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?