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Copyright Charles HAMEL 20 Avril 2011 Page 1 sur 23 

Release version 1.1 June 2011 

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. 

SPHERE = 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. 

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 multi- 

STRAND) 

* 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.



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.



ABoK #2218 DISSECTION : 

Fig 1 

Fig 2 

Fig 3 Fig 4 

But is that a reality perceived or just wishful thinking ? 

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 

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.



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 

Fig 5 

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 .



Fig 8 Fig 7 

Fig 9 

There is something that we must 

examine (and do) in the bottom right 

hand corner of Fig 7 

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 .



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 14 

FiG 13 is the correct cylindrical diagram to use.



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



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.



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.



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.



FIG 4 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



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.



Fig 3 

Fig 4 

The two coding O1-U1 (U1-O1) that can be applied. 

This #2217 is full of lessons



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 NESTED- 

BIGHTS.) 

Fig 6 is a simili- #2217 WITH ONE SLIGHT MODIFICATION x=7 instead of x=5 ( 56 FACEs ) 

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 NESTED- 

BIGHTS CYLINDRICAL KNOTS.) 

Fig 6



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.



FIG 4 

FIG 4 is the only way to obtain a ‘real’ symmetric 

nested-bights cylindrical knot with 4 BIGHTS- 

NESTS , 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 

shows the 

trick used to 

make one 

strand route 

flow into the 

other strand 

route. 

FIG 5 BIS 

FIG 5 

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?)



FIG 7 

FIG 6 

Let us, as a « lesson » and mental (s)training, 

continue with the aberrant orientation.



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.



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



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.



ABoK #2232 

FIG 1 

FIG 2 

FIG 3



FIG 4 

FIG 5

Copyright Charles HAMEL 20 Avril 2011 Page 23 sur 23 


FIG 6 A TWO-STRAND IMITATION OF THE #2232 = A STANDARD HERRINGBONE- 

PINEAPPLE - ONE SET of component THK is empty the other SET contains 2 THK 3L 4B 

FIG 8 Still you can use just O1-U1 

FIG 7

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