Mathematical formulae to solve Turks heads

Hello fellow "knot-heads".
I've become addicted to tying Turks Heads. In my research of techniques and
methods I stumbled upon the informative book, The Art of Knotting and
Splicing, by Cyrus L. Day. On page 190 of the Third Ed. the author mentions
having been shown a mathematical formula for solving Turks Heads of any size
on paper before tying them. Mr. Day also mentions that it is an acquaintance
of his, a Mr. L. G. Miller, who developed the formula. I've not been able to
find any books by Mr. Miller that would describe or explain his formula nor
have I found any references in other knotting books with additional
information on this matter.
Does anyone know of the formulae to which Mr. Day referred or where I might
turn for further research in this matter???

Thanks For Your Help,
All Tied Up

I have a chart on my web site that might help you. It's not the mathematical
formula you are looking for, but it may help you in tying turks heads of all
sizes.

http://mywebpage.netscape.com/Tejay480/images/home.html

T J Bartruff

Hi T J,

I was playing around a little with coming up with a mathematical
expression to express the chart, but I stopped when I noticed that
your chart and the chart in the Ashley Book of Knots isn't exactly the
same in a few spots.

I don't deal much at all with Turk's Heads, so do you know anything
about the discrepancies?

Thanks,
roo

"T J Bartruff" wrote in message .. .
I have a chart on my web site that might help you. It's not the mathematical
formula you are looking for, but it may help you in tying turks heads of all
sizes.

http://mywebpage.netscape.com/Tejay480/images/home.html

T J Bartruff

What discrepancies are there? I thought they were the same, please tell me
where I went wrong?

Thanks Roo
TJ
"roo" wrote in message
m...
Hi T J,

I was playing around a little with coming up with a mathematical
expression to express the chart, but I stopped when I noticed that
your chart and the chart in the Ashley Book of Knots isn't exactly the
same in a few spots.

I don't deal much at all with Turk's Heads, so do you know anything
about the discrepancies?

Thanks,
roo

"T J Bartruff" wrote in message
.. .
I have a chart on my web site that might help you. It's not the
mathematical
formula you are looking for, but it may help you in tying turks heads of
all
sizes.

http://mywebpage.netscape.com/Tejay480/images/home.html

T J Bartruff

Ah, I thought maybe you did some independent review and found
different results, but from your response, it looks like you intended
to have an exact copy of the ABoK chart.

Here are a few discrepancies I found in the format of (Leads,Bights):
12,11
14,13
14,14
17,22
25,13

As you can see, I was mostly looking at prime numbers and found
strange things that blew my theory apart. If these are just
transcription errors then I think I could describe the chart in a
mathematical expression;

==For non-trivial cases, a Turk's Head is impossible if the number of
leads and bights have a common divisor other than 1.

I think that seems to work.

Cheers,
roo

P.S. This is assuming the originial question regarded this chart and
not something like rope length required, etc.


"T J Bartruff" wrote in message . ..
What discrepancies are there? I thought they were the same, please tell me
where I went wrong?

12,11 Can be tied, both my chart and Ashley agree
14,13 Can be tied, both my chart and Ashley agree
14,14 Can't be tied, both my chart and Ashley agree
17,22 Can be tied, my chart is wrong, but 22,17 is right on my chart
25,13 Can be tied, both my chart and Ashley agree

The chart in Ashley's is hard to read, but for the most part as long as you
don't have a common divisor in the bight and leads you should be OK.

T J Bartruff

TJ,

I must apologize. Before you even posted your webpage, I printed out
another chart with your initials on a replica site that was made long
ago when you were switching URL's and someone else made a copy because
it was such a great resource. I didn't realize the chart on your page
is more up-to-date. I didn't even know where your offical page was.

As an aside, it looks like maybe you made your chart in Excel. If you
ever want to make a huge chart someday, you can use the Greatest
Common Divisor function (GCD) by adding the Analysis ToolPak under
Tools... Add-In's... Analysis ToolPak. That way you could
automatically generate a chart by filling down a cell formula with a
IF GCD (of bight cell & lead cell) 1, then "X", ELSE "O"
[paraphrasing Excel language]. Something to that effect. It works
pretty slick.

Just in case there are people out there who aren't fond of math who
tie really big Turk's Heads, there are some easy Greatest Common
Divisor calculators online:

http://ccins.camosun.bc.ca/~jbritton...m/jbgcdlcm.htm

http://www.spjc.edu/clw/math/whiteg/lcmgcdcalc.html

G'night,
roo



"T J Bartruff" wrote in message . ..
12,11 Can be tied, both my chart and Ashley agree
14,13 Can be tied, both my chart and Ashley agree
14,14 Can't be tied, both my chart and Ashley agree
17,22 Can be tied, my chart is wrong, but 22,17 is right on my chart
25,13 Can be tied, both my chart and Ashley agree

The chart in Ashley's is hard to read, but for the most part as long as you
don't have a common divisor in the bight and leads you should be OK.

T J Bartruff

An unnecessary comment:
The previous mentioned greatest common divisor (GCD) is the number of strings needed to make the turk's head.
The best thing seems to chart the GCD. If you want the classical turk's heads look for a one in the chart. If you want a multi color turk's head, look for the number of colors you like.
And it will not take up much more space.

/patrick

Interesting. I placed such a chart for download in .xls and .txt
format on a temporary web page in case anyone finds it helpful or
would like to place it on an appropriate website, such as TJ's:

http://www.geocities.com/roo_two/GCD.html

With the .xls chart you can tinker with the ranges or expand the chart
by filling down and filling right. If anyone has problems, you might
read in this thread about activating the GCD function in Excel, but
perhaps it won't be an issue.


Patrick Andersson wrote in message ...
An unnecessary comment:
The previous mentioned greatest common divisor (GCD) is the number of strings needed to make the turk's head.
The best thing seems to chart the GCD. If you want the classical turk's heads look for a one in the chart. If you want a multi color turk's head, look for the number of colors you like.
And it will not take up much more space.

/patrick

Hey roo
Can't get the GCD.xls link to work, the GCD.txt link works fine though.
Great work.

Thnaks
TJ