TITLE: Fair but irregular polyhedral dice
QUESTION [59 upvotes]: I am interested in determining a collection of geometric conditions that will guarantee that a convex polyhedron
of $n$ faces is a fair die in the sense that, upon random rolling, it has an equal $1/n$ probability of
landing on each of its faces.
(Assume the polyhedron is composed of a homogeneous material; i.e., it is not "loaded.")
There has been study of what Grünbaum and Shephard call isohedral polyhedra,
which always represent fair dice: "An isohedron is a convex polyhedron with symmetries acting transitively on its faces with respect to the center of gravity. Every isohedron has an even number of faces."
It is clear such a polyhedral die is fair.
Here is an example of the trapezoidal dodecahedron, an isohedron of 12 congruent faces,
from an attractive web site on polyhedral dice:

          

But a clever argument in a delightful paper
by Persi Diaconis and Joseph Keller
("Fair Dice." Amer. Math. Monthly 96, 337-339, 1989)
shows (essentially, by continuity) that there must be fair polyhedral dice 
that are not symmetric.
For example, there is no reason to expect that equal face areas is a necessary condition
for a polyhedral die to be fair.
Nor is it reasonable to expect that the distance from each face to the center of gravity
of the polyhedron is alone a determining condition. 
Rather it should depend on the dihedral angles between faces, the likelihood of one face
rolling to the next—perhaps a Markov chain of transitions?
My question is:

Is there a collection of geometric conditions—broader than isohedral—that guarantee
  that a (perhaps asymetrical, perhaps unequal-face-areas) convex polyhedron represents
  a fair die?

Sufficient conditions welcomed; necessary and sufficient conditions may be too much to hope for!
Speculations and literature leads appreciated!

REPLY [2 votes]: This is a tentative to solve the questions that arose in Matt Fayer's answer and comments, regarding the impossibility of a fair experiment.
Appealing to Information Theory, it is established that randomness is equivalent to the lack of information. More formally, uniform measures are the one that maximize Shannon's entropy. Conversely, complete knowledge is equivalent to Dirac measures. From this point of view it is clear that a perfect machine, where perfect means indeed complete knowledge, can do whatever you want and, in particular, is able to land a die always on the same face (this is a Dirac measure!). 
After these observations, it seems to me clear that a fair throwing can be simulated imposing the lack of knowledge: the person who throw the die must be blinfolded; (s)he has to throw the die uniformly (i.e., as already suggested, chosing the Haar measure on $SO(3)$) - this can be forced using heavy gloves, but also more naturally using die with a uniform structure (I will come back to this point in a moment). On the other hand, a person who bet on the die is not allowed to use any other tool but his own eyes, in the case of uniform structure (in case of heavy gloves, he's blinfolded either). In this case, the Nash equilibrium of the ideal game between the two players (who throws and who bets) is the uniform measure on the number of faces of the die; i.e. the experiment is fair.
A comment about uniform structures: this probably brings to classical interpretation more or less discussed in other answers: I am thinking of a cube; more specifically, keeping this analogy with the real-world, I am thinking to something with the following property: if you close your eyes and you are allowed only to touch the die, then you cannot distinguish two different faces and the edges are trascurable; i.e. since the set of human beings is finite, take $\epsilon$ small enough such that every imperfection of the die is smaller than $\epsilon$.
P.s. maybe I should write this somewhere, but an important observation to keep in mind is that fairness, randomness, knowledge and so on are subjective. Something can be really random iff it is perfectly isolated.