I'm trying to find a solution to the next problem:
If I have a rectangular $2D$-die with uniform density such that each side has a certain probability $P1,P2,P3,P4$ respectively.
I want to find a bijection between this die and a $2D$-square die with displaced center of mass.
I started by making a drawing, but I don't know how to proceed. Any help would be appreciated.
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$\begingroup$ Please clarify what you mean by a side having a certain probability: is it the probability of the side landing "face-up", or something else? I'm also not sure what you mean by bijection in this case, what is being mapped to what? $\endgroup$– sortaiCommented Apr 29 at 9:14
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$\begingroup$ What I mean is that if we have a 2D rectangle with faces 1,2,3,4 with the probabilities of falling to the ground P1,P2,P3,P4 respectively. How could you create a bijection such that a 2D square has the same probabilities on each face but with the center of mass displaced? $\endgroup$– EloyCommented Apr 29 at 9:17
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1$\begingroup$ What I want is to understand where the center of mass would be. $\endgroup$– EloyCommented Apr 29 at 9:17
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$\begingroup$ Understood. How are you modelling the dice? To be physically accurate you'd need to run actual physics simulations; do you have a formula for the probabilities of the faces of the square die, given the position of the baricenter, or is that part of the problem? $\endgroup$– sortaiCommented Apr 29 at 9:29
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1$\begingroup$ For physical modelling, you can imagine turning the lamina through an arbitrary angle, placing it on the ground, and seeing which way it would fall. (This ignores things like angular momentum, moments of inertia, different ways of "rolling" such a die, and toppling, but should give a good start.) $\endgroup$– Chris LewisCommented Apr 29 at 13:15
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