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The impossibility of certain constructions in Euclidean geometry, such as squaring the circle with straight-edge and compass is usually shown by using algebraic methods. I am wondering if there are "purely" geometric proofs of the impossibility of these constructions.

I searched the internet and could only find this blog entry terrytao's geometrical proof where a "more geometrical" proof of the impossibility of trisecting an angle by straight-edge and compass is presented. As far as my understanding goes, algebra is however still needed ("Lemma Two" in the blog entry).

I am asking this question also as there were some eminent early modern mathematicians, for example Johannes Kepler, that rejected on philosophical grounds altogether algebraic methods in geometry. So I wonder if today there would be a kind of impossibility proof that would be accepted by someone like Kepler.

I also remember vaguely but cannot find the reference right now that Christiaan Huygens presented a purely geometrical argument against the claims of Thomas Hobbes having squared the circle by straight-edge and compass. But I do not know if that would amount to a kind of geometrical proof of the impossibility of squaring the circle by straight-edge and compass.

Any comment is appreciated.

corto
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1 Answers1

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"The impossibility of the quadrature of the circle wold follow if it were shown that the square root of $\pi$ is not algebraic. This was done by the German mathematician C.L.F. Lindemann, in 1882. The proof requires methods that are not algebraic."

Source: Modern Algebra: An Introduction 6th edition by John R. Durbin

You should reference the Gelfond-Schneider Theorem as well. As far as I know, there are no "purely geometric proofs" available.

Daniel Robert-Nicoud
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  • See also http://math.stackexchange.com/questions/103786/direct-proof-that-pi-is-not-constructible. – lhf Dec 28 '13 at 13:31