The five platonic solids have all faces regular Polyhedra. They are also all convex. But there are other properties they satisfy. Like each vertex should have the same number of edges meeting there. Let's say we relax all other criterion and only require that the solid be convex and that it have all faces regular. How many such solids can exist? One example is pasting two tetrahedra together along a face to get a solid with 6 sides and triangular faces.
How many convex solids have the same faces
3dgeometrysolid-geometry
Related Solutions
This statement is not true. At least, there are shapes for which it no longer makes "intuitive sense" that they would be fair dice, though rigorously proving unfairness would require a more formal specification of what it means to roll a die.
As an example, consider the snub disphenoid:
The general issue here is that intuitive "fairness" comes from a polyhedron being face-transitive, or isohedral, so that we cannot distinguish any two faces from one another. Thus, if spun with enough random noise that any orientation of the shape is equally likely, we shouldn't expect any one face to be favored over another, since we can't even write down a property possessed by one face and not another without making reference to a fixed orientation or location on the solid.
But "having all congruent faces", or being monohedral, is not in general enough to guarantee face-transitivity; one can find other counterexamples among the non-uniform convex deltahedra, like the triaugmented triangular prism and the gyroelongated square bipyramid. I believe these are the only counterexamples with regular polygons as faces, but using non-regular faces we can find examples like the pseudo-deltoidal icositetrahedron, the rhombic icosahedron, and the Belinski dodecahedron.
Best Answer
Convex polyhedra with regular faces which are not Platonic are called Johnson solids. There are 5 Johnson solids with the same faces. It happens they are all triangular faced (can you see why?)
So together with 5 Platonic solids, you have 10.