Hidden geometrical treasures (Q354149)

The author has found several objects in museums around the world (a wooden die in Gyeongju, the ancient capital of Korea, produced sometime during the 7th--9th centuries, an ivory goblet in Copenhagen, probably made in Nuremberg around 1650, spiked polyhedra in an ivory object, probably made in Dresden around 1600) that are octahedrally and icosahedrally symmetric roundest polyhedra (polyhedra that are solutions to the following isoperimetric problem: among all polyhedra with a given surface area and a given number of faces, which one has the maximum volume?).