At one point I spent some time in a very similar space, modelling how you would make a fair 4-sided or 5-sided die, by varying the depth of a d6 and appropriately labeling the faces. You would end up with 2 square faces and 4 rectangular faces, and vary the aspect ration of the rectangles to get fairness.<p>So a d4 would have 1 & 2 on the square faces, and 3 - 3 - 4 - 4 on the rectangular faces; this would require flattening the die.<p>A d5 would have 1 & 1 on the square faces, and 2 - 3 - 4 - 5 on the rectangular faces; this would require stretching the die to even out the probabilities.<p>I did an analysis similar to this, using the solid angle subtended by the faces, but the "bouncing" model is obviously much more rigorous and I'm curious to apply it to this case as well.