This was what I wrote for ARIANE User's Manual but today I see the need to make additions to that. I should have added : ALL the BIGHT-RIM belonging to the ...
PROPOSAL OF ADDITIONS OF TWO NEW TYPES OF NESTED-BIGHTS CYLINDRICAL KNOTS :
ABERRANT and HYPER-ABERRANT First let us refresh our minds about NESTED-BIGHT CYLINDRICAL KNOTS nomenclature as formalised by SCHAAKE and TURNER.
There exist : -
REGULAR SYMMETRIC
-
REGULAR ASYMMETRIC
-
IRREGULAR ( symmetry and asymmetry is not really a criterion in such cases)
*** ON BOTH KNOT EDGE ALL the BIGHT-NESTs WITHOUT EXCEPTION have ONE BIGHT on the MOST EXTERNAL BIGHT-RIM. BIGHT-RIM N° 1. (Criterion ONE). *** ON A GIVEN KNOT EDGE ALL THE BIGHT-NESTs have the same number of BIGHTs. This number is the same for both edges in symmetric and different in asymmetric.(Criterion TWO corollary of Criterion ONE). ***In EACH BIGHT-NEST OF A KNOT ALL the BIGHTs (PINs) belonging to it are PERFECTLY ALIGNED and INSIDE A BIGHT-NEST THERE IS NO PIN BELONGING TO ANOTHER BIGHT-NEST. (Criterion THREE). ALL THE CYLINDRICAL KNOTS WITH BIGHT-NESTs THAT COMPLY WITH THOSE THREE CRITERIA ARE SAID TO BE “REGULAR”. ONE MISSING CRITERION MAKES IT AN “IRREGULAR”.
This was what I wrote for ARIANE User’s Manual but today I see the need to make additions to that. I should have added : ALL the BIGHT-RIM belonging to the SAME BIGHT-BORDER are NOT separated by the intercalation of a BIGHT-RIM belonging to the OPPOSITE BIGHT-BORDER but this was not necessary as ALL the NBCK as formalised by Schaake were compliant with that ‘rule’.
Note : all along this document we will be using the HORIZONTAL cylinder frame of reference with BIGHT BORDERS
at TOP and BOTTOM.
In this frame of reference and in a normalised system the ODD numbered HALF-PERIODS (HP) are going from BOTTOM-RIGHT to TOP-LEFT and the EVEN numbered HP are going from TOP-RIGHT to BOTTOM-LEFT
Fig C HOW TO “COUNT” ON THE GRID ( IRREGULAR here )
Fig D
IRREGULAR NBCK
Charles HAMEL
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Fig E IRREGULAR NBCK
You will, of course, have noted that for the REGULAR NBCK whether SYMMETRIC or ASYMMETRIC
on each BIGHT-BORDER the distance separating each BIGHT-RIM from its
proximate neighbour(s) is always 2 “UNIT”
For the IRREGULAR NBCK the minimal distance inter-BIGHT-RIM is 1 “UNIT” an can be equal to 2 “UNIT” or be greater than 2 but in such a case always a multiple of 2. More than that all the inter-BIGHT-RIM intervals are not equal one to any of the others and each BIGHT-BORDER has its own sequence of inter-BIGHT-RIM intervals. Each successive interval ( going from the innermost BIGHT-RIM to the outermost one ) is either equal to the preceding one of is the double of it. The greatest BIGHT-RIM interval of one BIGHT-BORDER is at most the double of the greatest BIGHT-RIM interval of the other BIGHT-BORDER.
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Now let us explore a new country :
ABERRANT : I am using this word in a strictly descriptive value, in no way demeaning, and meaning that there is some particularity ( ties) that make what is refered to depart, deviate from the normal usual types used as nomenclature.
Source for the ABERRANT and HYPER-ABERRANT NBCK : The Globe Knot Cookbook by Don Burrhus - self-published. ABERRANT : SIX grids Page 164 - 165 - 166 - 167 - 168 - 169 HYPER-ABERRANT : FOURTEEN grids Page 172 - 173 - 174 - 175 - 176 - 177 - 178 - 179 - 180 - 181 - 182 - 183 184 - 185 Now let us see about the ABERRANT ones, a new type I am proposing for the nomenclature.
Fig 1
This way of drawing a knot diagram is senseless and meaningless ( intellectual evaluation and not affective judgement ) for any observation or analysis purpose, it is just good enough for some knot tyers who like “pretty drawing” over “technical drawing”. ( and don’t you dare think that I am baying after Burrhus ! my good friend Norbert TRUPIANO aka NONO does this sort of grids when he invents a knot and I very much love Norbert and have a lot of respect for his intelligence)
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Discarding the high rate in prevalence of lack of efficient power of observation and analysis acumen around us it remains the fact that with such diagrams it is about impossible to perceive characteristics that will “ jump to your face” when studying an isometric diagram.
This probably explain why almost no one will have seen that in Fig 1 there is NOT THREE BIGHT-RIM at the TOP and ONE at the BOTTOM but TWO BIGHT-RIM on each BIGHT BORDER.
The tell tale sign is that a BIGHT belonging to ONE BIGHT BORDER has it concavity oriented toward the OPPOSITE BIGHT BORDER.
The ABERRATION seen in Fig 2 is that of having a Schaake’s distance ‘x’ ( SPACING in Ariane) that *is not* cut through by the horizontal middle line that is equidistant from the two most external BIGHT RIM. (Fig 2 bottom left side figure)
Fig 2
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Fig 3 As is quite evident in Fig 3 the ‘distance x ‘, that is here the 3 ‘UNIT’ ‘spacing’ between the two innermost BIGHT-RIM in each BIGHT-BORDER, is well above the horizontal line of the middle of the height in this grid.
Usually persons not seeking variation for the sake of variation and augmenting the tally of ”””new””” knots change the size of ‘distance x’ (SPACING in Ariane ) but not the spacing separating TWO BIGHT-RIMS ON THE SAME BIGHT BORDER. ( 2 ‘UNIT’ at the TOP, 8 ‘UNIT’ at
the BOTTOM ) which make that the greatest BIGHT-RIM
interval of one BIGHT-BORDER is more than the double of the greatest BIGHT-
RIM interval of the other BIGHT-BORDER ; in fact it is quadruple ( still a multiple of 2 ) Only one BIGHT-BORDER (TOP) has “true” NESTED-BIGHT.
Fig 5 Fig 4
The two Fig 4 & Fig 5 grids above illustrate the normal way of keeping the NESTED BIGHT CYLINDRICAL KNOTS TYPE while resizing the knot ( in number of LEADS: augmentation of
the SPACING from Fig 4 to Fig 5). Both BIGHT-BORDER have “true” NESTED-BIGHT”.
Charles HAMEL
18 mai 2012
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Page 8 sur 14 Fig 6
From this starting point
(REGULAR SYMMETRIC
NBCK)
If one augments – hence if one is not complying with the rules of the usual nested-bight cylindrical knots - the distance between the two lower BIGHT RIMS then one obtains
ABERRANT NBCK and destroy the “true” NESTED-BIGHT in the modified border.
Fig 7 is not ‘obviously’ aberrant and could still be considered as IRREGULAR ( missing criterion three – see page 1 ). But BOTTOM BIGHT BORDER does not have “true” NESTEDBIGHT.
Most of the ‘spacing” area ( more than 2/3) is well above the horizontal midline.
Fig 3 shows the aberration more clearly but the aberration is already evident in Fig 8 The lowest limit of the spacing zone is above the level of the horizontal middle line.
Fig 7 .
Fig 8
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The cheap trick played in Globe Knot Cookbook with 27SSC (p 164) 40QUC (p 167) 48 TTG (p 168)
36WSA (p 165)
42UYQ (p 166) , 52ZQB ( p 169) allows us to extend the
conclusions about one to the others in the “parentage” group !
Fig 9
In Fig 9 slicing off the light violet area of 48TTG yields 40QUC while slicing off the light blue area of 48TTG yields 36WSA. Slicing off the yellow area of 36WSA *and* modifying the lowermost crossings yields 27SSC.
Fig 10
In Fig 10 slicing off the light blue area of 56ZQB yields 42UYQ.
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Lets us now look the
HYPER-ABERRANT NBCK.
Fig 11 ( quoted from BURRHUS ) This is the ‘original’ grid leading to this proposal of HYPER-ABERRANT NBCK
Fig 12 Fig 12 is the grid traced using ARIANE ( non public version ) It is SYMMETRIC ( same number of BIGHT-RIM for both BIGHT-BORDER ) but the inter-BIGHT-RIM intervals
are ‘queer” : 2 ‘UNIT’ for interval Bight-Rim N°1 Bight-Rim N°2 on each BIGHT-BORDER BUT 3 ‘UNIT’ ( 3 is certainly NOT a multiple of 2 ! ) for interval Bight-Rim N°2 Bight-Rim N°3 on each BIGHT-BORDER
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Fig 13 You see that the third (innermost) BIGHT RIM of each BIGHT BORDER is
lying between two BIGHT RIM belonging to the opposite BIGHT BORDER :
hyper aberrant as this make for an “inverted” spacing : the topmost limit of the spacing is made by a BIGHT-RIM belonging to the BOTTOM BIGHT BORDER and vice versa.
Here again the inter-BIGHT-RIM intervals are ‘queer” : 2 ‘UNIT’ for interval Bight-Rim N°1 Bight-Rim N°2 on each BIGHT-BORDER BUT 7 ‘UNIT’ ( 7 is certainly NOT a multiple of 2 ! ) for interval Bight-Rim N°2 Bight-Rim N°3 on each BIGHT-BORDER
Fig 14 In Fig 14 I am pointing to the overlapping of the two BIGHT BORDER due the fact that the innermost BIGHT RIM of both BIGHT BORDER has been
inserted between two BIGHT RIM belonging to the OPPOSITE SIDE BIGHT BORDER.
Yellow plus blue yield green and the overlapping is the “green zone”. It is like tiles overlapping.
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Fig 15
Fig 16
80DGU dissected Again the cheap trick played in Globe Knot Cookbook with 80DGU in page 178 allows us to extend the remarks that it is possible to make on this knot to several other ones which are just very simple derivations of it : ** Take off the yellow part of 80DGU and you get 72SYX in page177 ** Take off the yellow and blue parts and you get 64XQZ in page 176 ** Take of the yellow, blue and red parts and you get 56WYA in page 175 Here we can see something quite paradoxical : the interval between BIGHT-RIM N° 2 of ONE BIGHT BORDER and BIGHT-RIM N°
3 of the OPPOSITE BIGHT-BORDER is 3 ‘UNIT’ while the interval between BIGHTRIM N° 2 and BIGHT-RIM N° 3 in each BIGHT-BORDER is a huge 18 ‘UNIT’.
Charles HAMEL
18 mai 2012
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Fig 17
It is the same cheap trick with 48MQN in page 174, 48JWR in page 173 and 42UMD in page 172
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18 mai 2012
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REGULAR NBCK
ABERRANT
HYPER ABERRANT
SYM
ASYM
METRIC
METRIC
yes
no
yes
yes
yes
no
yes
yes
have the same number of
yes
no
yes
yes
In ALL the BIGHT-NEST the pins are perfectly
yes
no
yes
no
no
no
yes
yes
no
no
no
yes
no
no
yes
no
no
no
no
yes
Same number of BIGHT RIM on each BIGHT BORDER ALL BIGHT-NEST
on a BIGHT BORDER have a BIGHT
on the most external BIGHT-RIM ALL BIGHT-NESTS BIGHTS
aligned and there is no pin belonging to a neighbouring BIGHT-NEST The innermost BIGHT RIM of a BIGHT BORDER may be pushed beyond the middle line The innermost BIGHT RIM of one BIGHT BORDER may be pushed to lay between two BIGHT RIM belonging to the opposite BIGHT BORDER The innermost BIGHT RIM of ONLY ONE BIGHT BORDER LINE
is on the same side of the KNOT MIDDLE
as the OPPOSITE BIGHT BORDER is
The innermost BIGHT RIM of BOTH BIGHT BORDER is on the same side of the KNOT MIDDLE LINE as the OPPOSITE BIGHT BORDER is
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