Doubling the cube with the help of a parabola

Question

Looking into the intersection of abstract algebra and geometry, it's well known that it is impossible to double the cube with ruler and compass, since $\sqrt[3]{2}$ is not constructible. However, I have read that if one is given a parabola in $\mathbb{R}^2$, it is indeed possible to double the cube. Out of curiosity, how is this done?

One thing I have noticed that given a parabola $y=(1/2)x^2$, then the circle centered on $(a,1)$ will meet the parabola at $2\sqrt[3]{a}$ as the x-coordinate of intersection, implying we can find $\sqrt[3]{a}$. Is it then possible to find $\sqrt[3]{2}$ in $\mathbb{R}^2$ if we're given any arbitrary parabola?

Answer 1

The calculation that you made is very close to the proof you seek. Given a parabola, it is not hard to construct first of all its axis of symmetry, and then the focus and directrix. In particular we can construct axes so that the parabola has equation of the shape $y=ax^2$. We can construct $a$ since the focus is at distance $1/(4a)$ from the vertex.

Now construct the circle with center $(a^2,1/(2a))$ and passing through the origin. This meets the parabola at point with $x$-coordinate a root of the equation $$(x-a^2)^2 + (ax^2-1/(2a))^2=a^4 +1/(4a^2)$$ The equation simplifies to $a^2x^4 -2a^2x=0$. The roots are $0$ and $\sqrt[3]{2}$.

Answer 2

Note that all you need additionally to duplicate the cube beyond straightedge and compass is a compass that makes right circular cones and a "big straightedge" that makes planes--because then you can make a cone and slice a parabola off of it to use for some regular straightedge and compass work to get the cube root of 2. It seems like those ought to have been allowable tools for a 3-dimensional construction.

It seems weird to me that it's standard to tell students that you can trisect an angle if you are allowed marks on the straightedge, but not to tell them this.

I don't know whether this is an answer, but it won't fit in a comment and it doesn't quite fit as an edit to the question. Just call it a lamentation that teaching ignores solid geometry.