Watching videos about Reuleaux polygons, it is neat to see that the triangle is not the only polygon that creates a shape of constant width. Actually, a shape of constant width can be created from any odd sided polygon by rounding the edges.
When moving to 3D, a solid of constant width can be made by making a solid of revolution from Reuleaux polygons, but more surprisingly there is a solid of constant width that is derived from a Platonic solid, specifically by rounding a tetrahedron (AKA Meissner tetrahedron).
Can there be any solid of constant width constructed from other Platonic solids? Can a Meissner octahedron be made? If yes, what is the condition for the number of faces? If not, why is the tetrahedron the only one?
I have seen some models of Reuleaux dodecahedra but I am not sure if it is a true shape of constant width. Example:
taken from grabcad.