No, because the group G of rotations of the 2-sphere that preserve the set of all "dimples" X (so for all g in G, g(X) = X) on the sphere has to have a member g(x,y) in G for all x, y in x such that the action of g(x,y)x = y; that is, any point can be transformed into any other point by a rotation that nonetheless preserves the whole lattice. This is called face-transitivity* and there are a finite number of recursively enumerable sets of face-transitive solids: the bipyramids, the trapezohedra, the prisms and antiprisms (as log dice), and the Catalan solids. Bipyramids and trapezohedra are quite impractical at high face counts because they take on a biconical shape whereas spheroids are ideal. Log dice are fair if they roll a lot, but they're not as cool.
Or a bag filled with numbered tiles, but a dice is a dice not the general category of random generators.
I have seen both your suggestions used in games so fair enough, and the cylinder reminds me of the inelegance of the 10 sided dice, but that had no end face to land on like a cylinder.