.

The very low level similarity acceptable is : made of two interlocking thumb knots more or less convolutedly intertwined.

THE END

Printer friendly Table 3

ZEPPELIN HUNTER ASHLEY ALPINE BUTTERFLY Blue knot CW CW CW CW orientation

Blue knot S S S S chirality

Red knot CW CW CW CCW orientation

Red knot Z S S Z chirality

Over Equality Equality S S All chirality minus/plus minus/plus Type of blue K OVERhand OVERhand OVERhand OVERhand

Type of red K UNDERhand OVERhand OVERhand OVERhand

PAIR OF OPPOSITE OPPOSITE SIDE BY SIDE BY TAILS SIDES SIDES SIDE SIDE Number of 9 11 11 11 “holes”

Number of 10 12 12 12 Xings

Nb of Xings in 6 6 7 7 SPart SET NB of Xings 0 6 0 0 Inbetween SET Nb of Xings 2 0 4 5 above SPart SET Nb of Xings 2 0 1 0 under SPart SET