One platonic solid can be inscribed in another platonic solid in several ways. How to find, in general, the largest platonic solid that can be inscribed in another one?
e.g. one way to inscribe a dodecahedron in an icosahedron would be: the 20 vertices of the dodecahedron pointing at the centre of the 20 faces of the circumscribing icosahedron. If the side length of the icosahedron is $1$, the side length of the inscribed dodecahedron would be $\frac{1+\sqrt 5}{6}\approx 0.54$. Alternatively, if we allow the above dodecahedron to rotate $36°$ upon any of its 5-fold axes and scale up by $\frac{3(4\sqrt5-5)}{11}$, this yields a larger dodecahedron with side length $\frac{15-\sqrt 5}{22}\approx 0.58$, which also fits perfectly in the icosahedron.
The question is, how do I know if this is the largest one?