Constructions involving Straightedge, Compass and A Conic

A paper “On Points Constructible from Conics” by Carlos R. Videla, published in Math. Intelligencer came to my attention recently that answered a question that a few of my colleagues had asked back in my undergraduate algebra course and later in graduate school. We know from Galois theory that some ancient Greek open problems of the duplicating the cube, trisecting an angle, constructing a 7-gon, and squaring the circle are impossible when limited to a straightedge and compass.

It turns out that if one extends the notion of constructibility to include any type of conics whose foci are constructible points and whose directrix is a constructible line, then these problems can be answered in the affirmative (except of course squaring the circle, since pi is transcendental). It turns out that this extended notion allows not just for square roots but cube roots as well. Therefore a number is constructible if and only if the degree of the field extension over the rationals is 2^n3^m. So we see that as a natural generalization of the theorem about constructible polygons, the n-gon will be constructible provided that n=2^r3^s p_1\dotsb p_t, where p_i-1 factors into powers of 2 times powers of 3. It seemed strange at first that this would involve cube roots, since conics are zeroes of degree-2 polynomials. But the method is to exploit the asymmetry of the non-circles to get a linear equation is equal to a quartic, providing cube roots.

In addition to this paper, there was another paper recently posted to arXiv (arXiv:1210.8046) that proves that only one conic (any nondegenerate, noncircle will do) is enough to obtain the constructibility with all conics, and hence the classical Greek problems are solved in the affirmative with the addition of a single conic (say an ellipse).


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