SOME BASIC TERMINOLOGY IN CASA KNOTS

All TH are tubular knots. It means that the start and the end of such a knot coincide. In addition, when one counts the number of bights, the end of the final bight coincides with the start of the first bight. Furthermore, when I make the calculations of the disposition of the first wrap, I use the numbers on the picture below that are in red. That is, I do not use the bights, but the crossings of the rope. However, the use of this pattern is characteristic for one thing – when the working end crosses the standing end, for instance in position 1 (in the lower or upper part of the picture), it means that bight No2 is started; when the working end crosses position 2, then bight No3 is started, etc. To summarise: when the working end crosses the rope in pos Z, then bight No (Z+1) is started. THIS IS ONE OF THE DIFFERENCES WITH THE WAY SIDNEY WOOD ON HIS WEB DOES THE CALCULATIONS FOR BUILDING CASA KNOTS. His web is accessible on http://www.taylortel.net/~stwood/index.html.

Another thing to be mentioned is the direction in which the knot is built – it starts from the lower

right-hand side and continues in the direction shown by the red arrow.

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CALCULATION OF THE LOCATION OF THE FIRST BIGHT IN A CASA KNOT

LET THE KNOT I DESCRIBE IS P PARTS AND B BIGHTS.

As per Sidney Wood on his website - http://www.taylortel.net/~stwood/lesson2.html

1. If P is greater than B (ie P>B), then the first wrap finishes at a position R determined by the

following formula: R = P – X*B + 1 , where X is the number of whole times B is repeated in P.

Furthermore, X shows the number of times the running end crosses the first half of the first

wrap.

Eg 1:Let’s have a look at 11x9 casa knot – R = 11 – X*9 + 1

X = 11/9 = 1 R=11-1*9+1=2+1=3; ie the first

wrap finishes at bight No 3 having

in mind that bight No 1 is located

at the position of the standing end

of the casa knot (see Pic 1)

Pic 1 – 11x9 casa knot

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Eg 2: Let’s have 11x4 casa knot: - R= 11 – X*4 +1

X = 11/4 = 2 R = 11 – 2*4 + 1 = 4; ie

the 1st wrap finishes at

bight No 4 and there are

2 crossings during the 1st

wrap (see Pic 2)

Pic 2 – 11x4 casa knot

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Eg 3: 15x4 casa knot – R = 15 – X*4 + 1

X = 15/4 = 3 R = 15 – 3*4+1 = 15-12+1 = 4 The first wrap

finishes at bight No4 and there are 3

crossings during the first wrap (see

Pic 3)

Pic 3 – 15x4 casa knot

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2. If P is smaller than B (P<B), then R = P + 1

Eg 4: Let’s have 13x19 casa knot P<B, then R = 13+1=14, ie the first wrap finishes at bight No 14.

(here the standing end is also located at bight No 1) (see Pic 4)

Pic 4 – 13x19 casa knot

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However, the question of where exactly the first half of the first wrap finishes, is also of great

importance. This is not as difficult as it seems. And in explaining this, I rely on S. Wood’s web page.

Considering the fact that the casa knot diagrams are drawn on a flat surface and that the casa knots

themselves are made around a mandrel, when starting making a casa knot for the 1st time there

might arise the question – “Where is the start and where the end of the knot?” I would like to say

that the end of a casa knot is where its start is, ie where its standing end is. It means that the wrap

passing through the final bight, is not the finish of the casa knot (see Pic 5 – 9x5 casa knot)

Pic 5 – 9x5 casa knot

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In order one not to be confused during the reading of the text (hopefully it is in clear language), the

numbering of a casa knot starts at the standing end of the knot (ie this is bight No1). On the other

side of the knot, the numbering of the bights starts from the bight which is located on the

perpendicular cutting through the casa knot and passing through the standing end, ie through bight

No1. (This is the situation with an even part casa knot in which the pairs of bights on the both sides

are on the same perpendicular. In an odd part casa knot the bights on the both sides of the casa knot

are moved by half a bight – see http://www.taylortel.net/~stwood/lesson2.html ;see also Pics 3 & 6)

Casa knots will be divided in two parts:

1. Those with P<B, and

2. Those with P>B.

1. P<B – The casa knot is a symmetrical one which means that the 1st and 2nd halves of whichever

wrap are ALWAYS equal. This, in turn, means that the halves of all wraps are located on the

line bisecting the stretch connecting the starting and final points of each wrap. (see Pic 6).

Furthermore, the distance between two neighbouring bights on a same side is the same, ie R

is a constant.

Pic 6

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ODD PART CASA KNOT: Let’s have an odd part casa knot. It means that R = P+1 is an even

number and R/2 is a whole number. Ie each wrap finishes on the bight R while it changes its

direction on the bight R/2 on the other side of the casa knot.

Eg 5: Have a look at 9x14 casa knot R = P+1 = 9+1 = 10; R/2 = 10/2 = 5. Having in mind the

above-said as well as the fact of the symmetry of the knot, the first wrap finishes on bight No10

while its half on the other side of the knot is bight R/2 = 5, ie bight No5 located on the other side

of the casa knot. (see Pic 7).

Pic 7 – 9x14 casa knot

EVEN PART CASA KNOT: R= P+1 which is an odd number, ie R/2 = P/2 + 0.5 bights. P/2 is a

whole number. However, the distance between starting and final bights of each wrap is an even

number.

5 4 3 2 1

Have a look at the line above. It is divided into four equal parts:

1-2 2-3 3-4 4-5

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Now consider that the numbers are actually a part of a casa knot and that the line drawn is actually

a part of the bights and that R = P+1 = 5, ie P=4. The bisection of this line is located exactly on bight

No3 or (P/2+1).

If one considers the more general scenario that if the line above has (2*Z+1) points, located

equidistantly of each other, then they divide the line into 2*Z equal lines and the middle of the line is

on point Z+1. However, on an even part casa knot, two opposite bights on the opposite sides of the

casa knot are located on the same perpendicular. The conclusion is that the middle of each wrap is

actually bight (Z+1) = P/2+1.

What is the difference between the odd and even part knots? If one applies the reasoning for the

location of the bisection of the even bight casa knot on an odd part casa knot, it appears that the

bisection of the odd part casa knot actually falls behind half a bight. This is compensated by the fact

that the bights on the different sides of the odd part casa knot are moved by half a bight. (see the

beginning of p.7)

2. P>B R = P – X*B + 1 .It means that the second half of the first wrap passes its first half.

(see Pic 8 below - It’s the same as in Eg.1)

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Pic 8 - 11x9 casa knot

We’ll have a look at Eg.1 – 11x9 casa knot R = 11 – X*9 + 1

X = 11/9 = 1 R=11-1*9+1=2+1=3

However, we can’t say that the half of the first wrap is located on bigh No3 on the other side of the

casa knot. In order to determine where the half of each wrap is, one has to have in mind the

following constant characteristics of the casa knot:

1. The casa knot is a symmetrical knot, meaning that all the halves of the wraps are equal;

2. The halves of the wraps lie on the bisection of the line connecting the start and finish of

each wrap

Having in mind the above-said, it is enough to find the middle of the line connecting the start and finish

of each wrap (see also Pics 6 &7). However, as has been said on p.2 that if P>B, then R = P – X*B + 1,

where R is the bight where the first wrap finishes and X is the number of times B is repeated in P. On

the other hand, if the casa knot were a flat knot, then the distance between the start and finish of each

wrap would be (P+1) bights.

If P is an odd number, then (P+1) is an even number and the middle of this line is on the (P+1)/2

bight. (the line of thought is the same as on p.8 discussing odd part casa knots whose P<B). Therefore

the middle of the first wrap is on the (P+1)/2 bight.

If P is an even number, then (P+1) is odd number and (P+1)/2 is not a whole number. However,

following what has been said elsewhere (even part casa knots on p.8) the middle of this line is on the

(P/2+1) bight. Therefore the middle of the first wrap is on the (P/2+1) bight.

Another issue worth mentioning is when the numbers (P+1)/2 or (P/2+1) are greater than the

number of the bights:

Eg 6: 37x16 casa knot – 37 is an odd number (P+1)/2 = (37+1)/2 = 19>16. The middle of the first

wrap cannot lie on bight No19 by the simple fact that the knot consists of 16 bights. What we do is to

follow the formula – R`=[(P+1)/2] – B*X` This is the bight on which the half of the first wrap lies.

(X` is the number of whole times B is repeated in either [(P+1)/2] or [P/2 + 1]). Therefore, in the

37x16 casa knot the half of the 1st wrap lies on bight R`=[(37+1)/2] – 16*1 = 19-16 = 3rd bight (see Pic

9 below)

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Pic 9 – 37x16 casa knot

Eg 7: Let’s have 50x7 casa knot 50 is an even number 50/2 +1 = 25+1=26>7. The middle of the

first wrap cannot lie on bight No 26. In order to determine the middle of the first wrap, we have to

follow the formulas:

R` = [P/2 +1] – B*X = 26 – 7*X`

X`=[P/2+1]/B = 26/7 = 3 26 – 7*X` = 26 – 7*3 = 26 – 21 = 5 , i.e. the middle of

the first wrap finishes at bight No 5 (see Pic 10 below)

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Pic 10 – 50x7 casa knot

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In case X’ is a whole number, ie it has no decimal part (eg in 15x4 casa knot X’=[(15+1)/2]/4 = 16/4 =

4), it would mean that R’ = 0. However, there is no bight numbered 0. In such a case R’ would be

equal to the number of bights, ie in 15x4 casa knot R’=4 (see pic 11)

Pic 11 – 15x4 casa knot

T H E E N D

Hope this helps!!!!!!!!!!!!!!!!!!!!!!!!!!

Ognyan

http://www.esnips.com/profile/7f59d312-7c89-4b4c-99f5-497359b93848

e-mail: ognyo1@yahoo.com

ogny0@mail.bg

Many thanks to:

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Sidney Wood and his web - http://www.taylortel.net/~stwood/index.html

The authors of the gridmaker, Tim and John Allwine - http://khww.net/gridmaker/

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