A regular 6 sided die, which is geometrically a cube, implicitly represents a discrete uniform density over the set {1,2,3,4,5,6}. After tossing the die, any face of the cube has the same probability of ending face-down on the ground.
My (handwavy) questions are the following: Do all discrete probability distributions have an associated "geometrical-die"? If so, is the relationship one-to-one or can multiple die represent the same pmf? Given a pmf how does one construct such a die, and vice versa?