Mark Dominus says that "For some reason I've been trying to construct a die whose faces come up with probabilities 1/21, 2/21, 3/21, 4/21, 5/21, and 6/21 respectively." He has not been able to do this exactly, although it wouldn't be too hard to come up with an approximate solution given his claim that "the probability that the hexahedron will land on face F is not proportional to the area of F, but rather to the solid angle subtended by F from the hexahedron's center of gravity."
This sounds right -- if you throw a die in the air then it'll rotate around its center of gravity, and you expect that whichever face is pointing "down" when it hits a table for the first time is the face which will eventually settle as "down". (Note that because the faces of a general polyhedron are not necessarily parallel, I'm considering the "down" face, not the "up" face.) But I'm not entirely sure if it's true, because if the die comes close to landing on an edge it might take a funny bounce; I suspect the system is a little more chaotic than Mark's representation of it, and the only real way to solve the problem is experimentally. Unfortunately, the "experimental" part would require tossing a die a very large number of times, and would be incredibly boring. Stereotypically, one gets one's grad students to do that. But I'm a grad student and I wouldn't do it.
What makes Mark's problem difficult is the lack of symmetry; each face has to be different. Let's say, hypothetically, that I wanted to make a die which had two faces which come up with one frequency p and four with another frequency q; clearly 2p + 4q = 1. If q = 0 this is just a coin; if q = 1/4 it's some kind of four-sided stick; my guess is that we can smoothly vary q from 0 to 1/4 by taking a wide variety of rectangular boxes with two of the three dimensions the same.
Similarly, if we want a die whose sides come up with frequencies p, p, q, q, r, r we should be able to do it by taking rectangular boxes with all three dimensions different.
If we're willing to go away from the rectangular box, but have a shape which is still topologically a cube (by this I mean all six sides are quadrilaterals, meeting three at each vertex) I think we can get dice where four faces have the same frequency and each of the other two has a different frequency by using a truncated square pyramid; clearly the four trapezoidal sides each come up with the same frequency, by symmetry. There are two degrees of freedom in constructing such a truncated pyramid (say the ratio between the height and the side length of the base, and the ratio between the side length of the top and the side length of the base) so there's no obvious "dimensionality" reason why this wouldn't work. (A degenerate case of this has five of the six faces having the same frequency.)
And it's probably possible to get three faces with one frequency and three with another by "stretching" the three faces around a single corner, although I'm having trouble picturing how exactly one would do that.
But all of these constructions take into account the symmetry of the cube; I'm assuming that faces that "look the same" are landed on with the same frequency. Mark can't take advantage of that symmetry in his problem.
I also suspect that designing exotic dice like these is difficult, or perhaps impossible, because the probabilities might not be "stable"; they might depend on how hard one throws the die, which defeats the purpose of dice, which is to be a source of randomness that can't be controlled by the person throwing them. Wikipedia offers some oddly shaped dice, though, and seems to claim there exists a fair seven-sided die in the shape of a pentagonal prism, which suggests that stability isn't as much of an issue as I suspect.