Reference prior post: Less the Answers, a Couple of Related Questions
The extent of validity for non-integer generalization of an + bn = cn are represented by point B of triangle ABC.
For integer values of exponent n, triangles of conformance are found in the (Re)x, (Re)y plane.
Non-integer exponent triangles exist in (Re)x, (Im)y planes where the angle of the (Im)y plane is defined as increment of 1/π.
B(Re)x = cosC × a
B(Re)y = cosπn × sinC × a
B(Im)y = sinπn × sinC × a
Thus, B rotates about the (Re)x axis tracing a spiraling, discontinuous spherical path.
Loss of continuity from exponent extremity occurs where triangle ABC of a:b = 1 approaches equilateral or a:b ≠ 1 the isosceles, as well the degenerate limits of ∠C = π and 0 where B becomes coincident with the (Re)x axis.
Link to Geogebra animation below.
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