Nautile
aka Charles Hamel's personal pages

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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 .

Just to get an idea of the width of the landscape have a look at this illustration from S & T.

A-PASS SHPK :

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 S

boundary 1 to the right bight boundary a2

The TYPE associated with a2=0 is denoted S

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

"

.....REGULAR NESTED CYLINDRICAL BRAIDS

-1) Standard Regular nested Cylindrical Braids

-2) Semi-Standard Regular nested Cylindrical Braids

-3) Perfect Regular nested Cylindrical Braids

-4) Semi-Perfect Regular nested Cylindrical Braids

-5) Compound Regular nested Cylindrical Braids

-6) Semi-Compound Regular nested Cylindrical Braids

......FOUR Herringbone Pineapple Knots Sub-Families

and SIX Broken-Herringbone Pineapple Sub_Families.

Herringbone Pineapple Knots :

-1) Standard Herringbone Pineapple Knots

-2) Semi-Standard ">Herringbone Pineapple Knots

-3) Perfect Herringbone Pineapple Knots

-4) Semi-Perfect Herringbone Pineapple Knots

Broken Herringbone Pineapple knots

-1) Standard Broken Herringbone Pineapple knots

-2) Semi-Standard Broken Herringbone Pineapple knots

-3) Perfect Broken Herringbone Pineapple knots

-4) Semi-Perfect Broken Herringbone Pineapple knots

-5)Compound Broken Herringbone Pineapple knots

-6) Semi-Compounded Broken Herringbone Pineapple knots"

Dizzy are not you ?

Now the important part straight from 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"

See this illustration from S & T

********************************************************

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.

Left edge and Right edge essential coding straight from S & T

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)

B

x = cA + δ where in (3)

δ= |2 (l

c = ( 2m - 3 ) + ( 2 (l

L

L = Sum L

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)

Url : http://charles.hamel.free.fr/knots-and-cordages/