It should be noted a change in convention as the use of n to represent non-Diophantine exponents has created some confusion. Thus, the use of ℝ going forward in this respect.
FIRST:
Could the immunity of the Pythagorean (et al.) Theorem to the a : b ratio be the implication of Fermat in his marginal note of x n + y n = z n concern?
i.e., Based on the following, does the a : b proportion play a role, and can perhaps that be generalized to compositions that do not fulfill the Triangle Inequality?
For any triangle ABC where *c is the long leg, an exponent ℝ exists that will satisfy a ℝ + b ℝ = c ℝ of non-Diophantine concern.
*Alternatively, a ℝ + b ℝ = c ℝ can be re-written b ℝ – a ℝ = c ℝ at the isosceles inversion to maintain constancy of convention. This is necessitated as ∠C changes between the degenerate triangle limits of zero and π radians, the transition from obtuse scalene to acute scalene. Constrained by this equation, as triangles of a : b < 1 approach isosceles, ℝ becomes extreme. In the case of a : b = 1, the equation is invalid at the equilateral where c fails to be the long leg.
The function:
from the equation:
provides all a : b ratio curves are subject to intersection at (Xℝ=2, YC=π/2) and termination at the degenerate limit of (Xℝ=1, YC=π) and the isosceles limit where (Xℝ→∞, YC=0).
The rotating or pivoting of the function about the integer exponent 2 seems in concurrence with proof of Fermat’s Last and subsequent implication that the a : b proportion may be significant.
To illustrate, I have created a Geogebra animation here: Why does the Pythagorean Theorem work? – GeoGebra
SECOND:
An update to this question found here.
Regarding a more fundamental polygonal unit of area and in reference to the diagram below:
Why does the Fractal Dimension ln 4 / ln 3 of the Koch Curve satisfy ℝ for the Pythagorean analogy
a ℝ + b ℝ = c ℝ where ∠C is 2π / 3 radians?
Less the equilateral, there seems disparity between the two. One of scale and detail used for characterization and one of area expressed in exponential terms other than square. While having explored the latter prior (Can Area be Expressed in Terms Other Than Square Units?), my question of coincidence remains unreconciled.
