the freedom to make critical
remarks, there cannot exist sincere
flattering praise - Beaumarchais.
palimpsest in case you have not (many don't, so
...) the word in stock is a written upon
in order to be reused is scrapped and then
parts of the old
text still visible
a warning : I don't
see the least practical
interest of this
algorithm-diagram for a knot
tyer as exposed by HALL.
Now SCHAAKE is
certainly another kettle of fish from what I heard.
No need of the cumbersome recipe as shown by HALL,
in HALL) may be, but without any real "power" as an run of
practicality for the
compare it 10 times to
cylinder + mule for making a true THK (so
(U-O) and one strand as defined in THK OR NOT
THK) you are not
used to make.
With the mule you just have to use 2 colours to mark the crossings as
seen on the
finished knot and there you have your numberless complete
algorithm (just remember
that the colour coding CHANGE MEANING for a
given colour depending on you going
on an odd numbered
half-period or an odd numbered one ! ).
Despite what one can believe you can do *any* mixing ( GDC permitting )
and *any* coding without resorting to mathematics.
You will need to carefully adapt what I write here using
my researches and my frame of
reference if you want to return to HALL's
words relating what he thought he had
understood from SCHAAKE (never
could get hold of Schaake's writings at the moment
of writing - Added
2008 Dec 14th : I now have THE BRAIDER )
Look at the table at the bottom of page 7 which give the equivalence
*** HALL's frame : a
mandrel held horizontally, so LEAD at top and
BIGHT on the left and right.
*** NAUTILE's frame : a
cylinder held vertically, so LEAD on the left
and BIGHT at top and bottom. Never
forget that important point when discussing the topic.
I am keeping the 7L 5B THK
; I used it in the past
trying to make head and tail of what
was for me not much more than
HALL's giberrish. ( but then I have no great mastery of the
English language )
follows suppose you have read and understood (minimal
requirement) all of
THK pages from 3 onward, at least.(except page 8 which is the
cruel --not by intent
but by inevitable result-- dissection of
HALL's writing. Dissection because for me it
was a corpse, no life
there, otherwise it would have been "surgery").
Adding to your reading THK
ARE NOT BRAID , THK
OR NOT THK ,
AND THK ( Part 1 - Part 2 ) will surely be a plus.
By the way this was intellectual myopia not to go for a nomenclature
not dependent on the
frame of reference horizontal or vertical but
a frame intrinsic to the knot itself and so
that stays "constant".
INTER-BIGHT line, alignment , coding....( row on the mandrel, column on
the cylinder )
PARALLEL BIGHT line, alignment, coding...( column on the
mandrel, row on the cylinder )
To go from one BIGHT to its immediate
neighbour on the same
have to "step"
two ROWs of crossings.
When 'stepping' two
then we have gone from
being on the COLUMN of one
BIGHT to the COLUMN of its immediate neighbour on the same RIM. ( but in
interval we stepped over the COLUMN of a BIGHT on
other RIM hence the 2
Another perspective is :
--- a one
COLUMNadvance is one
half BIGHT advance.
--- a two COLUMNs
advance is one BIGHT
We have all seen that many times when looking at diagrams of cylinder
without being really conscious of it.
vector invoked above is the vector journeying the cordage
route goes when
cordage is being laid :
--- for the
first half-period of each period ( yellow
line ) from Bottom-Rightto
or as "
or rather an upward
moving obliquely right to left.
Each get the same digit ( PNHP
respectively ) on both of its extremities.
the LEADs ROW numbers that goes with it are read
zero to (L+1) ) by
the vector as it goes along, so of coursein the direction
followed by it : from Bottom to Top ( Hall will
have fromLefttoRight for that )
( For Hall the vector will be : from
--- for the second
half-period of each period ( blue
Top-Right to Bottom-Left
or as "
, a slash or a downward going arrow,
right to left.
They get the different digits ( PNHP
respectively ) on each extremity.
the LEADs ROW numbers that goes with that are read
zero to (L+1) )
by the vector as it goes along, so of coursein the direction
followed by it : from
Bottom ( Hall will have fromRighttoLeft)
( for Hall the
vector will be from
MIND THESE DIFFERENT DIRECTION FOR READING.
THAT EXPLAINS THE WAY THE DIGITS ARE USED IN THE "algorithm"
As of now I
will desist from giving the equivalence in HALL's mandrel
Do it yourself if you prefer the clarity of his system to
the clarity of mine, to each his wants
The LEADs themselves (half-period they are) are numbered
sequence for all of them ):
are all EVEN
and the "
are all ODD
No EVEN can cross another EVEN,
No ODD can cross another ODD,
Only if they are of different parity ODD/EVEN , EVEN/ODD can they cross.
There are two lines of digits, two at the top rim and two at the bottom
one in black
and the other in blue.
for the PNHP
for the ANB.
That was addressed in page-7.
By the way : take a second look at how
to compute the BIGHT NUMBER from the
gives the numbering of BIGHT in
the order of their making on
( as soon as half a period is "on the pin"
then it is
numbered ( from zero to (B-1) )
for the ANB
the artificial numbering of BIGHT done in circular fashion, on each
using as start point the first half-period SPart-Wend segment
which get 0 at both
Then on each rim the numbering proceed from right to
In a circular
forget that a RIM represented on the diagram by a straight line is in
fact a circle, hence
the use of modulus to make use of the "rewinding"
or "carriage return".
Now we know where we stand so to speak.
I do hope all that comes 5/5, loud and clear.
Let us forge on still
trying to shed some light on HALL's words.
5 BIGHT so 5 PINS
Those PINs we will represent with '#' characters. ( pay
, latter, in another context,
'#" may mean row of crossing)
Start point is on the bottom of leftmost side on a mandrel
( on the right
side of lower rim
on cylinder )
# # # #
The "stepping " number is the " 3 " HALL tell us without any
qualms about total
hard justification being given.
L/B= nB +r 7/5 == (1*5) + 2
5/4 == (1 * 4 ) + 1
B-r=v 5 - 2 = v 3 = v
4 - 1 = v
3 = v
I just happened to recognize that ((minus L) modulus B) goes as
(minus 7) modulus 5 = 3
Seems a better idea that the above formulas given with the 5L4B
for the 5L4B ( -5) mod 4 = 3
Note (-L) mod B can also be written in it equivalent[ B - (L)mod B]
which I find easier to
"visualize". B-r = B - (L)mod B
hence r = (L)mod 7 mod 5 ==2
5 mod 4 == 1
(-7) mod 5 = 3 === 5 - (7) mod 5 = 5 -
2 = 3
(+L)mod B == B - ( (-L) mod B ) == B -[ B -(L)mod B] == B - B + (L)mod B
If you are bothered by the (-L)mod B try to think "time pieces"
B = 24 (hours)
L = 6 ( hours )
so (-6) mod 24 == 18 means 18 hours ( 6 hours minus of 24h,
6 hours before midnight
is 18:00 hours
we number in a circular fashion on the line, progressing from left to
) till lack
of PINs to STEP on either force us to "rewind the
( or do what people my generation and with an Underwood typing
machine know as
return') to the left for a new left to
) bout or to stop altogether
empty slot. -------------------
1 3 #
0 2 4
1 3 #
All of the PINs
# now possess their individual digit.
The numbers are the BIGHTs accompanying number as shown by PNHP ( blue
Let us accept to call that sequence the
'periodic' is self-evident, as it was done with the circular motion on
a cylinder rim.
Complementary was less evident to justify.
Here follow no less that three different aspects of the same
For each digit apply the ( 5 - x ) operation
For the suite 0 2 4 1
result is 5 3 1 4
but just as 24h is
also 0h on the clock 5 being "over the clock mark is put down as 0
Or for each digit you get its MODULUS B
(5 - 0 ) = 5 5
mod 5 == 0
(5 - 3 ) = 2
2 mod 5 == 3
(5 - 1 ) = 4
4 mod 5 == 1
(5 - 4 ) = 1
1 mod 5 == 4
(5 - 2 ) = 3
3 mod 5 == 2
Or it can be seen as a complementary modulo [ B - (
-L) mod B]°°°== 5 - 3 = 2
that is L mod B == 7 mod 5 == 2
your "stepping" is then 2 instead of 3
°°° (+L) mod B == B - ( (-L) mod B ) == B -[ B -(L)mod B] == B - B +