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I occasionally get essentially this question from non-mathematicians, some of whom are very pleased with their appeal to the Intermediate Value Theorem. But I always reply that the whole discussion is absolute nonsense: the only possible notion of a fair die is an isohedral one, because for any other die, it depends how you throw it.

What do we mean by saying that a die is equally likely to land on any of its faces: what is the random element? In theory, if we could accurately measure the speed and spin of the die as it is released as well as coefficients of friction and restitution etc, we could work out in advance which face the die will land on, so in this sense there is no randomness. The randomness only arises when you choose how to throw the die: the thrower selects from a continuum of "possible throws", and we assume that he samples from some probability distribution on this continuum. Morever, there is an action of the rotation group of the die on this continuum, and what we mean by saying that an isohedral die is fair is that we assume the probability distribution the thrower uses is invariant under this action. There is no sensible further assumption that we can make about the probability distribution which will ensure that two faces of the die which are not in the same orbit under the rotation group will occur equally often. However "obvious" it is that an octahedral octagonal coin is very unlikely to land on its edge, and a pencil is very unlikely to land on its end, there's no meaningful way to find a happy medium between these. So there is no meaningful mathematics to be done here. (Feel free to substitute "pure" for "meaningful" in the last sentence.) If you wanted to make a machine to throw a die in a standard way, then you'd have no chance of making a fair die, because with a perfect machine the die would always land on the same face.

Incidentally, the above discussion suggests that even isohedrality is not enough for a fair die - we require there to be only one orbit of faces under the rotation group of the die, not the full symetry group.

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I occasionally get essentially this question from non-mathematicians, some of whom are very pleased with their appeal to the Intermediate Value Theorem. But I always reply that the whole discussion is absolute nonsense: the only possible notion of a fair die is an isohedral one, because for any other die, it depends how you throw it.

What do we mean by saying that a die is equally likely to land on any of its faces: what is the random element? In theory, if we could accurately measure the speed and spin of the die as it is released as well as coefficients of friction and restitution etc, we could work out in advance which face the die will land on, so in this sense there is no randomness. The randomness only arises when you choose how to throw the die: the thrower selects from a continuum of "possible throws", and we assume that he samples from some probability distribution on this continuum. Morever, there is an action of the rotation group of the die on this continuum, and what we mean by saying that an isohedral die is fair is that we assume the probability distribution the thrower uses is invariant under this action. There is no sensible further assumption that we can make about the probability distribution which will ensure that two faces of the die which are not in the same orbit under the rotation group will occur equally often. However "obvious" it is that an octahedral coin is very unlikely to land on its edge, and a pencil is very unlikely to land on its end, there's no meaningful way to find a happy medium between these. So there is no meaningful mathematics to be done here. (Feel free to substitute "pure" for "meaningful" in the last sentence.) If you wanted to make a machine to throw a die in a standard way, then you'd have no chance of making a fair die, because with a perfect machine the die would always land on the same face.

Incidentally, the above discussion suggests that even isohedrality is not enough for a fair die - we require there to be only one orbit of faces under the rotation group of the die, not the full symetry group.