1,777 research outputs found
A Rigorous Geometrical Proof on Constructability of Magnitudes (A Classical Geometric Solution for the Factors: √2 and √(3&2))
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
Geometric constructibility of cyclic polygons and a limit theorem
We study convex cyclic polygons, that is, inscribed -gons. Starting from
P. Schreiber's idea, published in 1993, we prove that these polygons are not
constructible from their side lengths with straightedge and compass, provided
is at least five. They are non-constructible even in the particular case
where they only have two different integer side lengths, provided that . To achieve this goal, we develop two tools of separate interest. First, we
prove a limit theorem stating that, under reasonable conditions, geometric
constructibility is preserved under taking limits. To do so, we tailor a
particular case of Puiseux's classical theorem on some generalized power
series, called Puiseux series, over algebraically closed fields to an analogous
theorem on these series over real square root closed fields. Second, based on
Hilbert's irreducibility theorem, we give a \emph{rational parameter theorem}
that, under reasonable conditions again, turns a non-constructibility result
with a transcendental parameter into a non-constructibility result with a
rational parameter. For even and at least six, we give an elementary proof
for the non-constructibility of the cyclic -gon from its side lengths and,
also, from the \emph{distances} of its sides from the center of the
circumscribed circle. The fact that the cyclic -gon is constructible from
these distances for but non-constructible for exemplifies that some
conditions of the limit theorem cannot be omitted.Comment: 9 pages, 1 figur
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