the comment might have become a bit confusing in the end.

i guess what im really find interesting is that in the right triangle the self similar areas defined by the two sides add up to the area defined by the hypotenuse, such that one might say that the area is split into the two dimensional components, but the lenghts of the sides obviously dont add up taxicab-like to the lenght of the hypotenuse. yet in the end there is this beautiful relationship of the area to the euclidean norm/the hypotenuse.

well in the end i think that i should probably stay humble and let those patterns be as awesome as they are, instead of speculating all too much. sure its nice to think about those patterns, but i just looked at a pair and it was of such beautiful complex structure, that i just felt like i should spend my time rather on using the patterns to construct something nice. the patterns are there anyway and for the pythagorean theorem i actually have quite a nice intuition already, thanks to your article