A cubus simus is an Archimedian body consisting of 6 squares and 32 equilateral triangles. Although the surface area of the 6 squares contributes only 30% to the total surface area, when I roll my cubus simus (glued together from card-board) in about 40% of my attempts the cubus simus lands on a square. Must the probabilities for a side be proportional to the surface area of a side (in this case, my cutting and glueing is poor) or can the probabilities deviate from the surface area of a side? Has anyone seen a study on this subject? (Not necessarily for a cubus simus, the same question arises for the probability of a football to land on a black pentagon or a white hexagon.)
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A generic shape will not have face probabilities proportional to surface area. Take a coin as an example - the probability to land on the coin's edge is much less than the proportion of the edge's surface area.
For a discussion of fair dice by their symmetries I would recommend 'Fair Dice', a short article by Persi Diaconis and Joseph Keller. There are also a couple Numberphile videos on the subject.
Once you move beyond face-symmetric shapes, computing face probabilities is difficult because they vary based on the assumed physics. Some faces are more stable than others, and the probabilities will vary based on the surface material they are thrown on (sand vs carpet vs hardwood), the density of the dice, and the way that they are thrown.
