Copyright Charles HAMEL 20 Avril 2011
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DISMANTLING SOME KNOTS DIAGRAMS AND REBUILDING THEM (ABoK #2216 ; #2217; #2218 ; #2219 (1391) ; #2222 ; #2232 ) AS THEY SHOULD HAVE BEEN DRAWN (IMO) IN THE FIRST PLACE :
AS CYLINDRICAL DIAGRAMS. Why “as it should have been” ? Simply for the fact that I do think that it is quite aberrant and inefficient to draw as a flat mat what is supposed to be a ‘VOLUME’ COVERING (core inside and the knot is just what covers the core area or surface) that will need quite a lot of fairing and dressing to be put on the volume to be covered. = AREA, SURFACE of a globe/ball. Not my perspective but Mathematics’ perspective! GLOBE (the globe of a breast !) or BALL is VOLUME. VOLUME can be: sphere, ovoid, octahedron, tetrahedron…anything that can be properly covered by the knot. SPHERE
In some other cases instead of a covering it could be a GLOBE / BALL knot (no core inside ; the knot itself is the whole volume).
Please, please ! do not do as ignorant persons do ; see the pompous asinine quote under. [open quote] You commented on one.../…projects in the "So-and-So" album but you need someone to teach you the difference between a Turks Head and a Globe knot; that's a Globe knot! …/… And if you want to learn the difference between the two, I know a guy that sells a great set of books you can use to learn about them. [end quote]
That dull lesson giver does not even know what a ‘globe knot’ is, he does not even realise that in fact he is speaking of spherical covering and is making a bad confusion about those two types of knots applications. Spherical (or other shape surface) covering knots are not a particular knot type but rather some particular FUNCTION, USE or APPLICATION of some knots that are quite diverse in types. ▲Most are NESTED BIGHTS CYLINDRICAL KNOTS (single or multiSTRAND) * REGULAR SYMMETRIC * REGULAR ASYMMETRIC * IRREGULAR SYMMETRIC * IRREGULAR ASYMMETRIC
▲SOME TURK’S HEAD KNOTS can indeed serve as satisfying spherical covering in particular dimensions. ▲A Monkey’s Fist can be use either as a globe knot or as a spherical covering. It is certainly not with the book pointed to that someone will learn not to confuse globe knot and spherical covering. The book’s author does not seem to make a difference between them. My intent here is not so much to offer new diagrams as to show how anyone can “explore and find new tricks” : screen captures serve as illustrations but my exploration were made “by hand” and the software used only after the results had been verified.
Copyright Charles HAMEL 20 Avril 2011
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Lets us state some points. *** Most often the cylindrical diagram is easy to deduce once the coloured markers are in place, that despite the aberrant clockwise move that is shown on some of the flat diagrams that on a vertical cylinder translates into the aberrant BOTTOM-LEFT to TOP-RIGHT for ODD numbered HALF-PERIODS. ( to know the meaning I attached to ‘aberrant’ please read page 7 and beginning of page 8 of my .pdf on Pins notation at http:/charles.hamel.free.fr/knots-andcordages/PUBLICATIONS/KnotNotatio n-V1.pdf This aberration is corrected in all but one of the cylindrical diagrams and ODD numbered Half-Periods go from BOTTOM- RIGHT to TOP LEFT.
*** I also dispense with what is, in my opinion at least, a formidably inefficient way of indicating the nature of the crossings which is more hinder than help for the knot-tyer and may have some usefulness only if used with a cord laid on a printed diagram.
Ashley went for ‘proprietary formulation’ alas his utterly bizarre crossing ‘cryptography” : - is cumbersome, more hinder than help for the knot tyer - is quite error prone - does not allow the immediate perception of the “PATTERN” made by the crossings. All in all “ a work not really done and certainly not to be done” ( ni fait, ni à faire as goes the French expression to qualify a somewhat less than ideal job) A parté : immediate rejection without a carefully and honest open-mind critical evaluation (as for academic works) is bad but I do wonder for sure if parroted and medullar ( automated, by-passing the brain higher structures and functions) praise and admiration without same examination is not worse. Ashley’s work is a tremendous effort, irreplaceable, but is far from lacking in sore points !
We begin with a “shock” treatment using #2218 first. After that first dive in cold water we take a more reasonable learning gradient : #2216, #2222, #2217, #2219 (#1391), #2232
Readers who prefer to avoid any possibility of brain sprain will best go directly to #2216, read till they are finished with #2217 then go back to #2218 and finish with #2219 , #2232.
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Copyright Charles HAMEL 20 Avril 2011
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ABoK #2218 DISSECTION : Fig 1
Fig 2 Lets us dispense with the way of indicating the nature of the crossings which is more hinder than help. (Fig 2 ) A first glance immediately shows an intimation of possible SYMETRIC REGULAR NESTED-BIGHTS CYLINDRICAL KNOT, 4 BIGHTS-NESTS , 2 BIGHTS-PER NEST, 8 LEADS, x=6
Fig 3
Fig 4
But is that a reality perceived or just wishful thinking ? A first quickly made grid, using ARIANE the Claude HOCHET’s program that makes grids of Nested-Bight Cylindrical Knots, shows that 2 STRANDS are needed to make such a knot with a 4 BIGHTS-NESTS, 2 BIGHTS per NEST, distance x = 6 and 48 CROSSINGS. Fig 5.
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Copyright Charles HAMEL 20 Avril 2011
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Fig 5 It is not so much the 48 CROSSINGS that make it “no go” than the 2-STRANDs. A PINEAPPLE BUT NOT A STANDARD HERRINGBONE-PINEAPPLE !
Fig 6
The first point to solve, if we are to succeed in our making of a cylindrical diagram, is that the need of two STRANDS must be nullified and transformed to single strand-ness so to speak ; the 2 strands must be reunited by making the route of one flows into the route of the other .
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Fig 8
Fig 7
There is something that we must examine (and do) in the bottom right hand corner of Fig 7
Fig 9
Fig 10 Fig 9 is equivalent to Fig 10 but is much more efficient for spherical covering practical tying. Remains to see if Fig 11 and Fig 12 are equivalent to Fig 8 .
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Fig 11
Fig 12
Obviously the answer is NO. None is equivalent to Fig 10. So better leave that trail that was looking for a simplification by re-entering. In Fig 12 (Fig 14) ? is a newly created crossing and ?? stands for crossing #23 GHOST ! It has 47 FACEs instead of 48 but it may (though I doubt it) be worth a try as for some ASYMETRIC IRREGULAR NESTED-BIGHTS CYLINDRICAL KNOTS.
Fig 13
FiG 13 is the correct cylindrical diagram to use.
Fig 14
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Copyright Charles HAMEL 20 Avril 2011
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ABoK #2216 Fig 1
Fig 2
Fig 3
In the first second of looking at that diagram it is evident that this is a REGULAR SYMMETRIC NESTED-BIGHT CYLINDRICAL KNOT drawn in an absurd flat form.
It is quite easy, once the coloured PINs are in place, to transform it into a cylindrical diagram.
Fig 4
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Fig 5
Fig 6
Fig 6 shows a grid of a 4 BIGHTS-NESTS, 2 BIGHTS per NEST, 24 FACES, 5 LEADS, x=3 so it is “just like” #2216. NO! not at all. We have to comply with a correct OFFSET between the TOP and BOTTOM BIGHTS-NESTS.
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Fig 7 shows the correction to be made
Fig 8
Fig 8 shows the cylindrical diagram that is, by and large, a lot more helpful and descriptive for any knot tyer than the cumbersome flat diagram made by ASHLEY. The flat form is uneasy to use and also deforms the knot so much that it will need a lot more adjusting and dressing to be put on a ball that the one coming off of a cylinder.
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ABoK #2222 FIG 1
FIG 2
FIG 3
Note that there are SIX BIGHTs on one KNOT EDGE and THREE BIGHTs on the other KNOT EDGE== ASYMMETRIC
As there are NESTED-BIGHTs on one KNOT EDGE this leads to ASYMMETRIC NESTED BIGHTS CYLINDRICAL KNOT.
Putting in the PINs makes things clear and the obvious use of “usual” crossing (rather than the queer Ashley’s crossings) offers the knot-tyer a better and more immediate ‘visual acquisition’ of the PATTERN.
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Copyright Charles HAMEL 20 Avril 2011
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FIG 4
June 2011
FIG 5
FIG 6
It is easy with the use of colours to verify the congruence of the crossings of FIG 5 and FIG 6
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ABoK #2217 Fig 1
Another example of aberrant drawing under a flat mat form of what would have been better made under the form of a cylinder diagram because (quite evident to see) this is a SYMMETRIC REGULAR NESTED-BIGHTS CYLINDRICAL KNOTS. 40 FACEs 4 BIGHT-NESTS, 2 NESTS per BIGHT, just as #2216 but while x=3 for #2216 here x=5 and that makes a difference. Fig 2
This isometric cylindrical diagram is easy to deduce once the coloured markers are in place.
Copyright Charles HAMEL 20 Avril 2011
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Fig 3
Fig 4
The two coding O1-U1 (U1-O1) that can be applied. This #2217 is full of lessons
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Fig 5
Fig 5 is #2217 with O2-U2 ( or vice versa ) applied == herringbone PATTERN (NOT, repeat NOT, herringbone KNOT - a STANDARD HERRINGBONE KNOT DOES NOT HAVE ANY NESTED-BIGHTS, despite
what some person considered by himself and his friends as an “ex-spurt” put on his web pages. STANDARD HERRINGBONE KNOTS have all their BIGHTs along a unique BIGHT RIM on each KNOT EDGE so it is absolutely impossible to have NESTEDBIGHTS.)
Fig 6 is a simili- #2217 WITH ONE SLIGHT MODIFICATION x=7 instead of x=5 ( 56 FACEs ) Fig 6 It can also take the O2-U2 and get a Herringbone PATTERN as it is easy to verify (this suppleness of exploration is one of the great advantage of ARIANE, Claude HOCHET’s program for NESTEDBIGHTS CYLINDRICAL KNOTS.)
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ABoK #2219 FIG 1
FIG 2
FIG 3
Fig 2 shows the crossings in place and Fig 3 shows, between the “wasp” crossings, the aberrant points in the O1-U1 pattern as N° 38 and N°37 crossings are U1-U1 and NOT U1-O1 or O1-U1.
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Copyright Charles HAMEL 20 Avril 2011
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FIG 4
FIG 4 is the only way to obtain a ‘real’ symmetric nested-bights cylindrical knot with 4 BIGHTSNESTS , 2 BIGHTS per NEST, x=10. Unfortunately for our purpose it is TWO-STRAND so we will need a trick to make it 1-STRAND.
FIG 5 FIG 5 shows the trick used to make one strand route flow into the other strand route.
FIG 5 BIS
The orientation of winding in the cylindrical diagram is aberrant so we would need some geometrical manipulation to make it “normal” as in FIG 5 BIS in which ODD numbered HALF-PERIODS go from BOTTOM-RIGHT to TOP-LEFT Hopefully every reader will have noted that to get the correct orientation the INNER KNOT EDGE is put at the TOP and the OUTER KNOT EDGE at the BOTTOM. Plus the colours that read clockwise on FIG 7 are (in circular order) Yellow – Dark beige – Orange – Yellow and Dark violet – Mauve rose – Dark red – Blue are to be put IN THAT ORDER BUT WRITING RIGHT to LEFT. Or write them LEFT to RIGHT but READ THEM ANTI-CLOCKWISE (this must reminds you of Schaake’s Bights Algorithm. NO?)
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Copyright Charles HAMEL 20 Avril 2011
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FIG 6
Let us, as a « lesson » and mental (s)training, continue with the aberrant orientation.
FIG 7
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Copyright Charles HAMEL 20 Avril 2011
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FIG 8
FIG 9
FIG 9 is the way Ashley put the crossings.
Following is the “making of” of ABoK #1391 as a practical example for training purpose.
Copyright Charles HAMEL 20 Avril 2011
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THE MAKING OF OF ABoK 1391 as PRACTICAL STEP BY STEP EXAMPLE FIG 1
By calling this knot a TURK’s HEAD, ASHLEY shows, as alas in other instances, a staggering lack of efficient observation, depth of structured analyse and knowledge. He was just “following what I was told by tradition”, thoughtlessly, inattentively, without any of the critical spirit of the trained explorer. This knot has absolutely nothing in common with a THK as defined by perfectly known ( for many centuries) cycloids equations. See my features on the mathematics of THK. It is a SYMMETRIC NESTED-BIGHTS CYLINDRICAL KNOT light years from looking like any Regular Knots Row and Column coded O1-U1 known as THK. FIG 2
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Make a hand drawing, rough sketching is quite all right, mark the PINs and crossings, make a rough diagram under it and explore the HALF-PERIODS. Verify and finalise.
FIG 3
Make a neater diagram using an isometric graph paper.
FIG 4
If you are lucky enough to have a licensed copy then explore with ARIANE.
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Copyright Charles HAMEL 20 Avril 2011
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ABoK #2232 FIG 1
FIG 2
FIG 3
Copyright Charles HAMEL 20 Avril 2011
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FIG 4
FIG 5
June 2011
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Copyright Charles HAMEL 20 Avril 2011
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FIG 6 A TWO-STRAND IMITATION OF THE #2232 = A STANDARD HERRINGBONEPINEAPPLE - ONE SET of component THK is empty the other SET contains 2 THK 3L 4B
FIG 7
FIG 8
Still you can use just O1-U1
