Nautile aka Charles Hamel's personal pages
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Without the freedom to make critical remarks, there cannot exist sincere
flattering praise
- Beaumarchais.

Tom HALL Introduction to Turk'sHead Knots,  privately published, can be
obtained (despite all I may express here you would be throwing the baby with the
water of the bath if you don't buy this book !  discard the silly theorizing he gives
but by all means do use the practical side )
at Martin COMBS '

this is my OPINION honestly given and NOT INCONTROVERTIBLE FACT.

I am no great admirer of the *manner* the brains has of expressing themself that is for sure,
but that does not prevent me from admiring the craft the hands deploy and anyway
from respecting the human being separated from the work.

This is used with the sincere intent of being a FAIR QUOTE of the 3 pages ( out of a 141
pages book)  evoking the coding .

IMO it is done in a manner rarely understood by readers if I take on account the many
questions raised and asked to me from here and there.

The "recipe" given is not really profoundly (only half-baked) justified on mathematical or
logical ground.

It stays, alas, at the, most usual in knots quarters alas, simplistic level of a regurgitated,
handed down recipe.
Even that regurgitation is not even done at the clearest level IMO (that is not entirely the
fault of Tom HALL, it is the way it is with "recipes", they have no real flesh and brain, they
are only a dismantled deformed skeleton.)

Coding as explained by Tom HALL of Wyoming (USA) ( Tom HALL is the pseudo of

We will see much later how we can use what has been exposed in the pages preceding this
one to clarify the matter a bit.

 As it stands in the book it is an hopeless task, for me at least ( and I know for sure for
several knots tyers ! ) , it must be erased and built anew if I want to stand a chance of
 really understanding it.

Very important to remember all the while :
HALL/HICKEY use horizontal mandrel as frame of reference and Nautile uses cylinders
held vertical.

This bring huge changes as can be easily seen with the 2 frames being compared
(already addressed in the page before that one : see table at bottom of page 7)

Please make a careful note of the fact that HALL is numbering BIGHT in his drawing  p27
in continuous sequence from one RIM to the other : THE NUMBER SEQUENCE IS
SHARED BY THE 2  RIM OF BIGHTs ( he call them cycles just as he calls PARTs what
I call LEAD ) while Nautile has been numbering BIGHT or "summit of period"
independently for the 2 RIMs.

Hall in page 18 use a 5L 4B - here this THK is treated in the 2 frames of reference.
L/B = B*n + r

L modulo B             5 modulo 4 = 1

the remainder of 5 / 4 is = 1  or if you want we can  write 5 - ( 4 * 1) = 1

if instead of 4 it is 2 then
5 modulo 2 = 1
remainder of 5 / 2 = 1  or if you want ( - (2 * 2 ) = 1

B - (L modulo B) = 4 - (5 mod 4) = 4 - 1 = 3  in fact this corresponds to L - 2 (as in
computing the JUMP )

Having FOUR bight to contend with you put
               .    .    .    .      
four points in line on which you STEP ( circular permutation or modular progression ) 3 by 3
staring from the leftmost point that is marked as  "ZERO"

I will use"#" instead of a point to avoid an ugly presentation

#       #        #        #
4 BIGHT to place. and counted not from 1 to 4 but from 0 to 3.

0                          1
#       #        #        #      third step toward right (→) from zero
now you make AS IF
0                          1        0             0                           1
#       #        #        #    rewind to    #       #        #        #

0                          1       
#       #        #        #
0                          1        0                           
#       #        #        #      #        #        #        #    third step toward right (→) from zero
0                 2         1        
#       #        #        #      
0                 2         1       0        3 
#       #        #        #        #        #        #
0       3         2        1        
#       #        #        #       

Then out of the vast blue yonder, without so much as a polite introduction,
on page 19- line 6, like a magician pulling a white coney from his hat the bolt strikes :


Well , err, please to meet you I am sure, but who the hell are you, where do you come

This is the trouble with handing down recipes !

Well it is complementary ( using 3 ) of  3   0   1    2
because when you add  the two lines you realize they complement to 3
0   3    2     1
3   0    1     2
3   3    3     3

this 3   0   2    1 you magically get by reading as in Arabic from RIGHT TO LEFT
the  0   3  2    1  ( recipe recipe recipe again as this is not a "reasoned" justification
but just a sleight of hand that "works")

All of a sudden ( I always give a mental somersault when being so victimized by
unreasoned assertion ) appear
\  /  \  /
said ( without any justification, just stark naked and unashamed affirmation :
this is "the coding of our 5 Part 4 Bight  so-called """casa""" knot".
Casa is just a not really useful and notreally intelligent  "proprietary" label for a THK
in particular for a "foundation THK " or "base THK" in an assembly of THK.
\  /  \  /  is supposed to represents "the coding of a  so-called """casa""" knot
is Over-one, Under-one, so the coding will be alternating slashes and back-slashes".

I don't know about you but I fail to grasp with the explanation given why  U
Under-one  Over one == \  /  \  /  being use in such a way even if I understand
that a true THK is Over one- Under Oner ( or vice versa )

Once again, handed down recipe oblige, without any procedural justification, we just
get a Grand Master interjection :   THAT IS THE WAY IT IS!

We are told to put  NOT    0  3  2  1  as we would have naively believed but rather
3  2  1  0 ; that is you put 0 at the other end !
not  3  0  1  2 but 0  1  2  3 that is you push 3 at the other end.
No justification given either for putting one Above and the other Under.

" at the bottom of this coding we write the cyclic- number ( me , that is 3  0  1   2 ) going
from left to right
( me : OK , usual way to read)  starting at the second number ( me :
why is that ? justify please, well sort of justification just follow ). The zero (0) would be
over the left hand bight column that does not have a coding so we don't need it
. On
long knots with more parts than bights we just keep repeating the cyclic bight numbers
 ( ???
AH ! so 0   1   2   3 were Bight related and not LEAD related  ???)  until all the codings
have a number over them.
( why is that ? no justification given again, no need in a recipe!).
On a narrow knot with less parts than bights we just start with the second cyclic
( me : why again ? ) and go until we run out of codings. The rest of the cyclic
numbers are not used
( me : again , why is that ,)

      3       2   1       0
   \    /   \    /
        0    1      2    3

and with that you get the soothing : "that completes our algorithm-diagram"


I am not a complete idiot and I heartily dislike handed down half-baked recipes.
I want true "grande cuisine" and that my intelligence be addressed rather than my
 memory or my bowing to the fallacy of "authority".


Then on page 20 to 23 Author goes trying to explain.
 IMO it would have been better to explain first and illustrated with a numbered
example in a second phase, that is the right way for training people or so I  
believed all the years I taught(medicine, diving ) and was led to believe by seeing
every other instructors applying that sequence.

Teaching cooking before teaching  how to buy at the market and how to make a
fire is the way of too many.

Then when you have failed preparing an edible meal  you are taught about buying
and making a fire, but in another location ( with another THK : no wonder people
get lost so easily )

Page 20
" the top row of cyclic bight-numbers are used for the LEFT to RIGHT half-cycles so they
are read by going from left to right
( me I don't' see why LOGICALLY one follow the other
, ). The bottom row of cyclic bight-numbers are used for the RIGHT TO LEFT half cycles,
so they are read by going from right to left
( again I don't see why LOGICALLY...)
The knot is tied in an upward direction, so the left to right half-cycles cross the coding from
lower left to upper right..... The right to left half cycles cross the codings going from
lower-right to upper left... 

Then the hilarious part " To make it easier to see we have written a "U' and "O" for the
unders and overs above the top cyclic bight-numbers and below the bottom cyclic numbers

Now " the best way to see how to use the algorithm-diagram is to do an example"
OH ! I thought it was the whole point of the preceding page  18 & 19.
How silly I am .

"This time we will ......7Parts 4 Bights  so-called """casa""" knot."

Here goes again L/B = B*n  +r  

B-r = v n     v = 4 - 3 = 1 so "our count value" is 1 ( me : read your stepping value is )

then " we mark off four (4) dots for the number of BIGHT we have, and count off our
cyclic bight-numbers    0   1   2   3"

"   Then we mark off our  so-called """casa"""-codings marks for seven (7) Parts" so 7 LEAD
"Which will be six (6) marks, (P-1)"

Note that (L-1) * B we know already as being the number of crossings ( or facets or faces,
or of "holes") in the true THK

   \   /   \   /   \   /
"going from left to Right we write the cyclic bight-numbers above the coding marks. Then
going from right to left we write the cyclic bight-numbers under the coding marks.

In fact Hall use twice 'cyclic" when one is cyclic ' and the other the "complementary-cyclic' if 
I did not miss something ; that does not make for easy understanding !
        1   2 3   0 1   2
         \   /   \   /   \   /
         2 1   0 3   2 1
"Now to make the overs and unders easier to see we write (O) or (U) above and below the
cyclic bight-numbers for the overs (O) and  (U)
        U  O U O U O
        1   2 3   0 1   2
         \   /   \   /   \   /
         2 1   0 3   2 1
        OU   OU  OU

"Starting on the left side of the algorithm diagram we do half-cycle #1 which is a free run (me :
what tell you that in the diagram ???) . Note : All ODD numbered half-cycles go from LEFT
TO RIGHT, so we use the TOP of the ALGORITHM-DIAGRAM.
Half-cycle #2 going from right to left using the BOTTOM CYCLIC-NUMBERS.
We are looking for cyclic bight-number zero (0). Wherever we find a zero (0) we look at the
coding to see if it is an under or an over. ( me why ? justification ? ). Where we have the "U's"
and "O's" under the cyclic bight-numbers we can just mark it with a line here we have
an Over(0)

        U  O U O U O
        1   2 3   0 1   2
         \   /   \   /   \   /
         2 1   0 3   2 1
        OU   OU  OU
"Half-cycle #3 ......

        U  O U O U O
        1   2 3   0 1   2
         \   /   \   /   \   /
         2 1   0 3   2 1
        OU   OU  OU

I will leave you do the rest till the end all by yourself, I find this too boring for my patience.

Now I would like to attempt to make that just a wee bit neater and clearer.

I will prefer to stay with *my* frame of reference, as I find it too boring to try to make a
neater version of what HALL point to.  
I am sure that the source idea (SCHAAKE) must have been high level, may be above
HALL's, may be above mine and that if was incompletely digested and assimilated.
That is the only explanation I can find.

Plus using another frame of reference will pull you out of the "ruts" you were set in by
HALL's text.

Sorry but you will have to comply or to desist ;  most sorry  ( not really ;-)  ) but not enough
to change tack !

I gave the mean to "translate" one frame into the other at the bottom of the previous
page 7 in : CAVEAT

TO ME! Please note that and keep it well in mind

if you don't want to use your brain the undoubtly HALL recipe may be just what you want to
learn (
even if itis hardly understandable du to essential parts being overlooked),  a half-baked recipe to write an algorithm and a coding.
I  suppose that many will just want that so...

HAIL HALL !  (will rather HAIL SCHAAKE myself)

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

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