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TURK'S HEAD
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Added 2010 June 26th

THE SEQUENCE RULE IN THE HERRINGBONE KNOTS

 - It is important in explaining why, before Schaake discovered it, no one ever published a Standard Herringbone knot with more than  TWO PASSES.

I did some computation of the probability of successful outcome when not observing the sequence rule . Not observing == using a fair coin and spinning it to decide which way to go.

For those liking illustration rather  than articulated words just have a look.

Up to PASS-3 included nothing can go wrong SEQUENCE-RULE-wise, even if you
make them in a random order.  

So up to PASS-3 you get 100% success       4 in 4 possibilities

BUT after that things change drastically for a randomly made sequencing :

PASS-4   33.33%   success       in fact 4 in 12 possibilities
PASS-5     8.33%   success       in fact 4 in 48 possibilities
PASS-6     1.67%   success       in fact 4 in 240 possibilities
PASS-7     0.28%    success      in fact 4 in 1440 possibilities
PASS-8     0.04%    success      in fact 4 in 10080 possibilities

Easy to compute : you will NEVER get more than 4 'success' in all the opened possibilities in a given PASS-N !

Open possibilities for PASS-(N)   = Open possibilities for PASS-(N-1) *  (N-1)

open possibilities for PASS-6 =  48 * 5 = 240

Why never more than 4 success ?

--- Once a Mistaken Knot has been made ALL its successor will be mistaken even if after that the Sequence Rule is complied with.

--- If the knot in generation PASS-(N-1) is a success then only ONE of its descendants will also be a success as there is only one way to be right and complying with the rule.


As  PASS-3 has 100% success that is 4 successful outcome out of 4 possible outcome
it follows that you will be stuck with only 4 success in the whole of any of the successive generations.

PASS-2 = there are  2 ways to make the PASS-2        ( think modulo and disposed on
a circle circumference)   
 2 - 1   ( anti-clockwise)    and   1 - 2 ( clockwise )

PASS-3 = there are  4 ways to make PASS-3          
 2 - 3 - 1     3 - 2 - 1        1 - 2  - 3       1 - 3 - 2

PASS-3 = there are 12 ways to make PASS-4
2 - 3 - 4 - 1     3 - 2 - 4 - 1        1 - 2  - 4 - 3       1 - 3 - 4- 2
2 - 4 - 3 - 1     3 - 4 - 2 - 1        1 - 4 - 2  - 3       1 - 4 - 3 - 2
4 - 2 - 3 - 1     4 - 3 - 2 - 1        4 - 1 - 2  - 3       4 - 1 - 3 - 2


only   2 - 4 - 3 - 1       3 - 4 - 2 - 1      1 - 2  - 4 - 3       1 - 3 - 4- 2 may hope to give birth to a successful descendant all the other lineages being already doomed to failure.

It is plain to see that with    A     A      A      A   disposed on a circle there are 4 ways to add a fifth  'A'

A   A   A     A     A             A     A   A    A     A  
A    A   A    A     A             A     A   A    A     A

there are as many 'slots' as there are PASSes already in place ( problem of posts and intervals on a circular fence ) and so on...

 so PASS-5  has 12 ( number of possibilities in PASS-4)  time  4  = 48 and only 4 can be 'success'


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Added 2010 June 26th  

POUNDING  THE NAIL OF THE ESSENtial DIFFERENCES BETWEEN A
Standard
HERRINGBONE Knot
and a Standard HERRINGBONE-PINEAPPLE Knot.  
See last topic in Turkshead_25 to understand some allusions.

Another illustration lay here compare those two STRUCTURALLY VERY DIFFERENT sorts of knots ( examples given :  PASS-5 size)  :

Only 2 ways to get  the HERRINGBONE right

120  ways ( A!) or factorial A ) to get  the HERRINGBONE-PINEAPPLE right



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Added 2010 July 3rd

ANOTHER POUNDING ( dedicated to 'les tÍte de bois' *** / 'the heads made of wood' as goes the French popular expression) ON THE NAIL OF THE ESSENtial
DIFFERENCES ( NOT SEMANTIC ! ) BETWEEN A
Standard
HERRINGBONE Knot
and a
Standard HERRINGBONE-PINEAPPLE Knot.

***une tÍte de bois or head of wood is a person rather...... 'thick-headed, and mulish'.
see topic of June 6th , last one in Turkshead_25. in order to understand the background.

As shown in this illustration  using just elementary observation of  FACTS, 'plain to see as the nose in the middle of the face'. 

Only someone totaly devoided of any power of observation will insist on persevering in the propagation such  an idea  as saying that the differences between
Standard
HERRINGBONE Knot
and a
Standard HERRINGBONE-PINEAPPLE  Knot
 are 'just semantics'.  ( yes he spelled it with 's' which just show that the poor guy does not even have a good handle on in own mother tongue  as with 's' it pertain generally to the science (a noun)  and is not an adjective ***)

What an astronomical level of...of....of...
I cannot even find a word strong enough and still polite.

This is not a question of "freedom of  speech" .   (freedom of speech is abused when used to throw stupid things in the air )
In any case  free speech need to be severely curbed (or so I think) when it crosses the limit and allows the deliberate propagation of utter stupidity that 30 seconds of attentive observation and the use of  a modicum of intelligence would have avoided, and when the end result in polluting other's minds.

Well YES ! You are right !   I AM STILL ANGRY !

I cannot stand so-called 'ex-spurts',  more ignorant than the persons wanting to 'learn',
propagating their own blissful absence of solid knowledge, no beg your pardon!  it is much more vicious than that : they do not propagate "absence of knowledge" they propagate 'warped, distorted, false knowledge' .
 
That, in my book, is a crime just as attempted murder is a crime because it is attempting to murder the intelligence, the learning stance, the acquisition of knowledge in their readers.

*** semantics : the whole body of study  while if one wants to refer to ' one word and the meaning
attributed to it  - which is clearly the semantic content of that guy's words ;-) - it is without 's'.

semantics: the whole body  of study of meaning in a language

semantic ( or semantical)  : the meaning value/content attributed to a word

It seems to me he does not even know the meaning of nest / nested.
Just to be sure I went to my trusted friend in such cases : The Cambridge International Dictionary of
English

nest [SET]..... a set of  things that are similar but of different size and have been designed to fit inside each other.

Operative words here are: DESIGNED TO FIT INSIDE EACH OTHER

 Not  knowing the semantic value of words in his own language is poor communication and very poor  
teaching !
So here Wooden Head  goes netherwards to an abyss of black ignorance and decided that in Standard
HERRINGBONE K.)  the bights ( of knots of identical size)  which are in fact staged along a single rim,
behind or in front of the others as is the case,   are 'nested' .

How does that measure on the scale of .... ( chose your semantic value for the '...'   ;-) )?




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Added 2010 July 3rd

KNOTS DIAGRAMS : COMPARISON  ISOMETRIC GRID versus SQUARE
ORTHOGONAL GRID

This illustration shows two representations of the cordage route of  a same knot :
one on  an ISOmetric grid
 the other on a  SQUARE ORTHOGONAL grid.

I find that the ISOMETRIC diagram  is the one respecting the 'relative proportion' between circumference in BIGHTs and 'length' in LEADs.



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Added 2010 July 4th
NESTED BIGHTS and NO-NESTED BIGHTS

PHASE SHIFT AND SCALING as neither a simpleton's explanation nor
semantics bla bla bla ( French for Blah blah blah  ;-)   ) or semantical personal quirk but as evident, objective, immediately perceptible FACTS.

No need of being trained in rocket science or of being of  uncommon intelligence.

Simple average sense of visual observation is more than sufficient, unless, alas for you, you happen to be one of a certain ex-spurt (see 2 topics above) misguided and uncritically admirative disciples that was led astray and being happy with being taken for a ride (out of hand half a dozen of names immediately sprang to my mind coming from the past)


*** NESTED-bights :  there is NO PHASE SHIFT between the components, you can easily align the radii (more than one radius makes for that bizarre spelling)

*** NO-NESTED-bights as in Standard Herringbone knots : there is a PHASE SHIFT between any two components (here only two), you cannot align the radii.

A 'pretty' image for those not liking words.

*** NESTED-bights : there is a SCALING in the components : they do not all have
identical dimensions (here 3 sets of dimensions so NOT a HERRINGBONE PINEAPPLE that have only 2 sets -one being 'empty' if need arise- ) but this scaling allows the 'nesting' in absence of phase shift.
SCALING implies : several bights rims = some 'outer', some 'inner' .In case of  all components being of the same size in L & B the role of the SCALING is taken by TRANSLATION. ( sliding or going crab-like)

*** NO-NESTED bights : NO SCALING needed as the phase shift takes care of the
'distribution and spacing' along a single bight rim.



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Added  2010 September 19th

PUTTING A MIS-ATTRIBUTION STRAIGHT as I dislike seeing someone - the
more so someone who cannot defend his property- he is dead- robbed ( even if involuntarily and with no intention to do harm by an otherwise nice and honest person )  of his work to see it attributed to an author who was not clear enough in his writing to unambiguously attribute the credit of the labelling of Enlargement Processes to the actual author of that labelling.

I saw that and let time pass hoping such a distressing fact would be corrected with time.
Alas !
see here

The author of the post declares "....is Tom HALL's "enlargement process #1..."

Most sorry but that is taking away from the real author of the studies and creator of the
nomenclature of the enlargement processes : namely GEORG SCHAAKE.
Enlargement Process #1  and #2 refer to the SIDE of the SPart-WEnd vector the
enlargement is done : right side for #1 and left side for #2. This labelling is Schaake's
in THE BRAIDER.

Tom  HALL was just using it in one of the books he committed ( Gallicism;  some book can be said "il a commis ce livre"  , just as the same way we say he " il a commis un meurtre") on the Turk'sHeads knots without, *apparently if I believe this posting*,  being clear enough in the attribution of the credit of the nomenclature used.for all readers to get it right first time.

HALL privately published Introduction to Turks-Heads in 1996
while SCHAAKE's Regular Knot Tree and ENLARGEMENT PROCESSES
was  published by the DEPARTEMENT of MATHEMATICS and STATISTICS of the
UNIVERSITY of  WAIKATO in 1991 (FIVE YEARS EARLIER!)
and in  1992  it was
An introduction to Evolution Processes



JUST READ THE BRAIDER !
As we say in France better address yourself to God than to one of his minors saints!

Wrong attribution put right .
As they also say in France " don't undress Paul to dress Jacques "



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Copyright 2005 Sept - Charles Hamel / Nautile -
Overall rewriting in August 2006 . Copyright renewed. 2007-2014 -(each year of existence)

Url : http://charles.hamel.free.fr/knots-and-cordages/

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