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
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)).
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?