Without
the freedom to make critical
remarks, there cannot exist sincere
flattering praise - Beaumarchais.
The PINEAPPLE (for short) Knot (PK) GROUP is but only a part
of a larger set :
let us
say a 'class' ; as in Linnean classification.
a CLASS with one
visually evident
characteristic : NESTED
BIGHTS
( bights that are not fully bights because they do
not lie smack on the frontier or the rim, but
are like
seaweeds left lower and lower on
the slant of the cove beach by an ebbing tide.
Lets us denote those NESTED-BIGHTs KNOTS by :
REGULARLY
NESTED BIGHT (RNB) CYLINDRICAL KNOTS .
Just two diagrams for a swift "dismembering" of the notion of PINEAPPLE diagram
one diagram
two
In a rather careless and unreasoned
manner (my opinion , you
may hold another one)
too many
persons unthinkingly, unREASONably, label ""pineapple"" any knot with
regularly nested
Bight.
A bit like :
Pineapple == a hammer
Nested Bight == a striking tool
so they call any striking tool a hammer.
Does not that sound
real funny (this time : funny as in, stupid )?
All
PINEAPPLE are
RNB Knot ( RNBK) but this is
not
a reciprocal relation so
all RNBK are NOT STANDARD PINEAPPLE.
Once again we need to fall back on hard, solid ground : the STRING RUN
of
the knots
will provide that.
(shadow == cordage route or string run for S
& T as I discovered ; see my explanation
on *shadow*
elsewhere in
this web space : SHADOW is "fixed" , STRING RUN can be
"dynamic" )
Just as with Linnean classification (which is rather using *rank*
or *order* variables rather
than *nominal* ones and
presuppose
a "gradual evolution" from one form to the other) we
will try
to follow a rational (we will forget though about Kingdom,
Phylum,
Division, Order,
Genus) and make use of Class and sub-class with Family and sub-Family
following in that
S & T very clear writing. (There is another defender of the
Linnean in knots it is Friend
Frank Charles BROWN of Tasmania - myself I would be leaning more
toward a cladistics
approach )
Using the cordage route with only the presence
of
crossings being acknowledged without
any indication of their nature,
that is without the so called coding / code we will get down to
sorting
into Knots Families and Sub-Families
according to their 'string
run'.
In S&T words all text in green is a direct
quote you should read the original text rather
than this rather ham-fisted summary )
" ....writers
recently seem to call any knot belonging to the Regular-Nested Bight
Cylindrical
Braid-class a Pineapple Knot. Then
the wonder why they are
unable to find the
relationships that govern these knots. It should be obvious
that we have to start at the
beginning,
and that it is with the study of the string-runs of braids ; this will
lead to braid
classes, each of which, in general divide into
braid-sub-classes. Each of these braid
sub-classes may in turn divided
into braid families ; and each of these may divide into braid
sub-families. Only
after
this classification has been completed can we start with the study
of
the consequences imposed by the different weaving patterns : the coding.
This
study will lead to knot classes, which in general divided into knot
sub-classes.
Each of these knot sub-classes may in turn divide into
knot families ; and each of these may
divide into knot sub-families"
Exactly
what conclusion I had arrived at before meeting with S
& T
writings even if I
don't
really like their use of "braid" around those
knots the conclusion still remain internally and
externally
valid and rock solid in logical grounds.
As usual in sciences it is best to go 'general to
particular'.
So from general RNBK to particular HPK
REGULAR NESTED
BIGHT KNOTS ( RNBK )
will give SIX "slots"
Standard RNBK
Semi-standard RNBK
Perfect RNBK
Semi-Perfect RNBK
Compound RNBK
Semi-Compound RNBK
EACH
have their OWN QUITE
DISTINCTIVE MATHEMATICAL
CHARACTERISTICS that differentiate
OBJECTIVELY
between them and which is
certainly not an unreasoned, whimsical or
unclear distinction even if most cannot even begin
to get that point.
The Standard Herringbone Pineapple Knots mathematical
formulation covers 5
printed pages plus 47
just for worked examples.
From this point I am abandoning, for the sake of "common ground"
building, what I
had built as my own nomenclature , another reason of doing so
is because theirs is
better
and
clearer
all over than what I had not as sufficiently deeply pondered before
meeting
S & T .
an ODD H-P ( H-P = half-period for me, half-cycle for S
& T ) reach to the opposite side
of the knot where the
even H-P starts and it can makes its 'run' 1, or 2, or 3.... or
A-1 , or A.
Imagine the seaweeds line abandoned by the ebbing tide !
There is only one degree of liberty in this serial choice : the first
choice.
As soon as the first choice of Pass 1 is made you are doomed to follow
a pre-determined
sequence from which you may not escape.
So as it is quite plain to see this will provide a sure way to classify
SHPK by
TYPES.
"...notation S1
to denote the TYPE of SHPK which has a half-cycle running from the left
bight boundary 1 to the right Bight boundary 1
Similarly S2 denotes SHPK will have a half-cycle running from the left
bight boundary
1 to right bight boundary 2.
Hence in general Sa2 denotes SHPK
which have
a half-cycle running from the left bight
boundary 1 to the right bight
boundary a2
The TYPE associated with a2=0 is denoted SA
"
Another by S & T
"... strongly
advised against using such restrictive methods ( my note = memorizing
"sequence patterns" ), since they
invariably lead, sooner or latter, to misconception
and erroneous conclusions.
For example, there exists literature which deals only with the two
types ( me : that indeed
do no more that speaking about two types
really ) of the 2-PASS SHPK ; and it wrongly
refers
to four types, when there are only 2 . There are ,
indeed, in case of
2-PASS
SHPK, two ways to braid each type, so there are four methods of
braiding them ;
but only two different types of these knot
result . This is the
kind of misconception
that can arise when very restrictive
set of members of an infinite family is studied or written
about."
How wonderfully put!
Going on with S & T
"...the standard SHPK are
generally referred to in the literature as Pineapple
Knots
( PK );however
this term is much too loose
I really cannot
put it
better.
******************************************************** General nature
REGULAR NESTED BIGHT BRAIDS
(Chap 2 )
" DEFINITION :
A Regular-Nested Cylindrical Braid is a cylindrical braid in which the
circular edges
consist of nests
of bights such that each
nest contains the same number of single
bights, stacked uniformly....
the number of bights per nest will be indicated by A, and the number of
nests per
circular edge will be indicated by B*. hence the total number
of bights per circular
edge is equal to A.B*,
and this will
be denoted by B"
******************************************************** PINEAPPLE KNOT CLASS
are defined by means of the essential
coding of
its members ( Chap 3 )
This knot class
have many sub-classes, one being the HPK sub-class which
is
divided into two families
:
- the HPK Family
- the Broken-HPK Family ( see appendix 2 )
"We now present
a definition of PINEAPPLE KNOTS. The reader should consult
Figures 7
and 8 , to help in understanding the details involved>
Consider the two half-cycles ( one slanting upwards and the other
downwards ) which
issue from a bight-point on bight-boundary k . On these
half-cycles, the first
( k
- 1 ) intersections they each make ( with half-cycles
issuing from bight-boundaries
1 to (k-1) inclusive) are
all under-crossings.
The essential coding is
the crux of the definition for knots of the pineapple Knot
class.
In practice, this weave is one that creates cylindrical braids
with hemi-spherical ends.
The knot-class
of ¨Pineapple Knots can be divided into many sub-classes, one of which
is
the Herringbone Pineapple knot sub-class.
The general definition
of herringbone coding is as follow :
Herringbone coding is a row-coding; it consists of sets, each of n adjacent rows,
every
row within a set having the same coding; the coding in adjacent sets
alternate throughout.
The overall effect in a Herringbone coded knot is of a succession of
stacked 'V's, arranged
with their apices pointing towards the left and
right bight-edges.
the Herringbone Pineapple sub-class in turn can be divided into two
major families, one
which has the herringbone coding, and the other
which has a broken-herringbone coding.
The members of the family which have a herringbone coding can be
divided into four
sub-families; whereas those in the family of the
broken-herringbone coding can be divided
into six sub-families"
To get the
whole story just buy from Dr J. TURNER Book 4/1 1991
BRAIDING - STANDARD HERRINGBONE PINEAPPLE KNOTS
just over 200 pages of
articulated explanations made painstakingly clear.
********************************************************
STANDARD
HERRINGBONE ¨PINEAPPLE KNOTS
(Chap 4 )
"...The SHPK are characterized
by the following relationships :
Δ = 0
(1)
Bcomponents and Lcomponents
are coprime (2)
x = cA + δ where in (3)
δ= |2 (li
+ ri
)|Mod A and (4)
c = ( 2m - 3 ) + ( 2 (li
+ ri ) -δ) / A
(5)
Lcomponents = ( x
+ 4A - 2 (li
+ ri )) / A
(6)
L = Sum Lcomponents = x +
2A - 2 (7)
IF AND ONLY IF conditions (1), (2) and (5) are ALL met, then the SHPK
will result.
They are necessary and
sufficient conditions for this type of knots. ********************************************************
Chap 5 : all mathematics THE ALGORITHM-TABLES FOR THE
STANDARD HERRINGBONE PINEAPPLE KNOTS
goes on from page 34 to page 85 ( limits included )
If you want to write a program "doing" those then you cannot escape
mastering that and
Chap 5
Good luck if you are of those who find S & T not clear !
******************************************************** " It is
convenient to divide the Standard HPK into TYPES; this enabled the
braider to
specify
unambiguously a particular Standard HPK. "(Chap 6 )
CLASSIFYING THE STANDARD HERRINGBONE PINEAPPLE KNOTS INTO TYPES
page
86 to page 92
********************************************************
Then follow
APPENDIX 1 (p 93 to p 133) : THE REGULAR-NESTED CYLINDRICAL BRAIDS
APPENDIX 2 ( p 134 to p 188 ) : THE HERRINGBONE PINEAPPLE KNOT SUB-CLASS
APPENDIX 3 ( p189 to 191 ) : MODULAR ARITHMETIC
APPENDIX 4 (p 192 to 194 ) : RELATIONSHIPS
BETWEEN BRAIDING ALGORITHM
CALCULATIONS
the rest is Bibliography This
book is well worth studying if one is to fully understand and
truly know
what one is tying or not tying : beware : you may not be tying what you
believe
you are tying !
A last cynical word :
there are fortunes*** to be made with a number of
ex-spurts (they come in all sort
of flavour ;
again thank
you Jimbo for introducing me to this spelling of the word quite
a while ago now) but there is none to be made with SCHAAKE and
TURNER.
***
This is in reference to one of the greatest French philosopher
and
moralist of the
Siècle Des Lumières :
" there are persons with whom it would be easy to make a great fortune
:
simply buy them for what they are actually worth and sell
them
for what they
are
supposed to be worth"
Copyright 2005 Sept - Charles
Hamel / Nautile -
Overall rewriting in August 2006 .
Copyright renewed. 2007-2012 -(each year of existence)
