HOW TO COMPUTE THE ORDER IN WHICH BIGHT ARE IN FACT LAID
BY THE RUNNING
IFF (IF and ONLY IF of Logic) you really want to study the topic I
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
First some remarks :
as far as the route followed by the cordage during the making of a THK
there are several ways to represent its cylindrical form
in diagrammatic drawing.
my remark about the route
followed by the cordage (will be denoted as cordage
route 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.
consistent I will not figure the nature of
crossings but just there mere
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
--- a drawing representing the periods aligned one after the
other (unrolled so to speak) in
the order of their appearance during the
the knot, so representing a time
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
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
sequence of type of crossings as you see them following the SPart-WEnd
vector in the finished
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
knot is as follow.
are numbered using a same starting line on both rim
the SPart-WEnd vector, the
one going from BOTTOM-RIGHT
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
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
necessity of having 'zero' available.
On each rim from this '0' continue numbering from RIGHT to LEFT ( ←
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 :
got the same digit at each of
leave us with yet a third way of sequencing the BIGHTs, quite
principle to what my H&L system does for other knots.
numbered ( one sequence for each rim, not the same sequence shared
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
The PERIODs, put one after the other, are in the order they appear in
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
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
* PNCB for
the count of complete BIGHTs
* PNHP for the count
as soon as one HALF-PERIOD is present
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.
is immediately apparent if lines are drawn as they are in this picture
the offset between the 2 RIMs when the count is done from the start
There will be some tiny bit of correction to be made somewhere to keep
track of that fact.
LES DENTS DE LA MER, TEETH IN THE SEA or rather JAWS !
Just having a bit of fun and wondering if this can be of use and for
This is about "unrolled periods" and numbering systems
The PERIODs are unrolled as they were laid that is from RIGHT to LEFT ( ←) (
also follow this rule when made on the cylinder
the knot or its diagram
This unrolling of the diagram
representing the cylinder form obviously change the way
are strung along compared to the diagram it is normal form)
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
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
made only if the
PERIOD has both its HALF-PERIODs laid on.
numbering is from RIGHT
at '0' on both the TOP and
BOTTOM RIMS : using the
SPart-WEnd first HALF-PERIOD as starting line.
is the numbering as soon as one BIGHT as one HALF-PERIOD at least
is the numbering of BIGHTs in their order of appearance in
making but only
when a BIGHT has got its full complement of HALF-PERIOD.
is the "anatomic" numbering : the order in with a finger staring on a
random BIGHT in
the finished knot running clockwise on the
rim will encounter them.
Copyright 2005 Sept - Charles
Hamel / Nautile -
Overall rewriting in August 2006 .
Copyright renewed. 2007-2012 -(each year of existence)