If you were to maximize the volume of a truncated octahedron while keeping it in completely inside a given sphere, what percentage of the sphere's volume would it take up?
This question is an extension of a larger one I've been wondering. What is the largest space-filling polyhedron of any kind that can be inscribed in a sphere? Space-filling meaning, can perfectly tile the 3D plane in Euclidean space. Would, for example, a rhombic dodecahedron maximized in a sphere take up more space than the maximized truncated octahedron?
More concretely, imagine you have spheres of a valuable material you need to pack. You can pack them more densely by cutting them in a more packable shape, thus saving more of the material, but you lose whatever you cut off. Is there a space-filling polyhedron you could cut each sphere into that contains more than ~74.048% (maximum sphere packing density) of the original sphere?