Nautile
aka Charles Hamel's personal pages

PAGE 9

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Without the freedom to make critical remarks, there cannot exist sincere

flattering praise - Beaumarchais.

PALIMPSEST* of HALL's prose

* palimpsest in case you have not (many don't, so ...) the word in stock is a written upon

parchment that in order to be reused is scrapped and then re-written, with parts of the old

text still visible

First 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.

It is a less interesting tool than enlargement processes of all ilk are, and that by leaps and

bounds in my opinion.

It is interesting, indeed quite a lot, on theoretical grounds but certainly not IMO for any

practical purpose for an amateur; it is a million time easier in comparison to use a cylinder

and a mule to do ANY TRUE THK.

No need of the cumbersome recipe as shown by HALL, intellectually fascinating (incipiently

in HALL) may be, but without any real "power" as an run of the mill practicality for the

'lambda tyer".

Just compare it 10 times to cylinder + mule for making a true THK (so absolutely

limited to O-U (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 ) of Lead/Bight

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 between :

*** HALL's frame : a mandrel held horizontally, so LEAD at top and bottom and

BIGHT on the left and right.

*** NAUTILE's frame : a cylinder held vertically, so LEAD on the left and the right

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 )

With this THK I will attempt to ease the passage from one system to the other, in other

words show how the "mandrel" and the cylinder compare.

This one is just the precedent picture but with its full complement of numbering.

For persons wanting to continue to slog along with HALL his 5L 4B THK is here.

What follows suppose you have read and understood (minimal requirement) all of my

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 ,

MATHEMATICS AND THK ( Part 1 - Part 2 ) will surely be a plus.

I will treat the topic using

- a cylinder diagram of 7L 5B without any crossing type . This is the one I will use mainly.

Either make a colour printing of this picture or keep it opened in another tab or window for

frequent consultation.

or

- its copy with the type of the crossings indicated for those anxious to have them.

BIGHTs are figured on the TOP and BOTTOM RIMs. ( They get a green colour

identification like this : B(4) which is the fifth BIGHT ( counted from zero ! ) .

It is a bit like the "anatomical" number in red Roman digits elsewhere in one of my drawings

but this is in a reasoned way and is much different in building.

It is the TEMPORAL BIRTHING ORDER and *not* the direct SPATIAL

POSITION SUCCESSION ORDER as in the "anatomical".

LEADs ( or rather the ROWs of crossings they built ; note that what is

ROW on the mandrel is COLUMN on the cylinder and what is COLUMN on the mandrel

is ROW on the cylinder. )

are numbered on the right side and on the left side, in differing directions. ↑ ↓

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".

example :

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 rim we have to "step"

two ROWs of crossings.

When 'stepping' two ROWs then we have gone from being on the COLUMN of one

given BIGHT to the COLUMN of its immediate neighbour on the same RIM. ( but in the

interval we stepped over the COLUMN of a BIGHT on the other RIM hence the 2

ROWs)

Another perspective is :

--- a one COLUMNadvance is one half BIGHT advance.

--- a two COLUMNs advance is one BIGHT advance. see here

We have all seen that many times when looking at diagrams of cylinder knots, probably

without being really conscious of it.

This is just another perspective on what we have already addressed :

there are L BIGHT *STEPs* between the two lower extremities of a given PERIOD, and

the length of each HALF-PERIOD vector is separated in as much "portions or parts" as

there are LEADs and the whole "run of the 2 HALF-PERIODs "sides" is ( 2 * L )

The vector invoked above is the vector journeying the cordage route goes when the

cordage is being laid :

--- for the first half-period of each period ( yellow line ) from Bottom-Right to

Top-Left (B.R.T.L) or as " ↖", anti-slash, or rather an upward going arrow,

moving obliquely right to left.

Each get the same digit ( PNHP ; ANB respectively ) on both of its extremities.

so the LEADs ROW numbers that goes with it are read and recognized (numbered

from zero to (L+1) ) by the vector as it goes along, so of course in the direction

followed by it : from Bottom to Top ( Hall will have from Left to Right for that )

( For Hall the vector will be : from Bottom-Left to Top-Right )

--- for the second half-period of each period ( blue line) from Top-Right to Bottom-Left

(T.R.B.L) or as " ↙" , a slash or a downward going arrow, moving obliquely

right to left.

They get the different digits ( PNHP ; ANB respectively ) on each extremity.

So the LEADs ROW numbers that goes with that are read and recognized (numbered

from zero to (L+1) ) by the vector as it goes along, so of course in the direction

followed by it : from Top to Bottom ( Hall will have from Right to Left) ( for Hall the

vector will be from Bottom-Right to Top-Left).

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 reference.

Do it yourself if you prefer the clarity of his system to the clarity of mine, to each his wants

and desires.

The LEADs themselves (half-period they are) are numbered (one continuous single

sequence for all of them ):

the " ↙" 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 rim,

one in black and the other in blue.

blue for the PNHP and black 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

half-period number.

PNHP (blue) gives the numbering of BIGHT in the order of their making on the cordage

route

( as soon as half a period is "on the pin" then it is numbered ( from zero to (B-1) )

black is for the ANB the artificial numbering of BIGHT done in circular fashion, on each rim,

using as start point the first half-period SPart-Wend segment which get 0 at both extremities.

Then on each rim the numbering proceed from right to left ( ← ).

In a circular fashion ↻. (modular)

Don't 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".

The cordage route can be "unrolled" in space to represent its temporal sequence, as we

already did in a preceding page, only this time it is for the 7L 5B .

Here is the summary in image.

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 attention , 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 )

0

# # # # #

The "stepping " number is the " 3 " HALL tell us without any qualms about total absence of

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

So we number in a circular fashion on the line, progressing from left to right ( → ) till lack

of PINs to STEP on either force us to "rewind the clock"

( or do what people my generation and with an Underwood typing machine know as

'carriage return') to the left for a new left to right ( → ) bout or to stop altogether for

want of empty slot.

-------------------

0 1

# # # # #

-------------------

0 2 1

# # # # #

-------------------

0 2 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 digits)

Let us accept to call that sequence the complementary periodic sequence.

'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 justification in fact.

--------------------

For each digit apply the ( 5 - x ) operation

For the suite 0 2 4 1 3 the result is 5 3 1 4 2

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 + (L)mod B

0

# # # # #

--------------------

0 1

# # # # #

--------------------

0 1 2

# # # # #

--------------------

0 3 1 2

# # # # #

--------------------

0 3 1 4 2

# # # # #

This last result 0 3 1 4 2 is labeled as : the periodic sequence.

it is the sequence in which the cordage route meet with the BIGHT pins in a

TEMPORAL SEQUENCE.

so

complementary periodic 0 2 4 1 3

+

(direct) periodic 0 3 1 4 2

= 0 5 5 5 5

observe that 0 5 5 5 5 == 0 0 0 0 0 as 5 becomes 0 (rewinding the counter

which counts 5 points on a 0 to 4 scale.)

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/