A Rigorous Geometrical Proof on Constructability of Magnitudes (A Classical Geometric Solution for the Factors: √2 and √(3&2))
Keywords:Delian Constant, Pythagorean Factor, Constructible Magnitudes, Constructible Points, Rational Numbers, Irrational Numbers, Euclidean Geometry, Doubling A Square, Constructability
In the absence of a comprehensive geometrical perceptive on the nature of rational (commensurate) and irrational (incommensurate) geometric magnitudes, a solution to the ages-old problem on the constructability of magnitudes of the forms and (the square root of two and the cube root of two as used in modern mathematics and sciences) would remain a mystery. The primary goal of this paper is to reveal a pure geometrical proof for solving the construction of rational and irrational geometric magnitudes (those based on straight lines) and refute the established notion that magnitudes of form are not geometrically constructible. The work also establishes a rigorous relationship between geometrical methods of proof as applied in Euclidean geometry, and the non-Euclidean methods of proof, to correct a misconception governing the geometrical understanding of irrational magnitudes (expressions). The established proof is based on a philosophical certainty that "the algebraic notion of irrationality" is not a geometrical concept but rather, a misrepresentative language used as a means of proof that a certain problem is geometrically impossible.
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