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TURK'S HEAD
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ENLARGEMENT of THK / turks-head knot / turk's head knot STEP BY STEP in
slow sequence
This is specially for Friend Jimbo aka and self named The Wretched Knottyer of
Lower-Eastern Cordage. Salut mon Ami.

2 slideshows
this one show a blow by blow ( 12 steps) 4L B Right enlargement
this one is the precedent ( 7 steps) BUT WITH THE MULE SYSTEM

A slideshow of 19 pictures of a RIGHT side enlargement ( real big coloured cordages )
large format ( slow uploading)
1024 pixels format : reasonable speed

The secret of the mule unveiled : the mule is the dummy "shadow splitting the tracks"
you are doing step after step all along the "laying the track" phase.

A good safe way to learn making enlargement. Better not use it when you have mastered
enlargements which are really dead simple once you have "seen" it" and the mule system,
a very old trick, whose inventor is lost in the night of time.

This PICASA slideshow is of the Left enlargement of a 4L 5B to a 6L 7B in photos



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ENLARGEMENT of THK / turks-head knot / turk's head knot :
THE SECRET OF THE MULE !


This is almost never caught on first time around when explained with words only or with
poorly sequenced drawings (my really charitable opinion on what I have seen on the Net,
not a fact).

This 'secret'  has been unveiled in sequential diagrams and in photographies given in the topic
just above.
.Here it is again for the Lazy One.
The mule system in diagrams
The mule system in photographies

and now...the mule system in words :

The mule system  is a dummy splitting of the tracks you make alongside the 'laying for keep'
: Doing so you "lock" your added BIGHTt and you have an easy to use "disambiguating"
device.

At the end of the track laying phase you do the 'splitting the tracks' phase for keeps with the
help of the mule that is then 'sacrificed" after having shown the way.

But first (that will make you feel how much a mule may be useful tool) the enlargement
high points.


--- first stone tripping your feet  in enlargement is :
LEFT / RIGHT ?  (lateralisation given relative to the SPart-WEnd vector)
which one to choose?

Any will do get you an addition of BIGHTs and LEADs BUT you will not get an identical
expansion with one compared with the other.

One add more L & B than the other.
Which one ? It all depends on the starting THK !

-- second stone : Laying track is the more difficult of the two phases.
Splitting the tracks is relatively easy as you have two lines to guide you, they are very
distinctive in their crossing behaviour.
They are adjacent to each other and are doing identical OVER and UNDER crossings when
going from one KNOT EDGE or BIGHT-RIM to the other, so they are marring the nice
U-O or O-U  periodicity with either O-O or U-U.

Those pairs of O-O and U-U you must *split* to get again a nice O-U-O ( respectively
U-O-U ).

 'Retournons Ó nos moutons', 'let us come back to our sheep' as they say in France.

Laying : starting with a finished ONE PASS THK atthe very point whenre you are ready to
"double" it.
Consider the frame of reference for orientation as the one used in my diagrams.

Please do *as if*  running a course parallel to the SPart-WEnd vector (play at following
the leader as in doubling)  making the same type of crossing  O or U that the SPart-WEnd
vector does BUT THAT ONLY as far as the frontier.

There, at the BIGHT RIM,  two points are the keystone of a successful THK
ENLARGEMENT.
They must be fully understood and memorized to open the road to brilliant success.

--- Point One is : as you are stepping "out of " the THK into the great void you will want the
security of home and you will return as soon as possible inside the THK.
SPart-WEnd vector goes out stepping over one BIGHT, then immediately gets back
stepping over the next ( immediately next) BIGHT, always using the SPart-WEnd vector as
guide  but with two twists often overlooked and not understood :

AT THE FRONTIER no more nice alternating O-U ( U-O) but  a O-O ( or U-U as is the
case ):  TWICE IN SUCCESSION YOU MAKE THE SAME TYPE OF CROSSING.

--- Point Two : you must become agile at doing what politicians do as easily as breathing  
that is YOU CHANGE SIDE.
If on going out you were on the RIGHT (LEFT) then you return in on the LEFT (RIGHT)
side of the SPart-WEnd vector. (a vector is an oriented arrow : it as a point and a feathered
end so there is no trouble whatever your position may be or the position of the arrow
-relative to each other - you will always know were  the RIGHT side of the arrow is and  
where the LEFT is. Feathered extremity is the SPart and point is the WEnd ).

Do that at each frontier ( bight rim, bight border, knot edge )  and you are (almost) out of
the woods.

Here a little 'apartÚ' quite adapted to the topic seeing how many of us were lost first time out
in the vast wilderness of THK enlargement .
Story goes that Davy Crockett said :
I can’t rightly say I’ve ever been lost, but I’ve been mighty perplexed for two or
three days runnin'
-- Davy Crockett



We all like to pretend that enlargement has *never* been a problem for us.
Well I have new for you : there are several mathematical facts behind enlargements
and the "doing it "part is simple enough ONCE YOU HAVE MASTERED IT.
Even then be honest and confess than from time to time, you go astray , oh! certainly not
lost but mightily perplexed may be.

The other stone that will trip you is not a "theoretical" one but a very practical one.

Doing even mammoth enlargement is (relatively) easy on a diagram, but with cordage that is
"une autre paire de manches" altogether ( 'another pair of sleeves' meaning that may be you
need to become someone else )

Cordage tend not to stay put,  the more so with a THK in the making and you may find that
for large addition of BIGHTs ( say you add 8 , that means you have 4 to keep in line till you
get to splitting them ) it is easier said than done as a BIGHT tend to slides "inside" the warm
 inside of the THK ; so you better keep them in line and quite often make the necessary
correction to have them staying where they are supposed to be.

Well, this tripping stone of the BIGHT sliding inside is pulverized by the use of a mule.

As you do in parallel your laying the track for keep and your dummy splitting the tracks with
the mule all is "locked" in place.

Some persons (the less butter you have, the more you need to spread it on your bread !),
will consider as an exploit to be making things difficult when they can be made simple and
will disdain the MULE.

Well in my world there is no dishonour in using it for high level enlargement or iduring the
whole learning phase.
Better put in your neurons a "clean program" than inserting "wrong lines of instructions" that
will makes bugs and need a re-writing. 

Australian Aborigines (the only rightful owners of the whole continent and its riches) are
said (don't know if that is true or not)  to change their self-attributed nickname and so from
"the one that try to do enlargements" when they judge they truthfully can change it to "the
one who does enlargements"

I tried to give you all I know or think I know and retained nothing.
I hope this will be enough to allow you, with a bit of personal effort, to change your name !



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ENLARGEMENT : use of"corridors"
(something I personaly found and proposed here
mainly for person wanting to do their own researches )

Illustration one
Illustration two

A matrix presentation of the corridors used for each type of enlargement.

For persons wanting to try theirs hands a finding 'new' facts here are :
first Excel worksheet  and   second worksheet to start you on the way of facts finding

About "corridors",  half of them  (frame of reference : cylinder held vertical as in my
diagrams  are going from BOTTOM-LEFT to TOP-RIGHT and half from
BOTTOM-RIGHT to TOP-LEFT.

The total of corridors in a THK is equal to 2 * B (twice the number of BIGHTs)

Now  for the ENLARGEMENTS illustrated in the 2 Excel worksheets linked just above
only a part of the sum total of corridors of a given knot is in use in a given enlargement .  
Here is a table
______________________________________________________________
THK                 sum total of
corridors
sum of used
in RIGHT
enlargement
BIGHT
added
sum of used
in LEFT
enlargement
BIGHT
added
3L  2B 4 2 1 * 2 2 1 * 2
3L 4B 8 6 3 *2 2 1 * 2
4L 3B 6 2 1 * 2 4 2 * 2
4L5B 10 8 4 *2 2 1 * 2
4L 7B 14 4 2 * 2 10 5 * 2
5L 3B 6 4 2 * 2 2 1 * 2
5L 8B 16 10 5 * 2 6 3 * 2
6L 5B 10 2 1 * 2 8 4 * 2
7L 3B 6 2 1 * 2 4 2 * 2
10L 7B 14 10 5 * 2 4 2 * 2

This diagram made for a very good friend shows the use of corridors in the enlargements of
a 4L 5B

Whatever the side on which you choose to do your enlargement, at the end of the laying
track phase and and the end of the splitting track phase the path followed must close on
itself, which makes it a circuit,  and that is governed by mathematical consideration of
"periodicity"

In this illustration think about "CORRIDOR" and not of the cordage route of the THK to be
enlarged. (cordage route and SHADOW which is the correct topological concept are
similar)
On the way UP  you *may not* play phantom and go through blue walls and on the way
DOWN you *may not* go through yellow walls.

Only the 'laying track' phase is shown up to its closing at the start of the second tour where
begin the "splitting tracks" phase, as it was just to show the principle of closed path at work.

Added  in a 2010 revision for misspelling :
ENLARGEMENTS typology (Type I and Type II , Right, Left) I learned much later is
SCHAAKE & TURNER hard work and HALL used SCHAAKE's typology in his book
on THK published several years after SCHAAKE's published his research.
So don't get the wrong idea some managed to get in their head : The ENLARGEMENT
naming is certainly not HALL's .


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THK ENLARGEMENT   (from MAT perspective this time) 

Drawing this enlargement , a simple, low level one, in  MAT diagram really tasked my sense
of geometry and perspective

Here is the result.

This one shows the "corridors" of a quite simple mat : 5L 3B

This one show a 5L 3B mat enlarged Left to a 9L 5B : the SPart is shown highlighted in
yellow,  red is lying the track and  green is splitting the tracks.
I dispensed with putting the crossing as this only muddle the affair.

This one show the 9L 5B with the 5L 3B which is still hiding inside  same colour coding as
above).

Compare all this with the diagrams for cylinder given previously.

I will certainly never try to make this enlargement or a bigger one in real cordages on a
MAT THK.
Believe me, cylinders are the easy way, must come from the fact that we were once
arboricole apes and loved trees !




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A FEW FORMULAS VALID ONLY FOR |L-B| = 1 or SQUARE THK


Let
alpha = (L - B + 1) / 2    and beta = ((L -B - 1) * -1) / 2

for L > B   ( L-B = +1)   alpha = 1   beta = 0
for L < B   ( L-B = -1)    alpha = 0   beta = 1
then

RIGHT ENLARGEMENT  
Number of ADDED LEAD is
( alpha * 2) + ( beta * ( L - 1) *2)

L - B = +1    Lead added =   2
L - B = -1    
Lead added =   ( L - 1 ) * 2

Number of ADDED BIGHT is
( alpha * 2) + ( beta * ( B - 1) * 2)

L - B = +1    Bight added =  2
L - B = -1     Bight added =   ( B - 1 ) * 2
LEFT ENLARGEMENT
Number of ADDED LEAD is
( alpha * ( L - 1) * 2 ) + ( beta * 2)

L - B = +1    Lead added =  ( L - 1 ) * 2
L - B = - 1    
Lead added =  2

Number of ADDED BIGHT is
( alpha * ( B - 1) * 2) + ( beta * 2)

L - B = +1    Bight added =  ( B - 1 ) * 2
L - B = - 1    Bight added =   2

Here is an Excel sheet
  (only if you have Excel on your computer) :
just enter the number of LEAD and the number of BIGHT in the green cells ( A2 & B2 )
and you will get the computing in the yellow cells.
There is no check to verify if your entries comply with  |L-B| = 1 so if your keying in is for
something else than a SQUARE THK you will be in the condition to observe : GIGO
(Garbage In , Garbage Out)




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A FACT valid for any THK  

Let  L(righ)t  , L(left) , be LEAD added  by Right & Left enlargement to the L and B of
the starting THK
B(right)   ,  B(left)  are for BIGHT added...

then
Sum total of L added by R + L enlargement = L(right) + L(left) = 2 * L
Sum total of B added by R + L enlargement = B(right) + B(left) = 2 * B

example

starting THK        10L    3B                                 3L    5B
Right                    +6      +2                                +2      +4
Left                     +14     +4                                +4     +6
sum                     +20     +6                                +6    +10
                         (2*L)    (2*B)                         (2*B)   (2*L)
(Thank you Doug Varney aka Knotworkn for spotting one error in one of those examples .)



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Thinking about ENLARGEMENT AGAIN

*** Added BIGHT seems to be a  function
of pre-existing BIGHT,
of the existing LEAD,
of the added LEAD
as the more LEAD pre-existing the less BIGHT are added so it is probably 1/L
influence.the more BIGHT pre-existing the more BIGHT are added so it will be
probably a direct B influence.

*** Added LEAD seems to be a function
of  pre-existing LEAD,
of the existing BIGHT,
of the added BIGHT
as the more BIGHT pre-existing the less  LEAD are added so  it is probably 1/B
influence.
the more LEAD pre-existing the more LEAD are added so it will probably be a
direct L influence.

I think that this will need an algebraic equation using several unknowns with integer
coefficients, for which of course  integer solutions are required ( as LEAD and BIGHT can
only be added in 'integer number'
Now the difficult part is to find that equation and to work out an easy way to get solutions.

Delta  is the "growth gradient ) δ

δB   →   f ( B ; L ; δL )
δL   →   f ( L ; B ; δB )



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UNROLLING OF TIME SEQUENCE COMPARED WITH SPATIAL
SEQUENCE FOR ENLARGEMENT PROCESSES


This on the 4L 5B THK with RIGHT and LEFT ENLARGEMENT

Picture one : SPATIAL SEQUENCES compared

Picture two : TEMPORAL SEQUENCES compared

This clearly show that enlargement is a true LINEAR EXPANSION IN TIME visible as 
an INTRICATE EXPANSION IN SPACE and that the expanding sequence is really a
prolong of the starting THK.



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EXPLORATION OF EXTENDING ENLARGEMENT TO OTHER KNOTS
THAN THK


I made an attempt with Roy Chapman 18 faces knots and am waiting his practical
verification ( I think there is a little mistake : lower rim bight 1 )

I did another one on the Ocean Plat ( ABoK #2243)

Here is how it comes out in draft mode , the crossings were verified and it is OK
I will have to draw it proper.

Added 2008 Octobre 23rd  a week or so after the posting about this exploration
Pragmatic proof by Donald B. WRIGHT ( always a very neat  knot tyer, perfectionist ! )
Don made this mat from the drawing with the crossings indicated and his opinion on it is

[open quote]

I tried it and it worked. Very very hard to adjust and dress.smiley Use bigger grid
next time. (at least 4X6) 1.8mm cord.

[end quote]
 

Added  in a 2010 revision for misspelling :
I learned  latter that in The Braider SCHAAKE made theoretical and practical exploration on MATS ENLARGEMENTS

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Copyright 2005 Sept - Charles Hamel / Nautile -
Overall rewriting in August 2006 . Copyright renewed. 2007-2012 (each year)

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