Nautile aka Charles Hamel's personal pages
MENU
 _______________
TURK'S HEAD
PAGE 7
Page1   Page 2  Page 3   Page 4  Page 5  Page 6    Page 7   Page 8   Page 9   Page 10
Page 11   Page 12   Page 13  Page 14   Page 15  Page 16   Page 17     
Page 18     Page 19  Page 20  Page 21   Page 22    Page 23   Page 24   Page 25   Page 26
 _______________


ABOUT HALF-PERIODs

We will be using the printing you  made for the topic just before this one.

For now (compare with "CORRIDORS spoken of  before) we will just look at the
numbering of the HALF-PERIODs as we see them in the intact knot and *not* in "unrolled"
form.
 numbering and orientation of the half-periods.

There are B PERIODs so (2 * B) HALF-PERIODs.

Half are  (first HALF-PERIOD of each PERIOD) are going ( frame of reference : the way
I orient my diagrams ) "ANTI-SLASH" oriented from
 BOTTOM RIGHT to TOP LEFT  the " "
and
half (second HALF-PERIOD of each PERIOD) are going "SLASH" oriented
from TOP RIGHT to  BOTTOM LEFT  the "  

Note that
all
the B.R.T.L are numbered ODD along their course.
all the T.R.B.L are numbered EVEN along their course
NO ODD CAN CROSS ANOTHER ODD  
NO EVEN CAN CROSS ANOTHER EVEN  
ONLY CROSSING DONE IS BETWEEN ODD AND EVEN           

This will be put to use when attempting to find an algorithm.


Note that the Number of the BIGHTs ( Bight-number PNB in YELLOW in the last diagram
given)  to which a given HALF-PERIOD is "attached" is given by

--- if ODD numbered HALF-PERIODs or "" cordage route going from BOTTOM
RIGHT
to TOP LEFT
PNB(upper) = (half-period - 1)/ 2     
PNB(lower) = (half-period - 3) / 2  this is the "origin" Bight of the half-period

--- if EVEN numbered half-period or " ↙" cordage route going from  TOP RIGHT to 
BOTTOM LEFT
PNB or PNHP (upper) = (half-period - 2) / 2   this is the "origin" Bight of the
half-period

(lower) =( half-period - 2) / 2
   

Careful when using the Tom HALL frame of reference MANDREL HELD HORIZONTAL
equivalent to CYLINDER turned Pi/2 radian (90°) clockwise ; validity must be verified.
I will not do it, not sufficiently interested.


You will have noted that you can get the "number"
of the "" T.R.B.L
or the "" B.R.T.L 
quite easily by using the different order numbering ( except the anatomical one or DNB )
either on the top rim for the ""  or the bottom rim for the ""

Look at the last picture given.

digits in GREEN
(B - ANB (upper) * 2) + 1
(2 * B) - (2 * ANB(lower))

digits in BLUE
(2 * PNHP) + 1
2 * B - ( 2 * PNHP(lower))

digits in YELLOW
(2 * PNCB(upper)) + 1
(2 * PNCB(lower)) + 2

I consider the YELLOW or PNCB to be "more general", more "economical" ( Ockham's
razor) because they use only one variable PNCB while the other two need "B", the BLUE
or PNHP need "B" only once.

So I guess it will be either  PNCB or PNHP that will be the more useful, and for now my
heart belongs to PNCB.

so we can have
    2    1    0    4    3          ANB  
   /      /     /     /    / 
 going from upper right to lower left (2nd half period of a period)
 
1    0    4    3    2    offset of one with slant "/"

4    3   2    1    0
  \    \    \     \     \    
 going from lower right to upper left (1st half period of a period)
4    3    2    1    0    
this has "correspondence" or offset zero

if you mix the two

     1      2     3      4     0                
    /  \  / \   /  \   /  \   /  \    the first half period are "congruent"
 0      1    2      3       4      0      



    3    4    0    1    2         PNHP  

    /     /    /     /     /   going from upper right to lower left (2nd half period of a period)
  4    0   1   2    3     offset of one with slant "/"

1   2   3    4    0
 \    \    \      \    \       
going from lower right to upper left (1st half period of a period)

1   2    3    4   0       this has "correspondence" or offset zero

if you mix the two

       4      3        2         1        0             

     /   \   /  \    /  \     /  \     /  \         the first half period are "congruent"
   0      4       3        2        1       0    


   3    4    0    1    2         PNCB
  /     /     /     /     /  
going from upper right to lower left (2nd half period of a period)
3    4     0    1    2   this has "correspondence"

1   2    3    4    0  
 \    \     \     \     \    
going from lower right to upper left (1st half period of a period)

0    1    2    3    4    offset of one, but the slant "\" is different from the 2 above
   
if you mix the two

     4        3      2       1      0             

    /  \    /  \   /  \   /  \   /  \        the second half period are "congruent"
 4       3       2      1      0      4  

I now believe that it may be a better choice to take on account the first half period laid ( a in
a real laid cordage ) than the second laid in other word the "begun" BIGHT rather than the
"completed" BIGHT.



CAVEAT
remember the huge importance of  "frame of reference" :
mine is the diagram of the cylinder knot as seen on a cylinder held vertical
but Tom HALL's is the diagram of a cylindrical knot as seen on a mandrel held horizontal
Going to "mandrel" is turning the vertical cylinder by Pi/2 radian or 90° CW

so for exactly the "same item" they don't label it in an identical manner !

SO BEWARE WHEN YOU READ SOMETHING THAT YOU FIND IS NOT
COHERENT WITH SOMETHING ELSE YOU READ ELSE WHERE
it may be a mistake but FIRST CHECK THE CHOSEN FRAME OF REFERENCE

cylinder VERTICAL so diagram HORIZONTAL mandrel HORIZONTAL  so diagram VERTICAL
TOP RIGHT
BOTTOM LEFT
RIGHT BOTTOM
LEFT TOP
 UPPER RIGHT LOWER LEFT  LOWER LEFT UPPER RIGHT
 LOWER RIGHT UPPER LEFT LOWER RIGHT UPPER LEFT

Juts draw a diagram of a cylinder held vertical, annotated Left, Right, Up, Low,  and 
and turn it by Pi/2 radian or 90° clockwise   ↻  and draw your own conclusions



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

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