Author Topic: History of Turk's Head Knots  (Read 7546 times)


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History of Turk's Head Knots
« on: December 08, 2012, 11:51:04 AM »
The Legend of Euclid and turk's head knot's.  :)
There is no evidence, but there are reasonable grounds for believing  ;)

The ancient Romans knew these knots. For example traced mosaic:

(Roman Theatre in Bosra, Syria, built in the second quarter of the 2nd century )

Interestingly, by manipulating this knot we can find a reduction algorithm in a natural way.
Surprisingly, it's nothing else but Euclidean algorithm :-)
It's possible so that Euclide discovered his algorithm by reducing leads (L) and bights (B) in THK.

Observing problem for the case of a more general multi-strand (for example 3 strand (S)).

This knot THK has 6 bights (B) , 9 leads (L) and 3 strands (S) (separate thread):

You can reduce it, without changing the amount of strands,
using the Euclidean algorithm for the number of B and L.

The reduced knot has 6 B, (9-6) = 3 L and S 3 separate thread:

We can reduce further without changing the amount of S.
(6-3) = 3 B, 3 L, and 3 separate S:

Next step is impossible, we have reached the minimum.

You can get to the algorithm yourself by cutting the knots THK.
 This is here:

for: B<L ;
6 B,   (9-6) L,   3 S

for: B>L; 
(6-3) B,   3 L,   3 S

Perhaps Euclid had come up with the idea of their algorithm in the same way as I showed you?

 I have used the Euclidean algorithm in generating these diagrams:
 Then I noticed that they are topologically similar to an ,occurring in many forms, ancient mosaics.

 There were probably used logarithmic spirals, and I applied the arithmetic spirals.
 Then it turned out that Leonardo da Vinci had a similar idea.
 Here is a copy of the emblem by the master Durer himself :

Euclid lived during the reign of Ptolemy I and wrote an work made comprised of 13 volumes called Elements.
As ruler during his reign (323-283 BC), Ptolemy personally sponsored Euclid,
but found his work to hard to comprehend. Once, when Ptolemy asked of Euclid
if there was no shorter road to geometry than the Elements,
he replied, "Sire, there is no royal road to geometry."

He found the royal road to THK? :),_King_of_Egypt-SPL.jpg