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Ciekaw
Posts:
27
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: http://en.wikipedia.org/wiki/Turk 's_head_knot#Leads_and_bights
The ancient Romans knew these knots. For example traced mosaic: http://upload.wikimedia.org/wikipedia/commons/f/f4/Bosra_mosaic.jpg (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 multistrand (for example 3 strand (S)).
This knot THK has 6 bights (B) , 9 leads (L) and 3 strands (S) (separate thread): 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, (96) = 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. (63) = 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
What do you think about my hypothesis about Euclid's algorithm?
Regards, Ciekaw



