new method for Turk's Head Knot

[Ruby]

like the famous  Fibonacci sequence equation F[n]=F[n-1]+F[n-2]
there is an equation for golden ratio THK:
TF[n]=TF[n-3] + double + split

[Jan van Leiden]

Hi Ruby,

One final reflection on your equations.

You probably know there are many other ways to generate sequences of regular knots with, for example, a U1O1 pattern.

Ever heard about the Masurel Method? Its two starts produce all p/b = 4/b THK.

I found it in KM51, pp67-69, December 1995.

[Jan van Leiden]

so the direction is always changing,  left right left right.....

Indeed, and that is the first part that messes up the linear equations.

like the famous  Fibonacci sequence equation F[n]=F[n-1]+F[n-2]
there is an equation for golden ratio THK:
TF[n]=TF[n-3] + double + split

If you insist on retaining your U1O1 pattern, you can lay and split tracks, and your Fibonacci Equation will hold.

For other patterns, the story may well be different.

[Ruby]

hi, just checked this article,

yes, this is the easiest linear enlargement


apply LeftSplit or RightSplit of any 2-parts THK to get any 4-parts THK,

left for 4n-1 , right for 4n+1

begin with simple  THK as 2-parts  or 1-bight, we can get quite a lot THK by only apply LeftSplit or RightSplit linear enlargement.

linear enlargement is especially useful for small THK.
for 55 THKs below 10, only 3 can not be made like this: 75 85 97, and their mirror.

[Jan van Leiden]

Hi Ruby,

The point is that Masurel's Method leads to 4/b that have no direct connections in the regular knot tree via Schaake and Turner's fundamental enlargement methods.

The Masurel Method has two starts, and indeed yields two 'linear' sequences:

• 4/(4n - 1)
• 4/(4n + 1)

Where n indicates the number of tucks in the initial Multiple Overhand Knot.

In the regular knot tree, however, you cannot get from for example 4/9 to 4/17, short of untying.

Here are the paths:

• 1/3 - 2/5 - 3/7 - 4/9
• 1/5 - 2/9 - 3/13 - 4/17

It is an interesting exercise to find the location of these 4/b grid diagrams in the regular knot tree.
And subsequently the p/4 grid diagrams.

 

[alana]

hi, do you have an easy method to tie THK like 7Leads x 9Bights or 9B x 7L ?

I mean, tie them easily like basic 3 Leads x 2 Bights, around your fingers,
without any grid or pin or mandrel tool.

i've got nothin'!  i need a pin board,

so 7x4 is as far as i've got tying in hand,
(in 2-pass if i consult my notes)

glad you mentioned golden raito,
i have tied 5x8 somewhere,
cant find pic,
and - i needed pin board to do so -
but proportionate and appealing

i found my 5x8, except it's a 7x8 2-pass ...!
(in case anyone wanted to see how it looks)