which is the most difficult THK

[Ruby]

hi,

for bights and leads under 10 (max 9), which one do you think is the most difficult turks head knot THK to tie?

after some practicing, I find that they are all pretty easy. it is difficult to find the most difficult one.

with one meter shoelace in hand, it is easy to tie all these THK.

at first 5x7 5x8 7x9 and 7x5 8x5 9x7 seem to be difficult,
but it turns out that they are easier than some other one, say 9x8

my favorite one is the golden ratio one with 8 Leads x 5 Bights.

for bights and leads exceed 10, maybe too large to tie them in hand.

[Pat_Ducey]

A standard leads x bights turks head is pretty basic. Even if you want to tie one with a specific count of leads or bights, it is readily attainable using standard expansion methods {ABOK chapter on turks heads}. Where I find the most difficulty is in complex turks heads like the tee, cross, pear shaped turks heads and turks head mats. This is where the math gets a little fuzzy as the count of leads and bights becomes more of a challenge to define.
The attached photo is a turks head tied as a cylinder, then flattened. I call it a 24 bight mat, but with 4 bights at the center, is it really a 24 bight turks head?

---

[Ruby]

@PatDucey post:2

A standard leads x bights turks head is pretty basic.

yes, very simple.

@PatDucey post:2

Even if you want to tie one with a specific count of leads or bights, it is readily attainable using standard expansion methods {ABOK chapter on turks heads}.

yes, with tools like grid pin and mandrel, not by hand.

look at Ashley #1351 for 5L x 8B, one can not follow those vertical waves ︴︴ without pin.

I like to tie all of them directly around my fingers.

@PatDucey post:2

Where I find the most difficulty is in complex turks heads like the tee, cross, pear shaped turks heads and turks head mats. This is where the math gets a little fuzzy as the count of leads and bights becomes more of a challenge to define.

that is another story. I’d like to try them after mastering commonTHK. sounds difficult to do them without tools.

@PatDucey post:2

The attached photo is a turks head tied as a cylinder, then flattened. I call it a 24 bight mat, but with 4 bights at the center, is it really a 24 bight turks head?

beautiful line art design.

though there exists the over-under pattern.
actually I can hardly call it a THK.
maybe THK-based? or THK-variation THK-family.

since working ends change direction randomly, there seems to be no meaning to count bights or leads.

---

[Ruby]

for common THK in flat or disk mat sinnet,
I like those spirograph shape, that is to say, moving a little circle inside a big fixed circle, and any point in the little circle forms a curve, like THK.

every spirograph is a common THK, if you mark those crossings with the over-under pattern.

or should I say, every common THK is a spirograph.

like the Law of the Common Divisor,
the simplest fraction form of the radius (tooth) ratio of these two circles is THK leads and bights.

for example, try it online here https://nathanfriend.io/inspiral-web/

with typically out fixed circle 105 tooth,
and inner circle 63, ratio is 3:5 (213=63, 215=105), and then you get a spirograph of THK 3x5

if inner circle is 84, then ratio is 4:5, and then you get a spirograph of THK 4x5

if inner circle is 75, then ratio is 5:7(155=75, 157=105), and then you get a spirograph of THK 5x7

63 : 105 → 3 x 5
84 : 105 → 4 x 5

30 : 105 → 2 x 7
45 : 105 → 3 x 7
60 : 105 → 4 x 7
75 : 105 → 5 x 7

40 : 105 → 8 x 21
50 : 105 → 10 x 21
60 : 105 → 12 x 21
80 : 105 → 16 x 21

24 : 105 → 16 x 35
36 : 105 → 16 x 35
48 : 105 → 16 x 35
72 : 105 → 16 x 35

32 : 105 → 32 x 105
52 : 105 → 52 x 105
64 : 105 → 64 x 105

and for out circle with 96 teeth：
24 : 96, → 1L x 4B square
30 : 96, → 5L x 16B
32 : 96, → 1L x 3B triangle
36 : 96, → 3L x 8B
40 : 96, → 5L x 12B
45 : 96, → 15L x 32B
48 : 96, → 1L x 2B ellipse
50 : 96, → 25L x 48B
52 : 96, → 13L x 24B
60 : 96, → 5L x 8B
63 : 96, → 21L x 32B
64 : 96, → 2L x 3B
72 : 96, → 3L x 4B
75 : 96, → 25L x 32B
80 : 96, → 5L x 6B
84 : 96, → 7L x 8B

---

[Ruby]

one intresting thing of THK is you can use old trick for new unkown knot,
something like Mathematical Induction.

for example, if you know a method which expands 3x2 to 5x4, and it expands 5x4 to 7x6
then the same method presumably can expands 7x6 to 9x8, leads+2 and bights+2

previously I found a method to expnd 4x3 to 8x5,
and today I out of curiosity extend this method a bit,
and I got a 12 x 7 THK for the first time. so the pattern seems to be leads+4 and bights+2

I can guess that if I continue this method I can get a 16L x 9B THK, with 99% possibility?

---

[Pat_Ducey]

I would argue that the 24 bight mat is a turks head. It is one continuous line that begins and ends at the same place. It goes around a common center. It started life tied in hand as a 24 bight x 5 lead turks head. I passed another line to double the plies of the original knot, only every other time I got to one end of the cylinder, I skipped a few bights ahead with the working end before re-entering the pattern. When the working end got back to the origin, I removed the original knot and I had a turks head cylinder with 24 bights on one end and 2 layers of 12 bights on the other end. Repeat the process with every third pass to the outer 12 bights, and you end up with an added layer of 4 bights. Start pulling out slack and it makes a nice turks head mat.