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Added 2013 June 11th




SOME RANDOM THOUGHTS ABOUT SPHERICAL COVERS  ( AND
CALCULATORS = ALL QUITE USELESS AS PRECISION TOOL AND EVEN FOR MOST AS "APPROXIMATE GUESSTIMATE TOOL" ! )

During my vacation in Brittany I made some spherical coverings in a "testing perspective" as deliberately opposed to aesthetic perspective.


In my experience, ALL, repeat ALL calculators are, in diverse degrees,  HOPELESSLY out of kilter outside a rather narrow range ! if only because they only take in account the number of CROSSINGS and leave aside the Number of FACES and do not take LEAD, BIGHT-NEST, BIGHT and tighnteness of setting and Poisson modulus of the cordage....

I made mine using a data base of many such knots made by several tyers in widely different core diameter and cordage diameter so it is a bit better but still there are "ranges" where it is nearly as hopeless as any of the others.








THREE SPHERICAL COVERS / COVERINGS 210 CROSSINGS  - grids
specifications taken from Schaake and drawn using ARIANE



2-PLY 210 CROSSING  180 FACE
2-PLY 210 CROSSING 150 FACE  first CODING PATTERN
2-PLY 210 CROSSING 150 FACE with another CODING PATTERN

ALL are made using the same CORDAGE ROUTE:
--- minimizing as much as possible the polar aperture ( important for this test targetting the maximum of surface covered ) : important , when not testing it is best not to try to minimize the polar aperture at the detriment of the covering between the two polar circles ). To  "reveal" what happen it is important to minimize the polar apertures, while for aesthetics it is better to favour the clossest covering even if this mean  greater polar aperture.
--- with stiff ( do not flatten, do not become thin when stetched ) polyester braided rope 2.5mm diameter
--- on 39 mm rubber beach ball.
--- 2-PLY 

One is 150 FACE / FACET , the other 180 FACE / FACET

My personal calculator which take in account crossings AND facets tell le that 40mm for the 150F and 41mm for the 180F are the right dimensions.

Still I hold that ALL calculator are "silly" and works only in a quite narrow range of number of crossing ( if you do not take in account Face then from the very start it is hopeless ) , face/facet, number of PLY, cordage diameter, material of cord...

To prove that it is silly not to take in account the number of  Facets and to work only with the number of Crossings in building a calculator look attentively at the picture :

BOTH have 210 crossings
                                          BUT the CODING PATTERN is such that
one has 150F and the other 180F.
That difference in number of FACE and CODING make a whole difference.

It is plain to see that one is 'crowded' along the equator ( while being loose at the poles ) while the other is sort of  'crowded' at the pole and loose between the "two tropics".
As the number of crossings is the same then this is a direct result of the difference in the Number of FACE and/or the CODING PATTERN and/or the repatition of zone in that.

It is ABSOLUTELY IMPOSSIBLE to perfectly cover a sphere with 'tiles' , there exist only APPROXIMATE solutions.
Here one core should be like that and the other like that to be really adequate.

GRID for 180 F
ARI for 180F

GRID for 150F
ARI for 150F
This 150 FACE has another rendering when made in a softer and thinner cordage.
(  in knots-craft-ornamental_2 ). This shows that a cordage as soft ( flatten easily and
become much thinner when sterched ) as the green one may give a better "correction
factor" than a stiff one like the white polyester one can yields..


Intrigued I decide to make another 210/150 with identical cordage and core but another
CODING PATTERN.
GRID second 150F
ARI for second 150F

Now let us compare those two 210/150 differing only by the CODING PATTERN.

It is apparent that this second coding leads to a less compact ( 'crowded' ) covering at the poles than the first one and that in the inter-tropics zone it gives a better covering than the first.
So NO calculator, absolutely NONE of  them is able to give a precise idea of that prior to the actual making of the knot.

Unless... unless one try to see an explanation in the repartition of the Crossings and the Faces in the inter-tropic zone ( I used latitude proportionality ) in the zones between polar circle and tropic line. Here is a visual representation.

zones   180 FACE  1039 150 FACE
first pattern   1047
150 FACE
second pattern  1049
Northern
 "temperate"
60 crossings
15% of Xing are 'half' FACES
crowding 2 out of 3
15% of Xing are 'half' FACES
TIGHT
crowding : tightest (1 rank) of the 3
0% of Xing have a partial or full FACE
character

crowding: least (3 rank ) of the 3
Inter-tropic
90 crossings
1/3are 'half' FACE
1/3 are 'full' FACE
crowding: tightest (1 rank)  of 3
1/3 are 'full' FACE
crowding : least (3 rank)  of the 3
1/3are 'full' FACE
crowding 2 out of 3
Southern "temperate"
60 crossings
15% of Xing are 'half' FACES 15% of Xing are 'half' FACES
TIGHT
0% of Xing have a partial or full FACE
character
LAX





THREE OTHER SPHERICAL COVERS / COVERINGS 210 CROSSINGS  - grids with Schaake'scordage route but my own coding pattern and drawn using
ARIANE

2-PLY 210 CROSSING  135 FACE "my-pattern"
GRID 210X-135F "my pattern"
ARI "my-pattern"


2-PLY 210 CROSSING 210 FACE   "natural" aspect when not hard set
2-PLY 210 CROSSING 210 FACE  "globular" when tightened
GRID 210F-135F pattern O1-U1
ARI O1-U1


2-PLY 210 CROSSING 135 FACE "my-other-pattern"
GRID 210X-135F "my-other-pattern"
ARI "my-other-pattern"

Comparing those last three grids
Comparing the last three knots


zones   35 FACE "my-pattern" 210 FACE
O1-U1
135 FACE
"my-other-pattern"
Northern
 "temperate"
60 crossings
25% of Xing are 'half' FACES
25% are 'full' FACE
second tightest of the three
100% of Xing are single crossing FACE
tightest of the three
25% are 'full' FACE

Inter-tropic
90 crossings
2/3are 'half' FACE
abit lesstight than "my-other-pattern"
100% of Xing are single crossing FACE
tightest of the three
1/3are 'full' FACE
1/3are 'half' FACE
second tightest
Southern "temperate"
60 crossings
25% of Xing are 'half' FACES
25% are 'full' FACE
second tightest of the three
100% of Xing are single crossing FACE
tightest of the three
25% are 'full' FACE

Comparing the SIX grids











vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv


An example of how wrong, even the best -after mine of course !- a calculator can be
when taken outside its optimal range:

Michel SINCE's pineapple
with  
cord 1.5mm,
1-PLY,
6-PASS
5184 CROSSING  
864 FACE



ACTUAL MEASURE OF THE CORE IS 79 mm  

Ariane calculator  ( by far the best after mine ) gives a core of 60.93mm.
This is  hopelessly MISTAKEN by over 1/5th in the wrong direction : minus 22.9% ( the setting is already VERY tight in this knot on a 79 mm core, it will be quite immpossible on a 61 mm core )
Reducing the 79 to 61 will reduced the area from  19596,74 to 11683,94 or minus 40,4 % HOPELESS ! ( area of a polar cap is quite easy to compute instead of Pi*4*R*R for the sphere you use Pi*4*R1*R2 where R1=R and R2 = h of height of the cap so Pi*4*R*h ). The polar caps in the 79 occupy  % of the whole sphere so it it , again, quite hopeless to reduce the sphere used for one 40% less in area.  [ cap have h=2.5mm   R = 79/2=39.5 so for ONE cap area = Pi*4*39.5*2.5  = 1240,3
for 2 caps = 1240,3*2=2480,6  as sphere = Pi*4*39.5*39.5= 19596,74 the "part" of the caps is   2480,3/19596,74  so 12,66  %) so even making the two polar apertuer completel  dissapear WHICH IS ABSOLUTELY IMPOSSIBLE WITH SO MUCH BIGHT-NEST we would NEVER ATTAIN the minus 40.4 hence the HOPELESSLY WRONG qualifier.

ANYWAY ANY CALCULATOR WHICH DOES NOT TAKE INTO ACCOUNT
*** A CORRECTION FACTOR FOR THE CROSSINGS
*** THE EXISTENCE OF FACES RATHER THAN ELEMENTARY CROSSINGS
*** A CORRETION FACTOR FOR THE FACES
are FROM THE VERY START HOPELESS AS A TOOL.



In this range  only mine happen to apparently gives a correct answer.
using all the 10 "correction modules" the diameter is 64mm to 102mm ( nine measures are
64,  79,  80,  81,   91,  90,  91,  92,  100,  102 )
the usual best measure gives 79 mm and the two usual next best give 80, 90 and
the average of all 11 proposals is 89,8
eliminating the smallest and biggest proposals  ( 9 remaining ) then the mean is 90,6
the mean of the usual 3 best is 90,7
the single usual best is spot on : 79
note that going from 79 to 90 (11mm or  a 13.92% augmemtation )  will make the polar aperture a lot larger and the cordage less
tightly put on the core as this will produce a  31.39 %  augmentation of the area available

It is easy to see that there is no "linear" relation

30 mm  41mm ( + 11)  leading to an area augmented from 2826 to 5278,34
30 mm    41 = 1,37 * 30 = (2*0.685)*30  so area is multiplied by 1,87              

30 mm   34.2 mm ( +13,92%)  leading to an area augmented from 2826 to 3672,67
30 mm   34,2mm = 1,14 * 30 = (2 * 0,57) * 30 so area is multiplied by 1,3

100 mm  111 mm ( +11)   leading to an area augmented
100 mm   111 = 1,11 *100 so area is multiplied by square root of 1,11 or 1,054

100 mm   113.9 mm ( +13,92%)   leading to an area augmented
100mm     113.9 = 1,139 * 100 so area is multiplied by square root of 1,139 or 1,067

say you double, triple, quadruple..... the core diameter ( also the radius )

30 mm              60 mm (30 * 2)                   90 mm (30 * 3)                120 mm (30 * 4)
2826                 11304                    25434                 45216
1                        1*4                        1*9                    1*16
If the diameter (radius) is multiplied by 'n' then area is multiplied by 'n*n'
To double the area you have to multiply the diameter (radius) by square root of 2
to triple the area you have to multiply the diameter (radius) by square root of 3

The area augment along the square of the multiplication of the core diameter.
the radius(diameter) augmenet by the square root of the area multiplicating factor.


If you knot is quite well "seated" on its core , like Michel Pineapple then if you que the 1-PLY a 2-PLY you will need twice the surface ( forgetting the problem of the polar aperture that did not need any augmentaton , but this would lead us into calculation of "cap" area... Any calculator that works "linearly" is from the start hopeless, the more so if it does not take in account the difference between a CROSSING and a FACE ( AT LEAST 2 crossings )**.
to have twice the are then you need a core which has time square root of 2 the radius
of the first core 79 mm  so 39.5 mm radius
39.5 * square root of 2 = 55,86 radius that is 111.7 mm diameter



** Ideally the calculator should take in account onle FACES and ALL thier "categories":
1 crossing FACE ( ordinary CROSSING )
2 crossing FACE
3 crossing FACE
4 crossing FACE

n crossings FACE

and allow the proper correction factor in the computation !





CALCULATORS AT WORK

For each Ariane calculator gives a core of 40.88 mm.
My own calculator - not published, not distributed except to few friends and in a slightly older version-  gives 40mm (lowest)  for the 150 FACE and 41mm (lowest)   for the 180
FACE.
2% variation leads to 39,2 to 40,8
5% variation ( a bit too large ) leads to 38 to 42

My calculator (present version) gives for the length of cord PER PLY in the finished cover :
423 cm compared to an actual  422 cm ) for the 180F
405 cm for the 150F for an actual 352 cm ( wrong by giving "too much" -safety ! -
at +15,1% )

In this range 40mm or about both calculator are about equivalent but for the length of cordage they cannot be compared and this is a crucial point .




CONCLUSION : calculators are hopeless, they are just good for a slightly better
prediction that "nose guesstimate" ( mesure vue de nez ) because even when, like mine, they take, as they absolutely must, take in account FACE as well as CROSSING they are hopelessly ignorant of the distribution of the different zones in the CODING PATTERN.

If you want some adaptability stay well away from stiff cord like this polyester that does not flatten and does not thin like a polyethylene cord ( green) which thank to is ""plasticity"" allow a rather needed ""correction factor"".



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Added 2013 April 21rt


FALSE COLOURS or FALSE FLAG ! BEWARE !

MIND YOU DO NOT THROW THE BABY WITH THE WATER :
the nomenclature there is egregiously mistaken but inventiveness is present as certainly as the nomenclature is absurd.
By all means explore this Knoopenzo.
It should not be a waste of your time as far as cordages structures ( as opposed to knots names) are concerned .


THIS IS ABSOLUTELY, CERTAINLY, BEYOND ANY SHADOW OF DOUBT ,
NOT, REPEAT 
NOT, A TURK'S HEAD KNOT, ( Regular Cylindrical Knot  )
let me call it  A TURK'S HEAD NOT to get it a name that it is impossible to contradict !

http://www.knoopenzo.nl/knots55/0.htm
I  have had this site in my LINKS page for years and IIRC it was Theo Slijkerman who took the initiative to propose that each of us put the site of the other in our respective links.

I waited, and waited, and waited stupidly keeping the hope that the webmaster would read at least my pages on THK and  realize how monstrously he is misleading his readers about the nature of what he is showing.

Now is the time for reckoning, I can no longer stay accomplice of such misdirection leading too many to think that anything circular can be call a Turk's head knot.


Any one with the tiniest grasp of the real THK characteristics ( mathematical ) will realize that this is NOT a Regular cylindrical Knot, much less a Semi-regular Cylindrical Knot or a Nested-Bight Cylindrical Knot ).
This particular CYLINDRICAL KNOT is in a special slot entirely of its own so PLEASE ! PLEASE ! DO NOT CONTAMINATE OTHER PERSONS WITH EGREGIOUSLY MISTAKEN NAMING.

The NAME  (THK) corresponds to a VERY PARTICULAR STRUCTURE so a name is not just a name but a short hand summary for a very particular conjunction of cordage route and coding pattern.

I know a number of dull mind ( I immediately thought of  a particular French idiot which is the epitome of that absurd stance : don't bother about using a proper nomenclature ) that say" as long as my knot is good I don't care if the name I give is false".

Imagine your surgeon or you car mechanic giving arbitrary name to instruments/tools and organs/parts when others are doing the same in a different manner or are using a proper catalogue !!!

I know " a rose by any other name will smell like a rose " :
well just try to order a bath perfume to be made for you as "special order"'  just stating : " manure perfume" and thinking "they will understand I mean rose " and see if it will smell of rose when you get it !

Saying a sea gull is a cow is not simply misnaming, it is confounding a bird with a mammal !
Very different ""structures"  and "behaviour" and not only a name.


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Added 2013 April 3rd


DUMB IGNORANT ( all ignorant are not dumb ! ) PLAYING AT "SAVANT
AND CONTAMINATING NAIVE IGNORANT WITH FALSE KNOWLEDGE

 
Each and every knot-tyer should study TURNER and SCHAAKE works before
writing stupid posts.

See page 75 of BRAIDING REGULAR KNOTS
See page 77 of BRAIDING REGULAR KNOTS
See GAUCHO from The Braider by Schaake      BIG       Small
See HeadHunter and FAN from The Braider by Schaake     BIG       Small



Culprit shows a 2-STRAND knot saying it is a FAN.

First strictly understood FAN are REGULAR CYLINDRICAL KNOTS so SINGLE-STRAND but let us forget that point.

Second : there can be NO LEFT or RIGHT FAN knot as both side are identical so how do you decide on RIGHT and LEFT
See this illustration about the utter stupidity of such a concept.

Three : some dumb ignorant speaks of GAUCHO-FAN, there is no such thing : either it is GAUCHO or it is FAN but GAUCHO-FAN is nonsensical or absurd.



It is not wrong not to know, what is wrong, deeply wrong is to open one's big mouth when being an ignorant pretending to be a savant and so contaminating other persons with a false knowledge.  Myself I can be some times mistaken but I am usually quick to make a correction and then to warned all that I pushed in the wrong direction by being mistaken.
Never take, from any one, what is given to you as "absolutely true" , at best it is "absolutely honest" but honesty and sincerity does not make TRUTH.

Such persons should read MODULARITY     http://tinyurl.com/ck5zl74  ( extracted from last topic down the page in Turks head_27 )to be a bit less prone to contaminate others wit  his false-knowledge.

What is shown is a
(U2-O2) 2 times , U1 one time    
so NOT a GAUCHO, NOT a FAN, and NOT a HEAD-HUNTER  KNOT 


ALL REGULAR CYLINDRICAL KNOTS SO SINGLE-STRAND !
(control that with MODULARITY in case I mistyped )

(
Ux - Ox) m times +  (Ux)  n times == TURK'S HEAD  with x=1 and n=1


and with x greater than 1

(Ux - Ox) m times == GAUCHO

(Ux - Ox) m times , (Ux ) one time == HEAD-HUNTER 

and  mirror Head-Hunter knot ,  (Ox ) one time , (Ux - Ox) m times

(U(x-1) - Ox-1) one time, (Ux - Ox) m times , (U(x-1) - O(x-1)) one time == FAN

I will let alone AZTEC, AZTEC-FAN and HYBRIDS without name  that you  can
study in MODULARITY. http://tinyurl.com/ck5zl74  

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Let us tackle again the case in point but with no abstract formulas.

Let us chose 2-PASS


KNOT SHOWN is U2 - O2 - U2 - O2 - U1  


GAUCHO ==     a "round number" of  ( O2-U2 ) or of (U2-O2)
U2 - O2 - U2 - O2  or  U2 - O2 - U2 - O2  - U2 - O2

FAN ==    ONE time (U1) +  a "round number" of ( O2-U2 ) + ONE time (O1)  
so FAN ==  U1 - O2 - U2 - O2 - U2 - O1  
or
ONE time (O1) +  a "round number" of de U2-O2) + ONE time (U1)
so  FAN ==  O1U2 - O2 - U2 - O2U1  

The knot shown   U2 - O2 - U2 - O2 - U1   is at best  an AMPUTATED FAN so you see
it is absurd , in order to express the side that was amputated, to say RIGHT FAN or LEFT FAN as if you turn  a RIGHT by 180 it is now a LEFT !      O1 - U2 - O2 - U2 - O2

The knot  shown  could be seen as a variation of FAN or a variation of HEADHUNTER or
a variation of GAUCHO : hence I see it as a "NO NAME REGULAR CYLINDRICAL KNOT an you can have it in four flavours

ONE time (U1) +  a "round number" of ( O2-U2 )
so    U1 - O2 - U2 - O2 - U2

a "round number" of ( O2-U2 ) + ONE time (O1)
so    O2 - U2 - O2 - U2 - O1
of course those two above are  just the same knot turned 180 hence the absurdity of trying
to call it RIGHT FAN

ONE time (O1) +  a "round number" of (U2-O2)
 so  O1 - U2 - O2 - U2 - O2

a "round number" of ( U2-O2) + ONE time (U1)
 so  U2 - O2 - U2 - O2 U1
of course those two above are  just the same knot turned 180 hence the absurdity of trying
to call it LEFT FAN




HEADHUNTER would be

ONE time (U2) +  a "round number" of ( O2-U2 )
U2 - O2 - U2 - O2 - U2

a "round number" of ( O2-U2 ) + ONE time (O2)
O2 - U2 - O2 - U2 - O2
of course those two above are  just the same knot turned 180


ONE time (O2) +  a "round number" of U2-O2)
O2 - U2 - O2 - U2 - O2

a "round number" of ( U2-O2) + ONE time (U2)
U2 - O2 - U2 - O2U2
of course those two above are  just the same knot turned 180

I hope that now I am clear ? ( just hope I did not mistyped something ! )

The knot shown is a "NO NAME  REGULAR CYLINDRICAL KNOT"




Same  profoundly dishonest act of intellectual contamination
if for this knot someone were to be ignorant enough to call it a GAUCHO Knot or a FAN knot; or a HEADHUNTER knot.

It is  a (U1-O1) + 9 time (U2 - O2 ) + (U1-O1)


a GAUCHO would be  11 time (U2 - O2 )

A FAN  would be   (O1) +  9 time (U2 - O2 )  + ( U1)

A HEADHUNTER would be 10 time (U2 - O2 )  + ( U1)


Again a NOTHING, NO NAME  REGULAR CYLINDRICAL KNOT if I am not
mistaken


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Copyright 2005 Sept - Charles Hamel / Nautile -
Overall rewriting in August 2006 . Copyright renewed. 2007-2014 -(each year)

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