Nautile
aka Charles Hamel's personal pages

PAGE 4

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I have added quite a number of pages to my work on THK mathematics to use a big word .

I would very much like to verify their validity against SCHAAKE 's works that I could not got hold of :

books and pamphlets ( not "The Braider" magazine that I can get ) at the lowest cost possible , DVD or

CD , paper printing are not necessary. This in 2008September.

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)

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/