Nautile aka Charles Hamel's personal pages
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


IFF (IF and ONLY IF of Logic) you really want to study the topic I suggest you make a
printing of the drawings linked to in this page before reading ( don't look at those drawings
till after having made a first read).

On the second run, pace your reading and study the drawings jointly with reading and
drawing your *own* diagrams to really grasp things and make them clear for you.

First some remarks :

--- as far as the route followed by the cordage during the making of a THK is concerned
there are several ways to represent its cylindrical form in diagrammatic drawing.

Recall my remark about the route followed by the cordage (will be denoted as  cordage
or CR  and is indeed the 'shadow')  being immensely more important than the record
of the type of crossing when this record is done on the finished route.

To be consistent I will not figure the nature of crossings but just there mere existence.

We need to represent this cordage route. (CR)

This can be made in different manners :

   --- just a simple grid or space sequence

   --- a grid highlighting the periods (cycles for some) so making an inter-meshing of the
space sequence with the time sequence.

   --- a drawing representing the periods aligned one after the other (unrolled so to speak) in
the order of their appearance during the making of the knot, so representing a time

As for the numbering of the BIGHTs ( LEADs as such do not need to be numbered here ;
may be latter if we touch the mathematics of crossings ), it can be made in several different
manners, each bringing its particular kind of data that your brain will transform in information
bringing you new knowledge.

Pictures will (hopefully) make clear  what my words will have left obscure.

   ---  one manner, denoted in Roman digits in the pictures, is the equivalent of recording the
sequence of type of crossings as you see them following the SPart-WEnd vector in the
finished knot, in the knot in its fixed terminal state.
This is quite a pure loss of time as far as "information" on how the knot come in existence is
This just represent the order in which a finger running along the chosen rim encounter them.

   --- another manner, a particular one, recorded on the finished knot is as follow.
BIGHTs are numbered using a same starting line on both rim  the SPart-WEnd vector, the
one going from BOTTOM-RIGHT to TOP-LEFT . Each extremity of that segment will
get '0'.  '0' ,  not '1' , as we will number from 0 to (B-1) and not from 1 to B.

BOTTOM RIM '0' position is the beginning of the first PERIOD and TOP RIM  '0' position is the  HALF-PERIOD mark. (end of the first HALF-PERIOD and beginning of the second
HALF-PERIOD of the first PERIOD)

The reason of doing things like that ?  :
the fact that we will need MODULUS, hence the necessity of having 'zero' available.

On each rim from this '0' continue numbering from RIGHT to LEFT ( ), the way the
cordage is laid along, till you arrive at B-1 then  begin another sequence if you want
(doubling or even enlargement).

Having done all that :
---each BOTTOM-RIGHT to TOP-LEFT segment got the same digit at each of its
See picture 

---That leave us with yet a third way of sequencing the BIGHTs, quite comparable in
principle to what my H&L system does for other knots.
BIGHTs are numbered ( one sequence for each rim, not the same sequence shared between
the 2 RIMs) along the cordage route as they appear in the process of making the knot on a

See picture ONE and  TWO : here with only a half-period attach to it a BIGHT is numbered
but here picture one  and picture two : the BIGHT is numbered only when it is completed
with its two half-period actually laid.

I am not sure yet which is the more useful way though the first seems to make better sense
somehow, feels more elegant.

Note that we can also number in a unique sequence the order of the BIGHT  in the
zig-zagging circuit between the 2 RIMs.

PERIOD, (BIGHT) will be numbered  "as if they had been coloured coming off the spool
immediately prior to being threaded in the knot".
See picture
The PERIODs, put one after the other, are in the order they appear in material existence,
this order makes their label.

We need name for the different numbering sequences.

--- first in Roman digits is "the dead numbering" or the numbering of dead bight.
It will be denoted by DNB

-- second is "intellectual or  artificial " numbering of  bight : denoted by ANB

- third is the "practical or birth order" of the bight and will be denoted
        * PNCB  for the count of complete BIGHTs
        * PNHP for the count  as soon as one HALF-PERIOD is present

Now the stage has been set up.
Play should begin any time now : nous attendons juste que les trois coups soient frappés ;
we are just waiting for the 3 strokes to be stricken.

In this picture something was left out :
the oblique LEADs that were crossed along the way.
Each crossing being a "degree" as in a flight of stairs steps.

Something is immediately apparent if lines are drawn as they are in this picture :
it is the offset between the 2 RIMs when the count is done from the start line.
There will be some tiny bit of correction to be made somewhere to keep track of that fact.



Just having a bit of fun and wondering if this can be of use and for what ?

You will recognize a particular representation of a THK diagram already seen :
the one with the PERIODs strung along one after the other



Before going on I suggest you print that diagram.

This is about "unrolled periods" and numbering systems

The PERIODs are unrolled as they were laid that is from RIGHT to LEFT ( ) ( numbering
also  follow this rule when made on the cylinder form of the knot or its diagram
This unrolling of the diagram representing the cylinder form  obviously change the way
numbers are strung along  compared to the diagram it is normal form)

You will observe :

that the ""  HALF-PERIOD ( first HALF-PERIOD of each PERIOD ) get different
number at each of their extremity EXCEPT when using the numbering of the Bight in their
order of appearance in the making, numbering being applied as soon as even only one
half-period in a full period is laid on

that the ""  have the same  numbering at each of their extremity ONLY in the case of the
numbering of BIGHTs in their order of appearance in the making is made only if the
PERIOD has  both its HALF-PERIODs laid on.

GREEN numbering is from RIGHT to LEFT ( ) starting at '0' on both the TOP and
BOTTOM RIMS : using the SPart-WEnd first HALF-PERIOD as starting line.

PINK is the numbering as soon as one BIGHT as one HALF-PERIOD at least

BLUE is the numbering of BIGHTs in their order of appearance in the making but only
when a BIGHT has got its full complement of HALF-PERIOD.

RED is the "anatomic" numbering : the order in with a finger staring on a random BIGHT in
the finished knot running clockwise on the upper rim will encounter them.

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

Url :