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added 2011 Feb 9th ( forgot it  and only added it 2011 March 15th )


"  GENERALISED ENLARGEMENTS OF KNOTS  "  
NO LESS !!!

This page started with the question 'pushed' towards me, 
may be a week or two ago, 
guess just to stir my bad temper



At first I found the question to be a touch dull, not really clearly formulated despite all
outward appearances to the contrary.
I thought it was uninteresting first because of its 'gauche' formulation, second because its
answers are so readily obtainable, either with minimal basic personal pondering coupled
with experimenting or/and by reading some of the so despised "theory".  
All the 'necessary and sufficient' indications are readily obtainable on the Net.


To take 'le contre-pied' of one of the silliest
sentence written by Ashley - a gem
needing a full dissertation to completely show its despairing thoughtless absurdity- :
no amount of so-called experience or fingers gymnastics can hope to replace
some minimal theory and activity within the brain cells ! (said 'theory' does not even
 need to be "in words or in mathematics, it just need to be there inside the head ".
 I know such persons)

On second thought I thought, oh let us have fun : Ask and you shall receive !

To steal from a work I have admired since my teens, D'Arcy Thomson's,
there IS  indeed
a great need to jar any unthinking complacency.


I will not comment directly each of the answers I am said were given to the question as
they range from  " being beside the point of the question" to acceptable per se but not really
answering the question, passing thought the idiotic, the meaningless, the mumbo-jumbo....
all flavours can be had it seems! 


This under is what I was given the original question accompanied by the gist of answers
received and apparently found wanting by at least my correspondent. (I can now add my
name to those holding this opinion that indeed they are wanting)



-----------------------------
                                            -----------------------------
THKs can be expanded, on the LEFT or the RIGHT....yielding new knots with more
 LEADs & BIGHTs
--- is there away to do that with globe knots?
        --- with other decorative knots?
            --- which knot types can be expanded?
                --- has someone developed the methods ?
                                            -----------------------------
-----------------------------


To begin with :

*** I am wondering about what the questioner could have had in mind ?
Is it strictly only the THK proper ? 
In other words is it  strictly only ONE VERY SPECIAL 
form of REGULAR CYLINDRICAL KNOTS
(single-strand, made on a THK cordage route), or
is the questioner calling THK ANY
Regular Cylindrical Knot
as do 99.99% of the ignorants at large on forums and web sites ?

I will be a good sport and consider, despite some pointers to the contrary,  that there is a
sound base of knowledge in the question and that indeed it is the *true* THK of the
REGULAR CYLINDRICAL KNOTS that is evoked in the question and nothing else.

***please! define what is meant by globe knots
or enumerate  those you are speaking
about.

If it is REALLY globe knots and not spherical covering then the question is soon
answered but I guess that under this "gauche" formulation in fact the question is about
spherical coverings ; in that case is it the single strand or the multi-strand one? May be
even it is spherical coverings AND globe knots ?
It is anyone guess so imprecise the question is due to nomenclature used or rather not used.

***please ! define
what is meant by decorative knots or enumerate .... : according to
Ashley? according to Lydia CHEN, according to...? 
Is it single-strand ?I is it multi-strand or  both ?


The answer CAN NEVER BE clearer than the question is ! 


============================================================


First  let me indulge myself with a quote that, by happenstance, I found Jan 12th IIRC in
SCHAAKE's parting writing.

This find is serendipitous !


[open quote]
...at various instances we might have made some scathing and, to some people,
inflammatory remarks
about the way braiding topics have been and still are being dealt with. Although these remarks are more than justified, especially  there where it concerns people who are supposed to be knowledgeable in their field...

...the purpose of those scathing and to some people inflammatory remarks referred to above should, however, be seen to stress the utmost importance in the necessity of
correcting false and misleading concepts, and are certainly not meant to offend
anyone, but rather to awaken those leaving in dreamland.


[end quote]
THE BRAIDER- appendix 2009 - in a final note concerning the material
dealt in the SIXTY ISSUES of THE BRAIDER.


A French philosopher was shorter in words but as intellectually violent !
"Without the freedom to make critical remarks, there cannot exist sincere
flattering praise" -Beaumarchais.



============================================================



I am going to attempt to do my utmost to be compassionate, charitable, indulgent and shall,
as usual, be using logical arguments.
Using logical and/or factual argumentation, mind you, not as it is alas so common, in so many
posts made on the net, commenting, answering with a heap of "belief", "hope","tautology",
"show of blissful self-satisfied ignorance".... and such types of empty non-answer but scant
argumented reasoning based on external reality instead on figments of imagination and
wishful thinking, not to mention abyssal ignorance.





"Expanded" does not appear to me to be a proper way to speak of the PROCESSES leading,
starting
with an ALREADY MADE REGULAR KNOT WITH A STRICT O1-U1 CODING ( a THK
PROPER)
, to another THK of greater dimension in LEADs & BIGHTs
.

The word 'ENLARGEMENT' as applied to THK seems to appropriately apply to the "intent'
behind the processes as persons are more interested in such a case with covering a larger
area than having a knot with more bulk, volume.
For some knots the intent may be rather focused on making them bulkier in which case the
variation is rather an expansion.

Still, communication was not broken as "expanded" may be deciphered by the alert minds
as alluding to :
ENLARGEMENT PROCESSES as formally defined by SCHAAKE & TURNER
about
20 years ago in a Waikato University NZ publication.


IMO it will be doing outstandingly well to abstain from equating ENLARGE , EXPAND,
MAKE BIGGER
when in a TECHNICAL DISCUSSION.

Thanks to SCHAAKE and TURNER's work 'ENLARGEMENT PROCESSES' is now a
TECHNICAL TERM  WHICH IS ATTACHED, HARD AND FAST, WITH  PARTICULAR PROCESSES
 APPLIED TO PARTICULAR KNOTS
.

A professional lifetime has shown to me that "not keeping a guarded and surveyed language"
in technical matters has a high potential of leading to bodily or intellectual disaster.

Words, in a technical discussion, between informed and educated reasonably intelligent
persons are to be used in a guarded fashion (not as a  exchange made on  street pavement),
with a severely narrow(ed) meaning, the more so when they are use in reference with a
procedure, a process, a method,  previously defined in a publication.

When a word is used to denote or invoke a particular "way of doing thing" that has
been duly formalised and published it is always preferable to not use "the elegant variation"
and make use of only one chosen and explained word : the 'original brand'.




Common ground building :

To expand = rather for "volume"   (3D). 

To enlarge  = rather for width and length (2D -  in enlargement only 2 dimensions are
 affected, namely LEADs and BIGHTs). 

To extend = rather unidirectional as in pulling on a rubber band or adding a prolong electrical
 cord..

...just my 2 cents of Euro, but after all I am just the ignorant foreigner barely managing to
express myself, and not so clearly at that, in English (more UK than USA I certainly hope !
even if that explain why some Americans seems to be unable to understand some spelling,
words and formulations)

To evolve= a gradual change and growth over time (opposite : devolve) of particular
'physical 'traits (as number of Leads or Bights for example and not simply size )

Growth=an increase of some stated quantity with time according to a very precise process.
**** see bottom of page for some more common ground building



####################################################################


------------------------
THKs can be expanded  enlarged, on the LEFT or the RIGHT....is there away to do that
with globe knots?....
------------------------

19 words used !  
3 "not so good" points in them.


We have seen the first of those points, here comes the second :

LEFT or the RIGHT........... OF WHAT ?  ;-D)  

As written here this has no logical meaning.

I could very well take it to means that one can make a larger knot being on the left or the
right side of the model knot ! (just  poking fun !) or expanding the right side (or the left)
of the knot;

If  a side is stated then elementary logic demands that the reference for it to be given.
This is so in my French mind's map of this world. (read bat's belfry 12 about the importance
 of frame of reference)


It is just making a saddening exhibition of ignorance of the reality lying behind the
 enlargement processes " la Schaake" to just say
: "THKs can be expanded  enlarged,
on the LEFT or the RIGHT..."

One must specify the reference used because what is the "RIGHT" of a Type I at the
very beginning of it ( respectively 
"LEFT" for a Type II) becomes a "LEFT"
(respectively a
"RIGHT")along the next Half-Period the SPart makes.

Only in a DOUBLING can the statement be limited to
, "on the LEFT or the RIGHT"
for the simple reason that in doubling there is no change of side till the next re-entering.

In an ENLARGEMENT PROCESS the side of the reference HP that is followed change with the
PARITY of the number of the Half-Period.

I devised this simple way of showing things.

In this illustration, for those needing pictures to understand, are the details of each
enlargement process
in a stylised fashion.

The reference is the First Half-Period (HP) made by the SPart.

All HP with an ODD number will be followed on the same side the HP1 was followed,
all HP with an EVEN number will be followed on the other side

So you see you get an alternation of
RIGHT/LEFT for a Type I, an alternation of
LEFT/RIGHT for a Type II hence the necessity of EXPLICITLY state not only
RIGHT or LEFT but also the side OF WHAT. It is the firs HP or SPart which is the
reference.



Now for the third not so good point :

globe (think : 'the globe of a breast' ) is, strictly speaking, for the VOLUME of a ball.

sphere is, strictly speaking, for the AREA of said ball  (the sphere enclose the volume
occupied by the globe so one may speak of the volume enclosed by a sphere but not of the
volume of the sphere unless one want to say that it is the volume of the covering that is to
be computed.
(similar mistakes are made about SPIRAL which is for a 2D curve and HELIX which is
for a 3D curve).



Common ground building again :

globe knot is A KNOT HAVING THE SHAPE OF A ROUND BALL, FORMING BY ITSELF,
WITH NO CORE,  THE ENTIRE VOLUME OF THE KNOT.
(example : a Monkey Fist with no core)

spherical covering is A KNOT SPREAD ON THE SURFACE OF A GLOBE AND COVERING ITS
AREA., SO THE PRESENCE OF A CORE IS UNAVOIDABLE HERE TO GET A KNOT KEEPING ITS
SHAPE
.
(ex: a monkey-fist with a core)





Only having been served a hefty helping of ignorance of the nature of those knots can explain
 the fact of speak of 'globe knot' and/or of 'spherical covering' as if it they were all belonging
to a UNIQUE STRUCTURAL CLASS encompassing all cases of such knots !


I have news for all of those who still insist on using awfully unclear and badly built
nomenclature (the funny "THK umbrella" and not so funny "complicated THK" sort of guys.
Guys who go on poisoning new comers' minds with junk appellations) :
spherical coverings (again globe knots is an idiotic appellation if a core is used)  are in
correspondence with a "function", an "application" made of SOME knots.
This application is not an intrinsic character of the structure of the knot (some spherical
covering may be use on an hexahedron rather that on a ball so the spherical is not 'intrinsic')
but attached to the particular use made of this knot.


Those knots usable as spherical covering may range from
Regular Nested Bights cylindrical Knots
to
Irregular Nested Bights cylindrical Knots ,
some
Regular Cylindrical Knots and some other knots not in those classes such as the
Monkey's Fist proper ABoK#2200 &
ABoK#2205 which can be either a globe knot
(no core) or a spherical covering (with a core) and can be single or multi-strand.  



So at best the question as it stands is, to me,  somewhat "gauche" and is mostly "undecidable"
as they say in mathematics.
At the very least it cannot be answered without some 'interpretation' from the reader (which
is never a good situation as the reader should not  have to 'interpret' at the risk of deforming,
but the fault rest on the questioner).
It would have been much better to specify, using an appropriate nomenclature, all the
"specific" classes and/or individual knots which are to be considered as addressed by the
question.




IMO the question would have been marginally less criticable formulated as


-----------------------------
                                            -----------------------------
Regular Cylindrical Knots with a strict O1-U1 (U1-O1) Column and Row coding known as
THK
scan be enlarged, by one of two perfectly defined processes, either on
the LEFT or the RIGHT
side of the SPart making the first half-period ....yielding new THK
with more LEADs & BIGHTs

--- is there a way to do that with globe knots (no core), with spherical coverings (with
core)
, belonging to each classes of knots that can be used in such a function ?
        --- with other decorative knots, in particular flat 2D ones such as shown in ABoK
or in Chinese, Korean, Japanese cultures?
            --- which specific knot or knot classes outside those above quoted can be
enlarged or be made to evolve ?
                --- has someone developed a formalisation of the methods used for THK
enlargement. Could those methods also be applied to other knots ?

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




The formulation of the original question seems to me to be pointing to the meaning :
" is there a way to enlarge, so-called 'globe knots' and 'spherical covering' belonging
to as yet
unspecified types or classes,  using the processes known for THK.

The processes known for going from one size of THK to another size of THK were
 formalised and published by Schaake & Turner.
I am saying that in the very improbable case (tongue in cheek if you must be told) there is
still some persons ignoring a fact dating from about 20 years!




The frame of reference, seemingly chosen, by the questioner, from which I will be speaking
 is :               ENLARGEMENT ' la Schaake'.
                 and none other.




If by some extraordinary fluke of life you are one of those rare in need of being 'enlarged'
about the formalisation of THK enlargement processes, please :

***  either buy THE BRAIDER ( 5US$ the CD :
this will be quite another thing than the simplistic ''tooling up' so many do in the hope that the
equation  "the more tools I own = the more talented I will be' can be verified ; buying this
CD will be having at hand a great occasion of learning and using your brain with profit !)
I have given a number of CD to friends BUT EACH ONE OF THOSE CD HAS BEEN PAID TO
Doctor John C. Turner,
putting the CD on the Net that I will not do as this would be stealing
from a dead Author and from an ageing one.

Here is an extract of Schaake's writing on ENLARGEMENT of Regular Knot.
I do not put it here to steal but to promote his work and boost Dr John TURNER selling of
THE BRAIDER
CD.
 

*** or -lesser quality it is mind you-  see two of my pages,  this page and that one , as a
 'practical only' second trail that should soon convince you to spend your US$5.




SUMMARY:

Making an enlargement is laying, with the very same strand that was used to made a
knot that could be considered a finished knot
, a track parallel to the SPart
(Standing Part) vector (directional arrow).
This track makes crossings of a type identical to those made by the "followed" length of
cordage. 
Upon re-entering (coming back to the start) it is following a course making a 'splitting of
the parallel tracks' by going 'in between', with one track on each side.
This is the first enlargement* (of the cordage route) in the process of enlargement. 
This 'splitting' passage, the second enlargement in the process, makes crossings of a type
opposite to those made by the "pair of tracks" thus re-establishing the O1-U1 coding, this
finishes the making of a 'new' THK now having a greater dimension in LEADs and BIGHTs
than the THK it comes from.

*In fact this is the process seen as a whole from the knot point of view.
If we now look at it from the cordage route (shadow) point of view one clearly see that the
complete process comprise TWO ENLARGEMENTS OF THE SAME TYPE (hence the
addition in EVEN number of Bights and of Leads in the whole process).
The first enlargement augment the cordage route and put out of kilter the coding of crossings.
The second enlargement augment again the cordage route in the same proportion the first
did AND put back the proper O1-U1 (U1-O1) code in the new THK.

Fast track to get a fast look :
track laying and splitting
Laying splitting
Right Left enlargement


####################################################################


'yielding"  
'to yield', verb. == to produce.
I do like this way of  saying things.

So it  means, in this case, to produce as result a 'larger' THK from a smaller one.
The "void", the "non existent" cannot yield.

So to "produce" one has first to make an investment, to put a "seed".
It does not come from "out of no where" (as in "made from a cordage without a previously
 made knot in it").
It does come from an already made knot that is to be 'pushed' to higher 'dimensions',  to be
 induced to GROW while retaining its identity.



Hence it is clear that we must push aside, throw on the junk pile of erroneous answering, all
the "really besides the
point " notions such as :

** starting from a 'free of knot' cordage to make a larger model, either using a bigger
 or a modified 
pattern/diagram.
but we have to add to the junk pile the case of a pre-existing structure to which one is
** adding STRANDs,
** adding PASSes, using a process that is not the Regular Cylindrical  Knot  process.
** doubling, tripling.




besides the point" notions such as :
** starting from a 'free of knot'  cordage to make a larger model,
Enlargement is not simply 'making a bigger model'.
I am using the Endless knot. ( for me a mat or 2D knot) to illustrate that point.

It can certainly be threaded in a bigger laying but that is all it will ever be : a bigger
model not an enlarged one ; a bigger model made with no utilisation of laying a double
parallel tracks then splitting those parallel tracks in a previously laid knot.

Any other Ornamental Chinese Knot (except those that are really THK) that I have
examined in Lydia CHEN's book could not, repeat
not, be subjected, by yours truly, to a
true enlargement process ' la Schaake' and still keep their 'general type'.  
They may be made bigger but "in the course of threading them from the very start, not after
having finished a smaller size and then enlarging it while keeping it in the same class of knots.
They can only be made in a bigger scale from the very start (diagram, pattern or idea),
not mid way in their making as an enlarged THK is.
Illustration one
Illustration two   
If it is is in fact a modification of the design that stay under the same label then, of course,
this disqualify it as being a possible 'enlargement'.
Possibly it could be seen as an "expansion" as the main intent, the focus seems to me a knot
of greater bulk, of greater volume.


Now taking the clover leaf and the Pan Chang from Lydia CHEN's book, the former
being a smaller scale model of the latter, you can easily see that IT CANNOT BE ENLARGED
(Nested and not only in one direction but in two directions !) it can only BE MADE in a
greater size.



In my experience (never forget that strong caveat about so called 'personal experience', it is
 'anecdote' , not "general')  :
making a count of the 'turning' going clockwise end the 'turning' going anti-clockwise is a
potential maker of "possibly possible" / 'not possible" as far as enlargement ' la Schaake
 goes.


***** If  one of the two counts end in an ODD number of turning then no enlargement is
possible.

***** If  the count are both EVEN but unequal then there is "possibly a possible"
enlargement.


***** If the count in both categories is equal then no enlargement is possible as in those
knots
with nested-bights (but is it the nested bight or the equality the acting factor ?)

***** If one count is equal to zero then no enlargement is possible unless  there is equality
 of 'bights' on both knot edge (top/bottom ; outer/inner ) THK can be enlarged  on both
side of the SPart but not
 ABoK#2360. (Ashley's drawing) .
That #2360 is not enlargeable ( cordage route ) 'a la Schaake & Turner' !
For those who want to verify here is a rough grid on square paper and here a more proper
diagram on an isometric grid. (one needs to know how to make such diagrams because
without them one will always stay a simple "handown-copier " of knot never fully understanding
 knots)


Of course a better manipulation of unicursal regular Hamiltonian circuits (single strands
knots are that ) could shed a better and surer light on the topic, but I learned that at school
when I was 15 or 16 and that is a very long way away in my past and I will leave that to
younger minds.


Here lie an example that, though it is *not at all* the ' la Schaake' enlargement of THK  is
 in somewhat a similar spirit : using the single strand BUT making use of both the SPart AND
 the WEnd, which is not  done (but could be mind you !) with the THK. I could almost be
 ready to put in the bag of enlargement but will not.
I chose rather, for the moment, to put it  in the bag of EVOLUTION :
single-strand, re-entering,  using both side of that strand and  the cordage route to bring out
the modification : the 'loops' are twice the number they were prior to the enlargement
illustration





besides the point " notions such as :
** doubling, tripling.
Obviously DOUBLING or TRIPLING a knot ( adding PLys or PASSes or Passages depending
on the word one want to use) will certainly augment its area ( persons are more interested in
 THK for covering an area than for having a given volume!)  BUT NOT ITS SIZE AS
 MEASURED IN LEADs &  BIGHT
; it is certainly  not an ENLARGEMENT ' la Schaake for a THK' .

Doubling, tripling is just using all over again the *very same* cordage route that built the knot,
it is not very different from using a bigger string from the beginning.




besides the point " notions such as :
** adding STRANDs,
Some repetitive pounding :
I must insist again and be clear on one point : the ENLARGEMENTs discussed here are
CERTAINLY NOT AN ADDITION,  NOR AN AUGMENTATION OF THE NUMBER OF STRANDS
 USED


We must keep the 'original' number of strand that made the knot that serve as
 "enlargement's  root".




This ENLARGEMENT ' la Schaake for a THK' here discussed is not a simple addition of
material but is about addition of material made in such a manner that if the knot is enlarged
it is because the structure of the knots goes from one pre-existing level to an upper level
on some ladder of construction characters (regular knot tree
in the case of Regular Knots).
 
It is GROWTH governed by that Regular Knot Tree.
This is not simply "fleshing up' the already laid cordage route as in doubling, it is making
appear a SPECIAL GROWTH of the existing cordage route while making use of this route
itself !





besides the points" notions such as :
** adding PASSes, using a process that is not the Regular Cylindrical Knot process.
Let us be clear Pineapple knots or ANY OTHER assembly of THK components CANNOT BE
ENLARGED
(this is quite self-evident when one know the enlargement processes and
 know the innards of  those Knots).

They can only be made in a LARGER MODEL by ADDING PASSes, in other words by ADDING
STRANDs


Multi-strand knots cannot be enlarged " la Schaake" for the same self-evident reason.

May be here is the place to speak about a knot that may be a globe knot (no core) or
a spherical covering (with a core) , the single-strand type Monkey's fist that is spoken
about in      knots_craft_ornamental_2.html  Palimpsest...

When finished, in the form that has the two tails emerging from the same place, we have a
(n)PLYs  knot made with a re-entering
single-strand (going back to the very point of
departure and joining the SPart.)  

Is it enlargeable " la Schaake" using the original strand ?
We may hope it could be since it is single stand and re-entering, no ?
Well ! We can make it bigger without adding any strand.

Yes... but it will be simply a sort of higher order "doubling" as you will just be adding
PLY(s) and doing that we will not be using the lay tracks/split tracks
process and so will not
be adding to number of so called FACEs in the knot.

We may add ODD number of PLYs : there is no RKT governing the addition.

With assemblies of interlocked THK ( Pineapple K , Herringbone K, other interlocked
assemblies of components THK ) the transformation of one THK component is forbidden
by the "locking" each component put on each of the other components.

Imagine a metaphor
--- enlarging a THK  is just like attempting to change a wheel in a belt drive : no special
problem to put in larger wheels , just adjust the belt length.
--- enlarging multi-strand is like trying to put a gear of a different diameter and different
"threading" in a box of gears.





Just do the successive isometric grids for an enlargement.
You will then visualise the addition of BIGHTs (in EVEN number) and of LEADs (in EVEN
 number)
This addition is *not* to be defined -as some may believe they are right to do- as a
symmetrical addition (it cannot be when ODD number of Bight and with the PINs shift
for those see : batsbelfry_14 ) but is rather a bilateral equal addition of BIGHTs, as for
the LEADs where the symmetry could be ?  
There are ODD-numbered Half-Period (so going from Bottom to Top, and EVEN numbered
Half-Periods so going from Top to Bottom while there are the order of the apparition of the
added bights to consider for the symmetry of the process!!! simple fact
of observation that any average brain can make, not theory, just digested and assimilated
experience.)
 Just look there and tell me where you see a symmetry (remember this is a cylinder
opened 
and put flat ! ) ?   here in photography so may be easier to 'get'.

Bilateral equality in the addition done is certainly not symmetry in the proper usage
of this word.

A proper way to state what happens is  "adding  BIGHTs in equal numbers on BOTH
KNOT EDGE while adding the LEADs that are necessary to do that addition of BIGHTs"
(not always it is the case that added BIGHTs = added LEADs).
Juts read my articles on the mathematics of THK Part one  and Part two written well
before I could have access to Schaake and Turner 's work.





####################################################################





Enlarging is taking a knot, that is pre-existing in the material world of cordage,
to make it GROW using "the tail" of the very same strand that built the "root knot" or
"seed knot".
This is a transformation by growth.



Having a 'larger' knot appears in "one go" from a larger grid is NOT a  transformation but a
formation.

Enlarging is a sedimentation, a gradual accumulation, in successive "waves" or "phases"
of growth, the transformation by growth of some pre-existing thing.

This time I am stealing from a French writer Henri BERGSON writing on Creative Evolution
 in 1907  : "there is also a RELATION OF CHRONOLOGICAL SUCCESSION....(....between the
species in which those forms  are materialised).  There lie a good parallel with 'enlargement
la Schaake'.


Enlargement is like the growth of a crystal by successive construction, nothing like the
crystallization of a "big one" all at once in a single act of creation.


For a larger model of a given knot its beginning is commencing at the same instant both in
space and in time.
 It is a" all or none" event ; either the knot is fully made, finished or it does not exist as such.
It is a bit like a Roman or a Gothic arch either it is fully finished and can stand up even after
the scaffold has been dismantled or it cannot because it  is still not an arch that "locks itself".
There is no intermediate phase where the knot may be considered a 'smallish' representation
of the projected one.



In an enlargement process the space is already occupied by a previous time line of
fabrication. A time line that has already ended in a fully finished knot when an additional time
line (successive or consecutive depending on the time lapse between two enlargement
-as for underwater diving procedure)  of fabrication is opened allowing, in space, some
addition to the structure of the knot. 
That newly fully finished knot, in turn,  may be subjected to an enlargement process.



The making of bigger and bigger creatures from bigger and bigger diagrams or by adding
material is DISCONTINUOUS but the successive enlargements from a"seed"  are a  
CONTINUOUS
making.
Enlargement of rank  n+1 inherit directly a big share of its substance from rank (n.)




####################################################################





I fear that now I am going to go on "battering down wide opened doors" as they say in
French but this may have escaped the attention of one or two persons, but of course not
the attention of all the exalted ex-spurts, even those not even knowing a proper nomenclature.

There transpire a very peculiar phenomena with the THK that  one can  easily observe.

Even if in biology the saying "ontology recaps phylogeny" is a figment of imagination under
the formulation used,  for the THK it seems true.


Start with a 3L 1B 

Using a
RIGHT side of the SPart or TYPE I enlargement go to a 7L 3B 
This add 4L and 2B

From this, using a LEFT side of the SPart or TYPE II enlargement make a 17L 7B
This add 10L and 4B (changing side changed the module)

From this, using  a
LEFT side of the SPart or TYPE II enlargement make a 27L 11B
This add and 4B (not changing side again : module is kept)

This succession  is hidden by the rigidity of the PINs positioning during the making of the
THK but  using some trick we can see that we could have stopped at intermediate stage
and still keeping a complete, finished knot.
The curious among readers will have look at the 7L 3B to see the 3L 1B it  is said to come
from and triumphantly  said "wrong Mister!".
Well! sorry!  sure someone is wrong there, but that is not me.


Annexe note :
From the 27L 11B another LEFT side of the SPart enlargement would lead to a 37L 15B 
This add 10L and 4B, each enlargement in the whole process of enlargement add 5L 2B.
BUT a 
RIGHT side of the SPart or Type I enlargement  would lead to a 71L 29B
This add 44L 18B  and then a RIGHT side of the SPart or Type I would lead to a  
115L 47B
This add 44L 18B
while a 
LEFT side of the SPart or Type II would lead to a  169L 69B
This add  98L 40B

Enlargements are MODULAR :  till you change the type of the enlargement the MODULE
 of enlargement
stay identical .

That is not so with the creation of a bigger model where there is no 'inheritance' from a
previously threaded cordage, it is a de novo creation. Nothing modular happen in the
making.



For "basic" thinking : enlarging is just keeping the "base model" of the car while adding
some new "options", it is done on the same assembly line!
Making a bigger model at one go is creating  a whole new model needing it own assembly
line.




####################################################################




This topic, as a by-product, hopefully, will show why THKs are not to be confused with any
 of the other Regular Cylindrical Knots***.
(***RK = SINGLE-STRAND knots made on a THK CORDAGE ROUTE but that can have an
ADDED  CODE PATTERN
that is not always the O1-U1  (U1-O1) of the 'true' THK).
It should also show why it is being egregiously mistaken, to the point of raw idiocy, to insist
on going on using such ''''funny'''' wording as : " THK umbrella", "complicated THK" and
other utterly meaningless naming. (no point of contact with an external reality readily
accessible by simple elementary visual observation).




####################################################################



THE BRAIDER n6 - 1996 May has a feature on enlargements, giving formulas.
It is not just a pragmatical collection of known knots as HALL's book is but is a general
theory
of the enlargement of THK

I used this theorisation to write my HP48GX programs and Excel Worksheets on
ENLARGEMENTs
, it also went into Rknot Builder created by Claude HOCHET which can do
ANY coding and is not sadly limited to column-coding.)

I wrote one HP48GX program which calculates the Regular Knot Tree (RKT) as it
was published  in THE BRAIDER n 7  in 1996 August (with follow-on in THE BRAIDER
n 8 and
THE BRAIDER n 9).


Schaake also published about MAT ENLARGEMENT in 
THE BRAIDER n 10, 11,12 & 13.
illustration one
illustration two


In 
THE BRAIDER  N 30 is published a feature on (perfect==single-strand)
SOME PERFECT REGULAR NESTED CYLINDRICAL KNOTS ENLARGEMENT . examples
For those having no understanding of the meaning of that name : those knots can be, some
of the time, the spherical covering put on a ball core (so a spherical covering and not a globe
knot)
Illustration one                  Illustration two                 Illustration three          




####################################################################



It could appear to some that an experiment on some basic knot is a good way to get THE
 solution.

ONE  PROBLEM, OF
FIRST MAGNITUDE, IS THAT THIS IS ENTIRELY RESTING
 ON THE EGREGIOUSLY MISTAKEN ASSUMPTION (no validity whatsoever) that
'spherical covering's are all belonging to a 
single homogeneous class of knots)

However, the experiment falls short, quite short, of the  useful whether it fails or it succeeds.

 
What would that experiment on one knot among many belonging to very different
structural classes prove? ABSOLUTELY nothing !

Fails : well that could be an authentic failure or it could be a sham failure with just
the experimenter not having been good enough to find a way.
Succeeds : well, so what ? this could be a fluke concerning this knot only (first verify the
validity of the procedure used, it is really dreadfully easy to make a slight mistake that
allow a "success" that, really,  is just a faulty way to do things).

From a logical point of view this isolated success or failure will apply only to the knot
experimented upon unless some 'general' conclusion can be formulated from it.
That one experiment on a random knot chosen in a range of structurally very different
knots belonging to different classes is a "falsely good" idea.

There is knowledge only if coming from the 'general' for the value of the 'particular' is quite
limited to being just an "anecdote", interesting may be but just an anecdote till it can be
generalised in manner having internal and external validity.



Here is a paradoxical truth :
Only one counter-example can disprove (falsify) an hypothesis, yet no amount of
accumulation of positive practical examples can prove the validity of an hypothesis.

Collect 1.000 examples that worked beautifully, this still prove absolutely nothing
as the 1.001 may happen to disprove that a general rule has been established by the fist
1000 positive results !

In sciences we accept only 'falsification' of hypothesis, we don't know how to prove an
hypothesis  to be 'true'.

An hypothesis that has been, with a properly controlled and accepted procedure,
experimentally falsified, even only once, is considered 'false'.
Not being able to prove something 'false' does not 'prove it is true".
Absence of proof is never proof of absence.


One HAS TO demonstrate with a logically established generalisation.

A logically established generalisation like Schaake did for Regular Knots and some
very particular
mats. (do not get the funny idea that mats """"""can generally""""""" (pincer
 needed again with this insignificant statement) be enlarged !)


Mats (so called 2D-knots), to be enlarged without fail, have to be equivalent to Regular
Cylindrical Knots. (see illustrations given in the digression).

Of course some mats that cannot be enlarged ' la Schaake' can indeed be made in a larger
 model starting with a cordage coming from the spool and using a larger diagram.





Let us quickly look a very tiny sample of the experiments I did about enlargement of
2-dimensional flat knots, mats or flat knots.


Some mats can be enlarged but not on both sides of the SPart.
Don WRIGHT superbly proofed in the cordage my expansion of the Ocean Plait.

This mat  I could enlarge ' la Schaake' *but* only on the 
LEFT side of the SPart.
Here is some finding without any firm conclusion reached.
This success does not prove that ANY 2-dimensional flat knots or mats may be expanded.
It established only that at least one, this particular one can.
Just one anecdote, interesting, even fascinating but with no possibility of making any
generalisation from it.



With a mat derived from the Jury Rig Mast knot (no rotational symmetry) it was impossible
for me to make an enlargement'  la Schaake'.
 Failure with Jury Rig Mast Mat
This alone falsify (establish as 'false') the 'all mats can be enlarged'
hypothesis. ( let us
forget all other formulation such as "generally", "usually"...which are meaningless)




Another (impossible) enlargement, this time of ABoK#2360 (rotational symmetry present)
also clearly disprove that *all* mats can be enlarged.
Beware about so-called rotational symmetry : always verify that indeed it is real and not
 a figment of imagination like in #2216

It is easy to test, that it is impossible to enlarge 
#2360 whether LEFT of SPart vector
or RIGHT of SPart vector.
It does not seem to me possible to, at the same time, enlarge it and keep it single strand or
at least I do not see how to do it !

THIS FAILURE WITH THE ABoK#2360 is quite sufficient to establish that a
ROTATIONAL SYMMETRY IS NOT SUFFICIENT to hope for a possible enlargement.
Abstain from making of a fluke happy positive event a general rule.

I think though  'some 'turning' in the cordage route' seems to have a role to play :
seems to me, from experience, that having opposite turns in a knot will kill the enlargement
(on a THK cordage route all gyrations go forward in the same direction. - no going back
no mixing of clockwise and anti-clockwise. This mixing is quite evident in the Pan Chang
cordage route) 



Saying 'mats can be enlarged', even with a prudent qualifier as 'most mats'  is a no-no !
SOME can be, the others have to be directly MADE IN A LARGER SIZE which is not at all
conceptually the same thing as enlarging them !



####################################################################



It would be not understanding the question (enlargement with a VERY PARTICULAR METHOD
 in  mind : " la Schaake") to imagine that, say, Matthew Walker Knots can be enlarged.

They can
certainly be made LARGER with a GREATER NUMBER OF STRANDs, that is why
one can qualify the MWK just made by appending the number of strands it is made of :
ex a 108S MWK but the naming method  says it all.
STRANDs need to be ADDED
So,
again, it is not  understanding what is LOGICALLY implied in the way the question was
formulated to think about that augmentation of the number of strands as doing an
enlargement.

Same thing for enlarging multi strand REGULAR NESTED BIGHT CYLINDRICAL
KNOTS such as HERRINGBONE-PINEAPPLE KNOTS. You may only ADD PASSes.



####################################################################



The enlargement of the THK cordage route is governed by the Regular Knot Tree.
(any of the Regular cylindrical knots : THK, gaucho, head-hunter, Samuel, Beverley,
Sedgewick, basket weave, helix, fan,... any what have you knot that are made on a THK
cordage route can be enlarged as far as the cordage route is concerned but this say
nothing of what will  happen to the coding pattern added on that route.



Enlargement ' la Schaake'  is a possibility, as far as I know, only for Column-coded
(horizontal mandrel frame of reference) and there are strict conditions to comply with if
one wants to meet with success in this enlargement).


*** Row-coded and Neither Row Nor Column coded will have their coded patterns
altered by addition of LEADs (hence addition of Columns - horizontal mandrel reference)
 and addition of BIGHTs(hence addition of Rows).
T
his alone plainly establish the silliness of equating all the Regular Cylindrical 
Knots with
the THK.


*** Column-coded (exclusive Column-coded without any Row code added -again
horizontal mandrel reference) can be enlargement IF AND ONLY IF COMPLYING (under
penalty of destroying them)  WITH STRICT PROCEDURAL MOVES also plainly establish
the idiocy of equating all the Regular Knots with the THK.



*** only the O1-U1 ( U1-O1) 
Row *andColumn coded Regular Cylindrical Knots
or  Turk's head Knots can take enlargements in their  stride with a simple procedure (make
tracks, split tracks) an addition in LEADs / BIGHTs.



It seems to me that here is the right place to hammer a nail = enlargement can be understood
as being 'of the knot itself' or 'of its cordage route or shadow'.
One comply with the original coding, the other does not take any account of the coding and
is just being attentive to enlarging the cordage route. ( ' la Schaake' is taking care of
complying with the coding while making the cordage route GROW)

A very good example in my opinion is ABoK#579

This last knot has exactly the shadow, the cordage route of a 3L 5B THK
if one take care
to make a re-enter.

ABoK#589 can have its cordage route enlarge very easily , with only one enlargement or
with two enlargement as in a full ' la Schaake' enlargement process :
first enlargement is to enlarge the cordage route but it kills the coding in a THK that needs
a second enlargement to re-establish the proper coding.
With #589 we can enlarge the knot and keep its coding with only one enlargement or
we can make two enlargement with keeping the coding or with changing it !
Funny trick that help to understand what 'a la Schaake' is and what it is not = its two
enlargement in one process, enlarging the cordage route *but with keeping* the original coding.

You may decide, as far as *only* the cordage route is concerned, to make only
one enlargement or two enlargements for a full process. Still that say nothing about the
coding of the crossings those enlargements are producing.

With only one enlargement you may have two choices : making the same type of crossing
than the reference HP (as in ' la Schaake) or making crossings of opposed type.
In *this* particular ABoK#579 only the same type will respect the original coding.

With the two enlargements you get several 'mix' of crossings.
*Only* one will respect the original coding and it is *not* following " la Schaake" mode.
The " la Schaake" (two enlargement in the same process, first making crossing of same type
and second making crossings of opposite type ) will not respect the original coding.

So you see : there MAY be possible " GROWTH(s)"  but they will not all comply with the
" la Schaake" THK mode : enlarging the cordage route *while still* complying with the
original coding.



####################################################################



Single strand knot is not a sufficient criterion even if it is a necessary one.

This single strand must necessarily re-enter the cordage route it is following (still if this is
 necessary  it is  not sufficient.)

This 're-enter'  means that the curve of the cordage route has to close on itself, has to re-enter,
the
Working End must come back to its Standing Part, to its starting point.
Hence the impossibility of enlarging say, a Bowline or a Fig-8 or any such other knot even
symmetrical ones !
Rotational symmetry or any other symmetry for that matter is not  a necessary condition.



####################################################################

A PARTING GIFT

Try your hand drawing the  TYPE II  ( LEFT side of SPART-HP1 ) on a FLAT
3L 4B cordage route ( no crossing , just the shadow of the route )
Try studying the 3L 5B example previously given ( #589) to get an idea

Here is a template that you can print for your dabbling.

Here is the "blue print" of the enlargement in isometric and non-isometric which will, again,
show the superiority of the isometric on any other grid for knot purpose.



Here is the solution shown, but you have at least 3 dozens of attempts at doing it yourself
before cheating. If you do it first time you area genius, if you do it in less than half-dozen
attempts you still are a genius, if you do it in a score (20) of attempts then you are still a
genius !

This should convince you to do it on an isometric grid diagram instead of on a tracing
of the mat form. SO EASY with ISOMETRIC.

####################################################################

Taking up the original question

THKs can be expanded, on the LEFT or the RIGHT...

--- is there away to do that with globe knots?
SEVERAL WILDLY DIFFERENT CLASSES OF KNOTS MAY SERVE AS
SPHERICAL COVERING, SOME MAY EVEN TRULY BE MADE INTO
GLOBE KNOT :
YES SOME 'special' ones CAN BE ENLARGED,  the other was not demonstrated
as being surely 'enlargeable'
.


---how about other decorative knots? TRY IT FOR EACH KNOT THAT IS
TEMPTING YOU BUT KEEPING IN MIND Charles Le Tmraire, KING of
FRANCE, motto borrowed a century latter by WILHELM of ORANGE the First :
"no need of hope to undertake, no need of success to persevere"       


---which knot types do people know can be expanded?
Some REGULAR CYLINDRICAL KNOTS among which THK, some MATS,
some REGULAR NESTED BIGHT CYLINDRICAL  Knots and, (if we accept to
take some liberty as in the SHAW's drawings, some ornamental flower knots. )  


--- has someone developed the methods ?
YES : SCHAAKE and TURNER for the THK processes !  
and for the flower knots :'tradition' with no formal theorising, just "practical recipes".            


===========================================================

[begin digression on history of the naming of the enlargement processes] xxxxxxxxxxxxxx
may be jump it !  
      
              
The PROCESSES of the ENLARGEMENT of THK (THK as strictly defined) were fully
studied and formalised by SCHAAKE, by none other, in particular *not* by Tom
HALL as some persons believe. Tom HALL, as I often do myself, is just using Schaake's
work)

History is explicit in the dates and implicit in the fact that Tom HALL, who was on the list
of people getting The Braider, does gives the address for contacting  Schaake. 
Tom HALL, in 1992, made pretty hand drawings for Braiding -Standard Herringbone
Knots
(SCHAAKE for ever after that one book with hand-drawing safely kept to the more
informative and clearer, and how so !,  isometric grids. Isometric diagrams are less pretty
visually though, alas this is not about "making pretty" but making clear!).


SCHAAKE with TURNER published :
 The Regular Knot Tree And Enlargement Processes -  pamphlet no. 4 by A.G.
Schaake and J.C. Turner. 38 pages of diagrams and formulas.  July 1991
Hamilton, n.z. : dept. of mathematics and statistics, university of Waikato, [1991]
38 p. : ill. .
illustration

First edition of Tom HALL, no University printing but self-published, Introduction To
Turk's-Head Knot
s was 1996 ;
FIVE years later !

Mind you!  Introduction To Turk's-Head Knots is a good buy for a curious learner (who,
IMO, will be well inspired to leave alone the lousy exposition made of Schaake's Bight
Algorithm. This algorithm will be best learned in THE BRAIDER or, faraway second, in this
web space. 
(Introduction To Turk's-Head Knots can be bought at  Martin Combs web shop :
http://www.angelfire.com/ak/skateworld/ )

[end digression]xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


===========================================================


**** Common ground building :

To expand = rather for "volume"   (3D). A "puffing up" ? as may be in expand this formula
so going from   ( 9x^2 + 1 ) ^ 4
to   6561x ^8 + 2916 x^6  + 486 x^4  + 36 x^ + 1

To enlarge  = rather for width and length (2D -  in enlargement only 2 dimensions are
 affected, namely LEADs and BIGHTs). A growth ?

To extend = rather unidirectional as in pulling on a rubber band.. A prolongation ? May be
with an idea of "being straighter" ?

...just my 2 cents of Euro, but after all I am just the ignorant foreigner barely managing to
express myself,and not so clearly at that, in English (more UK than USA I certainly hope !
even if that explain why some Americans seems to be unable to understand some spelling,
words and formulations)

To evolve= a gradual change and growth over time (opposite:devolve) of particular 'physical'
traits (as Leads or Bights for example and not simply size )

Growth=an increase of some stated quantity with time according to a very precise process.

[open quote]
We've seen that sloppy or misleading use of ordinary language can seriously limit our ability
 to create and communicate correct reasoning. As philosopher John Locke pointed out three
 centuries ago, the achievement of human knowledge is often hampered by the use of words
 without fixed signification. Needless controversy is sometimes produced and perpetuated by
an unacknowledged ambiguity in the application of key terms.We can distinguish disputes of
three sorts:


    * Genuine disputes involve disagreement about whether or not some specific proposition
is true. Since the people engaged in a genuine dispute agree on the meaning of the words by
means of which they convey their respective positions, each of them can propose and assess
logical arguments that might eventually lead to a resolution of their differences.

    * Merely verbal disputes, on the other hand, arise entirely from ambiguities in the language
used to express the positions of the disputants. A verbal dispute disappears entirely once the
people involved arrive at an agreement on the meaning of their terms, since doing so reveals
their underlying agreement in belief.

    * Apparently verbal but really genuine disputes can also occur, of course. In cases of this
sort, the resolution of every ambiguity only reveals an underlying genuine dispute. Once that's
been discovered, it can be addressed fruitfully by appropriate methods of reasoning.


We can save a lot of time, sharpen our reasoning abilities, and communicate with each other
more effectively if we watch for disagreements about the meaning of words and try to resolve
them whenever we can.

[end quote]
in http://www.philosophypages.com/lg/e05.htm you will find may think to ponder.



 _______________
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/