Final answer:
The problem of constructing a cube with double the volume of another with only a straightedge and compass relates to a classical problem in geometry that was proven impossible due to the mathematical constraints of Euclidean construction and advanced algebra.
Step-by-step explanation:
The question pertains to the impossibility of constructing a cube with double the volume of another cube using only a straightedge and compass. This has been shown to be impossible due to the constraints of classical Euclidean geometry and has its roots in the problems of doubling the cube, also known as the Delian problem. The challenge arises because the doubling of a cube's volume requires the cube root of 2, which cannot be constructed with a straightedge and compass, as was proven in the 19th century by the likes of Pierre Wantzel using advanced algebra known as Galois theory.
Moreover, the discussion involves understanding geometric transformations that maintain the symmetry of objects like a cube, reflecting some of the foundational ideas in geometry and algebra. For example, issues such as rotational symmetry and algebraic expressibility are touched upon, pointing to broader mathematical concepts beyond simple construction problems.