I have not been here for quite some time, but I have returned. Just for the record - not a newbie. And an apology for the form of this email 
Currently my attentions have bent toward the notorious TH - When I was last posting here I did not have much success with tying this knot, but nowâŚI am most successful. I am most curious about what methods are most widely known for tying turks heads. Though, thinking now about the different types, âirregularâ and âregularâ? I would call the Carrick Mat a 4b3p TH âregularâ.
A method that i developed on my own over the course of many hours and research - Not stating that it has not been invented yet, just that I independently discovered an easy(from my perspective) way to create guides to tie a TH of any number of bights with any number of parts. Taking into consideration the whole âcommon divisorâ rule - One would have the capability of tying a TH with more than 1 strand. Of course, I went to many differrent sources i.e. ABOK, Islamic Art, Geometry..
It all starts from a drawing. You donât even need to know how to draw!
Describing this is seeming more and more, a daunting task - or at least longer.
Geometry.
This is where it is all at. There is an example of a Square within a Square within a SquareâŚin the ABOK, in the Turks Head section. ;D
Just as a closed Overhand Knot is a simple TH - can be represented by a Triangle within a Triangle.
It goes on and on. Such as: in Star Polygons
A Pentagram a = 5b2p TH
Hexagram needs 2 strands = 6b2p TH
Though as we go on to 7 points and more there is more than 1 form for each star.
(7)Heptagram = 2 forms = (A) 7b3p & (B) 7b2p TH
(12)Dodecagram = 4 forms = (A) 12b5p, (B) 12b4p, (C) 12b3p and (D) 12b2p TH
A Star Polygon looked at from the perspective of it being a TH which has a common divisor - Simply becomes a series of Regular Polygons superimposed & rotated to varying degrees.
Now then, looking at Regular Polygons as sources of its hidden nature - I apologize if this subject has been posted already, I havenât been keeping up.
(7) Heptagon within a Heptagon within a Heptagon(doing this in my head) = 7b3p TH? Think so.
Just looking at the endless possibilities to the number of bights and parts. Really it comes down to illustrating that the TH drawn from the âMatâ State - is a type of logarithmic spiraling pattern. Check out this link, youâll see:
http://www.uwgb.edu/dutchs/SYMMETRY/log-spir.htm
Completing the Mat Guide brings in oneâs drawing skills. Starting anywhere on the drawing on a sheet of tracing paper - doubling the lines creating the over-under-over sequence(never was a big fan of following the circles where the line goes over/under). It makes quite a nice drawing and kept in good condition can be used again and again.
Something I became a non-fan of(was at one time) the looking at a diagram and then back at where I am with the rope, then back at the diagram and finding where I am with the rope, etc.. I have found that by using a clear sheet of âdry-erase friendlyâ plastic can be placed over the diagram and oneâs place can be kept by using a âdry eraseâ marker. More can be done I am sure of it. The more complex the knot like say a, 65b37p TH the addition of various colored pins, etc..
Though. If you donât really want to do all the research and drawing of such geometric shapes. Bare necessities = something round(or a compass to create the initial circle), a protractor, a pencil(or something to draw with and paper(or something to draw on). You will need to figure out where along the perimeter of the circle to place points for the given amount of Bights - For 10 bights every 36 degrees(360 degrees divided by 10); For 20 bights 18 degrees; 40 bights 9 degrees. The thing to note is, placement does not have to be exact because the is merely a guide. Each point is really an indication of a crossing(or part).
To make this one, initial circle with 40 points become a 40 Bight 677 Lead TH (for a ridiculous example) one would make a 676 exponentially larger circles around the initial circle - the points of every circle would not be able to mimic the prior or post circleâs points radially each would be slightly askew from the next. - Shit this is turning into a monster! I may need to post some pictures. 
This is how the spiral effect is created!
There is a way which I have found where you would not have to draw so many circles, just the initial one and it doesnât have to be perfect - it just has to fit your standards(I enjoy the crisp, clean look for the most part).
This way may need some photo assistance and another post which would be much shorter.
Hope this help,
Not hinder.
S. in ?