• we refer to a system of perpendicular axes and finite-degree vectors as a "cartesian" coordinate system for some reason, yet we instantly defer to a "euclidean" metric or "norm".

couldn't it have made sense to euclid to analogize prime subgroup ideals to the modern conception of perpendicular basis vectors? just like in flatland, all you really need to do to add 1 more line that only intersects with the existing basis at the point "1". the euclidean norm of a unit vector of any dimension n >= 1 is always = 1

we are taught the "euclidean" norm in terms of a square root. but pythagorus almost definitely wasn't thinking that way, not if he indeed made sure hippasus got his shit rocked for even daring to propose the existence of an irrational number.

i would assume the way a perpendicular basis in Rn (or even just Qn. in either case, we assume R2 for the moment) uniquely defines not just one, but two versions of transformationally-equivalent right triangles, each of which share one edge with an existing basis element (e.g. x or y), and another edge with the newly created shared vector that points in the direction the bases are going together.

i would suspect they'd employ the pythogorean identity there, where

• x2 + y2 = z2.

• x2 + y2 - z2 = 0 2 copies of right triangle created sharing norm line

THIS ALSO HOLDS FOR ALL HIGHER DIMENSIONS!!

in 3D:

• x2 + y2 + z2 - w2 = 0 4 copies of pyramid/triangular prism sharing/rotating around the norm line