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Topic: Euclidean algorithm and Turk's head knots
Replies: 2   Last Post: Nov 24, 2012 8:25 PM

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 Ciekaw Posts: 28 Registered: 11/24/12
Euclidean algorithm and Turk's head knots
Posted: Nov 24, 2012 6:54 PM

There is sometimes a need to generate one strand THK knots, of which
two characteristic parameters (leads and bights) must be relatively first:

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)
http://www.narval.republika.pl/obrazki/B6L9.jpg

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:
http://www.narval.republika.pl/obrazki/B6L3.jpg

We can reduce further without changing the amount of S.
(6-3) = 3 B, 3 L, and 3 separate S:
http://www.narval.republika.pl/obrazki/B3L3.jpg

Next step is impossible, we have reached the minimum.

You can get to the algorithm yourself by cutting the knots THK.
This is here:
http://www.narval.republika.pl/B6L9S3aB6L3S3.jpg
http://www.narval.republika.pl/B6L3S3aB3L3S3.jpg

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:
www.narval.republika.pl
Then I noticed that they are topologically similar to an ,occurring in many forms, ancient mosaics.
http://search.getty.edu/museum/records/musobject?objectid=7665
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 :
pl.wikipedia.org/wiki/File:Copy_of_emblem_of_school_of_Leonardo_da_Vinci.jpg