BIGHT NUMBER versus BIGHT ORDER
or keeping logical
planes well
separated :
structure,or
finished fixed state
&
process, ephemeral phase disappearing into the fixed state.
I was set onto this trail years ago by Tom HALL's book :
"
Introduction
to
Turk's-head Knots" in which he
succinctly evoked the CYCLE (I do prefer "PERIOD" if you don't mind).
Unfortunately the work from which he got his idea that he so sketchily
exposed without any
"justification made in a reasoned
fashion" has stayed unavailable to me at the moment of
writing.(2008 Sept).
In a lifetime worth of experiences I find that a most common
intellectual mistake is
confusion of logical planes.
Discussions about knots are rife with that if I must add
credence to my experience.
Present topic is an echo of
THK ARE NOT BRAID .
Only a dull teacher will think that they are"best described as
continuous braid"for beginners.
Beginning a teaching by teaching a mistake is certainly not good teaching.
"continuous braid" is an utter nonsense, as if a braid is not a
continuity ! plus they are NOT braid ANYWAY as clearly demonstrated in
my paper.
I hope I
made a convincing demonstration in it that there exist
essential
differences
between a BRAID
and a TRUE THK (
THK or NOT THK) which do not,
repeat, do not
share the same essence as
their geometry show
perfectly when studied.
In the space-time
distribution of
the route(s)
followed by the strand(s) in the process of making, each sort (braid /
THK) is indeed quite different from the other if you are not
blind to the visual
evidence, or
deaf
to reason.
Dumb
I will not pronounce about!
No one is that dumb anyway in the knot tyers community. ( in case you
missed it blind - deaf -dumb (in a different meaning ) stand for the 3
monkeys of wisdom ;-) )
For structure recognition the STRAND's route is more important in a knot that
their crossings are.
In fact the strand is following its
route along the SHADOW of the knot and in the actual
knot it obey the
crossing sequence program (as in my H & L system) or the
so-called
"""coding"""
That is my conjecture in the mathematical
sense of the word : something I recognize -and
verified as
true countless times- but cannot formally
prove)
The sequence of crossing built by reading their order of
appearance when following the
SPart-WEnd vector in a *
finished*
knot cannot really be of any great use to immediately
recognise a structure at a glance.
Please take good note that the two sort of sequencing are hugely
different : order in the actual making following the diagram
given by the SHADOW or CORDAGE ROUTE and
order read on a finished knot
and re-tracing the cordage route )
Just to show you :
What is the one you tentatively identify ( can put in a broad class
) in a thousandth of a second
OUOUOUOUOUOUOUOUOUOUOUOU
or this "
cordage route"
without
any
nature of crossing
represented which make it in fact
equivalent
to
the concept of SHADOW, taken from topology, exposed elsewhere
in these pages.
Second try ?
OUUOOUUOUOUOOU
or
this
route without any crossing type mentioned. (think SHADOW)
I
will bet that every one was much quicker recognising the class of the
route
or SHADOW
than in identifying with a knot's name the sequence of Os/Us.
Recording as my H&L
system is supposed to be used for) the order of their
appearance
on the route followed
during
the process of laying the knot is in fact as much or may
be
more
a recording
of the route ran over
by the STRAND than of the crossings per se.
The
route alone, without any
type
of crossing being specified, but just there
mere existence
being noted (SHADOW) , gives
more
information about the "class" of the knot ( I will say
out of hand 80%
at least) than the
crossings themselves taken in isolation (I will say 20% at
most).
There is less
degrees of freedom in the crossing sequence than in the route
; my guess is
that
it is one of the possible explanation of the above.
A knot is more swiftly assigned to a
downsized probable group of appurtenance by
looking
at its
SHADOW
(the diagram of
the cordage
route) than by
being
given the visual
sequence
of
the crossing in the finished knot.
BUT this has a big downfall : lazy minds will stay there and believe
that
identical shadow
cover similar knots.
How
false that can be
No wonder that so many are still erring in obscurity. ;-)
I suppose that the shadow is the explanation
why obdurate brains
still persist with the
greatest energy they can deploy in negating facts
and data and stay with the illusion that the
"THK umbrella"
(to steal what I find to be a
very funny expression) cover all sort of
NON-THK knot.
This is a case where
creed born from blind adhesion (unexamined by
sender and
by
receiver) to handed-down recipes (gospel if you like)
reinforce itself at each
presentation
of facts (the heathen miscreant's words for them The True Believers)
showing its mistaken
aspect.
The bright ones will only have been in a transitory phase
(which must
never be extended to
become a permanent state), called "error" and have
mistakenly confounded the shadow with
the complete knot nature for a
moment. Thanks to having an open-mind in autonomous (as in
not-programmed and not-blocked by creed) thinking mode they
can
change tack and
built new
knowledge.
[ illusion is an
inability to get the correct perception that will
persist even when warned about
it : it is just a
defect of
the system that is quite unable to get the true picture!
Whereas an
error
is simply
insufficient data having been furnished and or analysed so that it
could be built into
sufficient information to become acquired knowledge
; simply give or
point to the data
and the correction is readily made
by personal thinking]
The 'gestalt' of
the knot lies in its SHADOW or
"CORDAGE ROUTE" and
certainly not in the nature of
its
crossings and their
sequence.
One "proof" I can propose that the cordage route (SHADOW) is much more
important
than the crossing sequence as recorded, not as by my H
&
L system along the way of the
making, but on the finished
knot, following the SPart-WEnd
vector and tallying the crossing
as and when they are encountered is
the following fact follows :
while NO CROSSING has been altered in its
nature, no rethreading was done, we can
have 2 really
physically different knots that the topology recognise as being
equivalent (
which is quite a bit different from "identical" ).
Unchanged crossings but
different cordage route.
This the case of
the Fig-9 and #925 that I solved and show in another
page.
picture
one -
picture two -
picture
three -
picture four
No crossing physically re threaded and yet two quite individualized
knots :
The SHADOW (cordage
route) is more important than the crossings as usually
noted in the
order they are met in the finished knot. (quite different
from the
sequence of crossing as use in my H & L system which
is in fact the "crossings
route" so to speak )
So here we will attach ourselves to distinguish between :
--- the
order of BIGHTs as they can be seen
in
the finished knot (in
the illustrations these
number are denoted by Roman digits,
these
Roman digits though looking to be an "ordering"
variable are in fact - we
will see why latter - mostly mere "nominal" variable.)
from
--- the order of BIGHTs making
during
the process of laying the knot (denoted by
Arabic digits). This is not a "nominal" variable but a real "ordering"
variable.
Point "zero" is not arbitrary ( even if reasoned) as in the
first sort of ordering evoked immediately above.
THK 7L 5B has
[
I ; 5] [ II ; 3 ]
[
III ; 1] [ IV ; 4] [ V ; 2 ]
OUTer
ring
[ I ; 2] [ II ; 5 ] [
III ; 3] [ IV ; 1] [ V ; 2 ]
INner
ring
the
circular permutation , one among several
possible, is as they say in French " visible
comme le nez au milieu de
la figure" , "visible as the nose in the middle of the face".
5 3 1 4 2
3 1 4 2 5
1 4 2 5 3
4 2 5 3 1
2 5 3
1 2
and we can still do it in the "reverse" !;-) 2 4 1 3 5
5
2 4 1 3
3 5 2 4 1 ... I trust you have got it
by now.
the 3L 4B has
[ I ; 4] [ II ; 1] [ III
; 2] [ IV ; 3] OUTer ring
[ I ;
3] [ II ; 4] [ III
; 1] [ IV ; 2]
INner
ring
There is a phase offset between
INner
and
OUTer.
NB
: of course depending on the way (Clockwise or Anti or
counter-clockwise) you
make a drawing of
a closed
curve
or on the way you distribute the BIGHT OFFSET ( L/2)
in
cylindrical
diagrams between upper and lower) the under laying sequence stay
identical *but* the permutation you get immediately visible
is
different.
So you better standardise your practice so as to keep things
comparable and
understandable.
See the effect of the 2 different
distributions of STEPs among TOP and BOTTOM RIM.
5L 4B so STEP L/2 = 5/2 = 2.5 so 2 and 3 must be
use as you cannot be floating
between 2 pins.
Two ways :
2 on upper 3 on lower
3 on upper 2 on lower
This is not really important as long as you take great care when in
transaction with other
persons to make (
*and* keep)
your
choice EXplicit and not IMplicit .
Be consistent and congruent
all along your discourse :
always use the same frame of
reference knot diagram after knot diagram.(or warn the other person
you are changing it )
Personally I use :
-- for the spirograph drawn curve : CW if RC (Rolling Circle)
is signed minus and CCW if
signed plus.
--
for
the cylindrical diagram : I offset to the
LEFT
the numbering of the
BOTTOM RIM
and apply the smallest ( if this apply) of the L/2 to the
top rim. (this may change with time)
Say that 5 so L/2 = 2.5 which mean using 2 and 3, then I use
2 on upper and 3 on lower.
So the lines moves from
BOTTOM-RIGHT
to
TOP-LEFT
↖and
from
TOP-RIGHT
to
BOTTOM-LEFT
↙.
For
the 7L 5B in fact the structure outer rim order
I
II
III IV V is in correspondence
with the
process order
5 3 1 4 2
5 3 1 4 2
is 4 2 0 3 1 if you start with 0 instead
of 1 - remember your early days at
elementary school : to
count items
you can either count them 1 to n or 0 to (n-1)
That
just show the full importance of always minding about NOT
mixing
structure and
process : the route is not to be equated to the crossing sequence or
the
manner
of
tying per se !
The manner of tying a knot is an epiphenomenon only, that has
absolutely no
value to
classify or distinguish 2 knots that could be qualified
"different" just because of that
difference in the manner of tying
them. In other word : there can
be
several manners of tying a given knot : several recipes
arriving at
identical structures ).
Their common route
in the finished knot
is much more important that the mean of
obtaining the final common
route.
That is specially true if you want to compute a "distance"
between two
different routes for a
classification.
Classification of knot is
probably one silly endeavour, silly because it can be equated to
emptying the
seas and oceans using a pail without bottom !
The best (IMO) one can hope is, one of these days, to find a way to
conceived a fast
identifier of
knots to discriminate between them.
So
how can we, without actually making the knot, know in which order the
BIGHTs will
appear during the making of the knot ?
There we need the help of a bit of mathematics that will be seen
latter.
Before tackling that it is better to absorb, digest and
assimilate the full of MATHEMATICS
and THK
Part One Part Two
Those 2 Parts and what has been said elsewhere in those pages.
This is the absolute minimal needed I think.
A tip : if need be read again mainly
Page 3, but also
Page 2,
in particular the "corridors" use
and the PINs JUMPs or the PINs STEPs then
think by yourself about how to get the
processing order
of
BIGHTs when knowing the L & B of your THK.
NOT TO BE DISPENSED
WITH if you don't already have a working
grasp
of
what MODULO / MODULUS is :
Persons wanting a good
intro : just go over there
Some
quick formulas
I urge you to go to :
http://demonstrations.wolfram.com/
( FREE )
and use
http://demonstrations.wolfram.com/MultiplicationTableModM/
http://demonstrations.wolfram.com/ModularAdditionMultiplicationAndExponentiation/
http://demonstrations.wolfram.com/LCMGCDAndMOD/
http://demonstrations.wolfram.com/CreatingAGCDGrid/
http://demonstrations.wolfram.com/EuclideanAlgorithmSteps/
http://demonstrations.wolfram.com/FindingTheGreatestCommonDivisorOfTwoNumbersByFactoring/
http://demonstrations.wolfram.com/FindingTheLeastCommonMultipleOfTwoNumbersByFactoring/
http://demonstrations.wolfram.com/FactorTrees/
http://demonstrations.wolfram.com/ExtendedEuclideanAlgorithm/
http://demonstrations.wolfram.com/AreEulerNumbersPeriodicInModularArithmetic/
http://demonstrations.wolfram.com/ModulusCounting/
http://demonstrations.wolfram.com/FibonacciResiduesArePeriodic/
http://demonstrations.wolfram.com/PowerModIsEventuallyPeriodic/
http://demonstrations.wolfram.com/ComplexFibonacciResidues/
http://demonstrations.wolfram.com/EquivalenceClassesModuloM/
http://demonstrations.wolfram.com/QuotientsAndRemaindersWheel/
...to get a feeling
Here by
modulus
I mean the
measuring
stick
Without
understanding the use of modulus, THK anatomy and making cannot
really
be understood
in depth, even by the very best practical first order
practitioner of
the genre. (some
among the best are really lousy at theory !)
That does not mean someone applying the handed down
recipes will not produce stunning
items.
I can drive my car without being a mechanics or an engineer but I would
not boast of
"understanding" it in a profound sense.
A
spider can do stunning work without a brain, with just a good innate
program,
cockroaches can display highly complex behaviour (and adapted
to
survival too) using
only a few nervous ganglia, octopus do feats in
recognition of patterns and are quite adept
at "manipulation"
err "tentaclation" ;-)
I just happen to believe
that to be a knot tyers is to be using one's available brain as much
as
possible
in order to truly study and understand the deep nature of what one is
making
and not simply to have agile hands and fingers and myriad
of "unthought about "programs
(recipes) for knots.
Consider those 2 series of integers:
ODD parity -8 -6 -4 -2
0
2 4 6 8
EVEN parity -7 -5 -3 -1
1
3 5 7
modulus 2 is
used to built each one,
but this mean too that you can represent the 'ODD' series with '0' and
the 'EVEN' series
with '1', in other words with their modulus 2 ( any 'EVEN'
number divided by
2 has 0 has
remainder and any 'ODD'
number divided by two will leave 1 )
A PERIOD (cycle) is a
modulus
operation.
A bit like a counter starting
anew from 0 once the extremity of the modulus has been
attained.
Like a clock measuring the whole day-night span which by
convention finish at 24 which is
also the beginning ( 0 point ) of the
next period.
With modulus 12 you will use 2 (mod 12) to mean that it is 2 AM
or 2 PM.
This
leaves some
ambiguity but for a 24h nycthemeral cycle there is the need to
specify
the AM or
PM part of it.
Using modulus 24 , 14 (mod 24) will without doubt state that it is in
the afternoon so
equivalent to 2 PM.
If you know how to perfectly read and use a mechanical watch then you
are
making use of
modulus, even if you never realised that.
When, in the end of the afternoon, at 6 PM you say
that it is
18:00 then you are making use
of modulus
24. while with 6 PM you are using modulo 12.
When at 22:00 Saturday evening you decided to make your clock
ring in
8 hours, you can
immediately know that it will be ringing the next day,
Sunday, early morning at 06:00.
22:00 + 08:00 = 30:00
but 30:00 is way over the 24/0 hour mark
so
30:00
minus 24:00 = 06:00 or 30 divide by 24 than is 1 for the Integral Part
and 6 for
the Fractional Part. Here you keep the fractional part.
MODULUS IS NOT ALL THAT DIFFICULT :
( beware this is "correct" as far as results
obtained
go but is light years from mathematics
formalism ; but then seeing what
wonderful work the great Einstein made explaining relativity
in
Principles
of Relativity first English edition 1920 (1952) book (read
it it is a marvel of
intelligence that makes you for a while feels you
are intelligent).
I feel that the tracks of absence of formalism for the
sake of clear understanding was well traced and alas I never will begin
to 'reach his instep, much less his ankle' as they say in French )
Imagine you own a measuring stick which markings only appear
after it
has
measured the
whole length that is to be measured, leaving no
remaining length to be measured (zero) or a
length smaller than the
stick.
Modular is like those very good cook who are past master in the art of
making something
with the "left over" ;-)
If you are measuring something "negative", say the part of a
partially build wall
till to be built
as measured starting from the point the wall will
attained
when finished, you have to measure
a part of the already build wall at
the end of your measurement of the empty space.
You may not stay in the "minus", you cannot rest your measure below the
zero mark,
you
have to climb over it..
Your
measuring stick must, somewhere, somewhat, touch the built
part of the wall.
Then you have used of the modulus concept, the stick being the
modulo unit.
5 modulus 3 has 2 as result { 5 - 3 = 2
stop here as 2 < 3 }
8 modulus 3 has 2 as result { 8 - 3 = 5
5 -3 = 2 stop here as 2>3
-10
modulus 3 has 2 as result {-10 + 3 = -7 -7 + 3 = -4
-4 + 3 = -1 we cannot stop
here even if |-1| is
smaller than
|3| we must go over zero -1 + 3 = 2 we
can stop here
so we can state
5 == 8 == -10 modulus 3, that is 5 == 8
== -11 "in the frame of reference modulus
3"
You measure what is to be measured with length of stick after length
of stick till there is
either nothing left (zero) or the length left
is less than the length of your measuring stick.
The number of full length of measuring stick is discarded.
If measuring a "negative" then you go on laying length of stick after
length of stick till you are
either touching the zero mark or
have gone beyond.
If modulo is 3 then:
(-3) lead to result 0
(+3) lead to result 0
(-4) lead to result 2
(+4) lead to result 1
(-8) lead to result 1
(+8) lead to result 2
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/